An Introduction to Metric Spaces and Fixed Point Theory

An Introduction to Metric Spaces and

Fixed Point Theory

PURE AND APPLIED MATHEMATICS

A Wiley-Interscience Series of Texts, Monographs, and Tracts

Founded by RICHARD COURANT Editors: MYRON B. ALLEN III, DAVID A. COX, PETER LAX Editors Emeriti: PETER HILTON and HARRY HOCHSTADT, JOHN TOLAND

A complete list of the titles in this series appears at the end of this volume.

An Introduction to Metric Spaces and

Fixed Point Theory

MOHAMED A. KHAMSI WILLIAM A. KIRK

A Wiley-lnterscience Publication JOHN WILEY & SONS, INC.

New York / Chichester / Weinheim / Brisbane / Singapore / Toronto

This text is printed on acid-free paper. ©

Copyright © 2001 by John Wiley & Sons, Inc. All rights reserved.

Published simultaneously in Canada.

No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4744. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012, (212) 850-6011, fax (212) 850-6008, E-Mail: [email protected].

For ordering and customer service, call 1-800-CALL-WILEY.

Library of Congress Cataloging in Publication Data

Khamsi, Mohamed A. An introduction to metric spaces and fixed point theory / M.A. Khamsi, W.A. Kirk.

p. cm. — (Pure and applied mathematics (Wiley-Interscience series of texts, monographs, and tracts)) Includes bibliographical references and index. ISBN 0-471-41825-0 1. Metric spaces. 2. Fixed point theory. I. Kirk, W. A. II. Title. III. Pure and applied

mathematics (John Wiley & Sons : Unnumbered)

QA611.28 K48 2001 5I4'.32—dc21 00-068491

10 9 8 7 6 5 4 3 2 1

Contents

Preface ix

I Metric Spaces

1 Introduction 3 1.1 The real numbers R 3 1.2 Continuous mappings in R 5 1.3 The triangle inequality in R 7 1.4 The triangle inequality in R" 8 1.5 Brouwer's Fixed Point Theorem 10

Exercises 11

2 Metric Spaces 13 2.1 The metric topology 15 2.2 Examples of metric spaces 19 2.3 Completeness 26 2.4 Separability and connectedness 33 2.5 Metric convexity and convexity structures 35

Exercises 38

3 Metric Contraction Principles 41 3.1 Banach's Contraction Principle 41 3.2 Further extensions of Banach's Principle 46 3.3 The Caristi-Ekeland Principle 55 3.4 Equivalents of the Caristi-Ekeland Principle 58 3.5 Set-valued contractions 61 3.6 Generalized contractions 64

Exercises 67

4 Hyperconvex Spaces 71 4.1 Introduction 71 4.2 Hyperconvexity 77 4.3 Properties of hyperconvex spaces 80 4.4 A fixed point theorem 84

v

vi CONTENTS

4.5 Intersections of hyperconvex spaces 87 4.6 Approximate fixed points 89 4.7 Isbell's hyperconvex hull 91

Exercises 98

5 "Normal" Structures in Metric Spaces 101 5.1 A fixed point theorem 101 5.2 Structure of the fixed point set 103 5.3 Uniform normal structure 106 5.4 Uniform relative normal structure 110 5.5 Quasi-normal structure 112 5.6 Stability and normal structure 115 5.7 Ultrametric spaces 116 5.8 Fixed point set structure—separable case 120

Exercises 123

II Banach Spaces

6 Banach Spaces: Introduction 127 6.1 The definition 127 6.2 Convexity 131 6.3 £2 revisited 132 6.4 The modulus of convexity 136 6.5 Uniform convexity of the tp spaces 138 6.6 The dual space: Hahn-Banach Theorem 142 6.7 The weak and weak* topologies 144 6.8 The spaces c, CQ, t\ and ^ 146 6.9 Some more general facts 148 6.10 The Schur property and £j 150 6.11 More on Schauder bases in Banach spaces 154 6.12 Uniform convexity and reflexivity 163 6.13 Banach lattices 165

Exercises 168

7 Continuous Mappings in Banach Spaces 171 7.1 Introduction 171 7.2 Brouwer's Theorem 173 7.3 Further comments on Brouwer's Theorem 176 7.4 Schauder's Theorem 179 7.5 Stability of Schauder's Theorem 180 7.6 Banach algebras: Stone Weierstrass Theorem 182 7.7 Leray-Schauder degree 183 7.8 Condensing mappings 187 7.9 Continuous mappings in hyperconvex spaces 191

Exercises 195

CONTENTS vii

8 Metric Fixed Point Theory 197 8.1 Contraction mappings 197 8.2 Basic theorems for nonexpansive mappings 199 8.3 A closer look at ίλ 205 8.4 Stability results in arbitrary spaces 207 8.5 The Goebel-Karlovitz Lemma 211 8.6 Orthogonal convexity 213 8.7 Structure of the fixed point set 215 8.8 Asymptotically regular mappings 219 8.9 Set-valued mappings 222 8.10 Fixed point theory in Banach lattices 225

Exercises 238

9 Banach Space Ultrapowers 243 9.1 Finite representability 243 9.2 Convergence of ultranets 248 9.3 The Banach space ultrapower X 249 9.4 Some properties of X 252 9.5 Extending mappings to X 255 9.6 Some fixed point theorems 257 9.7 Asymptotically nonexpansive mappings 262 9.8 The demiclosedness principle 263 9.9 Uniformly non-creasy spaces 264

Exercises 270

Appendix: Set Theory 273 A.l Mappings 273 A.2 Order relations and Zermelo's Theorem 274 A.3 Zorn's Lemma and the Axiom Of Choice 275 A.4 Nets and subnets 277 A.5 Tychonoff's Theorem 278 A.6 Cardinal numbers 280 A. 7 Ordinal numbers and transfinite induction 281 A.8 Zermelo's Fixed Point Theorem 284 A.9 A remark about constructive mathematics 286

Exercises 287

Bibliography 289

Index 301

Preface

This text is primarily an introduction to metric spaces and fixed point theory. It is intended to be especially useful to those who might not have ready access to other sources, or to groups of people with diverse mathematical backgrounds. Because of this the text is self-contained. Introductory properties of metric spaces and Banach spaces are included, and an appendix contains a summary of the concepts of set theory (Zorn's Lemma, Tychonoff's Theorem, transfinite induction, etc.) that might be encountered elsewhere in the text. Most of the text should be accessible to reasonably mature students who have had very little training in mathematics beyond calculus. In particular a very elementary treatment of Brouwer's Theorem is given in Chapter 7. At the same time later chapters of the book contain a large amount of material that might be of interest to more advanced students and even to serious scholars.

Readers with a good background in elementary real analysis should skip Chapters 1 and 2, and those who have had a course in functional analysis should also skip Chapter 6. Those who have had a course in set theory will have little use for the Appendix. Most readers will find something new in the remaining chapters and they might find the inclusion of this other material helpful as well.

Although a number of exercises are included, only rarely are important de-tails of the major developments left to the reader. However in order to focus on the main development some peripheral material is included without proof, especially in later chapters.

Despite the fact that the text is largely self-contained, extensive bibliographic references are included.

In terms of content this text overlaps in places with three recent books on fixed point theory: Nonstandard Methods in Fixed Point Theory by A. Aksoy and M. A. Khamsi (Springer-Verlag, New York, Berlin, 1990, 139 pp.), Topics in Metric Fixed Point Theory by K. Goebel and W. A. Kirk (Cambridge Univ. Press, Cambridge, 1990, 244 pp.), and Nonlinear Functional Analysis and its Applications I: Fixed-Point Theorems by E. Zeidler (Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1986, 897 pp.). However, in addition to the inclusion of excercises, the level of presentation and the comprehensive development of what is known in a purely metric context (especially in hyperconvex spaces) is unique to this treatment. Among other things, it has been our hope especially to illustrate the richness and depth of the abstract metric theory. Also a number of Banach space results are included here which appear in none of the above books,

IX

X PREFACE

some by choice and some because they describe more recent developments in the subject.

Material on the general theory of Banach space geometry is drawn from many sources but one is worth special mention: Introduction to Banach Spaces and their Geometry, Second revised edition, by B. Beauzamy (North-Holland, Amsterdam, New York, Oxford, 1985).

This book could easily serve as a text for an introductory course in metric and Banach spaces. In this case material should be drawn selectively from Chapters 1 through 4 along with Chapters 6 and 7, and the Appendix as needed. A number of exercises have been included at the end of each of these chapters.

The second author lectured on portions of the material covered in the text to students of the I. C. T. P.- Trieste Diploma Program in Mathematics during May 1998. He wishes to thank them for providing an attentive and critical audience. Both authors express their deep gratitude to Rafael Espfnola for calling attention to a number of oversights in the penultimate draft of this text.

M. A. KHAMSI

W. A. KIRK

Iowa City December 2000

Part I

Metric Spaces

An Introduction to Metric Spaces and Fixed Point Theory by Mohamed A. Khamsi and William A. Kirk

Copyright © 2001 John Wiley & Sons, Inc.

Chapter 1

Introduction

1.1 The real numbers R

It would be an overstatement to say the real number system R is thoroughly understood despite its seeming simplicity. It is well known that R consists of both rational and irrational numbers, but beyond that questions arise almost immediately. A number is said to be algebraic if it is the solution of a poly-nomial equation with rational coefficients. All other numbers are said to be transcendental. The irrational number \/2 is algebraic since it is a solution of the polynomial equation x1 — 2 — 0. It is known (and these are deep facts) that π and e are transcendental. However, to this day, these are the only typi-cal numbers that are known to be transcendental; indeed it is not even known

7Γ

whether or not such basic constants as π + e, —, or Inπ are even irrational. (But e

surely no one really believes they are rational!) The best that can be said with certainty, at least now, is that they cannot satisfy any polynomial equation of degree eight or less with integer coefficients of average size less than 109. And apart from its listing to a few million places virtually nothing is known about the decimal expansion of π. It is possible, but not likely, that all but finitely many of the terms in its decimal expansion are in fact 0's or l's. (For other bizarre facts about π, see the recent article by Borwein, Borwein, and Bailey [15].)

Fortunately details such as the above, while curious, are not relevant for an understanding of what follows. Elementary mathematical analysis as it is usu-ally treated rests on basic properties of 1R that can be easily understood. One such property is the so-called least upper bound (lub) property—which deserves special attention because of the crucial role it plays in the development of anal-ysis. A nonempty set S of real numbers is said to be bounded above if there exists a number m such that for each number a; € 5 it is the case that x < m. Such a number m is said to be an upper bound for S. A number u is said to be the least upper bound or supremum (sup) of a set S if (i) u is an upper bound for S and (ii) u < m for any upper bound of S for which m ψ u. Notice by this

3



4 CHAPTER 1. INTRODUCTION

definition a set S can have at most one least upper bound (supremum). The sup axiom states:

(sup) Each nonempty set S of real numbers which is bounded above has a supremum.

The dual notions of bounded below and greatest lower bound or infimum (inf) are defined in the obvious manner, by replacing 'above' with 'below' and reversing the inequalities. The fact that any nonempty set which is bounded below has an infimum follows upon applying the sup axiom to the set —S — {—x : x € S}. Thus for a set S which is bounded below, inf S = — sup(—5).

Important Note: The terms 'least upper bound (lub)' and 'supremum (sup)' are usually used interchangeably, as well as the terms 'greatest lower bound (gib)' and 'infimum (inf)'.

The sup axiom is precisely the axiom that distinguishes the real numbers from the rational numbers. Consequences of the sup axiom are manifold. It assures, for instance, that any increasing [respectively, decreasing] sequence of real numbers which is bounded above [respectively, below] must have a limit. The following is an immediate consequences of this fact.

Proposition 1.1 If I\ D I2 3 I3 2 · · · is a descending sequence of nonempty closed intervals in R, then f] In Φ 0 .

Proposition 1.1 leads directly to what is called the Bolzano-Weierstrass The-orem. The relevant definition is this: If 5 is a subset of R, then a number p in R is said to be an accumulation point of S if every open interval which contains p also contains a point of 5 distinct from p.

Proposition 1.2 If[a,b] is a closed interval in R and if S Ç [0,6] contains an infinite number of points, then some point of [a, b] is an accumulation point of S.

a + b . Proof. Since 5 has an infinite number of points, an infinite number of points

I a + b] of S must lie in at least one of the half-intervals a, —-— or in

Select any one of the two which contains an infinite number of points of S and call it Ii. Now divide I\ in half and let h be one of the half-intervals of I\ which necessarily contains an infinite number of points of S. Continue by induction. Given In let 7 n + 1 be one of the half-intervals of /„ which contains an infinite number of points of S. In this way obtain a descending sequence of nonempty

0 0

closed intervals [a,b] D Ιγ D J2 2 ^3 Ξ? · · · · By Proposition 1.1 f| /„ φ 0.

b — a °° In fact, since the length of /„ is ——, it must be the case that f] In consists

1.2. CONTINUOUS MAPPINGS IN R 5

of a single point p. Any open interval containing p must contain /„ for some n and hence must contain an infinite number of points of 5; hence a point of S distinct from p. ■

A slightly more subtle fact which we take up in Chapter 4 is also true.

Proposition 1.3 Any family of closed bounded subintervals ofR, each two of which intersect, must have a point in common.

Another interesting consequence of Proposition 1.1 involves the 'cardinality' ofR.

Proposition 1.4 There does not exist a function defined on N whose range is all ofR.

Proof. Suppose / : N —> R and suppose /(N) = R. Then in particular / maps a subset of N onto [0,1], and it is easy to replace / with a new function / , which has the property /(N) = [0,1]. (Set f(n) = 1 if / (n) £[0,1] and f(n) — f(n) otherwise.) Consequently [0,1] = {x\,X2,···}, where / (n) = xn, n = 1,2,··· . Obviously it is possible to choose a nonempty closed subinterval I\ of [0,1] for which x\ $. I\. Having chosen / i , it is now possible to choose a nonempty closed subinterval I<i of I\ such that x<i £ I-^. (If χ-ι £ Ιχ, simply choose Ii = Ii.) Continuing in this way it is possible to choose a descending sequence Ιχ D h Ώ h 2 · ■ · of nonempty closed intervals in [0,1] such that Xn £ In, n = 1,2, · ■ · . By Proposition 1.1 there exists at least one number

oo

x € P| / „ . Since x φ xn for each n, x cannot be in the range of / . ■ n = l

In light of Proposition 1.4 R is said to be uncountable.

1.2 Continuous mappings in R

The concept of 'continuity' arises almost immediately in the study of calculus.

Definition 1.1 If S is a subset ofR and if f : S —» R, then f is said to be continuous on S if for each a € S,

Iim f(x) = / (a) . x—*a

In precise terms, this means the following: For each a E S and e > 0 there exists a number δ > 0 (depending on both a and ε) such that if x € S and if \x — a\ < δ, then \f(x) — / ( a ) | < ε.

Early in the study of calculus one also encounters two topological theorems on which much of the theory depends. The first is a consequence of a result pub-lished in 1917 by the Czech mathematician and philosopher Bernard Bolzano1

(1781-1848).

' I t appears that historically the mathematical community has been slow to fully recog-nize Bolzano's many contributions. In addition to the Intermediate Value Theorem and the


Theorem 1.1 (Intermediate Value Theorem) Let [a,b\ be a closed interval in R and let f : [a, b] —» R be continuous. Then for each number ξ which is between f(a) and f(b) there is a number c between a and b such that f(c) — ξ.

Theorem 1.2 (Maximum Value Theorem) Let \a,b] be a closed interval in R and let / : [a, 6] —> R be continuous. Then there is a number c G [a, b] for which f(x) < /(c) for each x G [a,b].

The proofs of each of the above theorems rest on the important sup axiom of R. To prove the Intermediate Value Theorem, suppose ξ is any number between f(a) and f(b); in particular assume f(a) < ξ < f(b). If f(a) = ξ there is nothing to prove. Otherwise the set

5 = { i e [ o , 6 ] : / ( x ) < 0

is nonempty and bounded above by b, so sup S exists. If c = sup 5 then it is easy to see that for each ε > 0, ( c - f , c ] f l 5 / 0 and this implies

lim f(x) = /(c) < ξ. X—*C~

But if /(c) < ξ, the fact that

Urn /(*) = /(c) x—*c+

shows that there must exist numbers x > c for which f(x) < ξ, and this con-tradicts the definition of c. Therefore it must be the case that /(c) = ξ.

The proof of the Maximum Value Theorem is a little longer. The first step is to establish the following:

Step 1. If / : [a, 6] —> [a, b] is continuous, then there exists a number M such that f(x) < M for each x e [a, 6].

Assume, for the moment, that Step 1 has been established (Exercise 1.1). Then the set

W = {y G R : y - f(x) for some x G [a, b]}

is bounded above so k — sup W is well defined. We wish to show that there is a number c G [a, b] such that /(c) = k. The proof is by contradiction. Suppose no such c exists; that is, suppose f(x) < k for each x G [a, b\. Then the function

1

9{x) = k - f{x)

Bolzano-Weierstrass Theorem, it seems that Bolzano discovered the modern definitions of convergent sequences and even the notion of a Cauchy sequence. See [140]; also [35], pp. 48-49.

1.3. THE TRIANGLE INEQUALITY IN R. 7

is defined and continuous on [a,b\. By Step 1 there is a number M > 0 such that g(x) < M for each x € [a, b]. But this is equivalent to the assertion that

for every x G [a, b], contradicting the assertion k = sup W. Already we are in a position to state our first fixed point theorem.

T h e o r e m 1.3 Let [a,b] be a closed interval in R and let f : [a,b] —» [a,b] be continuous. Then there exists a number ξ in [a, b] for which /(£) = ξ.

Proof. Introduce the mapping T : [a, b] —► R by defining T(x) = x — f(x) for each x € [a, b]. Then T is also a continuous mapping and since f(a) > a it must be the case that T(a) < 0. Similarly, f(b) 0. By the Intermediate Value Theorem there exists a number c € [a, b] such that T(c) = 0; whence /(c) = c. ■

x The rather trivial example f(x) = — shows that a mapping / : (0,1] —► (0,1]

need not have a fixed point. The only possible fixed point for such a mapping is the point 0 = /(0) , but 0 $ (0,1]. Similarly, the mapping / : R —> R defined by f{x) = x + 1 for each x € R cannot have a fixed point. Also, it is very easy to give examples of discontinuous mappings / : [0,1] —» [0,1] which fail to have fixed points.

1.3 The triangle inequality in R.

If a € R then the absolute value of a is defined to be the 'distance' between a and the 0: Thus

a if a > 0; —a if a < 0.

The triangle inequality in R asserts that for any three numbers a, 6, c 6 R

\a + b\ < \a\ + \b\.

This fact is totally transparent if one approaches it from the 'distance' point of view. Note that upon replacing b with —b and using the fact that \—b\ = |6| triangle inequality becomes

| α - 6 | < | α | + |6|.

Now think of \a — fc| as the distance between a and 6, and think of \a\ (respec-tively, l&l) as the distance between a (respectively, b) and 0. There are now only three cases to consider.

N = {


1. If both a and b are nonnegative, then clearly

|a - 6| < max{a, b} < a + b - \a\ + \b\.

2. Similarly, if both a and b are nonpositive, then

\a - b\ < max{|a|, |6|} < \a\ + \b\.

3. If one of a or 6 is positive and the other negative, then

\a-b\- \a\ + \b\.

1.4 The triangle inequality in Rn

It might seem strange to refer to the inequality of the previous section as the 'triangle inequality' since it pertains to points on a line. It is however a very special case of a more general fact that has been known virtually since the inception of rigorous mathematical thought. The triangle inequality involves one of the simplest (and most important) geometrical figures known—the triangle, and one of the most important properties of the triangle as it is understood in euclidean geometry is the fact that the length of no one of its sides exceeds the sum of the lengths of its other two sides.

It is possible to derive the triangle inequality in Rn from purely geometric principles. Consider the standard euclidean plane (which we shall denote R2). If A and B are points in R2 let \AB\ denote the distance between A and B. The triangle inequality in R2 now becomes the statement: For each three points A, B,CeR2

\AB\ < \AC\ + \BC\.

This statement has a rather nice geometric interpretation. Note that if the triangle Ù.ABC has a right angle at C then by the Pythagorean Theorem

\AB\2 = \AC\2 + \BC\2 ,

In this case one obviously has

\AB\ < y/\AC\2 + \BC\2 < j{\AC\ + \BC\)2 = \AC\ + \BC\.

One can now proceed to the general case by simply drawing an arbitrary triangle and carefully 'dropping perpendiculars'. There are only a few cases to consider, one of which is illustrated by Exercise 1.5.

Now consider the general n-dimensional euclidean space Rn. This is the space whose points consist of all ordered rc-tuples (xi,X2,·" ιχη) of real num-bers, with the distance between two such points x = {χχ,χ^,·-- ,χη)

a n d y = (î/i,2/2, · · · , yn) taken to be

d(x,y)= l ^ k i - ï / i l 2

1.4. THE TRIANGLE INEQUALITY IN R 9

Lifting the triangle inequality from R2 to Rn is not difficult since each three points of Rn lie in a two-dimensional subset (plane) of Rn which is itself (in terms of distances between points) a copy of R2. So if one has the triangle inequality in R2 the triangle inequality in Rn comes free.

There is an elegant algebraic approach to the triangle inequality in Rn as well. In explicit terms the triangle inequality in Rn asserts that for any three n-tuples, x = (xl,x2,--· ,*„) , y = ( Î / I , Î / 2 , · · · ,î/„), ζ = (2 ι , ζ 2 , · · - ,ζη) :

/ n \ 1 / 2 / n \ 1 / 2 / » \ 1 / 2

Some notation will facilitate the proof. For x = (χι,Χ2,··· >χη) and y = (2/1 ! 2/21 · · · i2/n) introduce the inner product

n

(x.y) = Σ Χ ί ^ '

and the norm

/ » X 1/2

ιΐχΐι = ( Σ > ? ) ·

Then (x,y) = (y,x), and (x, x) = ||x|| > 0. In particular for any real number t,

(x + ty, x + iy> > 0,

and a simple calculation (using linearity in the inner product factors) yields

( x , x ) + 2 i ( x , y ) + r . 2 ( y , y ) > 0 .

If y φ 0 one can set t — — (x, y) / (y ,y) and obtain

(x,y)2 < (x,x)(y,y>,

from which

| ( x ,y ) |< | | x | | | | y | | .

This is the well-known Cauchy-Schwarz inequality, and it in turn implies

||x + y||2 = (x + y,x + y) = ||x||2 + 2(x,y) + ||y||

2

< ||x||2 + 2||x||||y|| + ||y||2

= (llx|l + lly|l)2· This is the same as ||x + y|| < ||x|| -I- ||y|| which, on replacing x with x — z and y with z — y, becomes the triangle inequality; that is,

<f(x,y) < d (x , z )+d(z ,y ) .


1.5 Brouwer's Fixed Point Theorem In Section 1.2 we saw quite easily that a continuous mapping / : [a, b] —» [a, b] always has at least one fixed point. It is natural to ask whether one extend this fact to Rn and, if so, how? The answer is that such an extension is indeed possible, and a very natural one at that, but it is by no means easy.

The notion of continuity in R" offers no problem. Merely mimic the defini-tion in R.

Definition 1.2 If S is a subset of Rn and if f : S —> Rn, then f is said to be continuous if for each a € S,

lim/(x) = /(a). x—>a

In precise terms, this means the following: For each a € 5 and each ε > 0 there exists a number δ > 0 (depending on both a and ε) such that if x G S and if <f(x, a) < 6, then d(f(x), / (a)) < e.

An appropriate analogue for the closed interval [a, 6] is equally at hand. The interval [a, b\ is precisely the set of points of R whose distance from the midpoint

a — b —-— does not exceed . Thus if we let m = —-— and r =

[a,b] = {x e R: d(m,x) < r}.

then

The analogue of such a set in R" is a closed ball centered at a point m 6 Rn

of radius r > 0 :

B(m;r) = {x € R" :d(m,x) = | | m - x | | < r}.

Note also that an interval has an algebraic structure. Indeed, we have

[a,b] = {ta + (l -t)b:0< t < 1}.

We are now in a position to state one of the most famous and useful 'fixed point theorems' ever proved.

Theorem 1.4 (Brouwer's Fixed Point Theorem) Let B be closed ball in Rn. Then any continuous mapping f : B —» B has at least one fixed point.

Regarding the proof of Brouwer's Theorem, difficulties arise even in the case n = 2. We would like to show that if B is a closed ball in R2, for example, the unit ball:

B\ = {(xi,x2)eR2:x21+xl < 1},

then any continuous mapping / : B\ —» B\ has a fixed point. The trick in the case n — \ was the application of the Intermediate Value Theorem. Un-fortunately it is not obvious how to formulate an appropriate analogue of this theorem in Rn.

EXERCISES 11

There are several ways around this, and for excellent elementary discussions we refer to [57] and [143]. At the same time we shall revisit Brouwer's theorem in Chapter 7 and give detailed proofs of the cases n = 2 and n = 3, and these in turn will point the way to the general proof.

Brouwer's theorem has a long history. Ideas leading to the proof of Brouwer's theorem were discovered by Henri Poincaré as early as 1886. Brouwer himself proved the theorem for n = 3 in 1909. In 1910 Hadamard gave the first proof for arbitrary n, and Brouwer gave another proof in 1912. However in 1904 a result which is equivalent to Brouwer's theorem was published by P. Bohl. For further historical facts see, for example, [28], [106].

Exercises

Exercise 1.1 Prove Step 1 in the proof of Theorem 1.2.

Exercise 1.2 Show that a continuous mapping f : [0,1] —» [0,1] which satisfies f(f(x)) = x for each x 6 [0,1], and for which f(x) ψ x for at least one x € [0,1], must have exactly one fixed point.

Exercise 1.3 Does a continuous mapping / : R —> R which satisfies f(f{x)) = x for each x 6 R necessarily have a fixed point?

Exercise 1.4 Describe a continuous mapping f : [0,1] —> [0,1] for which

f(f(x)) — x and f(x) φ x

for more than one x € [0,1].

Exercise 1.5 Suppose A, B, C G R2 and suppose the line passing through C which is perpendicular to the line ê(A, B) joining A and B intersects l(A, B) in a point between A and B. Give a geometric proof that \AB\ < \AC\ + \BC\.

Chapter 2

Metric Spaces

A space is distinguished from a set by possessing attributes not possessed by a mere collection of elements, and a subspace is a subset of a space which is assumed to have inherited such defining attributes. A metric (distance) space suggests that given two points of the space there should be a real number that measures the distance between them. Accordingly, to discuss a 'metric' it is natural to begin with a pair (M, d) where M is a set and d : M x M —> R+ is a mapping of the cartesian product M x M into the nonnegative reals R+. If d(x, y) is thought of as the distance between two points x, y € M it is natural to assume that d satisfies for each x,y € M:

(i) d(x, y) = 0 <=> x = y; and

(ii) d(x,y) = d(y,x).

A pair (M, d) satisfying the above assumptions is called a semimetric space. These assumptions are in some sense minimal when one thinks of a distance. The semimetric spaces form a subclass of the important class of metric spaces defined below, yet it is doubtful whether semimetric spaces themselves offer sufficient structure to yield very deep results. However even at this point a number of useful concepts can be introduced, surely one of the most important being the concept of "limit". We begin by recalling that a sequence {rn} of real numbers is said to converge to a number r (written lim,,-.,» rn — r) if for each ε > 0 there exists an integer TV such that \rn — r\ < ε whenever n> N.

Definition 2.1 A sequence {xn} in a semimetric space M is said to converge to a point x £ M if limn-ux>d(xn,x) = 0. Thus for each ε > 0 there exists an integer N 6 N such that

n > N => d(xn, x) < ε.

In this case x is said to be the limit of the sequence {xn} o,nd we simply write lim xn = x.

n—*oo

13



14 CHAPTER 2. METRIC SPACES

Immediately something undesirable happens because, in general, there is nothing to assure that in a semimetric space the limit of a given sequence is unique. This defect can be remedied, however, by adding another fairly mild assumption.

Definition 2.2 The distance d of a semimetric space (M, d) is said to be con-tinuous if for each p,q € M the conditions lim pn = p and lim qn — q =>

n—>oo n—>oo

lim d{pn,qn) =d(p,q). n—»oo

Owing to the complete flexibility one has in assigning distances between points in a semimetric space it is clear that many bizarre examples of such spaces exist, and the preponderance of so many strange spaces diminishes the likeli-hood of really interesting theorems. In looking for an appropriate assumption to add, mathematicians long ago turned to the obvious model—the euclidean spaces—and added the fundamental inequality known to hold there—the trian-gle inequality.

Definition 2.3 A semimetric space (M,d) is called a metric space if it satisfies:

(iii) (The triangle inequality) For each three points x,y,z & M,

d(x,y) < d(x,z) + d(z,y).

The most natural metric space is, of course, the real line R with the absolute value metric: d (x, y) = \x — y\ for x,y ÇR. We have already verified the triangle inequality in Section 1.3. However even R has other interesting metrics. For example, identify R with the x-axis in R2 and let S be the circle in K2 with center (0,1) and radius 1. Draw a line from (0,1) to each point x on the rr-axis and let x' be the point where this line intersects S. Now for two points x, y on the x-axis define d$ (x, y) to be the length of the shortest circular arc on S which joins x' and y'. Then d$ is a metric on R with two interesting properties. First, ds (x, y) < π for each x, y E R; thus ds is bounded. On the other hand, as with the absolute value metric, if x,j/, z € R satisfy x < y < z, then ds (x, y) + ds (y, z) = ds (x, z). In particular, for a sequence {x„} in R, Ιητΐη-,οο |χη — x| = 0 « limn-.oods (xn,x) = 0. (Note that another way to describe this metric is to take ds (x, y) = | t an - 1 x — t a n - 1 y| for x, y 6 R.)

Having defined metric spaces1, what should it mean to say that two metric spaces are the same? Since the fundamental notion is distance, it makes sense to say that two metric spaces are the same if there is a (necessarily one-to-one) distance preserving mapping from one onto the other. Such mappings are called isometries.

1 These spaces were introduced by M. Fréchet in his thesis Sur Quelques Points Du Calcul Fonctionnel (Rendiconti del Circolo Matematico di Palermo, 22(1906), pp. 1-74) and called by him spaces of class (E). However Fréchet and his immediate successors did not develop the theory. The term metric space was introduced by F. Hausdorff (Grundzüge der Mengenlehre, Leipzig, 1914).

2.1. THE METRIC TOPOLOGY 15

Definition 2.4 Suppose (M, d) and (N, p) are metric spaces. A mapping T : M —* N is said to be an isometry if p(T(x),T(y)) = d(x,y) for each x,y £ M. IfTis surjective (onto), then we say that M and N are isometric. A surjective isometry T : M —> M is called a motion of M.

2.1 The metric topology A topology on a set X is any family T of subsets of X which satisfies the following simple axioms:

(1) 0 and X are in T.

(2) The union of any subcollection of T is a member of T.

(3) The intersection of any finite subcollection of T is a member of T.

Together the pair (X, T) is called a topological space.

A subset U of X is said to be an open set if U £ T. A closed set in X is a set whose complement is open. Thus B Ç X is closed if X\B £ f, where

X\B = {x£X : x<£B}.

If (X, J-) is a topological space then it is clear from the definition (and very elementary properties of sets) that:

(Ι') 0 and X are closed sets.

(2') The intersection of any subcollection of closed sets is a closed set.

(3') The union of any finite subcollection of closed sets is a closed set.

Most topological spaces, and especially those which arise naturally in the study of analysis, satisfy an additional assumption. A topological space X is said to be Hausdorff if given any two points x, y € X there are open sets U and V in X such that x 6 U, y € V, and UC\V φ 0 . A sequence {xn} of elements of a topological space X is said to converge to x 6 X (written lim xn — x) if given

n—»oo

any open set U containing x there is an integer TV such that for n > N, xn € U. The assumption that the space is Hausdorff assures that limits of sequences are always unique.

There are two natural ways of introducing the metric topology in a metric space (M, d). For x e M and r > 0 let

U(x;r) = {ye M : d(x,y) < r}.

U(x; r) is called the open ball centered at x of radius r. The metric topology on a metric space M is the topology obtained by taking as open sets the collection of all sets T in M which have the property S € T provided each point x £ S is the center of some open ball U(x\r) (for r > 0) which also lies in S. It is easy


to check that F is indeed a topology2, that T is Hausdorff, and that with this topology the topological notion of limit is consistent with the one defined in the preceding section: um x„ = x if given any r > 0 there exists an integer N such

n—»oo

that if n > TV, d(xn,x) < r. This gives rise to an important characterization of closed sets in a metric

space.

Theorem 2.1 A subset B of a metric space M is closed if and only if

{xn\ Ç B and lim i „ = i = > i E ß . (*) n—»oo

Proof. First assume that B is closed, and suppose {xn} Ç B with lim xn = n—»oo

x. Since B is closed M\B is open, so if x ^ B there is an open ball U = U(x; r) such that U Ç M\B. Since lim xn = x there exists an integer TV such that n—»oo

xn € U{x; r) if n > TV. Hence xn £ B for sufficiently large n and this is a contradiction.

Now assume B is not closed and assume that (*) holds. This is the same as saying that M\B is not open. Thus there must be some point x e M\B which has the property U{x;r) is not contained in M\B for any positive number r.

But if U(x; r) is not contained in M\B then U(x; r)C\B φ 0 . Since this is true for any r > 0 it must be the case that for each n = 1,2, ·· · there is a point xn € U I x; — ) Π·^· Thus the sequence {xn} lies in B. On the other hand, if

r > 0 then one can choose TV € N so that — < r and conclude that for n> TV,

d(xn,x) < r. Therefore lim xn — x, and since x £ B this contradicts (*). ■ n—»oo

Another efficient way of introducing the metric topology in a metric space is to first define 'closed sets'. Let (M, d) be a metric space. Call a point a: € M a limit point of 5 Ç M if there exists a sequence {xn} in S such that lim xn — x.

n—»oo Now define closed sets in M to be precisely those sets which contain all of their limit points, and take as open sets those sets whose complements are closed. In view of the preceding theorem the topology obtained in this way is indeed the metric topology.

If 5 is a subset of a topological space X then the closure 5 of 5 is defined to be the intersection of all closed subsets of X which contain S. It is easy to see that a set 5 in a topological space is closed if and only if S = S. Another easy consequence of the preceding theorem is the following.

Theorem 2.2 If B is a subset of a metric space, then x £ B if and only if there exists a sequence {xn} Ç B such that lim xn = x.

2 As required, 0 is an open set in this topology. The assertion "If i € 0 , then there is an open ball centered at x which lies in 0 " is true since its hypothesis is vacuous. This is a subtle fact of logic.

2.1. THE METRIC TOPOLOGY 17

A subset S of a topological space (X, F) is said to be compact if whenever S is contained in the union of a collection U of sets of T (i.e., when S has an open cover U) it is the case that some finite subcollection of U contains S. This definition applies to metric spaces as well, but in the case of metric spaces there is a characterization of compactness that is quite useful.

Theorem 2.3 A subset S of a metric space M is compact if and only if any sequence {xn} of points of S has a subsequence {xnii} which converges to a point ofS.

Proof. Suppose Sm is a compact subset of a metric space M and let {xn} be a sequence of points of S. Suppose no point of S is the limit of any subsequence of {xn}. This means that given any x G S there is an integer Nx € N and a number rx > 0 such that for n > JVX, xn φ U(x;rx). Since S Ç \J U(x;rx) and the

xes family {U(x;rx)} consists of open sets, there must exist a finite subcollection

k

{U(xi;rx.) : i = 1 , · · ,k} such that S Ç |J U(Xi\rXi). However this implies

that if N = max{Ni,- ■■ Nk} then χχ £ S. Since this is a contradiction we conclude that some subsequence of {xn} must converge to some point of S. ■

In particular, a closed interval \a,b] in R, indeed any closed ball in Rn, is compact in the sense of Theorem 2.3. In elementary analysis this fact is known as the Bolzano-Weierstrass Theorem. As noted earlier, for a closed interval this is a relatively easy consequence of Proposition 1.1 of Chapter 1.

The previous two theorems yield an important fact: A subset of a compact metric space is itself compact if and only if it is closed.

It is important to note that the sequential characterization of compactness is not true for general topological spaces, although it becomes true if 'sequence' is replaced with 'net' and 'subsequence' with 'subnet'. This is discussed further in the Appendix.

Finally, there is another important fact about compactness that will be used repeatedly in what follows, both for metric spaces and in more general settings. We state and prove the result for metric spaces here, but we shall later invoke the fact that it holds for any topological space. The general result is proved in the same way upon replacing sequential convergence with net convergence.

Theorem 2.4 Let M be a compact metric space, and let f : M —> R be a continuous mapping. Then there is a point XQ £ M such that

f(x0)=inf{f(x):x€M}.

Proof. It is easy to see that {/(x) : x € M} is bounded below, so m = inf{/(x) : x € M} exists. By the definition of infimum (inf) for each positive

integer n there exists x„ 6 M such that m < / (x„) < m+ —. By compactness of


M the sequence {xn} has a subsequence {xnk} which converges, say, to XQ € M. Since / is continuous, lim f{xnk) = /(ô)· But it must be the case that

k—»oo

m < lim / ( i „ J = /(zo) < lim I m H 1 — m. k—*oo fc—+00 V Tlfc /

Therefore f{xo) = rn. m

Definition 2.5 A metric space (M, d) is said to be totally bounded if given ε > 0 there exists a finite set of points {χχ,χι,-- ,xn} Ç M, called an e-net, such that given any x G M there exists i € {1,2, ■ · · , n} for which d(x, Xi) < c.

It is easy to see that compact spaces are totally bounded because if M is compact, then the union of the family {U{x\e)}x^M of open sets contains M; hence some finite subcollection of this family also must contain M.

A subset K of a metric space (or for that manner, any topological space) is said to be precompact if its closure K is compact. It is useful to know that precompactness is equivalent to total boundedness in many metric spaces. In fact one implication always holds.

Theorem 2.5 / / a subspace K of a metric space (M,d) is precompact then it is totally bounded (and in particular bounded).

Proof. Suppose K is precompact and let ε > 0. Since K is compact, hence totally bounded, there exists a finite set {xi, X2, ■ ■ ■ ,xn} Ç K such that if x 6 K then for some 1 < i < n, d(xi,x) < ε/2. However since each Xi € K, for each i there exists x\ € K such that d(xi,x'i) < ε/2. It follows that for each x £ K, d(x,x'i) < ε for some 1 < i < n, so {χ',,α;^, · · ■ ,x'n} is the desired e-net in K. ■

Sometimes the converse of the above is true as well, a fact we take up later (Theorem 2.12). In the meantime we turn to two other important facts about compact metric spaces..

Theorem 2.6 Let (M,d) and (N,p) be metric spaces with M compact. Suppose T : M —* N is continuous. Then T(M) is compact.

Proof. In view of Theorem 2.3 it only needs to be shown that any se-quence {yn} Ç T(M) has a subsequence which converges to a point of T(M). Suppose yn = T(xn), n = 1,2, · · · . Since M is compact {x„} has a sub-sequence {xnk} which converges to a point x € M. Since T is continuous, lim ynk = lim T(xnk) = T(x) € T(M). m

A:—»oo k—»oo

We conclude this section with another important fact about continuous mappings defined on compact spaces. A mapping T : (M,d) —> (JV, p) is said to be uniformly continuous if given ε > 0 there exists δ > 0 such that p(T(x),T(y)) < ε whenever d(x,y) < 6.

2.2. EXAMPLES OF METRIC SPACES 19

Theorem 2.7 Let (M, d) and (N, p) be metric spaces with M compact. Suppose T : M —> N is continuous. Then T is uniformly continuous.

Proof. Suppose T is not uniformly continuous. Then there exists e > 0 such that for each n € N there exist a;n,yn G M such that d(xn,yn) < 1/n while p(T(xn),T(yn)) > e. Since M is compact it is possible to select subsequences {xn,} of {xn} and {ynk} of {yn} for which

lim xnk = x and lim y,lk — y. k—*oo k—Hx

Since lim d{xnk,ynk) < lim Ι /η^ = 0 it must be the case that x = y, and k—*oo fc—*oo

since T is continuous,

lim T(xn„) = T(x) = T(y) = lim T(ynk). k—*oo k—*oo

However, this contradicts the fact that p(T(xnk) ,T(ynk)) > ε for each k. ■

2.2 Examples of metric spaces

In this section we collect some examples of metric spaces which commonly arise in the study of functional analysis. The first is quite trivial, but it (or some variation) is useful to keep in mind when thinking of counterexamples.

Example 2.1 ( The discrete space) Let S be any set and define for each x,y € S

*.»>-{?£ = " Φν-The absolute value metric in R extends in several natural ways to the vec-

tor space R" consisting of all n-tuples of real numbers. Perhaps the nicest geometrically is the extension which gives the euclidean spaces RJ?· F° r these spaces the distance ^ ( x . y ) between two points x = (x\,X2, · ■ · ,Xn) and y =

/» 2y/2

(?/i 12/2> ■ · · ,2/n) in Rn is taken to be I ^ \xi ~ Vi\ ) ■ When we think of Rn

os 0 metric space without further qualification, the underlying metric will always be the euclidean metric.

The euclidean metric on R" is only the beginning. A number of other metrics can be described where the underlying space is R". We begin by describing three of the most common. Each of these provides Rn with the same metric topology since in each instance metric convergenc is equivalent to coordinate-wise convergence.

Example 2.2 (R") This is the space whose elements consist of all n-tuples (xi,a;2, · · · ,xn) of real numbers, with the distance di(x.,y) between two elements x = (x1,x2,-·· ,Xn) and y = (2/1,1/2,··· ,2/n) taken as

n

di(x,y) = ^2\xi-yi\.


All the metric properties of this space are easy to verify. In fact, the triangle inequality for this space follows immediately from the triangle inequality in R with the absolute value metric.

Example 2.3 (R£J This is the space whose elements consist of all n-tuples (xi,X2,··· >£η) of real numbers, with the distance <ioo(x,y) between two ele-ments x = (χι,Χ2ι· ' · )^n) and y = (j/i,j/2r · · i2/n) taken as

<*oo(x,y) = max \xi -Vi\. l< t<n

The triangle inequality is easy to prove in this case as well because given x, y and z = (z\, 22, · · ■ , zn) one has for each i = 1,2, ■ · · , n,

\xi - Vi\ < \xi - Zi\ + \zi - Ui\ < max \xi - Zi\ + max \zi - Μ\ . 1<τ<η K t < n

Hence

max \χι — yi\ < max \xi — Zi\ + max \zi — yi\. Ki<n Ki<n 1<ί<η

Example 2.4 (Rp, 1 < p < 00) This is the space whose elements consist of all n-tuples (χχ, X2, · · · , xn) °f real numbers, with the distance dp(x, y) between two elements x = (xi, X2,· ■ ■ , xn) ο,ηά y = (yi, 3/2, · · ■ , Un) taken as

rfp(x.y) = ( ^2\xi-Vi\p I

Having already shown that Rn satisfies the triangle inequality for each of the metrics d\,d2, and rft» it is intuitively clear that the triangle inequality should hold in all the intermediate cases dp, 1 < p < 00. We omit the details (which involve extensions of Cauchy-Schwarz inequality).

Another important class of metric spaces is obtained by extending the R£ to infinite dimensions in a way that preserves the euclidean geometry.

Example 2.5 (The Hubert space (2) This is the space whose elements consist of all square summable sequences (xltX2,··-) of real numbers (i.e., each of

the sequences for which Σ xl converges), with the distance ^ ( x . y ) between

two such sequences x = ( x i , X 2 , · ) and y = (yi, 3/2, · - - ) taken as d2(x,y) = 00 \ l / 2

|> i -y i ) 2 J · For x = (xi, X2, ■ ■ ■ ) , y = (yi, 2/2, · · ■ ) S 2 introduce the inner product

(x>y) = ΣχΜ' i = l


oo 2

1/2

and the norm

ιΐχΗΣ*' By the Cauchy-Schwarz inequality,

n

t = l

and by letting n —» oo it follows that (x, y) is well denned. In turn, this proves oo

that the series Y^{x% — Vi)2 converges, so d2(x>y) is also well defined. i= l

What about the triangle inequality in i-p. Intuitively, the quickest way to see that the triangle inequality holds is to realize that any three (nonlinear) points in £2 he in a two-dimensional hyperplane (translate of a two-dimensional subspace) of £2 which is metrically identically to R2. To be a little more rigorous, given three points x, y, z 6 £2 one can assume z = 0 via the translation T(u) = (x—z). A rotation can then be use to place x on the line determined by the unit vector βχ = (1,0,0,0, · · · ) and another rotation will place y in the plane determined by ei and e2 = (0,1,0,0, · · · ). The subspace of £2 spanned by ejand β2 is clearly identically to R| .

Of course, the above analysis requires some understanding of the geometry of £2 which may seem to be begging the question. Another approach is to extend the Schwarz inequality to infinite sums and follow the analytic approach used in R£. This leads to ||x + y|| < ||x|| + ||y|| which as before, on replacing x with x — z and y with z — y, becomes

ek(x,y) <d 2 (x , z ) + d2(z,y).

Example 2.5 describes a space which is infinite-dimensional yet it has the property that each of its two-dimensional subspaces is a euclidean plane and thus rich in geometric structure. Angles are meaningful, the law of cosines holds, and so forth. We return to these facts in Section 6.3 of Chapter 6.

The other metrics on R" described above extend to infinite dimensions quite naturally as well.

Example 2.6 (£1) This is the space whose elements consist of all absolutely convergent sequences (x\,X2, ■ ■ ■ ) of real numbers {i.e., each of the series 0 0

Σ \xi\ converges), with the distance d\(x.,y) between two such sequences x — i= l (xi,x2,· ■ ·) and y = (ΐ/ι,ΐ/2,· · · ) taken as

rfi(x.y) = X]la;t -Vi\ i=\


The next example will be studied extensively in Chapter 4 because despite the fact that it is infinite-dimensional it shares many nice geometric properties with the real line R.

Example 2.7 ( oo) This is the space whose elements consist of all bounded sequences (χχ,Χ2, ■ ■■) of real numbers, with the distance <f<x>(x, y) between two such sequences x—(xi,X2,··) and y = (j/i, J/2i · · · ) taken as

dooCx.y)^ sup \xi-yi\. l<i<oo

In each of the two preceding cases the triangle inequality follows as in the finite-dimensional cases.

There is of course an infinite-dimensional analog of R".

Example 2.8 (êp, 1 < p < co) This is the space whose elements consist of all p-oo

summable sequences (x\,X2, ■ ■ ■ ) of real numbers (i.e., each of the series Σ \xi\P

converges), with the distance dp(x, y) between two elements x = (xi, X2, ■ ■ ■ ) and y = (yi,3/2.···) taken as

dP(x,y) = I 5^1*1 ~yi\P

\ i = l

Now we turn to some slightly more complicated spaces.

Example 2.9 (The space C[0,1]) This is the space consisting of all continuous real-valued functions defined on the closed unit interval [0,1], with the distance d(f,g) between two such functions f,g taken as

d(f,g)=8up{\f(t)-g(t)\:t€[0,l]}.

The metric axioms for C[0,1] are easy to verify. In particular, the distance ^ ( / . fl) is always defined because the function / — g is itself continuous on [0,1] and hence by Theorem 1.2 bounded. The triangle inequality is derived from the fact that for any f,g,h 6 C[0,1] and any t € [0,1] the triangle inequality in R yields

\f(t)-g(t)\<\f(t)-h(t)\ + \h(t)-g(t)\.

The triangle inequality for C[0,1] now follows from the fact that

sup {\f(t)-h(t)\ + \h(t)-g(t)\}< sup \f(t)-h(t)\+ sup \h(t)-g(t)\. t€[0,l] t€[0,l) t€(0,l]

•■if

In view of the fact that [0,1] is compact, if / £ C[0,1], then / is, in fact, uniformly continuous.


Example 2.10 (The space B[0,1]) This space consists of all bounded real-valued functions defined on [0,1] with the distance d(f,g) for f,g € B[0,1] taken as above:

d{f,g) = SMp{\f(t)-g(t)\:t S [0,1]}.

Clearly the space C[0,1] is a subspace (in fact, a closed subspace) of B[0,1]. Indeed here we can think of 'subspace' in both the metric and the algebraic sense.

There are other ways of assigning a distance to the collection of all continuous functions defined on [0,1].

Example 2.11 Let X be the space consisting of all continuous real-valued func-tions defined on the closed unit interval [0,1], with the distance d(f,g) between two such functions f, g taken as

d{f,g)= f \f(t)-g(t)\dt. Jo

It is easy to see that (X, d) is metric space using standard properties of the integral of continuous functions, such as the fact that for a function / which is continuous on [0,1] :

/ | / ( i ) | dt = 0 <* f(t) = 0 for each t £ [0,1]. Jo

Example 2.12 Let X be the space consisting of all continuous real-valued func-tions defined on the closed unit interval [0,1], with the distance d(f, g) between two such functions f, g taken as

1

d(f,g)=(j\f(t)-g(t))2dt

Perhaps surprisingly, the above space has geometric properties similar to those of (2 ■

In a more abstract vein there are also standard ways of constructing new metric spaces out of given ones.

Example 2.13 ( The bounded metric d!) Let (M, d) be a metric space, and de-fine the metric space (M,d') by taking for x,y € M

2 ·

A'tn. ,Λ - ^ ^ ) d{x<y)-TTdJ£J)


Clearly 0 < d'(x,y) < 1. Also d'(x,y) = 0 & x — y, and d'(x,y) - d'(y,x). There are two ways to see that the triangle inequality holds. One way is the direct computation:

d'(x,z) =d(x,z)/[l+d(x,z)] = 1 - 1 / ( 1 + d(x,z)] <l-l/[l + d(x,y) + d(y,z)} = d(x, y)/[l + d(x, y) + d(y, z)] + d(y, z)/[l + d(x, y) + d(y, z)} <d'(x,y)+d'(y,z).

Another approach is to consider the function / : [0,oo) —» [Ο,οο) defined by

f(t) = ^ - t . Then /(0) = 0, and for t > 0, f'(t) = — l — ^ > 0 and f"(t) =

- 2 j - rz < 0. Thus / is increasing and concave downward. This shows that

Example 2.13 is a special case of the following:

Example 2.14 (The metric transform φ) Let (M,d) be a metric space, and define the metric space (M, άφ) by taking for x,y € M

άφ(χ,ν) = Φ(ά(χ,ν)),

where φ : [0, oo) —» [0, oo) is increasing, concave downward, and satisfies φ(0) — 0.

Example 2.15 (The Hausdorff metric) Let (M,d) be a metric space and let ΛΊ denote the family of all nonempty bounded closed subsets of M. For A € M. and ε > 0 define the ε-neighborhood of A to be the set

Νε(Α) = {x € M : dist(x, A) < ε}.

where dist(x, A) = inf d(x,y). Now for Α,ΒΕΜ set yÇ.A

H(A, B) = inf{e > 0 : A C Ne(B) and B Ç Ne(A)}.

Then (M,H) is a metric space, and H is called the Hausdorff metric on M-

To see that H is a metric, first observe that if x € A then infy&A d(x, y) = 0 so x 6 Ne(A) for each e > 0. Therefore A Ç Ne(A) for each ε > 0 and H (A, A) = 0. On the other hand, if a; € Νε(Α) for each ε > 0 then for each n G N there exists

yn E. A such that d(x,yn) < —. It follows that lim yn = x, so x 6 A — A. n n—«oo Prom this it follows that if H(A, B) = 0 then A — B. Since it is obvious from the definition that for each A,B e M, H(A,B) = H(B,A) and H(A,B) > 0, H is a metric if it can be shown that the triangle inequality holds.

Now let A,B,C e M, let σ = H(A,C) and μ = H(C,B), and let p > 0.

Since A Ç Νσ , 0u(C), if a e A there exists c e C such that d(a, c) < σ + - .


Also, since C Ç N + u(B), there exists b e B such that d{c,b) < μ + £.

Therefore for any a £ A there exists b € B such that

d(a, b) < d(a, c) + d(c, b) < σ + μ + p.

This proves that A Ç Νσ + μ + p{B) for every p > 0. Hence

Ίηί{ε > 0 : A C Νε(Β)} <σ + μ.

Interchanging the roles of A and B in the above argument yields

inf{e > 0 : B Ç Νε{Α)} < σ + μ.

Therefore

Η(Α, Β)<σ + μ = H (A, C) + H{C, B).

m

A natural metric space which exhibits some of the anomalies of 'metric convexity' which will be discussed later is the two-dimensional spherical space «i?2,r- (This example extends analogously to its n-dimensional counterpart, but ^2,r is easier to visualize.)

Example 2.16 The elements of S2,r are the collection of all ordered triples 3

(xi,X2!;c3) of real numbers which satisfy ^2 xf = r2, r > 0. Given a pair

x = (xi,x-2,X3) and y = (3/1,2/2.2/3) of elements S2,r define the distance p(x,y) to be the smallest non-negative number for which

The above definition assigns as the distance between two points on the sur-face of the ordinary sphere of radius r in R3 the length of the shortest arc lying on the sphere which joins them. Since such an arc always lies in the plane which passes through the two points and the origin, in order to see that this formula gives the arc-length distance it suffices to note that if (x ι, χ^, Χ3 ) and (î/i, î/2,2/3 ) are points of a sphere centered at 0 6 R3 with radius r > 0 and if Θ 6 [0, π] is the angle between the vectors determined by these points, then Θ = s/r where s is the length of the arc subtended by the angle Θ. Hence cos (s/r) — cos Θ. On the other hand, from the 'dot product' of elementary vector analysis,

3

y ^ XiVi — r2 cos Θ. i=\

The proof of the triangle inequality for this example also involves a general-ization of the Cauchy-Schwarz inequality. Details may be found in [11].


We conclude this section with another abstract example. The motions (Def-inition 2.4) of a metric space M form a group under functional composition, and it is possible to assign a metric to this group as follows.

Example 2.17 (The group of motions) Let (M,d) be a metric and let M de-note the group of all motions of M. Fix p 6 M, and for Φ, Φ £ M. define

/>ρ(Φ,Φ) = sup{d(<ï>(x)!(x))e-d(p·*) : x € M}.

2.3 Completeness

Probably the first person to recognize the truly fundamental role completeness plays in those spaces which usually arise in analysis was the celebrated Polish mathematician Stefan Banach [7]. Indeed, in recognition of its importance Ba-nach took it as an axiom in considering what are now known as Banach spaces. It is, however, an entirely metric condition based on the notion of Cauchy se-quences.

Definition 2.6 A sequence {xn} î n a metric space (M, d) is said to be a Cauchy

sequence if for each ε > 0 there exists N € N such that if m,n > N, then d(xn,Xm) < e.

Definition 2.7 A metric space (M, d) is said to be complete if each Cauchy sequence {xn} in M has a limit.

Complete metric spaces are abundant. Each of the metric spaces of the Examples 2.1 through 2.10 of the previous section are complete. Indeed, if the underlying space (M, d) is complete, the spaces of Examples 2.13, 2.14 and 2.15 are complete as well. However some of these facts are not entirely obvious. Before turning to specific examples, we establish two important general facts.

A subset 5 of a metric space M is said to be bounded if S is contained in some ball B(x; r) of M.

Theorem 2.8 A Cauchy sequence in a metric space is always bounded.

Proof. Let {xn} be a Cauchy sequence in a metric space (M,d). Taking ε = 1 in the definition of Cauchy sequence it is possible to obtain an integer N such that if n > N, then d(xN,xn) < 1- Since the set {d(xi,Xj) : 1 < i, j < N} is finite, it has a largest member, say d. It now follows from the triangle inequality that for any two integers m, n 6 N, d(xm,xn) < d+ 1. In particular, { i n } C B ( i i ; d + l ) . ■

Another easy fact is the following. The proof is left as an exercise.

Theorem 2.9 If {xn\ is a Cauchy sequence in a metric space M and if {xn} has a subsequence {xnk} which converges to x E M, then lim xn — x.

2.3. COMPLETENESS 27

The diameter of a nonempty bounded subset D of a metric space is the number

diam (D) = sup{d(x, y) : x,y 6 D).

Note that we have diam (D) = diam (£>).

Theorem 2.10 (Cantor's Intersection Theorem) A metric space (M,d) is com-plete if and only if given any descending sequence {Dn} of nonempty bounded closed subsets of M,

Um diam (D„) = 0 => Γ\ΌηφΖ. n—»oo I ' n = l

Proof. Suppose M is complete and let {Dn} be a descending sequence of nonempty bounded closed subsets of M for which lim diam(Dn) = 0. For

n—»oo

each n select xn € Dn. Then given e > 0 there exists an integer N € N such that n > N => diam(£>„) < ε. If m,n > N, both xn and x m are in DN, then d(xn, xm) < ε. This proves that {xn} is a Cauchy sequence. Since M is complete there exists x G M such that lim xn — x. At the same time this shows that

n—»oo

x G D„ = D n if n > iV. Since the sequence {£>„} is descending, we conclude

oo

xef]Dn. n = l

To prove the converse, assume the condition holds and suppose {xn} Ç M is a Cauchy sequence. For each n let

Dn — \%n> χη+ι, ' " · }·

Then {Dn} is a descending sequence of nonempty closed subsets of M for which

lim diam(Z)n) = 0. n—»oo

oo

By assumption, there exists a point x € (~) Dn. Now let ε > 0 and choose n=i

ΛΓ € N so large that n > vV => diam (Z?n) < e. Then clearly for such n it must be the case that d(x,xn) < e. Hence lim xn — x, proving that M is com-

n—»oo

plete. ■

Remark 2.1 Note that in the statement of the above theorem, the condition oo oo Pi Dn ψ 0 could have been replaced with the condition f] Dn consists of

n—l n = l exactly one point.


Here is a variant of Cantor's Theorem due to Ascoli.

Theorem 2.11 A metric space M is complete if and only if the intersection of any descending sequence of closed balL· in M having radii tending to 0 consists of exactly one point.

As an application of Cantor's Theorem we have a partial converse of Theorem 2.5.

Theorem 2.12 / / a subspace K of a complete metric space (M,d) is totally bounded, then it is precompact.

Proof. (This proof is admittedly tedious and perhaps best understood by drawing pictures to illustrate the first two or three steps.) In view of Theorem 2.3 it need only be shown that any sequence {xn} in K has a convergent subse-quence. Its limit will necessarily lie in K proving that K is compact. Note also that if an infinite number of terms of {xn} are the same then we could select a subsequence of {xn} which is constant and therefore converges trivially. So by throwing away some terms if necessary, we may assume that each two terms of {xn} are distinct.

Since K is precompact there exist points {21,1,22,1, ·· · ,2n,,i} Q K such that each point of K lies within distance at most 1 from at least one of these points, that is,

~Kç{jB{ziy,\). t = l

In particular, for one of the ζ,,ι'β, say zj, an infinite number of terms of {xn} lie in B\ — B{z\\ 1). Pick one of them, say xn,- Similarly there is a finite set {21,2,22,2, · · ■ , 2n2,2} Q K such that

KcijBîy

Since an infinite number of terms of {xn}n>n, he in B\, for one of the zit2S, say 22, an infinite number of terms of { i n } n > n i lie in B\ Π B-i where Bi =

B I 22; — 1 . Pick one of them, say xn.2. At this point, notice that diam(ßi Π

B2) < 1· Proceed by induction. Suppose we have obtained points {xni, · · · , xnt} i

and {21,· ·■ ,2fc} such that for each 1 nk he in f) Bj. \ l ) i=i

Apply total boundedness to proceed to the next step. There is a finite set


{zi.fc+i.^.fc+i.··· ,Znk+uk+i} Q K such that

t= l x '

In particular, for one of the Zitk's, say z;t+i, an infinite number of terms of fe+i ' / i \

{xn}n>nk lie in f] Bj where Bfc+i = B ( zk+\\ , I . Pick one of them, say

k

xn j + 1. Now let Dk = Π ß j , k = 1,2, · · · Then {Dk} is a descending sequence i= i

2 of closed sets with the property that diam(Z3fc) < —, k — 1,2, · · · . By Cantor's

K

intersection theorem,

oo

Π D" = { } fc=l

for some zeJ l f . Since xn/,. G Dk it follows that lim xnir = z. ■ fc—»oo

The following general fact about completeness also is quite useful. We omit the very easy proof.

Proposition 2.1 Every closed subspace of a complete metric space is itself com-plete.

Here is another interesting fact about completeness. A subspace 5 of a metric space (M, d) is said to be dense in M if the closure of S is all of M (i.e., S — M). Any metric space M is isometric with a dense subset of a complete metric space called the completion of M. (The use of the article 'the' can be justified by showing that any two completions of M must be isometric.) One way to see this is to consider the space Mc of equivalence classes of all Cauchy sequences in M, where two Cauchy sequences {xn} and {zn} are said to be equivalent (written {xn} ~ {zn}) if hrn d(xn, zn) = 0. Let

n—*oo

[{*„}] = { { * n } Ç M : { z n } ~ { * » } }

and for x* = [{x„}], y* = [{yn}}, set

d*(x",y*) = lim d{xn,yn). n—*oo

It is easily verified that (i) the above limit does indeed exist; (ii) the space {Mc,d") is complete; and (iii) M is isometric to the subspace of (Mc,d*) con-sisting of all equivalence classes of the form [{x}], x € M.

We now look at some specific examples.


Example 1. Completeness of the space R of real numbers (with usual distance d(x, y) — \x - y\) is a fundamental consequence of the sup axiom. Suppose {xn} is a Cauchy sequence of real numbers and define the two sequences {un} and {vn} as follows:

un = sup{xfc : A; > n}; vn = inf{xfc : k > n).

Then {it„} is monotone decreasing and bounded below (by inf{xn : n > 1}), while {vn} is monotone increasing and bounded above (by sup{xn : n > 1}). Therefore both lim un — u and lim vn = v exist. (In standard notation, we

n—*oo n—*oo

would write u = lim sup a; and v = lim inf xn.) Also, since {xn} is a Cauchy n—*oo n—»oo

sequence, it is easy to see that

lim \un - vn\ = lim un - vn = 0, n—*oo n—»oo

so u — v. Now let ε > 0 and choose N € N so that if n > Λ then both \un — v\ < ε/2 and \vn — v\ < ε/2. Since vn < xn < un, both xn and υ lie in the interval [wn,un], and if n > TV this interval has length less than e. Therefore if n > N, \xn — v\ < ε. ■

Example 2. Completeness of R" follows almost immediately from completeness of R although some awkwardness in notation arises. Indeed, if {XJ} is a Cauchy sequence in Rn then each element Xj is itself an n-tuple:

Now let ε > 0 and choose N so that if i, j > N, d(xi,Xj) < ε. Then for each fc = 1,2, · · ■ ,n :

1/2

x k ~ Xk < Σ(4 - Ί)2 < e. U = l

so for each such k, {x].}^ is a Cauchy sequence as well. By the completeness of R there exists for each such k a number x& such that lim x'k = Xfc. Now let

i—*oo

x = (xj, X2, · · · ! χη)- For each k = 1,2,··· , n choose TVfc € N so that i > Nk implies

K-xfc|<-i=.

Then if i > max{NuN2, ■■■ , NN},

1 / 2 / / J \ \ l / 2 ( n \ >■/* / / 2\ \ 1

Ç(xt-xfc)2j <{n{^))

e.


The above argument extends to £2 although a little more work is needed. The argument begins as above. If {XJ} is a Cauchy sequence in £2 then each element Xj is itself a sequence of real numbers:

Xj = ( X j , X 2 , · ■ · ) .

Now let e > 0 and choose JV so that if i, j > N, d(xi,Xj) < ε. Then for each fc = l , 2 , · - · :

1/2

xk ~ ^fc < 5>ί,-*£)2 <ε < n = l

so for each such k, {xj.}î is a Cauchy sequence as well. By the completeness of K there exists for each such k a number x/t such that lim x\ = Xfc. Let

i—>oo x = ( x 1 , x 2 , · · · ) .

00 Now there are two things to show. (1) x 6 £2, that is, J3 x^ < 00; and (2)

n = l lim Xi = x. To this end, let ε > 0 and choose N € N so that if i,j > N then

Κχ,,χ^ζΚ-χΙ)2^.

Let M € N and rewrite the above in the form

0 0 M 0 0

£(*i-4)2 = £(*i-*i)2+ Σ (4-*i)2<e. fc=l fc=l fc=M + l

This in particular means that £ (^* — ^ ) 2 < ε · Now fix j and let i —♦ 00 to fc=l

obtain

M

Σ>* - 4)2 < e·

Since this is true for any M we can let M —♦ co and conclude

0 0

J2(xk-x{)2<e. fc=l

0 0 00

The above, along with the fact that £ (χ£)2 converges, implies that £2 x2

fc=l fc=l

converges. This proves that x 6 ^2- At the same time we conclude that j > N implies

d(x,x.,) < ε.


Thus lim x, = x. j—>oo

Example 3. We now turn to the space C[0,1]. Since this space consists of all continuous real-valued functions denned on [0,1], with the distance d(f,g) between two such functions / , g taken as

d(f,g)=sup{\f(t)-g(t)\:te\0,l}},

it is immediate that if {/„} is a Cauchy sequence in C[0,1] then for each t 6 [0,1] {/„(i)} is a Cauchy sequence as well. Since {/n(0} Ç R for each t, lim fn(t)

n—>oo

exists. Let fit) = lim fJt), t 6 [0,1]. We need to show that / e C[0,1] and n—>oo

that lim d(fn,f) = 0 . n—»oo

Assume it is already known that lim d(fn, f) = 0. To see that / is contin-n—oo

uous let to € [0,1] and let ε > 0. Since lim d(fn, / ) = 0 there exists an integer n—>oo

N £ N such that

sup \fN(t)-f(t)\<£

-. £6[0,1] ύ

Also, since fn is continuous at to there is a number δ > 0 such that if t € [0,1] and if |i — t0\<6 then

| / Λ Κ « ) - Λ Τ ( * Ο ) | < ! ·

Hence if |i — to\ < δ,

1/(0 - /(*o)l < 1/(0 - fN(t)\ + \fN(t) - fN(to)\ + |/jv(io) - f(to)\ < e.

This proves continuity of / . To see that lim d(fn,f) = 0, let ε > 0 and n—*oo

observe that since {/„} is a Cauchy sequence there is an integer TV such that if m,n> N then

sup | / „ ( 0 - / m ( 0 l < e . «€[0,1]

that is,

/ n ( 0 - e < / m ( 0 < / n ( 0 + e ·

Letting m —» oo we see that for any t € [0,1] and n > N,

/n (0 ~e< / ( 0 < /n(0 + e;

hence

1/(0 - /n(0l < ε. from which d(/„, / ) < ε. Since ε > 0 is arbitrary we conclude lim d(fn, f) = 0.

2.4. SEPARABILITY AND CONNECTEDNESS 33

2.4 Separability and connectedness Most, but not all, of the metric spaces described in this chapter share another characteristic whose presence is often helpful to know.

Definition 2.8 A metric space M is said to be separable if it possesses a count-able dense subset.

Thus if M is separable, then there exists a sequence { « ι , ι * 2 , · · - } ζ Μ such that ifi4 = {u j : i = l , 2 , · · · } , then A — M. This concept will not play a large role in what follows so we only make a few quick comments here. However, one of the deeper results of Chapter 5 (Theorem 5.12) depends very heavily on separability.

First, the rational numbers form a countable dense subset of the reals R; hence R is separable. More generally, the space l\ of Example 2.6, all of the spaces iv (1 < p < oo) of Example 2.8, and the space C[0,1] of Example 2.9 are separable. The space ί^ is not.

For the lv spaces (1 < p < oo) a countable dense subset is obtained by taking the collection of all sequences which have rational coordinates and for which all but a finite number of coordinates are 0. The latter restriction is essential to assure that the dense subset is in fact countable.

For C[0,1] the situation is more intricate. For given n G N let P„ denote the set of all functions defined as follows. If x = k/n, k = 0,1, · · · ,n, let f(x) be a rational number, and for (k — \)/n <x< k/n let / be the function whose graph is a line joining the point ((k — l ) /n , / ((k — l ) /n)) to the point (k/n, f (k/n)). Then the set P = |J P„ is countable, and moreover, P = C[0,1]. The proof

n>l of the latter fact hinges on the fact that any function in C[0,1] is uniformly continuous. Armed with this fact, let / € C[0,1], let ε > 0, and choose δ > 0 so that \f(u) — f(v)\ < ε/5 whenever |u — v\ < 6. Now choose n so that 1/n < 6. For each fc = 0,1, · · · ,n it is possible to choose a rational number Wk so that \f(k/n) — Wk\ < e/5. Now let g 6 P„ be the function for which g(k/n) = Wk-Then if x € [k/n, (fc + 1) / n ] ,

\f(x)-9(x)\ < \f(x)-f(k/n)\ + \f(k/n)-g(k/n)\ + \g(k/n)-g(x)\ < 2e/5 + \g(k/n)-g({k + l)/n)\ < 2e/5 + | /(fc/n)-/((fc + l ) /n) |+2e/5

< ε.

Having established the above, it follows that given any / € C[0,1] there exists g € P such that d(f,g) < ε. This implies P = C[0,1].

What we have just observed is that there is a countable family of piecewise linear continuous functions which is dense in C[0,1]. It is also the case that the family of all polynomial functions with rational coefficients is dense in C[0,1]. This collection is countable as well.


In fact, the space C[0,1] is more than just separable. It is universal with respect to separability in the following profound sense!

Theorem 2.13 Every separable metric space is isometric with a subset ofC[0,1]

It is possible to give an elementary (but quite detailed) proof of this fact; see, for example, [145], p. 192.

The following is an easy consequence of the fact that compact metric spaces z.:e totally bounded.

Theorem 2.14 Every compact metric space (M,d) is separable.

Proof. Let (M,d) be a compact metric space and for each n € N, let Pn

be a subset of M which is maximal with respect to the property: if x,y € Pn

then d(x,y) > 1/n (a simple induction argument suffices for this). Since M oo

is compact each of the sets Pn is finite, so P = (J Pi is countable. Also by i = l

maximality of Pn, if x ç M it must be the case that there exists p 6 Pn such that d(x,p) < 1/n. Now let e > 0 and choose n so that 1/n < e. It follows that if x 6 M there exists p 6 P such that d(x,p) < 1/n < e, and this proves that ~P = M. m

Finally, we mention another interesting fact about separability. If (M, d) is separable then the space of all nonempty bounded closed subsets of M endowed with the Hausdorff metric need not be separable. However, if only the collec-tion of all nonempty compact subsets is considered then the resulting space is separable (Exercise 2.23).

Another basic topological concept which arises often is that of connectedness.

Definition 2.9 A metric space (M, d) is said to be connected if M is not the union of two disjoint nonempty open subsets of M.

The above definition immediately leads to the conclusion that a metric space is connected if and only if the only subsets of M that are both open and closed are the empty set and M itself.

Obviously the real line R is connected, as is any interval and ray in R. Indeed, the Intermediate Value Theorem (Theorem 1.1 of Chapter 1) rests on the connectedness of intervals in R. In fact, a more general version of that theorem holds.

Theorem 2.15 (Intermediate Value Theorem) Let M be a connected metric space and let f : M —> R be continuous. Suppose x,y e M and suppose c is any point which lies between f(x) and f(y). Then there is a point z 6 M such that f(z) = c.

The proof is left as an exercise.

2.5. METRIC CONVEXITY AND CONVEXITY STRUCTURES 35

2.5 Metric convexity and convexity structures

If (M, d) is a metric space and if p, q 6 M then a point r 6 M is said to be metrically between p and q \î p ^ r φ q and ef(p, g) = d(p, r) + c<(r, q). A metric space is said to be metrically convex if given any two points p, q 6 M there exists at least one point r 6 M such that r is metrically between p and q.

Metric convexity is a fundamental concept in the axiomatic study of the geometry of metric spaces. However, in its most general form, it fails to satisfy one of the basic properties of convexity in the algebraic sense (in a linear space); namely, if A and B are two metrically convex subsets of a metric space M, then it need not be the case that AdB is metrically convex. To see this it is sufficient only to consider the ordinary unit circle S in K2 where the distance between two points is taken to be the length of the shortest arc joining them. The upper half circle and lower half circle of S are each metrically convex but the intersection of these two sets is easily seen to be a set consisting of just two antipodal points.

A subset 5 of a metric space M is called a metric segment with endpoints (or joining) p, q 6 S if there exists a closed interval [a, b] in the real line R and an isometry φ which maps [a, b] onto S with φ(α) = p and y>(b) = q. If a metric space M has the property that each two of its points are the endpoints of a metric segment, then clearly M is metrically convex, and it is natural to ask when the converse is true. The answer is surprisingly often. This is a fundamental result in the theory due to Karl Menger, one of the pioneers in the study of abstract metric spaces. For a proof see, for example, [12], [67].

Theorem 2.16 If M is a complete and metrically convex metric space, then each two points of M are the endpoints of at least one metric segment of M.

We shall be concerned almost entirely with complete metric spaces; hence with the stronger notion of convexity assured by Menger's Theorem whenever the concept arises. For reasons that will be clear later, the usefulness of metric convexity in a fixed point context is hindered by the fact that the family of all convex subsets of a metrically convex metric space need not be stable under intersections. A quick way to remedy this defect is to turn to a more abstract notion of convexity.

A family C of subsets of a set X is called an (abstract) convexity structure if

(1) Both 0 and X are in C.

(2) C is stable under intersections; that is, if {Da}aei is any nonempty subfamily of C then f) Da € C.

(3) C is stable for nested unions; that is, if {Da}aej is any nonempty sub-family of C which is totally ordered by set inclusion, then (J Da £ C.

«6/


Unfortunately, for our purposes, there is even a problem with this abstract formulation. In the study of metric fixed point theory closed balls turn out to play a fundamental role. A subset A of a bounded metric space M will be said to be an admissible subset of M if A can be written as the intersection of a family of closed balls centered at points of M. The family A{M) of all admissible subsets of M enters into the study of metric fixed point theory in a very natural way and is therefore the obvious candidate for the needed underlying convexity structure. However, while this family satisfies assumptions (1) and (2), in general, property (3) does not hold. This fact cannot be rectified even if the collection is enlarged to include all ball intersections rather than just intersections of closed balls. Nonetheless, the family A(M) will arise frequently in the ensuing chapters. Thus we shall exchange property (3) with the assumption that the convexity structure under consideration contains all the closed balls of M.

The structure A{M) has one very distinctive advantage when compared with metric convexity. Any subset A of a metric space is contained in the set

cov(A) = (~){B : B is a closed ball in M and B D A}.

Moreover cov(yl) € A{M). This provides a nice analog for the concept of 'closed convex hull' in functional analysis, with the family A(M) replacing the family of all closed (algebraically) convex sets.

This analogy does not quite work with metric convexity. It is natural to define the metric convex hull of a subset A of a convex metric space M to be a set which is closed, metrically convex, contains A, and for which none of its proper subsets has those properties. With this definition the existence of a unique metric convex hull for each subset A of a complete convex metric space which has unique metric segments is immediate. Merely take the 'metric convex hull' of A to be the intersection of all closed convex sets which contain A. However, problems arise in the general case since, as we have observed, the intersection of two metrically convex sets need not be metrically convex. On the other hand, if M is any compact metric space and if A Ç M, then A does have a metric convex hull, although it need not be unique. This is a classical result proved by Karl Menger in 1931. It rests on the following lemma.

L e m m a 2.1 Let C\ 2 C*2 =? · ■ ■ be a descending sequence of nonempty closed oo

metrically convex subsets of a compact metric space (M,d). Then f] Cn is n=l

nonempty and metrically convex.

Proof. The fact that the intersection is nonempty is immediate from com-oo

pactness. Suppose x,y E f] Cn with x φ y. Then in each of the sets C„ there n=l

exists a point zn such that

d{x, zn) = d{y, Zn) = -d{x, y).

2.5. METRIC CONVEXITY AND CONVEXITY STRUCTURES 37

(This uses the fact that x and y are actually joined by a metric segment which lies in Cn.) By compactness of M the sequence {zn} has a subsequence {zn„} which converges to a point z 6 M and since each of the sets Cn is closed,

oo

z G f) Cn- Since the metric d is continuous, 71 = 1

d(x,z) =d(y,z) = -d(x,y).

oo

This proves that p | Cn is metrically convex. ■ n=l

Theorem 2.17 If A is any nonempty subset of a compact and metrically convex metric space M, then A has a metric convex hull.

Proof. Let Σ denote the collection of all closed metrically convex subsets of M which contain A. Observe that Σ ^ 0 since M e Σ. Since M is compact, for each n e N there is a finite collection Un = {i/n(i),^n(2) >·■· Un(k„)} of open balls of radius 1/n which covers M. For each K £ T, there is a well-defined integer μη{Κ) {μη{Κ) < k(n)) such that K intersects μη(Κ) members of Un. Choose Ci € Σ so that

μ1(ΰ1)=Μ{μ1(0):0ΕΣ}.

Having defined Cn choose C„+i Ç Cn so that

M„+i(C„+i) - ΐ η ί { μ „ + 1 ( 0 : C € Σ and C Ç Cn).

oo

Then clearly C\ D C-i D ■ ■ ■ and A C C — f] C„. Also, by Lemma 2.1, C is 7 1 = 1

metrically convex. Suppose some H € Σ is a proper subset of C. Then there exists a point c 6 C\H and in turn there exists an integer n for which the open ball U(c; 1/n) Π C C C\H. In particular, some member U of ÏÀ2n lies in U(c; 1/n); hence

KnW < M2„(C) < M2n(^2n),

contradicting the definition of μ^Λ^η)- ■

It is possible to give a much quicker proof of the above theorem using Zorn's Lemma. (See the Appendix.) The proof just given is Menger's original and it predates the discovery of Zorn's Lemma. In fact, with Zorn's Lemma it is quite easy to prove the following.

Theorem 2.18 Let M be a metrically convex metric space, and suppose the intersection of every descending chain of closed metrically convex subsets of M is itself metrically convex. Then every nonempty subset A of M has a metric convex hull.


Proof. As above, let Σ denote the family of all closed metrically convex subsets of M which contain A, and order Σ by set inclusion. Then Σ Φ 0 since M e Σ, and by assumption every descending chain in (Σ, ~D) is bounded below by its intersection. By Zorn's lemma Σ has a minimal element, completing the proof. ■

Exercises

Exercise 2.1 Show that if a semimetric space {M,d) has continuous distance function, then

lim pn —P and lim pn = q => p — q. n—»oo n—*oo

Exercise 2.2 Suppose (M,d) is a semimetric space and suppose {xn} is a se-quence in M which converges to x € M. Is {x„} necessarily a Cauchy sequence? What if d is continuous?

Exercise 2.3 Give an example of two different metric spaces, each of which is isometric with a subspace of the other.

Exercise 2.4 Suppose (M,d) is a metric space and suppose {xn} is a sequence in M which converges to x 6 M. Show that {xn} is a Cauchy sequence.

Exercise 2.5 Let M be the real unit interval [0,1] and for x,y £ M define d(x,y) — \x — y\ . Show that (M,d) is a semimetric space with continuous distance which is not a metric space.

Exercise 2.6 Let S be any set of nonnegative real numbers which contains 0. Show that there is a metric space (M,d) such that given r 6 S there exist x,y € M for which d(x, y) 6 S.

Exercise 2.7 Complete the proof of Theorem 2.3, that is, show that if any sequence {xn} of points of S has a subsequence {xnk} which converges to a point of S, then S is compact.

Exercise 2.8 Show that if X is a compact metric space, and f : X —> R is a continuous function, then there is a point x\ £ X such that

f(xl)=snp{f{x):x€X}.

EXERCISES 39

Exercise 2.9 Suppose A is a precompact subset of a metric space (M,d) and suppose T : A —* M is continuous. Then is T uniformly continuous?

Exercise 2.10 Show that (Μ,άφ) is a metric space.

Exercise 2.11 Show that (M,p„) is a metric space.

Exercise 2.12 Show that if (M,d) is compact, then (M,pp) is compact (Ex-ample 2.17).

Exercise 2.13 Suppose a sequence {xn} in a metric space (M,d) satisfies the following condition. For each p € N and each e > 0 there exists N 6 N such that if n > N, then d(xn,xn+p) < ε. Is {xn} necessarily a Cauchy sequence?

Exercise 2.14 Let {xn} be a sequence in a metric space for which

oo

^2d(xi,xi+1) < oo. i = l

Show that {xn} is a Cauchy sequence. Is the converse true?

Exercise 2.15 Prove Theorem 2.11.

Exercise 2.16 Show that (Mc,d*) is a metric space.

Exercise 2.17 Show that (Mc,dm) is complete.

Exercise 2.18 Show that (M,d*) is dense in (Mc,d*).

Exercise 2.19 Show that the space of Example 2.11 is not complete.

Exercise 2.20 Show that if M is complete then the space (M,H) of Example 2.15 is complete.

Exercise 2.21 Show that the metric space consisting of all irrational numbers is separable.

Exercise 2.22 Show that any subspace of a separable metric space is separable.


Exercise 2.23 Show that the space of all nonempty compact subsets of a sep-arable metric space M endowed with the Hausdorff metric is separable.

Exercise 2.24 Let (M,d) be a complete metric space and φ : M —» [0, oo) an arbitrary nonnegative function. Assume that

ΐηΐ{φ{χ) + 4>(y) : d(x, y) > ε} > 0

for any ε > 0. Prove that each sequence {xn} in M such that φ{χη) —* 0, converges to one and the same point x € M .

Exercise 2.25 Let (M,d) be an arbitrary metric space and let A C M be a compact set. Let φ : M —> [0, oo) be an arbitrary nonnegative function such that

Μ{φ(χ) : d{x,A) >ε} > 0

for any ε > 0. Prove that each sequence {xn} in M such that φ(χη) —> 0, contains a subsequence which converges to some point x € A.

Chapter 3

Metric Contraction Principles

3.1 Banach's Contraction Principle

Banach's Contraction Mapping Principle is remarkable in its simplicity, yet it is perhaps the most widely applied fixed point theorem in all of analysis. This is because the contractive condition on the mapping is simple and easy to test, be-cause it requires only a complete metric space for its setting, because it provides a constructive algorithm, and because it finds almost canonical applications in the theory of differential and integral equations. Although the basic idea was known to others earlier, the principle first appeared in explicit form in Banach's 1922 thesis [7], where it was used to establish the existence of a solution to an integral equation. The underlying space was taken to be the space C[0,1] of Example 2.9 of Chapter 2.

Let (M, d) be a metric space. A mapping T : M —> M is said to be lipschitz-ian if there is a constant fc > 0 such that for all x, y 6 M

d(T(x),T(y))<kd(x,y). (3.1)

The smallest number k for which Eq. (3.1) holds is called the Lipschitz constant of T.

Definition 3.1 A lipschitzian mapping T : M —» M with Lipschitz constant k < 1 is said to be a contraction mapping.

Theorem 3.1 (Banach's Contraction Mapping Principle) Let (M,d) be a com-plete metric space and let T : M —♦ M be a contraction mapping. Then T has a unique fixed point xo, and for each x S M, lim Tn(x) = χο- Moreover,

η—*οο

d{Tn{x),x0)<Y^d(xiT(x)).

41



42 CHAPTER 3. METRIC CONTRACTION PRINCIPLES

Proof. Suppose T has Lipschitz constant k e (0,1). Then for each x € M,

d(T(x),T2(x)) < kd(x,T(x)).

Adding d(x,T(x)) to both sides of the above gives

d(x,T(x)) + d(T(x),T2(x)) < d(x,T(x)) + kd(x,T(x)),

which can be rewritten as

d(x, T(x)) - k d(x, T(x)) < d(x, T(x)) - d(T(x), T2(x)).

This, in turn, is equivalent to

d{x,T(x)) < (1 - k)-l[d{x,T{x)) - d(T(x),T2(x))}.

Now define the function φ : M —♦ R+ by setting ψ{χ) = (1 — k)~1d(x,T(x)), for x e M. Thus

d(x, T{x)) < φ(χ) - φ(Τ(χ)), χ£Μ.

Therefore if x G M and m, n G N with n < m,

m

d(T"(x) ,Tm + 1(x)) < Y^d(Ti(x),Ti+\x)) < φ{Τη{χ))-φ{Τ™+\χ)). i—n

(Note that the last inequality comes from cancellation in the telescoping sum.) In particular, by taking n = 1 and letting m —» oo we conclude that

oo

^ d ( T l ( x ) , r i + 1 ( x ) ) < ^ ( T ( x ) ) < c o .

This implies that {T"(x)} is a Cauchy sequence (Chapter 2, Exercise 2.14). Since M is complete there exists XQ£ M such that

lim Tn(x) = xo n—»oo

and since T is continuous

x0 = lim Γη(χ) = lim T n + 1 (x) = T(x0).

Thus xo is a fixed point of T. In order to complete the proof of the existence part of the theorem we need to show that xo is the only fixed point. Indeed, let y be a fixed point of T. Then from the previous result,

xo - lim Tn{y) = y.

3.1. BANACH'S CONTRACTION PRINCIPLE 43

Returning to the inequality

d(Tn(x),Tm+1(x)) < φ(Τη{χ)) - <£>(Tm+1(:r)),

upon letting m —* oo we see that

d(Tn(x),x0) < φ(Τη(χ)) = (l-k)-1d(Tn(x),Tn+1(x)).

Since (1 - k)-ld(Tn{x),Tn+l(x)) < - -d(x,T(x)), we obtain 1 — k

d(Tn(x),x0)<^d(x,T(x)).

While the preceding estimate on the rate of convergence of {Tn (x)} is sharp, in practical situations it might be necessary also to approximate the values of Tn(x). Here a natural question arises. If one replaces the sequence {T*1^)} with {yn} where yo — x and yn+i is 'approximately' T(yn), then under what conditions will it still be the case that lim y„ = xo? The following response

n—»oo

to this question was given by Ostrowski in [129]. (For a recent generalization of this result as well as related literature, see [76].)

Theorem 3.2 Let (M,d) be a complete metric space, let T : M —* M be a contraction mapping xuith Lipschitz constant k € (0,1), and suppose XQ 6 M is the fixed point of T. Let {εη} be a sequence of positive numbers for which lim εη — 0, let yo € M, and suppose {yn\ Ç M satisfies

n—»oo

d(yn+i,T(yn)) < e „ .

Then lim yn = XQ-n—»oo

Proof. Let yo = x and observe that

d(Tm+1(x)tym+i) < d(T(Tm(x)),T(ym)) +d(T(ym),ym+1)

<kd{Tm(x),ym) + em,

which implies

m

d(Tm+1(x),ym+1) < ^ f c m " ^ .

Thus

d(ym+i,*o) < d(ym+uTm+1{x))+d(Tm+l(x),x0)

m

< 5 ^ Ä m - i e i + d(Tm+1(x),a:o)· i=0


Now let ε > 0. Since lim e„ = 0 there exists N € N such that for m> N, η—·οο

£ m < £ . Thus

m N m

Y^kêi =Yjkm-iei+ Σ km~'£*

i=0 i=0

km-N

■NJ2kN-i=0

i=N+l

~î + ε 771

Σ i=N+l

k"1-*

Hence

i=0 x '

Since ε > 0 is arbitrary, and since lim d(Tm+1(x),xo) = 0, we conclude that m—>oo

lim ym+i = XQ. m m—>oo

The contraction principle also has a trivial, yet somewhat surprising, exten-sion. In this theorem the mapping T is not even assumed to be continuous!

Theorem 3.3 Suppose (M,d) is a complete metric space and suppose T : M —> M is a mapping for which TN is a contraction mapping for some positive integer N. Then T has a unique fixed point.

Proof. By Banach's Theorem TN has a unique fixed point x. Hence

TN+1(x)=T(TN(x)) = T(x),

so T(x) is also a fixed point of TN. Since the fixed point of TN is unique, it must be the case that T(x) = x. Also, if T(y) — y, then TN(y) = y, proving (again by uniqueness) that y — x. ■

The following is another extension of Banach's principle. We omit the proof since a more general result is taken up in the next section. However, the result itself is a nice excercise.

Theorem 3.4 Let M be a metric space that has two metrics d and p, and suppose p(x,y) < d(x,y) for each x,y 6 M. Suppose (M, p) is complete, and suppose T : M —» M is a continuous mapping with respect to the metric p and a contraction mapping with respect to the metric d. Then T has a unique fixed point in M.

In considering lipschitzian mappings an obvious question that arises imme-diately is whether it is possible to weaken the contraction assumption even a little bit and still obtain the existence of fixed points. In a broad sense the answer is no and here is an example. Begin with the complete metric space

3.1. BANACH'S CONTRACTION PRINCIPLE 45

C[0,1] and consider the closed subspace M of C[0,1] consisting of those map-pings / 6 C[0,1] for which / ( l ) = 1. Since M is a closed subspace of a complete metric space, M is itself complete. Now define T : M —► M by taking T( / ) to be the function in M obtained by setting

T(f)(t) = tf(t) t € [0,1].

If f,g e M then \T(f) - T(g)\ e C[0,1], so by the Maximum Value Theorem \T(f) — T(g)\ attains its maximum value at some point to € [0,1]. We then have

d(T(f),T(g))= sup \T(f)(t)-T{g){t)\=t0\f(to)-g(to)\<d(f,g). <€[0,1]

But if / φ g it must be the case that f(t) φ g(t) for some t G [0,1] and since / ( I ) = p(l) — li this in turn implies io < 1· Therefore, if f,g e M and f φ g,

d(T(f),T(g))<d(f,g).

Now suppose T(f) — f for / 6 M. This implies that for each t e [0,1], f(t) = tf(t). This implies that f(t) = 0 for all t e [0,1). On the other hand, / ( l ) = 1. This contradicts the assumption that / is continuous, so T can have no fixed point in M. Therefore the Banach Contraction Mapping Principle does not even extend to the following slightly more general class of mappings.

Definition 3.2 A mapping T : M —» M is said to be contractive if

d(T(x),T(y))<d(x,y)

for each x, y € M with x Φ y.

The next obvious question is whether there is a meaningful fixed point result for the contractive mappings. Now the answer is yes, but the class of spaces to which it applies is much more restrictive.

Theorem 3.5 Let (M,d) be a compact metric space and let T : M —» M be a contractive mapping. Then T has a unique fixed point XQ, and moreover, for each x £ M, lim Tn(x) = xo.

n—*oo

Proof. The existence of a fixed point for T is easy. Introduce the mapping ψ : M -* K+ by setting

φ(χ) = ά(χ,Τ(χ)), xeM.

Then ψ is continuous and bounded below, so φ assumes its minimum value at some point XQ s M. Since xo Φ T(xo) implies

φ(Τ{χ0)) = d{T{x0),T2(x0)) < d{x0,T(x0)) = ψ(χ0),

it must be the case that XQ = T(XQ).


Now let x € M and consider the sequence {d(Tn(x),xa)}. If Tn(x) φ xo,

d(Tn+1(x),x0) = d(Tn+1(x),T(x0)) < d(T"(x),x0),

so {d(Tn(x),xo)} is strictly decreasing (until perhaps it reaches xo). Conse-quently the limit

r = lim d(Tn(x),x0) n—*oo

exists and r > 0. Also, since M is compact, the sequence {Tn(x)} has a con-vergent subsequence {Tn*(x)}, say lim Tn*(x) = z € M. Since {Tn(x)} is

fc—»CO

decreasing,

r = φ , χ 0 ) = um d(Tn" (x), xQ) = lim d(Tnfc+1(:r),zo) = d{T(z),xQ). fc—»oo fc—*oo

But if z φ XQ then d(T(z),xo) = d(T(z),T(xo)) < d(z,xo) · This proves that any convergent subsequence of {Tn(x)} converges to xo, so it must be the case that lim Tn(x) = x0. ■

n—»oo

3.2 Further extensions of Banach's Principle

The strength of the Contraction Mapping Principle lies in the fact that the un-derlying space is quite general (complete metric) while the conclusion is very strong. The fixed point is unique and sequences of iterates of the original map-ping always converge to this fixed point. It seems that any theorem which should claim to be a genuine extension of the contraction principle should offer just as much, and indeed, many such extensions appear in the literature. The extensions we present in this section are easy to describe and include many of the others. They are in no way definitive, however.

The first extension we take up, which is due to Geraghty [63], was inspired by an earlier theorem of Rakotch [134]. We then consider extensions due to F. Browder and others.

Let <S denote the class of those functions a : R+ -+ [0,1) which satisfy the simple condition a(tn) —♦ 1 =*> tn —> 0.

Theorem 3.6 Let (M, d) be a complete metric space, let T : M —> M, and suppose there exists a G S such that for each x,y € M,

d(T(x),T(y))<a(d(x,y))d(x,y).

Then T has a unique fixed point z 6 M, and {Tn(x)} converges to z, for each x € M.

Proof. Fix x € M and let xn = Tn(x) , n = 1,2, · · ·. We break the argument into two steps, each of which illustrates something more.

3.2. FURTHER EXTENSIONS OF BANACH'S PRINCIPLE 47

Step 1. lim d(xn,xn+i) = 0 .

Proof. Since T is contractive the sequence {d(xn,xn+i)} is monotone de-creasing and bounded below so lim d(xn,xn+i) = r > 0. Assume r / 0 . Then

n—*oo by the contractive condition

- ^ - - < a(d(xn,xn+i)), n = l , 2 , · · · .

Letting n —♦ 00 we see that 1 < lim a(d(xn,xn+{)), and since a 6 S this in n—>oo turn implies r = 0. This contradiction establishes Step 1.

Step 2. ( i „ ) is a Cauchy sequence.

Proof. Assume l imsupd( i n ,x m ) > 0. By the triangle inequality

m.n—+00

d(xn,xm) < d(xn,xn+i) + d(xn+i,xm+i) + d(xm+\,xm),

so by the contractive condition

d(xn,xm) < (1 -a(d(a;„,a;m)))-1[flf(:rn,2n+1) + d ( i m + i , i m ) ] .

Under the assumption limsupci(:En,Zm) > 0 Step 1 now implies m,n—»oo

limsup(l — a(d(xn,xm)))~1 — +00, m,n—» oo

from which

limsupûi(ci(a;n,xm)) = 1. τη,η—· 00

But since a ζ S this implies limsupii(a;n,xm) = 0, which is again a contradic-m,n—>oo

tion.

Proof of Theorem 3.6 completed. Let x G M. Since M is complete and since {Tn(x)} is a Cauchy sequence, lim Tn(x) = z e M, and since T is continuous,

n—»00

T(z) = 2. Uniqueness of z follows from the contractive condition on T. ■

For the next result we let S' denote the collection of all monotone decreasing mappings ψ : R+ —♦ R + for which 0 (r) < r and for which φ is continuous from the right (i.e., rj J. r > 0 => ^( r j ) ~* VK1"))· This extension of Banach's Principle is due to Browder [25].


Theorem 3.7 Let (M,d) be a bounded complete metric space, letT : M —» M be continuous, and suppose there exists ψ G S' such that for each x,y E M,

d(T(x),T(y))(d(x,y)). (3.2)

Then T has a unique fixed point z, and {Tn(x)} converges to z, for each x 6 M.

Proof. This theorem is actually a special case of the previous theorem. First introduce the function φ : R —» [0,1) by setting ^>(0) = φ(0) and

φ(ί) = Ά for t > 0.

To see that φ is in the class S suppose φ{ίη) —» 1. Then {i„} must be bounded (otherwise, lim inf φ(ίη) = 0 ) . By passing to a subsequence we may assume

n—>oo

that tn —♦ to and we may assume further that either tn ] <o or tn I to. But since φ is continuous from the right, the assumption tn J, to leads to φ(ίο) = to, and since φ{τ) < r for r > 0 this in turn implies io = 0, that is, tn —> 0. On the other hand, if tn ] to we have, by monotonicity of φ,

Φ{ίο) > Φ{ίη) |

t ~ t ~> '

from which > 1 and this again implies i0 = 0. So, any convergent subse-

quence of the original sequence {tn} must converge to 0. It follows that tn —> 0, proving that φ is in the class S. Since

d(T{x),T(y)) < î>(d{x,y))d(x,y), x,y € M,

the proof is complete. ■

Subsequently, Boyd and Wong [17] obtained a more general result. In this theorem it is assumed that φ : R+ —» R + is upper semicontinuous from the right (i.e., rj [ r > 0 => lim sup φ(τ^) < 0(r)).

j—*oo

Theorem 3.8 Let M be a complete metric space and suppose f : M —► M satisfies

d (f (x), f (y)) <Φ(ά (x, y)) for each x,y 6 M,

where φ : R+ —> [0, oo) is upper semi-continuous from the right and satisfies 0 < φ(ί) < t for t > 0. Then f has a unique fixed point, x, and {fn (x)} converges to x for each x € M.

Proof. Fix x G M and let xn —Tn(x),n= 1 ,2 , · · · . We break the argument into two steps.


Step 1. lim d(xn,xn+i) = 0. n—»oo

Proof. Since T is contractive the sequence {d(xn,xn+\)} is monotone de-creasing and bounded below so lim d{xn,xn+\) = r > 0. Assume r > 0. Then

n—»oo

d(arn + 1,xn +2) < ψ(ά(χη,χη+ι)) =Φ· r < V(r) => r = 0.

Step 2. {xn} is a Cauchy sequence.

Proof. Suppose not. Then there exists ε > 0 such that for any k 6 N, there exist m/t > nk > k, such that

d(xmk,xnk) > ε. (3.3)

Furthermore, assume that for each k, m^ is the smallest number greater than n^ for which (3.3) holds. In view of Step 1 there exists ko such that k > ko => d(xk,Xk+i) < e. For such k we have

e S " \£mk i Xnk J S " (^mi. i ^τη^ — 1 ; T" " \%mk — li Xnk)

< d(xmk,xmk-i) +ε < d(xk,Xk-\) + ε.

This proves lim d(xmk,xnk) = ε. On the other hand,

)+d(xnk + l,Xnk)

< 2d(Xk,Xk-l) + i){d{xrrlk,

It follows that ε < V (ε) — a contradiction.

The proof is completed by observing that since {Tn(x)} is a Cauchy sequence and M is complete, lim Tn(x) = z e M. Since T is continuous, T(z) = z.

n—»oo

Uniqueness of T follows from the contractive condition. ■

Remark 3.1 Boyd and Wong also showed in [17] that if the space M is met-rically convex, then the upper semicontinuity assumption on ψ can be dropped. Matkowski has extended this fact even further in [117] by showing that it suffices to assume that ψ is continuous at 0 and that there exists a sequence ί„ | 0 for which ψ{ίη) < tn.

Since it is the explicit control over the error term that contributes so much to the widespread usefulness of Banach's principle, the following variant of the Boyd-Wong theorem due to Browder [25] is also of interest.

Theorem 3.9 Let X be a complete metric space and M a bounded closed subset of X. Suppose f : M —> M satisfies

d ( / (x) ,f{y))(d (x, y)) for each x,y € M,


where φ : [0, oo) —» [0, oo) is monotone nondecreasing and continuous from the right, such that ip(t) < t for all t > 0. Then there is a unique element x Ç. M such that {fn (x)} converges to x for each x 6 M. Moreover, if do is the diameter of M, then

and φη (do) —* 0 as n —> oo.

Another variant is due to Matkowski [118]. In this result the continuity condition on φ is replaced with another condition.

Theorem 3.10 Let M be a complete metric space and suppose f : M —> M satisfies

à ( / (x), / (y)) <Φ{ά (x, y)) for each x,y € M,

where φ : (0, oo) —» (0, oo) is monotone nondecreasing and satisfies lim φη (t) — n—*oo

0 for t > 0. Then f has a unique fixed point x, and lim d ( / " (x), x) = 0 for n—»oo

every x G M.

Proof. Fix x € M and let xn — Tn(x), n = 1,2, · · ■. As before, we break the argument into two steps.

Step 1. lim d(xn,xn+\) = 0. n—»oo

Proof. 0 < limsupd(a;n,a;n+1) < lim sup V'" (d{x,X\)) = 0. n—»oo n—·οο

Step 2. {xn} is a Cauchy sequence.

Proof. Since rpn (t) —► 0 for t > 0, V (e) < ε f°r any ε > 0. In view of Step 1, given any ε > 0 it is possible to choose n so that

d(xn+ï,xn) <ε-ψ(ε).

Now let

K (χη,ε) = {x e M : d (x, xn) < ε).

Ί\ινΛ\ίζ£Κ(χη,ε),

d{T(z),xn)<d (Γ (z), T (xn)) + d(T(xn),xn)

<ψ(ά(ζ,χη))+ά (χ„+ι, xn)

<Φ(ε) + (ε-φ(ε))=ε.

Therefore T : K(xn,e) —» Κ(χη,ε) and it follows that d(xm,xn) < e for all m>n. This completes Step 2.

The conclusion of the proof follows as in Theorem 3.8. ■


The key step in proving the existence of a fixed point in each of the previous results involved showing that given x € M, {T"(x)} is a Cauchy sequence (and then invoking continuity of T). It is possible to carry this idea much further. First we need some notation. For any mapping F : M —> M we use k(F) to denote the Lipschitz constant of F; thus

Note in particular that if F, G : M —> M are two lipschitzian mappings, then

k{FoG)<k(F)k(G).

Now let (M,d) be a complete metric space and suppose T : M —* M is lipschitzian. Fix x e M, and let xn = Tn(x), n — 1,2, ·■· . By the triangle inequality, if m > n, then

m - l

d{xn,Xm) < Σ, d(xi>xi+l)-i=n

oo

Consequently {xn} is a Cauchy sequence if Σ d{xi>xi+i) < °°> a n d since i= l

d(xi,xi+1) = d(T{x),Ti+l(x)) < k{T)d(x,T{x))

it follows that

oo oo

, Xi+l) < OO. i= l i= l

Also

k(Tm+n) < k(Tn)k(Tm).

Specifically [k(Tn)}1/n < k{T) and so

fcoo(T)=limsup[fc(Tn)]1/n

n—>oo

exists. This is all we really need at this point. However it is possible to say more. Replacing T with T? in [Α;(Γη)]1/η < k(T) and taking the pth root of both sides gives

\k(TPn)}l/pn < \k{Tp)]1/p , p = 1,2, · ■ ·

Also, a simple calculation shows that

fen tfc(r")l1/n =l. n—oo \k(Tn+1)]1^n+1


These two facts lead to the conclusion:

*οο(Γ)= lim [fc(Tn)]1/n = inf{[fc(T")]1/n:n = l , 2 , · · - } .

Now, by the Root Test for convergence of series, if / ^ ( T ) < 1 then Σ fc(Tl) < t = l

oo. This leads to the following theorem.

Theorem 3.11 Let M be a complete metric space and let T : M —> M be a continuous mapping for which /Coo(T) < 1. Then T has a unique fixed point z 6 M, and for each x e M the sequence {Tn(x)} converges to z.

Proof. In view of what was shown just prior to the statement of the theo-rem, all that remains is to show that the fixed point of T is unique. However, this follows from the fact that if ^ ( T ) < 1, then for n sufficiently large, k(Tn) < 1. ■

There is a point to the previous development. One might ask whether qual-itatively, Theorem 3.11 is stronger than Banach's contraction principle, and indeed the answer is no. To see this we introduce the concept of equivalent metrics.

Two metrics, p and d on a space M are said to be equivalent if there exist positive numbers a and b such that for each x,y € M

ad(x,y) < p(x,y) < bd(x,y).

From this it follows that if T : M —» M then

kd(T) < oo <* kp(T) < oo.

Moreover,

d(T(x),T(y)) < -p(T(x),T(y)) < -kp(T)p(x,y) < -kp(T)d(x,y). a a a

This implies kd(T) < -kp{T). Similarly,

p(T(x),T(y)) < bd(T(x),T(y)) < bkd(T)d(x,y) < -kd(T)p(x,y); a

hence kp(T) < -kd(T). Consequently,

\kp{T) < kd(T) < -akp{T),


and since lim I — n—oo V a

l / n 'a\1/n

= lim ( — ) = 1 we conclude

lim [kd(Tn)}l/n = lim (Μ τ ") ] 1 / η ·

Hence (kd)^ (T) = {kp)oo(T), that is, k^T) is the same for all equivalent metrics on M.

Now suppose (M,d) is a complete metric space and let T : M —* M be a

mapping for which fcoo(T) < 1. Then if λ € 0,

with

Therefore,

koo(T)J '

Xnd(Tn(x),Tn(y)) < Xnk(Tn)d(x,y)

lim [AnÄ(Tn)]1/n = A M T ) < 1. n—*oo

co

αθτ, ΐ / ) = ^ Ä n d ( r ( i ) , r ( » ) ) < oo, n=0

and moreover

d(x,y)<rx(x,y)< ^ A n f c ( T n ) d(xl2/). n=0

This proves that r\ and d are equivalent metrics on M.

Finally, for x, y G M

oo

Γλ(Τ(χ),Τ(ΐ/)) = Â n d (T" + 1 (x ) ,T n + 1 (2 / ) ) n=0

oo

= (l/A)£An+1d(r"+1(*),Tn+1(y)) n = 0

= (1/A) rx(x,y) - d(x,y)

<(l/X)rx(x,y).

Thus krx(T) < — < 1. This proves that T is a contraction mapping on the

metric space (M,rx). Also, since rx and d are equivalent metrics, ( Μ , Γ Α ) is a complete space. It is now possible to invoke Banach's original theorem to


conclude that T has a unique fixed point xo (with lim d(Tn(x),xo) = 0 for n—*oo

each x € M).

The preceding ideas lead to the following variant of Theorem 3.4.

Theorem 3.12 Suppose (M,d) is a metric space and suppose T : M —> M is a mapping for which (kd)^ (T) < 1. Suppose also that T is continuous relative to a metric p on M for which (M,p) is complete, and suppose p(x,y) < d(x,y) for each x,y 6 M. Then T has a unique fixed point XQ, and moreover lim Tn(x) =

n—»oo

xo for each x £ M.

Proof. There exists a metric d' on M which is equivalent to d and such that T : (M, d') —» (M, d') is a contraction mapping. Therefore, for each x e M, {Tn(x)} is a Cauchy sequence in (M,d'); hence {Tn(x)} is Cauchy in (M,d) and, since p < d, {Tn(x)} is in fact Cauchy in (M, p) as well. Therefore, relative to p, lim T"(x) = xo for some XQÇ. M and the conclusion follows because T is

n—>oo

continuous on (M,p). ■ We now turn to a principle of a different kind. In this result, the contractive

condition is imposed only at the first step. This paves the way for the results of the next section.

Theorem 3.13 Suppose M is a complete metric space and suppose T : M —> M is a continuous mapping which satisfies for some ψ : M —» R+,

d(x, T(x)) < φ(χ) - φ{Τ{χ)), χ 6 M. (*)

Then {Tn(x)} converges to a fixed point ofT for each x € M.

Proof. This is a piece of the argument used in the proof of Banach's contrac-tion principle. The condition (*) implies that {φ(Τη(χ))} is monotone decreas-ing and hence lim φ(Τη(χ)) — r > 0. By the triangle inequality, if m,n 6 N

n—»oo

and m > n then t n - l

d(Tn(x),Tm(x)) < Σ d(r(x),Ti+1(x)) < ψ(Τη(χ)) - φ(Τ'η(χ))

so lim d(Tn(x)1Tm(x)) = 0. Since M is complete there exists z e M such

m,n—*oo

that lim Γη(χ) = z and by continuity of T, z - T{z). ■

n—»oo

Remark 3.2 In the above result, one can obtain an estimate on the rate of convergence of (Tn(x)} by referring back to the inequality

m - l

J ^ d(T(x),Ti+\x)) < φ(Τη(χ)) - φ{Τ™{χ)).

3.3. THE CAPJSTI-EKELAND PRINCIPLE 55

This yields

d(Tn(x),Tm(x)) < φ(Τη(χ)) - ψ{Τη{χ)) < φ(Τη(χ)),

and ifT(z) = z, upon letting m —» oo we get

ά(Τη(χ),ζ)<ψ(Τη(χ)).

3.3 The Caristi-Ekeland Principle In view of the fact that continuity of T was essential to the proof of Theorem 3.13 it is remarkable to find that the result remains true without such an assumption if it is assumed that φ : M —► R is lower semicontinuous (l.s.c), that is, for any sequence {xn} c M, if lim xn = x and lim φ(χη) — r, then φ(χ) < r. This

n—*oo n~*oo new fact can be formulated in two equivalent ways.

Theorem 3.14 (E) (Ekeland, 1974) Let (M,d) be a complete metric space and φ: M - > R + l.s.c. Define:

x < y & d(x, y) < φ(χ) - <p(y), x,y € M.

Then (M, <) has a maximal element.

Theorem 3.15 (C) (Caristi, 1975) Let M and φ be as above. Suppose g : M -* M satisfies:

d(x,g(x)) < φ(χ) - <p{g(x)), x e M.

Then g has a fixed point.

(E) => (C). Proof. With M, φ as above, and g as in Theorem (C), define x < y <=> d(x,y) < ψ(χ) — <p(y), x,y € M. By (E) there exists x G M such that x is maximal in (M, <). But d(x,g(x)) < φ(χ) — ip(g(x)) => x < g(x). Hence by maximality, x — g(x).

(C) => (E). Proof. Assume (E) is false. Then Va: € M 3 g(x) 6 M such that x < g(x). It follows that

d{x,g{x)) < ψ{χ) - <p{g{x)), xeM. (*)

By (C) g must have a fixed point x. But by assumption, x < g(x)—a contra-diction.

Those familiar with the logical foundations of mathematics might note that the proof of implication (C) => (E) uses the Axiom of Choice, whereas the proof that (E) => (C) does not. In fact, these two theorems are equivalent only if one


assumes (as we do) the Axiom of Choice. (A proof of (C) can be given within the basic axioms of Zermelo-Praenkel set theory [116]. At the same time it is known that (E) is roughly equivalent to the Countable Axiom of Choice [32].)

We first prove Caristi's theorem using the principle of transfinite induction, and then we prove a more general version of Caristi's theorem using a Zorn's lemma argument. We do this merely to illustrate the methods involved, so nothing is lost in the overall development if the transfinite induction argument is omitted. Also, more constructive proofs could have been given in each instance.

The transfinite induction argument given below is a simplified version (due to Chi Song Wong) of Caristi's original proof.

Proof of Caristi 's Theorem. Let xo € M, let Ω denote the smallest uncount-able ordinal, let ß € Ω, and suppose for each a G ft with a < ß, xa € M has been defined so that

(1) μ < a => ά(χμ,χα) < φ(χμ) - φ(χα); (2) μ < a => χμ+ι = g(xh).

We need to define xp and show that (1), (2) hold for a < β.

Case 1. β — a + 1. Define x$ — g(xa). Then (2) holds by definition. Also from (*)

d(xa,xa+i) < ψ{χα) - ψ{χα+ι).

If μ < a, then by assumption

ά(χμ,χα) < φ(χμ) - φ(χα).

Hence

ά(χμ,χα+ι) < ά(χμ,χα)+ά(χα,χα+ι) < φ(χμ) - φ(χα) + φ(χα) - φ(χα+1)

This establishes (l).

Case 2. β is a limit ordinal. First note that (1) implies {φ(χα)}α<β is a nonincreasing net of real numbers. Since ψ is bounded below there exists r G R such that

lim ψ(χα) — inf ψ(χα) = r-a—0- a<0

Let ε > 0. Then there exists c*o < β such that if c*o < a < β then

φ{χα) <r + ε.

3.3. THE CARISTI-EKELAND PRINCIPLE 57

In particular, if c*o < μ < a < β, then

ά(χμ,Χα) < φ{χμ) - φ(χα) < ε.

This proves that {χα }<*</? is a Cauchy net. Since M is complete there exists x € M such that lim xa — x. Since lim φ(χα) = r and ψ is lower semicontinuous,

a—0- α-*β~

φ(χ) < r. Now define Xß = x. Then if μ < β, the definition of xp and lower semicontinuity of φ give

ά(χμ,Χ0) = lim cf(xM,xa)

< lim (ψ(χμ) - ψ(χα)) α->0-

= ν(^μ) - r

< ψ{χμ) - ψ{χ) = ψ{χμ) - ψ(χβ)-

Thus (1) holds for β, and (2) holds vacuously when β is a limit ordinal.

So {xa} is denned for all a 6 Ω. Since {< ?(a;Q)} is nonincreasing and Ω is uncountable there exists c*o 6 Ω such that {^(χα)}α>α0 is constant. In particular,

φ(χα+ι) = φ(χα),

from which

d(xa„, 9(xan )) = d(xao, xao+l ) < v?(xQü ) - φ(χαο+, ) = 0;

thusö(zQ o) = i Q 0 . ■

Next we illustrate a Zorn's lemma approach. The following theorem reduces to Caristi's theorem in the case that M = Y, f is the identity mapping, and c = 1. Here we need another definition. A mapping / of a subset A of metric space M into a metric space N is said to be closed if it has a closed graph; thus / : A —» N is closed if for {xn} Ç A the conditions lim xn = x and

n—»oo lim / ( i n ) = y imply x G A and f(x) = y. Applications of this result to n—»oo

nonlinear operator theory may be found in [48].

Theorem 3.16 Let M and Y be complete metric spaces and let g : M —» M be an arbitrary mapping. Suppose there exist a closed mapping f : M —» Y, a lower semicontinuous mapping ψ : f(M) —» K+, and a constant c > 0 such that for each x € M,

max id i a r . s ix J J . cd i / i x ) , / ^ ! ) ) )} < ^( / (x) ) - y>(/(<?(x))). (**)

TTien there exists x € M suc/i ί/ιαί g(x) = x.


Proof. Introduce the partial order > in M as follows. For x,y € M say that y > χ <=>

max{d(x, y),cd(f(x), f(y))} < φ(/(χ)) - <fi(f{y)).

Let {xa}ael D e a n y chain in (M, >), and for α,β € / set β > a <=> Χβ > xa. Then {φ{Ϊ{χα))}αξ.ι is a nonincreasing net in R + so there exists r > 0 such that

\imip(f(xa)) = r. a

Let ε > 0. Then there exists ao 6 / such that a > ao implies

r < tp(f(xa)) <r + e

and so for /3 > a > ao,

m a x i d ^ ^ ^ ) ^ ^ / ^ ) , / ^ ) ) } < <p(f{xQ))-<p(f(xß)) < ε. Thus {/(xQ)} is a Cauchy net in Y while at the same time {xa} is a Cauchy net in M. It follows that there exists y EY and x G M such that lim/(a:a) = y and

a l imxa = x. Since / is a closed mapping, f(x) = y, and lower semicontinuity of

a φ yields φ(/(χ)) < r. Moreover, if a, β € / with β > a, then

m a x { d ( i a , i ß ) , c d ( / ( i 0 ) , / ( i ß ) ) } < <p(f{xa)) - <p(f{xß)) <ψ{ί{χα)) -r

so taking the limit with respect to β yields

max{d(xa,x),cd(f(xa), f(x))} < <p(f(xa)) - ¥>(/(*))·

This proves that x > xa for each a £ I. Having thus shown that every chain in (M, >) has a upper bound we can

appeal to Zorn's lemma to conclude that (M, >) has a maximal element, say x. But by (**) g(x) > x; hence x = g(x)- ■

3.4 Equivalents of the Caristi-Ekeland Principle

There is a vast literature on this subject and a complete summary of the topic would be beyond the scope of this treatment. The following is an example of a seemingly more general formulation of Caristi's theorem which actually reduces to a simple application of the original result.

Theorem 3.17 Let M be a complete metric space and suppose φ : M —> R is lower semicontinuous and bounded below. Suppose ξ : R —» R is continuous, strictly increasing, concave downward, and vanishes at 0, and suppose g : M —> M satisfies:

ξ{ά(χ, g(x))) < φ{χ) - <p(g(x)), x € M.


3.4. EQUIVALENTS OF THE CARISTI-EKELAND PRINCIPLE 59

Proof. Observe that ξ o d is a metric on M (Example 2.14 of Chapter 2). Suppose {xn} is a Cauchy sequence in {Μ,ξ o d). Since ξ is strictly increasing and vanishes at 0, lim ξ(ά(χη,χπι)) = 0 implies lim d(xn,xm) — 0. Since

m,n—»oo τη,η—»oo

M is complete there exists x & M such that lim d(xn,x) — 0. By continuity n—>oo

of ξ this in turn implies lim ξ(ά(χη,χ)) = 0. Therefore {xn} converges to a; in n—>oo

(M, ξ o d), proving that (M, ξ ο d) is itself complete. It is now possible to apply Caristi's theorem directly to the space (M, ξ o d) to obtain a fixed point x for g, completing the proof. ■

Corollary 3.1 Let M be a complete metric space and suppose φ : M —» R is lower semicontinuous and bounded below. Suppose for some r € (0,1], g : M —» M satisfies:

d(x,g(x))r < φ(χ) -<p(g(x)), x € M.


Proof. Take ξ(ί) = tr, t 6 R, and apply the previous theorem. (It is shown in [4] that Corollary 3.1 is false if r > 1.) ■

It is also trivial to see that Caristi's theorem extends to set valued mappings.

Theorem 3.18 Let M be a complete metric space and let φ : M —» R be lower semicontinuous and bounded below. Suppose G : M —> 2M satisfies: For each x € M there exists u € G{x) such that

d(x,u) < φ(χ) — φ{η).

Then there exists x E M such that x € G{x).

We now take up one of the more recent developments due to Oettli and Théra [127]. This approach provides yet another proof of Caristi's theorem, one which uses neither transfinite induction nor Zorn's lemma.

Let (M,d) be a complete metric space and let / : M x M —» (—00,00] be an extended real-valued function which is lower semicontinuous in its second variable and satisfies

f(v, v)=0 for all v E M; (3.4)

f(u, v) < f(u, w) + f(w, v) for all u, v, w 6 M. (3.5)

Assume also that there exists vo 6 M such that

inf f(vo,v) > -00 . (3.6) v€M

Let

S0 = {v e M : f(vo,v)+d(vo,v) < 0}. (3.7)

Observe that SQ Φ 0 since by (3.4), VQ G SQ.


Theorem 3.19 Assume M, f and So satisfy the assumptions of the preceding paragraph. Let S Ç M have the property: For every v G SQ\S there exists v G M such that v φ ν and

f(v,v)+d(v,v)<0. (3.8)

Then there exists v* G SQ Π S.

Before proving this theorem we show how it implies Theorem 3.18. Suppose G : M —> 2M and suppose for each v Ç. M there exists v G G(v) such that

ά(ν,ν)<φ(ν)-ψ(ν), (3.9)

where φ : M —> R is lower semicontinuous and bounded below. Define / : M x M - t E b y setting

f(u,v) = ψ(ν) — <p(u), u,vÇ.M.

Thus (3.9) is equivalent to (3.8). Fix vo 6 M. Since < is bounded below,

inf f(vQ,v) - inf (φ{ν) - v?(v0)) > -oo .

Now let So = {v e M : f(v0,v) + d(w0,t;) < 0} and S = {v e M : ϋ € G(û)}-By assumption if t; € Sb\S there exists υ € G{v) (necessarily v φ ν) such that

d(v, v) < φ(ν) — φ(ν) = —f(v, v).

Thus (3.8) holds. So by Theorem 3.19 there exists v* € 5 0 n 5 ; thus vm G G(v*).

Proof of Theorem 3.19. This is a simple induction procedure. For each x G M define the set

S(x) = {v G M : / (x , u) + d(x, υ) < 0},

and define

7 ( x ) = inf f(x,v). vÇS(x)

Then clearly x G S(x) and 7(x) < 0 for each x € M. Now let So = S(vo). By assumption 7(^0) = 70 = inf f(vo,v) > -00 . Let n G N and suppose υη_ι has

»6JW

been defined with 7(υη_ι) > —oo. Choose vn G 5(υη_ι) so that

/(w„-l,l>„) < 7(υη-ΐ) + ~ ·

n Now let υ G S(vn). Then

f(vn,v) +d(vn,v) < 0.

3.5. SET-VALUED CONTRACTIONS 61

Since vn € 5(υη_ι)

f(vn-i,vn) + d(vn-i,vn) < 0 .

The triangle inequality and (3.5) now give

f(vn-i,v)+d(vn-i,v) < 0 .

This proves v € 5(t>„_i), that is, S(vn) Ç S(vn-i). Therefore,

7(υ„ )= inf f(vn,v) > inf (f(vn-i,v) - f(vn-ltvn)) v€S(vn) veS(vn)

> inf f(,vn-i,v)-f(vn-i,vn)

t)€S(«„_i)

= i K i - l ) - f(vn-l,Vn) > . n

Thus if υ £ S(vn) then d(vn,v) < -f(vn,v) < -^(νη) < - . Since vn e S(vn) n

this implies that the diameters of the sets S(vn) tend to zero and in turn that the sequence {vn} is Cauchy. Also, since / is lower semicontinuous in its second

oo

variable, each of the sets S(vn) is closed. Therefore there exists v* e Π S(vn); n=0 oo

indeed f) S(vn) = {v*}. We claim v* e S. If not, then by assumption there n=0

oo would exist v φ v* for which f(v',v) + d(v*,v) < 0. Since υ* Ε (~) S(vn) we

n=0 have

f(vn,v*)+d(vn,v*) < 0 f o r al ln.

Combined with (3.5) this yields

f(vn, v) + d(vn,v) < 0 for all n

oo

and this in turn implies v 6 f] S(vn). However this contradicts υ φ ν". Thus n=0

v* 6 S, completing the proof. ■

3.5 Set-valued contractions

Banach's Contraction Mapping Principle extends nicely to set-valued mappings, a fact first noticed by S. Nadler [125]. The key idea is the following: If A and B are nonempty closed bounded subsets of a metric space and if x 6 A, then given e > 0 there must exist a point y € B such that

d{x,y)<H{A,B) + e,


where H (A, B) denotes the Hausdorff distance between A and B. This is because the definition of Hausdorff distance (recall Example 2.15 of Chapter 2) assures that for any μ > 0

A C Νρ+μ(Β),

where p = H(A,B).

Theorem 3.20 Let (M, d) be a complete metric space, and let ΑΛ be the collec-tion of all nonempty bounded closed subsets of M endowed with the Hausdorff metric H. Suppose T : M —» M is a contraction mapping in the sense that for some k < 1 :

H{T(x),T(y))<kd{x,y), x,y € M.

Then there exists a point x € M such that x 6 T(x).

Proof. Select #o S M and xi € T(xo). By the observation immediately preceding the statement of the theorem there must exist x2 e T{x\ ) such that

d{xi,x2) < H(T(x0),T(Xl)) + k.

Similarly, there exists X3 € T(x2) such that

d(x2,x3) < H{T(xi)<T(x2)) + k2.

Proceed by induction to obtain a sequence {xn} in M with the property that for each i € N, Xi+\ 6 Τ(χ,) and for which

d(xi,xi+i) ^Η^χ^,^χφ + ^ < kd{xi-\,Xi) + k%

< fc[/f(T(xi_2),r(xi_1)) + k*-1} + fe< < fc2tf(xi_2,Xi-l)+2fci

< . . . < k% d(xo,x\) + ikl.

Therefore,

0 0 / 0 0 \ 0 0

2_]d(xi,Xi+i) < d(xo,xi) I y j ^ " ' ] + / J ^ ' * < °°-i=0 \i=0 / i=0

This proves that {xn} is a Cauchy sequence, so since M is complete there exists x € M such that lim xn = x. Also, since T is continuous, lim H(T(xn),T(x)) =

n—»00 n—»oo

0. Since xn € T(x n _i ) ,

lim dist(xn,T(x)) = lim inf{d(xn,y) : y £ T(x)} = 0. n—+00 n—»00

This implies that

dist(x,T(x)) = mî{d(x,y) : y e T(x)} = 0,

3.5. SET-VALUED CONTRACTIONS 63

and since T(x) is closed it must be the case that x Ç. T(x). ■

Two things are worth noting about the preceding theorem. First, in contrast to Banach's Theorem, it is not asserted that the fixed point x is unique. Indeed, it need not be. Also, since M is complete, (ΛΊ, H) is complete as well, but this fact is not needed in the proof.

There is an interesting stability result due to T. C. Lim that holds for set-valued contractions (hence ordinary contractions as well). We begin with a technical lemma.

Lemma 3.1 Suppose M and M. are as in the preceding theorem, and let Ti : M —> ΛΊ, i = 1,2, be two contraction mappings each having Lipschitz constant k < 1. Then if F(Ti) and F(T2) denote the respective fixed point sets ofT\ and T2,

i / (F( r ! ) ,F(T 2 ) ) < (1 - k)-1 sup HÇTiixlTtix)). xeM oo

Proof. Let ε > 0, choose c > 0 so that c Σ η^η < 1, and set n=l

ce £l-(TTfc)·

Select x0 € -F(T\), and then select x\ 6 X^zo) so that

φ ο , χ ι ) < Η{Τι{χ0),Τ2{χ0)) + ε.

Since H(T2(x\),T2(xo)) < kd(xi,xo) there exists x2 € Τ2(χ\) such that

d(x2,xi) < kd(xi,xo) + ke\.

Now define {xn} inductively so that x„+i € T2(xn) and

d(xn+i,xn) < kd(xn,xn-i) + knei, n = 1,2,··■ .

Then

d{xn+i,xn) <kd(xn,xn-i) + knei < k(kd{xn-i,xn-2) + Α;η-1ει) + fcnei = /c2CÎ(xn_i,Xn_2) + 2fcn£i.

Continuing in this fashion we obtain

<f(xn+i, x„) < kn d(xi, xo) + nkne\.

Therefore,

oo oo

Σ d(xn+1,xn) < fcm(l - fc)-1d(xi,i0) + Σ nknei-


Since the right side of the above tends to 0 as m —» oo, this proves that {xn} is a Cauchy sequence with limit, say z. Since T2 is continuous, lim H(T2(xn) ,T2(z)) = 0. Also, since xn+i £ T2(xn) it must be the case that

n—.00

z € T2(z), that is, z e F(T2). Furthermore, 00

n + l i Xn) n=0

oo < (1 - fc)-1d(xi,i0) + Σ n f c n £ i

n=l < ( l - f c ) - 1 ( d ( x i , x 0 ) + e) < (1 - k)-1(H(Tl(x0),T2(x0)) + 2ε).

Reversing the roles of 7\ and T2 and repeating the argument just given leads to the conclusion that for each 1/0 € F(T2) there exist y! € Τι(ι/ο) and u; € F(T\) such that

d(»o,ti;) < (1 -fc)-1(H(T1(yo),T2(2/o)) + 2e).

Since ε > 0 is arbitrary, the conclusion follows. ■

T h e o r e m 3.21 Suppose M and M are as in the preceding theorem, and let Ti : M —>ΛΊ, i = l , 2 , · · · be a sequence of contraction mappings each having Lipschitz constant k < 1. If lim H(Tn(x),To(x)) = 0 uniformly for x Ç. M,

n—00

then lim H(F{Tn),F(T0)) = 0. n—»00

Proof. Let e > 0. Since lim H(Tn(x),To(x)) = 0 uniformly it is possible to n—»oo

choose JV 6 N so that for n> N, s u p i e M H(Tn(x),T0(x)) < (1 - fc)e. By the lemma, H(F(Tn), F(T0)) < ε for all such n. ■

3.6 Generalized contractions

A thorough search of the mathematical literature would yield numerous ex-tensions of Banach's Theorem in which the contractive condition involves it-erates of the mapping T under consideration and/or the relationship between d (T (x), T (y)) and the five remaining distances determined by {x, y, T(x) ,T (y)} for any x,y in the domain of T. Many, but by no means all, of these results are subsumed by the approach of this section.

Again we assume (M, d) is a metric space with T : M —> M. For a; € M we adopt the notation

0(χ) = {χ ,Γ (χ ) ,Γ 2 ( χ ) ) · · · } ,

and we use 0(x, y) to denote 0(x) U 0(y) for x,y € M. Let φ be a contractive gauge function on M. By this we mean φ : M —> R+ is continuous, nondecreas-ing, and satisfies 0(s) < s for s > 0.

3.6. GENERALIZED CONTRACTIONS 65

As we have observed before, if φ(χ) — kx for x £ M and fixed k £ [0,1), then a mapping T : M —> M satisfying

d(T(x),T(y)) < <j>(d(x,y))

for all x, y £ M is a contraction mapping in the sense of Banach, and thus if M is complete T has a unique fixed point z £ M. It is a far-reaching extension of this approach inspired by Felix Browder that we take up here. We begin with the following fact due to Walter [157].

T h e o r e m 3.22 Let M be a bounded complete metric space and suppose T : M —> M satisfies the following condition. For each x £ M there exists n(x) £ N such that for all n > n(x) and y £ M,

d(Tn(x),Tn{y)) < tf>(diam(0(x,y))). (3.10)

Then there exists z £ M such that lim Tk(x) = z for each x 6 M. k—oo

Proof This is essentially the proof of Browder [27]. There are four steps. For x £ M we use the notation xk = Tk(x), k — 0,1,2, · · ■.

(1) If m = max{n(a;),n(y)}, then diam (0(xm,ym)\ < ^(diam(0(x,y))V

Proof. Suppose n>m and r > 0. Then any two elements (u, v) of 0(xm,ym) are of one of the forms: (xn,yn+r), (xn+r,yn), (xn,xn+T), or ( y n , y n + r ) . In the first case we have

d(u,v) = d(Tn(x),Tn(yr)) < tf>(diam(0(a:,yr))) < <^(diam(0(x,y))).

The other three cases follow by similar inequalities.

Next we define sequences {k(i)} C N and Ai C M by

fc(0) = 0; k(i + 1) = k(i) + max{n(xfcW),n(yfcW)}

and

A = 0 ( z f e ( i V ( i ) ) , 1 = 0 , 1 , 2 , · · · .

(2) diam(4 i + i ) < 0(diam(Ai)) for t = 0,1,2, · · · .

Proof. For i = 0 this is just a restatement of (1). Now let i be arbitrary and let ξ = xfc('), η = yfcW, and μ = max{n(xfc(i)),n(yfe(1))}. Applying Case (1) we have

diam(0(^,77")) < ^ (d iamO(i , t j ) ) . (3.11)

However ξμ = χΜΟ+μ = xHi)+m*x{n(xk^)MyHi))} = χ*(ί+ι). Similarly, rf =

yk(i)+ß = yk(i+i) T h u s ά\ΆΏΐ(0{ξμ,ημ)) = diam(Ai+i) and (3.11) coincides with (2).


Now let ai — diam(.Ai).

(3) lim ai — 0. i—»oo

Proof. Prom (2) and the fact that φ is decreasing we have ai+\ < φ(α{) < Oj. Thus {ai} is decreasing so there exists a > 0 such that lim ai = a < φ(α). Since

Ϊ—»OO

φ(α) < a if a > 0, it must be the case that a — 0.

(4) We have shown that Um d i a m ^ ) = lim diam(0(x f c ( i ) , / ( i ) ) ) = 0. This i—»oo i—»oo

clearly implies lim diam(0(xk,yk)) — 0. This implies that both {xfe} and {yk} fc—too

are Cauchy sequences and have the same limit, say z G M. Since y G M is arbitrary, we conclude that, in fact, lim xk = z for each x G M. ■

fc—»oo

Corollary 3.2 7/in addition to the assumptions of Theorem 3.22 T is contin-uous, then T(z) = z.

By strengthening the assumption (3.10) to require that n{x) — 1 for all x G M it is possible to conclude that T has a fixed point without assuming continuity.

T h e o r e m 3.23 Let M be a bounded complete metric space and suppose T : M —> M satisfies the following condition. For each x,y G M,

d(T(x),T(y)) < 0(diam(O(x,2/))). (3.12)

Then T has a unique fixed point z G M and lim T (x) = z for each x G M. k—»oo

Proof. By Theorem 3.22 there exists z G M such that lim Tk(x) — z for fc—»OO

each x G M. Assume z φ T{z). Then diam(0(z)) = a > 0. From this it is possible to select two sequences {p(k)} and {q(k)} such that 0 < p(k) < q(k) for which lim d(zp(~k\ zg(-h^) = a. Since lim zk = z there exists fco £ N such

fc—»OO fc—»OO

that for fc, I > fco, d(zk,zl) < a/2. Hence for some p with 0 < p < fco, it must be the case that p(fc) = p for infinitely many fc. Therefore there is a subsequence {r(fc)} of {q(k)} such that lim d(zp,zr(-k)) = a. If r(fc) = q infinitely often,

fc—»OO

then d(zp,zq) — a. Otherwise there exists a subsequence {s(fc)} of {r(fc)} with s(k) —» oo as fc —» oo and this implies d(zp,z) = a. In any case, there exist p, q > 0 such that d(zp,zq) — a. If p, q > 1 then (3.12) implies

d{zp,z") = d(r(zp-1) ,T(2<-1)) < φ(&&τη{0(ζρ-\ζ'1-1))) < ψ(ώαπι(0(ζ)).

Since a = d(zp, zq) = diam(0(z)) this gives a < φ(α).

On the other hand, if d(z, zq) = a, then since lim zk = z fc—»oo

a = lim d{zk,zq) < 0(diam(O(2fc_1,*,~1))) < <K<*) fc—»oo

EXERCISES 67

In either case we have a contradiction since φ(α) < a if a > 0. Hence a = 0, that is, 0(z) — {z}. m

Exercises

Exercise 3.1 Suppose M is a compact metric space and suppose T : M —> M is a mapping of M into M which satisfies d(T(x),T(y)) > d(x, y). IsT necessarily an isometry?

Exercise 3.2 Let (M,d) be a complete metric space, let M denote the family of all nonempty bounded closed subsets of M, and let C denote the subfamily of M consisting of all compact sets. Let Ti : M —* M, i = 1, ..,n, be a family of contractions. Show that there exists a unique nonempty compact subset X of M such that

n

X = \jTi(X). i = l

Hint: Consider the mapping T* :C —* C defined by

n

T'(X) = \jTi(X). i=l

Show that T* is a contraction. Application. Consider the real line R and the two contractions

ΓΙ (Χ) = | Ϊ ; Γ 2 (Χ) = | Ι + | .

Show that the compact X which satisfies X = Τχ(Χ) L)T2(X) is the well-known Cantor set (which is described in standard topological texts).

Exercise 3.3 Let (M, d) be a complete metric space and T : M —» M. Assume that for each ε > 0, there exists δ > 0 such that

d(x,Tx)<6 => τ(Β(χ,εγ) C B(x,e).

Prove that if d(Tn(x),Tn+1(x)) —► 0, for some x 6 M, then the sequence {Tn(x)} converges to a fixed point ofT. Note: The mapping T is not necessarily continuous.


Exercise 3.4 Let (M, d) be a complete metric space, and X an arbitrary metric space. Let T : M x X —> M be continuous in each variable separately. Assume aho that there exists a positive number k < 1 such that

ά(τ(χ,α),Τ(ν,αή < kd(x,y)

for any a E X and any x,y € M. Therefore, for any a € X, the mapping x >-* T(x,a) has a unique fixed point a*. Show that the mapping a >—* a* is continuous.

exercise 3.5 Let (M,d) be a complete metric space, and T„ : M —* M a sequence of lipschitzian mappings. Assume that each Tn has a fixed point xn

and the sequence [Tn} converges pointwise to a mapping T. Assume also that the sequence {L(Tn)} of their Lipschitz constants is bounded (by a positive number K). Show that

1. T is lipschitzian and L(T) < K.

2. If {xn} converges to xo, then xo is a fixed point ofT.

3. If K < 1, then {xn} converges to the unique fixed point ofT.

Show also that the boundedness condition of {L(Tn)} cannot be relaxed.

Exercise 3.6 Let (M, d) be a complete metric space and T : M —> M be such that

d(T(x),T(y))>kd(x,y)

for some k > 1 and any x, y S M, and assume that T (M) = M. Prove that T is one-to-one (or infective). Moreover, prove that T has a unique fixed point XQ, with limn^,00T~n(x) —» XQ for any x € M.

Exercise 3.7 Let (M, d) be a complete metric space and T : M —* M a contin-

uous mapping. Assume there exist an integer n and a positive number k < —

such that

d(T(x),T(y))<k(d(x,Tn(z)) + d(y,Tn(z)))

for all x,y,z G M. Prove that T has a unique fixed point.

Exercise 3.8 A metric space (M,d) is called ε-chainable if for any x,y 6 M, there exist xo = x,x\,··· ,xn,xn+i = y in M such that d(xi,Xi+\) < ε for 0 0, and there exists a positive number k < 1 such that

d(T(x),T(y)) < kd(x,y)

whenever d(x, y) < e. Show that T has a fixed point.

EXERCISES 69

Exercise 3.9 Let (M,d) be an arbitrary metric space and T : M —» M a mapping which satisfies:

d(T(x),T(y)) < d(x,y)

whenever x φ y. Assume that for some x ζ M, the sequence {Tn(x)} has a subsequence which converges to a point z € M. Show that z is a fixed point of T.

Exercise 3.10 Let (M,d) be an arbitrary metric space and T : M -+ M a continuous mapping. Assume that for some x € M, the orbit {Tn(x)} contains

a convergent subsequence {Tni(x)}. Show that ifd(Tni(x),Tni+1(x)j —» 0 then

T has a fixed point.

Exercise 3.11 ( Volterra Equation) Consider the integral equation

f{x) = X f K{x,t)f(t)dt + 4>(x)

Ja for a fixed real number X, where K(x,t) is continuous on [a,b] x [a,b]. Con-sider the metric space C[a, b] of continuous real-valued functions defined on [a, b]. Consider the mapping T : C[a, b] —* C\a, b] defined by

( Τ ( / ) ) (X) = \jX K{x, t)f{t)dt + φ(χ).

For f\, fa € C[a,b], show that

d{Tn{h),TnU2)) < l A r ^ ^ ^ d i / x . / a ) ,

where

M = max{|A'(a;,i)|; (ζ, ί) £ [a,b] x [a,b]}.

Then show that the above equation has a unique solution f(x).

Note: Recall

d( / i , / a ) = max{|/i(z) - / a ( i ) | ; x € [a,b}}.

forfuf2€C[a,b}.

Chapter 4

Hyperconvex Spaces

4.1 Introduction We motivate the study of hyperconvex spaces with a simple observation about the real line R1.

Propos i t ion 4.1 Let {Ia}açA be a family of bounded closed intervals of R1

each two of which intersect. Then

aÇA

Proof. Suppose the intersection is empty. Then by compactness there exist

{ / j , - " , / N + l } Ç { / a }

such that

N N+l

I^ÇÎi^Z while f] h = 0 . t = l t= l

Thus I and ijv+i are disjoint closed intervals in R1. Select any point x that lies strictly between them. (This is possible because the complement of IU IN+\ is an open set.) Then by the fact that any two members of the original family intersect,

xeUniN+i, i = i , · · · ,N.

This implies

N+l

xe f]li, i= l

which is a contradiction. ■

71



72 CHAPTER 4. HYPERCONVEX SPACES

We note in passing that the above fact is a special case of a famous result in geometry known as Helly's Theorem. Although it will not be needed in the sequel, we pause to state this result and to give an outline of a very elegant proof which is quite easy to visualize in R2.

Theorem 4.1 (Helly's Theorem) Let T = {Ka}açA be a family of bounded closed convex sets in Rn, and suppose each subcollection of T consisting ofn + 1 members intersect. Then

f]Ka^0. aÇA

Proof By induction. Suppose the theorem is true in R n _ 1 and false in Rn. Then there exists a family T — {Ka}a€A of bounded closed convex sets in Rn

having the property that each n + 1 members of T intersect while

n κα=0.

The basic line of argument follows the one given for Proposition 4.1. By com-pactness there must exist

{ΑΊ,· · ■ ,KN+\) Ç F,

for which

N ΛΜ-1

K=f]Ki^0 while f] Kt = 0. i= l i= l

Now let L be any hyperplane of Rn (a translate of an n — 1-dimensional subspace of Rn) which separates K and K^+i- (To produce such an L, let x and y be points of K and KM+\ respectively, such that

d (x, y) = inf {d (u, v) : u e K, v e KN+X}

and let L be the hyperplane through the midpoint (x + y) /2 of the segment joining x and y which is orthogonal to the segment joining x and y.) Now consider the family {Ki Π L}^. Since the intersection of any n members of {Ki}^=l also intersects KN+I the intersection of any n members of {Κί}^=1

must also intersect L. Hence the intersection of any n members of {Ki Π L}^ is nonempty. By the inductive hypothesis

N N

f]{Ki n L) = ( p | K?) n L = K n L Φ 0. i=\ t=l

This is a contradiction. ■

The argument of Proposition 4.1 extends to i^ (Example 2.7 of Chapter 2).

4.1. INTRODUCTION 73

Theorem 4.2 Suppose {Ba}açA ™ a family of closed balL· in Poo each two of which intersect. Then

f)Ba^0. a£A

Proof. Let

Ln = {x = (x!,X2,· · · ) e ioo : Xi = 0 if i φ n}.

Then Ln is an isometric copy of R1 and {J5Q DLn} is a family of closed intervals of Ln each two of which intersect. Thus for each n e N,

f | ( L n n ß a ) / 0 .

Select

xne f\(LnDBa) n = l ,2 , · - · . aÇA

Then if x = ( i i , i2 . · ■ · ) i* follows that

x e fl Ba. a£A

m

Another simple observation about i^ :

Proposition 4.2 Suppose B(xa;ra) and B(x0;r0) are two closed balls in (m. Then

B(xa;ra)nB(x0;r0) φ0

if and only if

"oo (Χα,Χ/3) <ra+r0.

Proof. One implication is completely trivial. If z € JB(xa;ra) (Ί B(x0;r0), then

doo(xa,Xß) < doo(Xa,z) +doo(z,Xß) <

Now suppose

doo(x«,X/3) <ra+r0.

Then if xQ = (χχ, χ-χ, ■ ■ ■ ) and x0 = (yi,3/2, · ■ · ) it must be the case that

\xi - Vi\ < ra + r0 z = 1,2,··· .


Let Zi = 2_i i = 1 2 , · · · . Then IXJ — zA < ra and \zi — yA < rg: where rQ+r0

^οο(Χα,ζ) < ra; ôo(x/J,z) < r0,

that is,

z 6 ß ( x a ; r a ) n ß ( x j 9 ; r / 3 ) .

Combining the above with Theorem 4.2, we obtain

Theorem 4.3 Suppose {B(xa;ra)}açA is any family of closed balls in ί,χ, each two of which satisfy

doo(Xa,X/3) < r Q + T>

T/ien Π B{xa;ra)yi0. a£A

Therefore, the space i^ provides a nontrivial example of a metric space which is hyperconvex. The general definition is given in the next section.

Before turning to more abstract considerations we take up another property of R1. Because this fact is a little deeper we give a detailed proof here and repeat the essence of the argument in a more general setting in the next section.

Recall that a subset L of a metric space is a metric segment if L is isometric with some real line interval [a, b] Ç R 1 . A nonexpansive retraction of a metric space (M, d) onto one of its subspaces S is a nonexpansive mapping R of M onto S (i.e., d(R(x),R(y)) < d(x,y) for each x,y G M) which leaves each point of S fixed. In this case S is said to be a nonexpansive retract of M.

Proposition 4.3 Suppose L is a metric segment in a metric space (M, d). Then there is a nonexpansive retraction R of M onto L. Equivalently, L is a nonex-pansive retract of M.

Proof. Consider the pairs (Tf,F) where F is a subspace of M for which L Ç F Ç M and Tp : F —» L is nonexpansive with Tp{y) = y for each y € L. Let M denote the collection of all such pairs, and partially order M as follows:

( T F , F ) x ( T c , G )

if and only if F Ç G and TG(z) = TF{z) for each z € F. Note that N is nonempty since (L, I) € TV, where / denotes the identity mapping on L.

If {(TFa,Fa)}açA is a chain in N relative to this partial order, take G = \JFa, and define TG ■ G -* L by setting TQ(X) = Tpa(x) for any a such that a

4.1. INTRODUCTION 75

x G Fa. Then TQ is well defined since if x € Fa Π Fß then, since either Fa Ç Fp or Fß Ç Fa, it must be the case that Tpa(x) = Tpß(x). With this definition we see that

(TFa, FQ) -< (TG, G) for each a e A.

Therefore, each chain in (Af, -<) is bounded above, so by Zorn's Lemma (λί, -<) has a maximal element, say (TH, H). All we need establish now is the following:

Assertion. H — M.

Proof. Suppose not. Then there exists z € M such that z £ H. Let H\ = H U {z}. We wish to define an extension 7//, of TH such that THX : H\ —> L and THI is nonexpansive. To this end, consider the following family of closed subintervals of L:

f - { / ( x ) n L ; X€H\,

where I(x) — 7//(x) — d(x, Z),TH{X) +d(x, z) . Suppose χ ι , χ 2 € H and label

them so that ΤΗ{χχ) < TH{x2). Then

TH(x2)-TH(xi) = l IMix) - 7V(x2)| < d(x1,x2) < d(xuz) + d(x2,z).

It follows that

TH(xi) + d(xltz) > TH{x2)-d(x2,z)

and consequently

/ (xi) n / f o ) ? ^ .

Therefore, each two intervals in the family T intersect, so by Proposition 4.1

η · ^ = Π {[TH(x)-d(x,z),TH(x) + d(x,z)]nL}ï0. xÇH

Select any point w 6 Π^", define 7tf,(z) = w, and let T//,(x) = 7//(x) for all x e H. Then since

7//, (z) € [ r„(x) - d(x, z), TH(x) + d(x, z)

for each x € H, we conclude that

ΙΤ,,,ΟΟ - Τ „ , ( χ ) | = |THl(*) - T „ ( X ) | < d(*,z).

Hence Τ#, is nonexpansive. This implies

(ΤΗ,Η)*^,^)


with Hi φ H, contradicting maximality of (Tu, H) and completing the proof. ■

Now let {B(xa; ra)}a€A be a family of closed balls in 4» and suppose

£>= f | B(xa;ra)^0.

As before, let

Ln — {x = (x\,X2, · · · ) G 4» : Xi — 0 for all i φ η}.

Then Dn — Ln Π D is a nonempty closed interval which lies in an isometric copy of the real line K1.

It is easy to extend Proposition 4.3 to the set D = Π-^η- To this end, n

suppose (an isometric copy of) D is contained in a metric space (M, d). Then by Proposition 4.3, for each n € N there exists a nonexpansive retraction Rn : M -* Dn. Thus

\Rn{x)-Rn{y)\<d{x,y)

for each x,y € M. Define R : M —► D by setting

Λ(χ) = (/*!(*), ß 2 ( * ) , · · ) ·

Then

doo(R(x),R(y)) = sup |Ä„(i) - Rn{y)\ < d(x, y). n

Also, since Rn(xn) — xn if xn 6 £>„, R(x) = x for each x € D. Therefore, R is a nonexpansive retraction of M onto D and we have the following.

Theorem 4.4 Lei {Ba} be a family of closed balL· in oo ojid suppose D — f]Ba φ 0. Let M be any metric space which contains (an isometric copy of) a D. Then there exists a nonexpansive retraction R of M onto D.

Finally, we turn to another elementary property of R1. If {Ia} is a family of bounded closed intervals of R1 for which I = Ç]Ia φ 0. Then I is itself a

a

closed interval, say I = [a,b], and if m = —-— and d = \a — b\, then / =

d d m--,m+- Thus I coincides with the closed ball B ( m; - ) . This fact

extends to ioo as follows.

Theorem 4.5 Let {Ba} be a family of closed balL· in οο ο,τΐίί suppose Σ) — f] BQ φ 0 . Let d = diam(D). Then there exists a point z £ D such that a

DCBUI

4.2. HYPERCONVEXITY 77

Proof. Again let

Ln = {x = (xi,X2, ■ ■ ■ ) € ί<χ> : Xi — 0 for all i φ η}

and use the fact that {BaDLn} is a family of closed subintervals in (an isometric copy of) R1. Thus for each n, Dn = f](Ba D Ln) is a nonempty closed interval

a in Ln. Moreover, if

d — sup{diam(.Dn) : n — 1,2, · · · }

and if zn is the midpoint of the interval Dn, then

DnC

Therefore, if z = (ζ\,Ζ2, ■ ■ ■), then z € D, and if x € D,

doo(z,x) = s u p | z n - z „ | < - .

This proves that D Ç B I z; - ) . ■

R e m a r k 4.1 All of the preceding arguments extend routinely to wider classes of spaces; for example, to the space Boo[0,1] of all bounded real-valued func-tions defined on [0,1]. (Again, distance is defined in the usual way: d00(f,g) — sup \f(t)-g(t)\.)

<€[0,1)

4.2 Hyp er convexity

The preceding observations should serve to motivate the following more general discussion. Let (X, p) and (M, d) be metric spaces. A mapping / : X —» M is said to be nonexpansive if for each x, y G X

d(f(x),f(y))<P(x,y).

A metric space M is said to be injective if it has the following extension property: Whenever Y is a subspace of a metric space X and / : Y -* M is nonexpan-sive, then / has a nonexpansive extension / : X —+ M. This fact has several nice consequences. For example, suppose M is injective and suppose M is a subspace of a metric space X. Then since the identity mapping I : M —> M is nonexpansive then / can be extended to a nonexpansive mapping R: X —> M. Since R is a retraction of X onto M we have the following.

Theorem 4.6 An injective metric space is a nonexpansive retract of any metric space in which it is metrically embedded.

d d 2 ' 2 n + 2


In light of the above it is clear that an injective metric space must be com-plete because it is a nonexpansive retract (hence a closed subspace) of its own completion.

It might seem that a class of spaces with such nice properties would be small, but, in fact, the opposite is true. We have already seen that the spaces R£, and £oo provide explicit examples of spaces which satisfy the conclusion of Theorem 4.6. We shall show that these spaces are, in fact, injective.

The injective envelope of a metric space M is an injective metric space M which satisfies (i) M contains an isometric copy of M and (ii) no proper injective subspace of M contains an isometric copy of M. Thus M is isometric with some subspace of any injective metric space which contains an isometric copy of M. It is known that every metric space has an injective envelope. This well-known fact, due to J. R. Isbell [75], will be discussed in detail in the last section of this chapter.

The injective metric spaces have been given an explicit metric characteriza-tion by Aronszajn and Panitchpakdi [3].

Definition 4.1 A metric space (M,d) is called hyperconvex if for any indexed class of closed balls B(xi;ri), i€ I, of M which satisfy

d(xi, Xj) <n + rj i,j e I,

it is necessarily the case that f) B(xi\ri) φ 0 .

Aronszajn and Panitchpakdi proved that a metric space is injective if and only if it is hyperconvex. This fact, which in light of the discussion of the preceding section is perhaps not too surprising, is taken up in the next section.

The term 'hyperconvex' suggests that the definition has something to do with convexity. This is indeed true.

It is easy to see that a hyperconvex metric space M is metrically convex. Indeed if x,y € M with x φ y let r\ = ad(x, y) and r^ = (1 — a)d(x, y), for any a € [0,1]. Then d(x, y) = ΤΊ + Γ2, so by hyperconvexity B(x; r\) ΠB(y\ r%) φ 0 . Let z be in this intersection. Then we have

d(x 1z) < ad(x, y) and d(y, z) < (1 — a)d(x, y).

In fact, the triangle inequality will imply

d(x, z) — ad(x,y) and d(y, z) — (1 — a)d{x,y).

4.2. HYPERCONVEXITY 79

Now suppose M is a metric space and suppose {B(xj;7\)}je/ is a family of closed balls in a metric space each two of which intersect. Then if z 6 B{xi] ri) Π B(XJ\ Tj) for i, j € / ,

d(xi, Xj) < d(xi, z) + d(z,Xj) < ri + τν,.

Therefore, if the space is hyperconvex,

Ρ | β ( χ ί ; Γ ί ) ^ 0 .

In particular, hyperconvex spaces have the binary ball intersection property (any family of closed balls, each two of which intersect must have nonempty inter-section).

Next suppose M is a complete metric space which has the binary ball in-tersection property, and suppose M is metrically convex. If {B(xi,ri)}i^j is a family of closed balls in M for which

d(xi,Xj) <ri + rj,

then there is a metric segment joining xt and Xj and clearly some point of this segment must lie in B(xi\ri) Π B{xy,rj). Therefore, f] B{xt\ri) φ 0 by the

t€ / binary ball intersection property.

We conclude, therefore, that for a complete metric space M the following are equivalent.

(1) M is hyperconvex.

(2) M is metrically convex and has the binary ball intersection property.

Completeness is not actually necessary for the above. Indeed, in the next section, we will prove that any metric space which satisfies the binary ball intersection property is complete.

At this point we introduce some notation which will be used throughout the remainder of this chapter and in later ones as well.

For a subset A of a metric space (M, d), set:

rx{A) = sup{d{x,y):ye A}, x G M; rM(A) = inî{rx(A) : x β Μ}; r(A) = inf{rx(j4) : x € A}; diam(.A) = sup{d(x, y) : x,y e A}; CM(A) - {xeM:rx(A) = r(A)}; C(A) = {xeA:rx(A)=r(A)}; cov(yl) =r f){ßi : Bi is a closed ball and Bi 2 A).


r/tf(A) is called the radius of A (relative to M), diam(.A) is called the diam-eter of A, CM(A) is called the center of A (relative to M), r(A) is called the Chebyshev radius of A, C{A) is called the Chebyshev center of A, and COV(J4) is called the cover of A.

4.3 Properties of hyperconvex spaces

We begin with a useful fact.

Proposition 4.4 Any hyperconvex metric space is complete.

Proof. Let {xn} be a Cauchy sequence in a hyperconvex space M and for each n 6 N set pn — sup{d(xn,xm) : m > n}. Then for m > n, d(xn,xm) < Pn — Pn + Pm- Thus by hyperconvexity there exists a point z in the intersection

oo

Ç\B{xn;pn). n=l

Since lim p„ = 0, then lim xn = z. m n—>oo n—»oo

Remark 4.2 The above proof suggests that the conclusion is valid if we assume that the collection of balls satisfies a kind of compactness property. Indeed, if M is a metric space such that any family of closed balls has a nonempty intersection provided any finitely many of them have a nonempty intersection, then M is complete.

The above proposition tells us even more about the structure of hyperconvex spaces. Since a hyperconvex space is metrically convex, each two points of such a space are joined by at least one metric segment. In topological terms this assures that hyperconvex spaces are always pathwise connected.

We now prove a technical lemma.

Lemma 4.1 Suppose A is a bounded subset of a hyperconvex metric space M. Then:

1. cow{A) = Ç]{B(x;rx(A)) :xeM}. 2. rx(cov(i4)) = rx(A), for any x e M. 3. rM(cov(A))=rM(A).

4- rM{A) = -d iam(A).

5. diam(cov(A)) =diam(A).

Proof. 1. Since B(x;rx(A)) contains A for each x 6 M it must be the case that

cov(A) Ç Γ\{Β(Χ;ΓΧ(Α)) : x € Al).

4.3. PROPERTIES OF HYPERCONVEX SPACES 81

On the other hand, if A Ç B(x;r), then rx(A) < r so B(x;rx(A)) Ç B(x;r). Hence

n{B(z;r2(A)) : z 6 M} Ç 5 (x ; r ) .

This clearly implies

cov(A) = D{B(x;rx(A)) : x e M} .

2. By 1 rx(cov(A)) = sup \d(x,y) : y 6 f) ß(z;rz(>l)) i . In particular,

y e cov(.A) implies y 6 B(x; rx(>l)), for any x € M. Hence cf(x, y) < Γ Χ ( Α ) . This proves rx(cov(A)) < rx(A). The reverse inequality is obvious since A C cov(i4).

3. This is immediate from the definition of r.

4. Let δ = diam(A) and consider the family ^ B I a; — 1 : a 6 A>. If a,b € A

then d(a, b) < δ = — + — so by hyperconvexity

£!SH) a€A ^ '

c c

If x is any point in this intersection then d(x, a) < - so rx(yl) < - . On the other

hand d(a,b) < d(a, z) + d(z, b) for any a,b e A and z 6 M so 6 < 2rz(A) from

which δ < 2TM{A). Therefore, δ < 2TM(A) < 2rx(A) < <5, proving r\i(A) = - .

5. Using 3 and 4, diam(A) — 2TM(A) — 2ΓΜ( cov(>l) J = diam(cov(A)).

Remark 4.3 Note that we have

rM(A)<r(A).

In general, we do not have the equality. But if A = cov(>l), that is, A is an intersection of balls in M, then we have

rM{A) = r{A) = - diam(A) .

We are now ready to prove the following.

Theorem 4.7 (Aronszajn and Panitchpakdi) A metric space is injective if and only if it is hyperconvex.


Proof. Assume (M, d) is injective. In order to prove that M is hyperconvex we need only to show that M is metrically convex and has the binary intersection property. Let x,y € M with x φ y. Let Y be the metric subspace of M consisting of the points {x,y} and let(X,p) be the metric space consisting of the points {x,y,w} (w an arbitrary element) with distance

p{x,y) = d(x,y); p{x,w) = p(y,w) = -d(x,y).

The identity mapping / : Y —* M is nonexpansive, so by injectivity of M it has a nonexpansive extension / : X —» M. Therefore,

d(x, y) < d(x, î{w)) + d(y, ï(w)) < p(x, w) + p(y, w) = d(x, y).

Since

d(x,ï(w)) =d(ï(x),î(w)) <p(x,w) = -d(x,y),

and

d{yj(w)) < -d(x,y),

it must be the case that

d(x,ï(w)) =d(y,î(w)) = -d(x,y),

and this proves that M is metrically convex. Now we show that M has the binary ball intersection property. Suppose

{Bi (xi ; ri )} iç. [ is a family of closed balls of M each two of which have nonempty intersection. Let Y be the subspace of M consisting of the points {x{ : i € /} and let Id : Y —» M be the identity mapping. As before, let X = Y U {w}, where w i s a n arbitrary element (of course, not in Y), and define the distance on X by extending d to

d(xj,w) = inf{r : B{xy,r) D Β(χί\τΐ) for some i € / } ,

j = 1,2, · ■ ·. To see that this is a metric one only need to check that for j , k Ç. I,

d(xj,Xk) < d(xj,w) + d(xk,w).

This is where the binary ball intersection property enters in. Note that since B(xj-,Tj) D B(xj;rj), d(xj,w) < rj for each j 6 / . Also, either d(xj, w) = rj or for some i φ j , B{xi\rî) Ç B(xj-:d(xj,w)). In either case, given j,k G / there exist i,n E. I such that

Β{χί·,η) Ç B(xj;d(xj,w)) and B{xn\rn) Ç B(xk;d(xk,w)).

Since B{xi\rî) Π B{xk\rk) φ 0 , it must be the case that

B(XJ\d(xj, w)) Π B{xk; d(xk, w)) Φ 0 ,

4.3. PROPERTIES OF HYPERCONVEX SPACES 83

from which d(xj,xk) < d(xj,w) + d(xk,w). Finally, since M is injective Id has a nonexpansive extension Id : X —► M.

Then we have Id(w) € M and

d(xj,Id(w)) < d(xj,w) < Tj for each j G / .

Therefore, îd(w) e Ç) β ( χ , ; π ) .

We now prove the reverse implication. This is a reworking of the proof of Proposition 4.3. Suppose (M, d) is hyperconvex, let Y be a metric space with T : Y —» M nonexpansive, and suppose Y is a subspace of a metric space X. Consider the family J\f defined as follows: M is the collection of all pairs (7>, F) where F is a subspace of X for which Y Ç F Ç X and TF : F — M is nonexpansive with Tp{y) = T(y) for each y € Y.

The family Af can be partially ordered by setting

( 7 > , F H ( T G , G )

if F Ç G and TG(z) = TF(z) for each z e F. Also, if {(TFa,Fa)}aeA is a chain in M relative to this order, take G = \J Fa and define TQ ■ G —> M by taking

7b(z) = TF„(x) for any a such that x £ Fa. Then clearly

(7>„, Fa) -< (TG,G) for each a G A,

so each chain in Λ^ is bounded above. By Zorn's Lemma M has a maximal element, say (ΤΗ,Η).

We assert that H — X. If not, then there exists z £ X such that z g H. Now let #1 = # U {z}. We want to define a nonexpansive extension of Tu on H\. Consider the family of balls {B(Tn{x);d{x,z))}xen. Note that for χχ,χ? G H,

d(TH(Xl),TH{x2)) < d(xi,x2) < d(xi,z) + d(x2tz).

Since M is hyperconvex it follows that

f l B{TH(x);d{x,z))?0.

Select we f] B(TH{x);d(x, z)) and define THl : Hx — M by taking T ^ x ) =

TH{X) ii X E H and setting T//, (2) = iu. Since

THl(z)e f]B(TH(x);d(x,z)), XÇ.H

it follows that

d{THi(x),THl(z)) < d(x,z) for each x € H.

Therefore, T//, is nonexpansive. This implies (ΤΗ,Η) -< {THX,H\) with H\ φ H, contradicting the maximality of (ΤΗ,Η). ■


4.4 A fixed point theorem In view of the intimate relationship which exists between the nonexpansive mappings and hyperconvexity via injectivity, it is perhaps not surprising that hyperconvex spaces should admit an interesting fixed point theory for such map-pings.

We begin with a basic definition.

Definition 4.2 A bounded subset D of a metric space is admissible if D = cov(D).

The collection of all admissible subsets of a metric space M will be denoted by A(M). Note that a set is admissible if and only if it can be written as the intersection of a family of closed balls in M. For this reason, the family A{M) is closed under arbitrary intersections. The admissible subsets of a hyperconvex space are interesting for another reason.

Proposition 4.5 Suppose (M, d) is a hyperconvex metric space. Then each set D G A(M) is itself hyperconvex.

Proof. Since D is admissible, we can write D = |~| B(xi\ri), Xi € M. Now

let {B(xa;ra)}a€A be a family of closed balls centered at points xa £ D for which d(xa,xp) < ra + rß, α,β 6 A. By the hyperconvexity of M,

Π B(xa;ra)^0. aÇA

Now consider the family of closed balls

{B(xa;ra): a € Λ } ( J { ß ^ i i n ) : i € / } .

Then for a e A and i,j € / ,

d{xa,Xi) < r, <ra+ri

and

d(xi,Xj) < d(xj, xa) + d(xa,Xj) < rj 4- rj,

so it again follows from the hyperconvexity of M that

( π β(*«;r«) ) n ( π β(χ- rA = ( n B^ r^ ) n ° * ®-\αζΑ ) \i€l ) \aÇA )

This proves that D is hyperconvex. ■

Theorem 4.8 If(M,d) is a bounded hyperconvex metric space and ifT:M—> M is nonexpansive, then the fixed point set Fix(T) of T in M is nonempty and hyperconvex.

4.4. A FIXED POINT THEOREM 85

Proof. We first show that Fix(T) φ 0 . This is another standard Zorn's Lemma approach. Let

F={De A{M) : D φ 0 and T : D — D.)

Then T Φ 0 since M G T and T is partially ordered by reverse set-inclusion. (A > B <* A C B).

First note that if C is a chain in T then C\C e A{M). To see that C € T we need only to show that C\C Φ 0 . To see that this is true apply the definition of hyperconvexity. First index C, say C = {Ca}aeA- Each set Ca can be written as the intersection of closed balls in M, say Ca = Π ■ο(α;«ο,;Γ»α)· Suppose

Ca Ç Cp, and select z € C a . Then for each i a 6 / Q , Î 0 € 7/3,

Φ » α > Zi„ ) < d (ü„ , z) + d{z, xiß ) < ria + riß.

Therefore by hyperconvexity of M,

π°«=n ( n B{xia;riaj)ï0-a£A αξ,Α ia€la

Since each chain in T is bounded above (by the intersection of the elements from the chain), T has a maximal element, which, of course, is a minimal element relative to set inclusion. Call this minimal element D. Our strategy is to show that D consists of a single point.

Since

T(D)C Γ) ß ( i ; r t ( T ( D ) ) ) χ6Λί

and since rx(T(D)) < rx(D), it must be the case that

cov(T(D))Ç f | B(x;rx{T(D)))ç f j B{x;rx{D)) = D.

In particular,

r(cov(T(£>))) Q T(D) C cov(T(D)).

This proves that T : cov(T(D)) -♦ cov(T(D)). Therefore, cov(T(D)) € T, and, since cov(T(D)) Ç D with D minimal, it must be the case that D = cov(T(D)). Thus

D= Π B(x;rx(T(D))), xÇM

which, in turn, implies rx(D) < rx\T(D)J for each x e M; hence

r I ( D ) = r I ( T ( D ) ) I x € M.


Now let δ — diam(D) and let

C{D)= [)B(X;6-

We have already seen that C(D) φ 0 by hyperconvexity lx,y £ D => d(x,y) <

δ δ\ δ — - + - 1. What is important here is the fact that

Dx = C(D)nD?0.

To see this we use the fact that D is admissible, say

D = f]B(xi;ri)

and consider the family

M = {B(xi;ri); i 6 /} (J {# U, Ç) \ ^ 4 c c

We have already seen that d(xi, Xj) < Ti+Tj for i,j £ I and that d(x, y) < - + -

for x, y € D. But also if i E / and x € D then by taking z e D w e see that

cf(xi,a;) < d(xi,z) + d(z,x) < r; + -.

So by hyperconvexity we conclude

Dx =Γ)Μ φ0.

Now let z G Di. Since T is nonexpansive, Γ Γ ( Ζ ) ( Γ ( £ > ) ) < rz(D). However, c

we have already seen that r?^ (T(D)\ = rT^(D), and since rz(D) = -

we obtain rx^(D) < —. Since the reverse inequality is obvious, we conclude

^T{z)(D) = - and, since T(z) e D we have shown T(z) € £>i, that is, T : Di —»

Z?i. Since Dj is admissible, this proves that Di £ T and so by minimality of D

we have £>i = D. But diam(Di) = - and diam(D) = 6. This can only happen if

6 = 0. We conclude, therefore, that D consists of a single point which of course must be a fixed point of T. We have proved the first part of the theorem, namely, that Fbc(T) φ 0.

We now show that Fix(T) is hyperconvex. Let {B(xi\ri)}i€i be a family of closed balls centered at points Xi 6 Fix(T) with the property d(xi, Xj) < ri + rj. Since M is hyperconvex the set

B^f]Bi(xi;ri)^0.

4.5. INTERSECTIONS OF HYPERCONVEX SPACES 87

Since B is admissible, B is itself hyperconvex. Also, if z € B then for each i £ I,

d{T{z),Xi) = d(T{z),T{xi)) < d{z,Xi) < rL

This proves that T(z) € B(xi\ri), that is, T : B —» B. Therefore, we can conclude from what we have shown above that T has a fixed point in B. Thus

te/

This proves that Fix(T) is hyperconvex. ■

4.5 Intersections of hyperconvex spaces

In general, the intersection of two hyperconvex subspaces of a given hyperconvex space need not be hyperconvex. Easy examples exist even in K^,. Let Hi denote the line segment joining (—1,0) to (1,0) and let Hi be the union of the line segment joining (-1,0) to (0,1) and the line segment joining (0,1) to (1,0). Then both Hi and Hi are hyperconvex subsets of the space R^, but Hi Π Hi consists of two distinct points. On the other hand, the intersection of two admissible subsets of a hyperconvex space is also admissible and again hyperconvex.

There are other situations where the intersection of hyperconvex spaces is again hyperconvex.

Theo rem 4.9 Let (M,d) be a bounded hyperconvex metric space and let {Hn} be a descending sequence of nonempty hyperconvex subspaces of M. Then H = oo P) Hn is nonempty and hyperconvex.

n = l

Proof. We begin by showing that H Φ 0, and to do this we introduce a new space.

oo

Let Ή = Γ] Hn be the metric space consisting of all sequences x = {xn} n = l

with Xi € Hit i = 1,2, · · · , and for x = {xn},y = {yn} € H set

d{x,y) =sup{d(xi,yi) : i = 1,2,·■·}·

Then H is bounded and hyperconvex. Now consider the shift operator T :H —» Ή defined by

Γ ({χ η } ) = { χ η + 1 } .

It is trivial to see that T is nonexpansive, so, by Theorem 4.8, T has at least one fixed point {a;n}. But this implies

Xn — X-n+l j ^ — 1, Λ, · · ■ .


oo

This, in turn, implies xn = x G f) Hn. n=l

To prove H is hyperconvex we use what we have just shown. Let {xa} C H and {ra} c K+ satisfy

d(xa,Xß) <ra + r >

Since {xa} C Hn and Hn is hyperconvex, (ΠαΒ (xa;rQ)) Π H n is a nonempty admissible (hence hyperconvex) subset of Hn for each n. Therefore {(naZ?(xa;rQ)) Π Hn}'^'_l is a descending sequence of nonempty hyperconvex subsets of M, so by what we just proved

(Ί~=1 [(naB {xa;ra)) n Hn] = (naB (xa;rQ)) n H φ 0.

This proves / i is hyperconvex. ■

Remark 4.4 Theorem 4-9 is actually true for arbitrary, rather than countable, descending families. This fact is an indirect consequence of Theorem 5.5 in the next chapter.

Definition 4.3 / / M is hyperconvex, then a subset H Ç M is said to be exter-nally hyperconvex relative to M, if for any collection {B(xa;ra)}aÂ of closed balL· in M (not necessarily centered in H) for which d(xa,xp) < ra + rß and

dist(xQ, H) < ra, it is the case that H Ç\ ( Ç)B(xa;ra)) φ 0 .

Theorem 4.10 Let M be hyperconvex and let H Ç M be externally hyperconvex relative to M. Then H C\D is hyperconvex for each D € A{M).

Proof. Let {Β{χα\τα)} be a collection of closed balls in M centered in H Π D for which d(xa,Xß) < ra + rß. Suppose D = f\ B(xi;ri), and let z €

H (ID. Then d(z;Xi) < r{ so dist(a:i,H) < rt. Trivially dist(xQ,H) < ra omce d(x{,Xj) < d(xi, z) + d(xj,z) <ri + rj and d(xi,xa) < rj < n + rQ, the family {B(xa;ra)} U {B{xi\ri)} satisfies the assumptions of external hyperconvexity for H so

( H O D ) f ) (ç]B(xa-ra) j = Hf] lÇ]B(xi]ri) j f | (f]B{xa;ra) j φ 0 .

4.6. APPROXIMATE FIXED POINTS 89

4.6 Approximate Rxed points The fundamental ideas in this section are due to Sine [147]. We begin with some more notation. For any set A in a metric space, let

Ne{A)= (Jß(a ;£) · αζΑ

Thus x e Νε(Α) if and only if there exists a € A such that d(x, a) < e. There is a nice characterization of Νε(Α) if A is an admissible set in a hyperconvex space.

Lemma 4.2 Suppose M is a hyperconvex metric space and let A € A(M), say A — p | B{xi\ rj). Then for each ε > 0,

i€l

Ne(A) = f]B(xi;ri + e).

Proof. Suppose y E Νε(Α). Then d(y,a) < e for some a € A. But for each

iel,

d(y, Xi) < d(a, Xi) + d(a, y) < n + e,

and this proves Νε(Α) Ç f] B(xi\ri + ε). (This implication does not require

hyperconvexity.) Now suppose y € f] B(xi;ri + ε) and let i E I. Hence

d(y,Xi) <u + e ,i e I.

Since A is not empty, we must have

d(xi,Xj) <n + rj ,i,j e / .

So by hyperconvexity of M

AnB(y;e) = (f]B(xi;ri)\ f\B(y;e) φ 0.

Consequently, y € ^ (^4) and the proof is complete. ■

We also need the following:

Lemma 4.3 Suppose M is a hyperconvex metric space and let A E A(M). Then for each ε > 0 there is a nonexpansive retraction R of Ne(A) onto A which has the property d(x,R(x)) < ε for each x E Νε(Α).


Proof. In particular, assume A = {~\ B(xi\ri). By Lemma 4.2 we know that

Ne(A) e A(M) and so Νε(Α) is itself hyperconvex. Consider the family T of all pairs {D,Rr>), where

AC DC Ne(A)

and RD is a nonexpansive retraction of D onto A for which d(x, RD(X)) < ε for each x g D. Note that (A, Id) € T so T ψ 0 . If one orders T in the usual way

( (D, RD) ^ (H, RH) if and only if D Ç H and Ru is an extension of Rp J then each chain in (JF, ) is bounded above, so by Zorn's Lemma T has a maximal element which we again denote (D, RD)- We need to show that D = Νε(Α).

Suppose there exists x € Νε(Α) such that x £ D, and consider the set

C=l f] B(RD(w);d(w,x))]f)( f]B(xi]ri)\f]B(x;e). \w&D J \iei J

First we show that C φ 0 , and in order to do this we need only to show that each two members of the family of balls used in defining C intersect. (This is because M is hyperconvex and thus has the binary ball intersection property.)

First if w\, w-i 6 D then

d(RD(wi),RD(w2)) <d{u>i,W2) < d(wx,x) + d(w2,x)-

This proves that

B(RD{w{)-d(Wl,x))nB{RD{w2);d{w2,x)) φ 0 ,

so each two members of the first family intersect. Also, for each w € D, RD(W) € A = P) B(xi\ ri) so each ball in the first family intersects each ball in the second

family. Since

xeNe(A) = P | ß ( x i ; r i + e)

we know that B(x; ε) ΓΊ Β{χι\ r^) φ 0 for each i e I. Finally, if w € D, then

d(RD(w), x) < d(RD(w), w) + d(w, x) < ε 4- d(w, x)

and this proves that

B{RD{w)\ d(w, x)) Π B{x; e) Φ 0.

We conclude therefore that A D C φ 0 . Now let u 6 C and define R' : D U {x} -> A by setting R'{w) = RD{w) if

w e D and Λ'(χ) = u. Then for w G D,

d{R'(x),R'(w)) = d{u,RD(w)) < d(u,w)

4.7. ISBELL 'S HYPERCONVEX HULL 91

so R' is nonexpansive. Also d(R'(x),x) = d(u,x) < ε. With this we conclude that the pair (£> U {x},R') contradicts the maximality of (D,RD) in (Τ,-<). Therefore, D = Ne(A) and the proof is complete. ■

For a mapping T : M —> M we use Fe(T) to denote the set of t-fixed points of Γ; thus

F,{T) = {x e M : d(x,T(x)) < e}.

Theorem 4.11 Suppose M is a hyperconvex metric space and suppose T : M —> M is nonexpansive. Then for any ε > 0, the set F£(T) is abo hyperconvex.

Proof. Clearly we may suppose Fe(T) φ 0 . For each i in some index set / , let Xi E FE(T), and let r{ > 0 satisfy

d(xi,Xj) < 7\ +rj.

We need to show that

(ç\B(xi-,ri)\Ç)Fe(T)ï0.

We know that J = f] B{xi\ n ) φ 0 by hyperconvexity of M. Also if x 6 J then

for each i Ç. I,

d(xi,T(x)) < d(T(x),T{xi)) + d{T{xi),Xi) < d(x,Xj) + ε < n + ε.

This proves that T(x) 6 NE(J). Now, by Lemma 4.3, there is a nonexpansive retraction R of N£(J) onto J for which d(.R(x), x) < ε for each x € Ne(J). Also since Λ o T is a nonexpansive mapping of J into J it must have a fixed point by Theorem 4.8. Suppose R o T(xo) = xo for ô 6 J. Then

d( io ,T( i 0 ) ) = d ( Ä o T ( i o ) , r ( i o ) ) < e.

and the proof is complete. ■

4.7 Isbell's hyperconvex hull

As we mentioned before, J. R. Isbell [75] showed that every metric space has an injective envelope. The injective envelope of a metric space M is an injective metric space M that contains an isometric copy of M and which is isometric with a subspace of any hyperconvex metric space which contains an isometric copy of M. In this section, we will discuss Isbell's ideas. (In light of Theorem 4.7 we may use the terms 'injective' and 'hyperconvex' interchangeably.)


Let M be a metric space. For any x £ M, define the positive real-valued function / T : M —► [0, oo) by

fx(v) =d(x,y).

Let us discuss some of the properties satisfied by these functions.

1. Using the triangle inequality, we get

d(x,y) < fa(x) + fa(y)

and

fa(x)<d(x,y) + fa(y)

for any x,y,a £ M.

2. Let / : M -» [0, oo) be such that

d(x,y) < f{x) + / (y) , for any x,y £ M,

and for some a £ M, assume that f(x) < fa(x), for any x £ M. In this case, we have f — fa- Indeed, first we have / (a) < / a (a) = 0, which implies / (a ) = 0. Using the above inequality, we get

fa(x) = d(x, a) < f(x) + / (a) = f(x)

for any x G M. Combined with the assumptions on / (x) , we get f(x) = fa(x), for any x € M. This is a minimality property for the pointwise order.

Using the above properties, Isbell introduced what he called the injective envelope of A, denoted e(A), for any subset A of M. The set e(A) is the set of all extremal functions defined on A, where / : A —» [0, oo) is said to be extremal if / satisfies

d(x, y) < f{x) + f(y) for all x, y in A

and, moreover, / is pointwise minimal with respect to this property, that is, if g : A —♦ [0, oo) satisfies

d(x, y) < g(x) + g(y) for all x, y in A

and g(x) < f(x) for all x £ A, then it follows that / = g. In particular, we have fa G e(A) for any a £ A. Consider the mapping e : A —> e(A), defined by

e(a) = fa for all a £ A.


The mapping e is an isometry. In other words,

d{e{a), e{b)) = sup |/0(x) - fb{x)\ = sup \d(a, x) - d(b, x)\ — d(a, b). xÇA χξ,Α

So, A and e{A) are isometric spaces and we may identify A and e(A) via a *-* e(a). Before we give some detailed properties of extremal functions, we will need the following lemma.

Lemma 4.4 Let A be a nonempty subset of M. Let r : A —> [0, oo) be such that

d(x,y) < r(x) +r(y)

for any x,y € A. Then there exists R : M —* [0, oo) which extends r such that

d{x,y) < R{x) + R{y)

for any x,y € M. Moreover, there exists an extremal function f defined on M such that f(x) < R(x) for any x G M.

Proof. This is left as an exercise at the end of the chapter.

Let us give some properties of extremal functions.

Proposition 4.6 The following statements are true.

1. If f € t{A), then f satisfies

f{x) < d(x, y) + f(y) for all x, y in A.

Moreover, we have

/ (x) = sup f(y)-fx(y) =d(f,e(x)). yeA

2. For any f 6 e(A), δ > 0, and x 6 A, there exists yeA such that

f(x) + f(y)<d(x,y) + 6.

3. If A is compact, then e(A) is compact.

4- If s is an extremal function on the metric space ί{Α), then soe is extremal on A.


Proof

1. Assume not. Then there exist xo,2/o 6 A, such that

d(xo,Vo) + f(yo) < f(xo)·

Set

9( . , = ί f{x) if x φ x0

\ d(x0,2/o) + /(3/o) ifx = x0·

It is clear that we have g(x) < f(x), for any x Ç. A. In particular, we have g(xo) < f(xo)- Let us show that for any x,y € A, we have

d(x,y) <g(x) + g(y)-

If both x and y are different from xo, we use the properties of / . So we can assume x — XQ and y φ XQ- We have

d(x, y) = d(x0, y) < d(x0, y0) + d(y0,y) < d(x0, y0) + /(y0) + f(y),

which obviously implies

d(x0,y) < g(xo) + g(y)-

The minimality behavior of / gives us / = g, which is a contradiction. Combining this inequality with the fundamental one (for extremal func-tions), we get

f(y)-fx(y)\<f(x)

for any y € A. The equality holds for y — x which clearly implies

/ (x) = sup f(y)-My) =d(f,e(x)). yeA

2. Assume not. Then there exist x € A and <5 > 0 (we may take it less than f(x)) such that for any y 6 A, we have

d(x,y) + 6<f(x) + f(y).

Set

h(z)

It is easy to check that

d{y,z) <h{y) + h(z)

for any y,z € A. Since h < f and h(x) < f(x), we get a contradiction with the minimality of / . This completes the proof of property 2.

if z φ x 6 if z = x.

4.7. ISBELL'S HYPERCONVEX HULL 95

3. From the property 1, we get

f(x)-f(y)\<d(x,y)

for any x,y € A. This implies e(A) c Lipi(A), where Lipi(A) is the space of all lipschitzian real-valued functions with Lipschitz constant less than or equal 1. Hence, all extremal functions are equicontinuous. Also, it is quite easy to show that pointwise-limit of extremal functions is an extremal function. Since A is compact, the Arzelà-Ascoli theorem (see Chapter 6, Exercise 6.6) implies that e(A) is compact.

4. Let s be an extremal function on the metric space e(A). Note that for any x,y E A, we have

d(x,y) = d{fx, fy) < s(fx) + s(fy) =«oe(x ) +soe(y).

Assume that s o e is not an extremal function on A. Then there exists an extremal function h G e(A) such that h(x) < soe{x), for any x £ A, and the inequality is strict at some point XQ. Define the function

ιν>-\ h(x0) itf = e(xo).

Let us show that t satisfies the inequality

d(f,g)<t(f) + t(g)

for any f,g £ e(A). Since t and s coincide almost everywhere and s is an extremal function, then we only need to prove the above inequality for g = e(x0) and / φ e(xo), that is,

d(f,e(x0))<t(f) + t(e(x0)).

For any δ > 0, there exists y € A such that /(xo) + f{y) < d(xo,y) + δ. If y = XQ, then we must have f(xo) < 1/2(5. Hence

<£(/, e(a:o)) - f(xo) <\δ + t(f) + t{e(x0)).

On the other hand, if y φ xo and / φ e(xo), then

d(f,e{x0)) + f(y) - à = f(xQ) + f(y) - δ < d(x0,y)

and

d(x0,y) < h{x0) + h{y) < h{x0) + s o e{y) = t{e{x0)) + t(e(y)).

Since s is an extremal function, then

t(e(y)) = s(e(y)) < s(f) + d(f,e(y)) = t(f) + f(y),


where we used the fact that / is an extremal function to get d(f,e(y)) = f(y). So we have

d(f,e(x0)) + f(y) - 6 < t(e(x0)) + t(e(y))

and

t(e(y))<t(f) + f(y).

Adding the above two inequalities, we get

d(f, e(*o)) + f(v) - <* + t(e(y)) < t(e(x0)) + t(e(y)) + <(/) + f(y),

which leads to

d(f,e(x0))-6<t(e(x0)) + t(f).

Since <5 is arbitrary, we get the desired inequality

d(f,e(x0))<t(e(x0)) + t(f).

The link between Isbell's ideas and hyperconvexity is given in the following proposition.

Proposition 4.7 The following statements are true.

1. e(A) is hyperconvex.

2. e(A) is an injective envelope of A, that is, no proper subset of e(A) which contains A (metrically) is hyperconvex. Moreover, any hyperconvex metric space H which contains A metrically and is minimal (i.e., any proper subset of H which contains A is not hyperconvex), is isometric to e(A).

Proof

1. In order to prove that e(A) is hyperconvex, let {/Q} be a family of points in e(A) and {ra} be a family of positive numbers such that

d(fa,fg) <ra+r0

for any a and ß. Define the mapping r : {fa} —* [0, oo) by r(fa) = ra. By Lemma 4.4, we extend r to the entire set e(A) such that

d(f,g)<r(f) + r(g)

for any / , g € e(A). Using the same Lemma 4.4, there exists h an extremal function on e(A) such that h <r. Using property 4, we know that hoe is an extremal function on A, that is, h o e 6 e(A). It is easy to see that

hoee f l B ( / ; r ( / ) ) c f | B ( / a ; r Q ) . f€e(A) a


Indeed, we have

Λ o e(x) - f(x) =ho e{x) - d{f,e{x)) < h{f) < r(f)

and

f(x) - h o e(x) = d(f, e{x)) - h o e{x) < h(f) < r(f)

for any x € A. Hence

d(f,hoe)<r(f)

for any / € c(A). The proof of our claim is therefore complete.

2. Recall that A and e(A) are isometric, and let if be a subset of e(A) such that e(A) C H. Assume that H is hyperconvex. Since e(A) is hyperconvex, there exists a nonexpansive retraction R : e(A) —> H. Let / € e{A). We have

d(R(f),e(x)) = R(f)(x) < d(/,e(*)) = / (*)

for any x e A. Since / is an extremal function, we must have R(f) = f. This clearly implies that H = e(A). So no proper subset of e(A) which contains e(A) is hyperconvex. On the other hand, let H be a hyperconvex metric space which contains A metrically such that no proper subset of W which contains A metrically is hyperconvex. Since Ή contains A met-rically, there exists an isometry i : A —* Ή. Using the fundamental exten-sion property of hyperconvex metric spaces, there exists Ri : e(A) —♦ Ή. a nonexpansive mapping which extends the mapping i. Since the mapping e o i _ 1 : i{A) —* e(A) C e(A), there exists R2 : H —» e(A) a nonex-pansive mapping which extends the mapping e o i " 1 . The composition i?2 ° Ri '■ t(A) -* e(A) is nonexpansive and extends the identity map-ping on e(A). The argument described above forces this mapping to be the identity mapping on e(A). In other words, the two maps R\ and R2 are inverse of each other. Since both are nonexpansive, then they are isometries. The proof of property 2 is complete.

So one conclusion from the previous investigation is that if M is any metric space, there is a hyperconvex space, h(M), such that M embeds isometrically in h(M) and the embedding is contained in no hyperconvex proper subset of h(M). The hyperconvex space h{M) is called the hyperconvex hull or injective envelope of M. Moreover, any two hyperconvex hulls of M are isometric. Whenever M is a subset of a hyperconvex space H, a hyperconvex hull of M, h(M) can be found so that M C h(M) C H (due to Theorem 5.5 in the next chapter). Thus h{M) is not unique.


Exercises

Exercise 4.1 Show that the space Ή in Theorem 4-9 is itself hyperconvex.

Exercise 4.2 Show that if M is hyperconvex and if D G A(M), then D is externally hyperconvex relative to M. (See Definition 4-3.)

Exercise 4.3 Verify the last two statements in Remark 4-3.

Exercise 4.4 Prove the statements of Lemma 4-4-Hint: Use the proof of Theorem 4-7.

Exercise 4.5 Suppose (M, d) is hyperconvex. Show that the metric space (N, p) is hyperconvex if and only if the space (M x N)oo is hyperconvex.

Exercise 4.6 Suppose (M,d) is hyperconvex. Show that the space of all nonempty closed subsets of M endowed with the Hausdorff metric is also hyperconvex.

Exercise 4.7 Let H be a hyperconvex metric space and suppose A is an exter-nally hyperconvex subset of H (see Definition 4-3). Show that Νε(Α) is exter-nally hyperconvex (in H) for each ε > 0.

Exercise 4.8 Let H be a bounded hyperconvex metric space and suppose {Ha} is a descending chain of nonempty externally hyperconvex subsets of H. Show that f] Ha is nonempty and externally hyperconvex in H.

a

Exercise 4.9 Let C be the bounded closed convex subset of lô defined by

C — {(xn)\ Xn 6 [0,1] and lim xn — 01. n—oo

Find a nonexpansive mapping T : C —> C with no fixed point. This will prove that the conclusion of Theorem 4-8 does not extend to any bounded closed convex subset of loo ■

Exercise 4.10 In this exercise, we will show that the conclusion of Theorem 4-8 does not extend to unbounded hyperconvex metric spaces. Consider the mapping T : loo —f 'oo defined by

T(XI,X2,-·) = (1 -Κ(χη)),Χΐ,Χ2,-··),

where X is a Banach limit. Check that T is an isometry with bounded orbits, and is fixed point free.

EXERCISES 99

Exercise 4.11 Let (M,d) be a metric space and let B(M) denote the collection of nonempty, bounded subsets of M.

(i) The Kuratowski measure of noncompactness a : B(M) —» [0,oo) is defined by

t=n

a(A) — Ίηΐ{ε > 0; A C M Ai with Ai € B(M) and diam(Ai) < ε}. i= l

(ii) The Hausdorff (or ball) measure of noncompactness χ : B(M) —> [0, oo) is defined by

i=N

χ(Α) = inf{r > 0; A C ( J B(xi,r) with Xi € M } . i = l

Show that if H is a hyperconvex metric space and A a bounded subset of H, then we have

a(A) = 2χ(Α).

Exercise 4.12 Let M be any bounded metric space and e(M) its hyperconvex hull. Show that

X(e(M))=x(M).

Hint: Consider the space λ[αι(,](Μ) of Lipschitzian real-valued functions defined on M with Lipschitz constant less than λ with values in the interval [a, 6], Show that

a(X[aM(M)) < 2λχ(Μ) .

Exercise 4.13 (A Selection Theorem) Let H be hyperconvex, and letT* :—+ S(H), where £(H) denotes the family of all nonempty bounded externally hy-perconvex subsets of H. Show that there exists a mapping T : H —» H for which T(x) € T* (x) for each x 6 H and for which

d(T(x),T(y)) <dH(T-(x),T*(y))

for each x,y 6 H. (Recall that du is the Hausdorff distance and S(H) is the family of externally hyperconvex subsets of H.) Hint: Use the ideas developed in the proof of Theorem 4-7.

Exercise 4.14 Let H be a bounded hyperconvex set, and for λ > 0, let T\ denote the family of all X-lipschitzian functions of H into H. Show that T\ is hyperconvex (relative to the metric p (/, g) = sup {d ( / (u), g (tt)) : u, v € H}).

Chapter 5

"Normal" Structures in Metric Spaces

5.1 A fixed point theorem

The behavior of the collection A(M) of admissible subsets of a hyperconvex metric space M was central to the proof of fixed point theorem for nonexpansive mappings in the previous chapter. Two facts about A(M) were essential to the proof. (1) The intersection of any descending chain of nonempty sets in A(M)

is nonempty and also in A(M). (2) For any D € A(M), r(D) = -diam(D).

The principal observation of this section is the fact that if (1) is assumed to hold for a bounded metric space M, then an assumption much weaker than (2) is sufficient to assure that every nonexpansive mapping T : M —♦ M has at least one fixed point. In Chapter 8 we discuss how this fact is frequently realized in Banach spaces.

Let (M, d) be a bounded metric space, and let A(M) denote the family of all admissible sets in M. (Thus D € A{M) <& D — Γ\{Β : B is a closed ball in M which contains £)}.) We now follow the definitions in the hyperconvex case:

rx(D) =sup{d{x,y):yeD}, xeM,

and

r(£>) = inf{rx(D) :x£D).

Also we use TM(D) to denote the number:

rM(D) = inf {rx{D) :xeM}.

Thus, in general, we have

- diam(£>) < rM(D) < r(D) < diam(D).

101



102 CHAPTER 5. "NORMAL" STRUCTURES IN METRIC SPACES

We have seen that if M is hyperconvex r(D) = TM(D) — - diam(£>).

As before, let

C(D) = {x e D : rx(D) = r(D)}

and

CM(D) = {x e M : rx(D) = rM{D)}.

Also, for A Ç M we denote

cov(A) = C\{B : S is a closed ball and B D A} = C\{B : B € A(M) and B D A}.

Definition 5.1 A(M) is said to be compact if every descending chain of nonempty members of A(M) has nonempty intersection.

It was noted in the previous chapter that the binary intersection property for closed balls implies completeness. Compactness of A{M) leads to the same conclusion.

Proposition 5.1 If M is a metric space for which A{M) is compact, then M is complete.

Proof. Let {xn} be a Cauchy sequence in M and let

rn = sup{d(xi,Xj) :i,j> n}.

Since {xn} is Cauchy, lim rn = 0. Also if F is a finite subset of N (the natural n—»oo

numbers), then

AF = n{B(xi;ri) : i € F} φ 0

since X S U P F € Ap. Compactness of A(M) implies

n { ß ( x i ; r i ) : i e N } ^ 0 .

Since lim r n = 0, this intersection must be a single point which is the limit of n—»oo

the sequence {xn}. ■ Definition 5.2 A(M) is said to be normal (or have normal structure) ifr(D) < diam(D) whenever D 6 A{M) and diam(D) > 0.

The assumption that A{M) is normal is equivalent to the following: If D 6 A(M) consists of more that one point then there is a number r < diam(D) and a point z £ D such that D Ç B(z;r).

At first the Definitions 5.1 and 5.2 may seem esoteric. However, we will see later that both hold in many Banach spaces. In fact, a quick examination of convex sets in R2 shows that the normal structure assumption is typical.

5.2. STRUCTURE OF THE FIXED POINT SET 103

T h e o r e m 5.1 Suppose (M,d) is a (nonempty) bounded metric space for which A(M) is compact and normal. Then every nonexpansive T : M —* M has at least one fixed point.

Proof. Compactness of A(M) and an application of Zorn's Lemma as in the proof of Theorem 4.8 of Chapter 4 yields the existence of a set D € A(M) which is minimal with respect to being nonempty and mapped into itself by T. Also, cov(T(D)) Ç D and T : cov(T(D)) -» cov(T(£>)), so minimality of D implies

D = cav(T(D)).

Now assume diam(Z)) > 0 and choose r so that r(D) < r < diam(Z3). It follows that the set

C={xeD:DC B(x;r)} φ 0 .

Also one can quickly check that

C=(f)B(x;r))f)D. xeD

This proves that C € A(M). Now let z € C. Then if x 6 D

d(T(z),T(x))<d{x,z) < r .

Therefore, T(x) 6 B(T(z);r) for every x e D; hence T(D) Ç B{T(z);r) from which cov(T(D)) Ç B{T{z);r). But D = cov(T(D)), so D Ç B{T{z);r). This proves that T(z) 6 C. Hence T : C —> C. However if z, w G C then d(z, w) < r, so diam(C) < r < dia.m(D). This proves that C is a proper subset of D. Since C e -4(M) and T : C —» C this contradicts the minimality of D. We thus conclude that diam(£)) = 0, so D consists of a single point which must be a fixed point of T. ■

5.2 Structure of the fixed point set

The development of this section will lead to the fact that Theorem 1.1 extends easily to finite commutative families of nonexpansive mappings (and with some-what more effort to arbitrary commutative families).

The following is the key to the structure of the fixed point sets of nonex-pansive mappings (and to the existence of common fixed points for commuting families of such mappings). A subset A of M is said to be a l-local retract of M if for each family {Bi}i€i of closed balls centered at A for which

iei

it is the case that A f] ( Q Bi ) ψ 0 . (It is easy to check that every nonexpansive v i€i J

retract of M is a l-local retract of M.)


We now describe several fundamental properties of 1-local retracts. Most of these results are taken from [90]. We begin with a direct consequence of Definition 5.1.

Proposition 5.2 If M is a metric space for which A(M) is compact and if {An} is a descending sequence of sets in A(M) for which lim diam(;4„) — 0,

71—>00 OO

then f] An — {z}-7 1 = 1

The following technical proposition collects several other properties needed in the sequel. These are analogous to ones already established in the hypercon-vex case.

Proposition 5.3 Let M be a metric space and let A be a nonempty subset of M. Then:

(1) cov(>l) = n{B(x;rx{A)) : x 6 A}; (2) rx(A) — rj:(cov(i4)) for every x 6 M; (3) r(cov(,4)) < r{A); (4) diam(cov(i4)) = diam(A).

Proof. 1. This is immediate since B(x;rx(A)) is the smallest ball centered at x

which contains A. 2. Since A Ç COV(J4) it follows from the definition of rx that

rx(A) < rx(cov(A)).

On the other hand, (1) implies r ^ c o v ^ ) ) < rx(A). 3. This is immediate from the definition of r and (2). 4. It obviously suffices to show that diam(cov(.A)) < diam(.A). Let z €

cov(i4). Then z € B(x; rx(A)) for each x 6 M. In particular, d(x, z) < rx(A) < diam(A) for each x e A; thus A Ç B(z;diam(.A)). Hence

cov(A) Ç B(z\diam(A)).

It follows that

diam(cov(j4)) < diam(A).

The next proposition, which shows that in the case of 1-local retracts nor-mality is a hereditary property, is fundamental to what follows.

Proposition 5.4 Let M be a metric space and suppose A(M) is compact and normal. Suppose N is a given subset of M which is a 1-local retract of M. Then A(N) is compact and normal.


The key to the proof of the above is the following lemma.

Lemma 5.1 Under the hypothesis of Proposition 5.4, r(cov(A)) — r(A) for each A € A(N).

Proof. Clearly we may suppose diam(.A) > 0. Since Proposition 5.3 implies r(cov(.A)) < r(A) we only need to show r(A) < r(cov(v4)). By assumption A e A(N), so A is of the form

A = Nf)(f){B(xi;ri):xieN}).

Also,

cov(A)çf]^B(xi;ri):xieN}.

Choose z e cov(,A) and let r = rz. Then

z e S = ( f l ß (x ; r ) J Q ( p { ß ( x i ; r i ) : Xi e N)). \xÇA /

5 is a nonempty set belonging to A(M). Since N is a 1-local retract of M and S is the intersection of balls centered in N, S Γ\Ν φ 0. Let w 6 SflN. Then

we (n {B{xi; n) : x{ € JV}) Π N,

that is, w € A. On the other hand, w 6 D{J5(a;;r) : x e A}, so rw(A) < r. It follows that

r(A) < r < rz(cav(A)).

Since z was an arbitrary element of cov(.A) the proof is complete. ■

Proof of Proposition 5.4- The definition of a 1-local retract assures that A(N) is compact. To see that A(N) is normal, let A € .A(iV) and recall that diam(cov(A)) = diam(A) by Proposition 5.3. By Lemma 5.1, r(cov(A)) = r{A). Since A{M) is normal, r{cov{A)) < diam(cov(A)), that is, r(A) < diam(i4). ■

We now have the following 'structure' theorem.

Theorem 5.2 Let M be a bounded metric space for which A(M) is compact and normal, and let T : M —* M be nonexpansive. Then the fixed point set FT of T is a nonempty 1-local retract of M, and consequently A(FT) is compact and normal.

Proof. The fact that FT φ 0 is immediate from Theorem 5.1. To see that FT is a 1-local retract of M let {Bj} be a family of closed balls centered in FT for which

S = f]Bi^0.


Then since T is nonexpansive, T : S —> S. Also, since S is admissible, A{S) is compact and normal. Therefore, again by Theorem 5.1, T has a fixed point in S, that is, FTDS Φ 0 . This proves that FT is a 1-local retract of M. The final assertion of the theorem follows from Proposition 5.4.

It is now possible to extend Theorem 5.1 to finite commutative families.

Theorem 5.3 Let M be a bounded metric space for which A{M) is compact and normal. Then every finite family Φ of commuting self-mappings of M has a nonempty common fixed point set F<j>. Moreover, Ft is a 1-local retract of M.

Proof. It suffices to show that F<j> φ 0. The fact that F<j> is a 1-local retract of M can be proved as above. Suppose Φ = {7Ί, · · · ,Tn}. By Theorem 5.2, if FT{ denotes the fixed point set of Tj, i = 1, ■· · ,n, then each family A(FTi ) is compact and normal. Since T\ and T-i commute, it is immediate that ^2 : &T\ —* FT\ · Thus Fj·, Π Fr2 φ 0 . Proceeding step by step one concludes

Π FT, Φ0- ■

5.3 Uniform normal structure

We now turn to the following question: Are there normal type conditions, weaker than hyperconvexity, that are sufficient to assure that the intersection of any descending chain of nonempty admissible sets in a metric space is nonempty (and admissible)? If the underlying metric space is complete the answer is yes, but the approach, due to Kulesza and Lim [105], is a little sophisticated. We continue with the notation of the previous section.

Definition 5.3 A(M) is said to be uniformly normal if there exists c € (0,1) such that r(D) < c diam(£>) for each D G A(M).

We shall see later that the family of admissible subsets of any bounded closed convex subset of a uniformly convex Banach space is uniformly normal.

Theorem 5.4 Suppose (M,d) is a bounded complete metric space for which A(M) is uniformly normal. Then A(M) is compact.

We first establish a more elementary fact. A(M) is said to be countably compact if every descending sequence {Dn} of nonempty sets in A(M) has nonempty intersection.

Lemma 5.2 If A{M) is uniformly normal, then A(M) is countably compact.

Proof. Let

fc0 = sup{r(D)/diam(D) : D € A(M) and diam(D) > 0}.

5.3. UNIFORM NORMAL STRUCTURE 107

Since M is uniformly normal, fco < 1. Now fix k € (fco, 1), and for D € A(M) set

A(D) = {x 6 D : d(x, y) < k diam(D) for each y e D},

that is,

A(D) = D f | I f ) B(y; k diam(£») I .

Clearly A(D) φ (δ and A(D) e A{M). Moreover, A(D) is a proper subset of D if diam(D) > 0.

Now let {D°} be a descending sequence of nonempty sets in A{M). We wish oo

to show that f] D° φ 0 . To this end, set n = l

D\ = cov M J A(D°)

/ oo S

D | = cov M J A(D°) î=2

DA = cov M J A(D°)

We assert that diam(D,\) < k diam(D°). To see this let x,y 6 (J Λ(Γ>°). Then

a; e A(D°) and y 6 A(D%) with, say, n °).

This proves that diam ( (J A(D°) J < fc diam(Z)°). The conclusion now fol-^ t=n '

lows from the fact that diam(cov(A)) = diam(.A) for any AC M. We now repeat the above construction in a step-by-step manner to obtain

the following:

£>? 5 Dl D ■■■ D D°n D · · ·

u u u D\ U

U

m

D

D

D\ U

U

m

D ■·

D ·■

• D

• 2

D\ U

u Dl

D . . .

D · · ■ .


We now invoke the completeness of M. Since diam(£)") < kn diam(Df ) —» 0 as n —» oo, the diagonal sequence {-D"+i}n>i has nonempty intersection by Cantor's theorem. We thus have

oo oo

0 Φ n D«+I ç n D -n=l n=l

completing the proof. ■

For the next step it is necessary to escalate the method of argument. The key idea in the proof of Theorem 5.4 is the following fact. Any uncountable chain of real numbers which is either strictly decreasing or strictly increasing is eventually constant. However, we need a preliminary fact which has to do with ordinal numbers.

L e m m a 5.3 If A{M) is countably compact but not compact, then there exists an uncountable ordinal Γ and a descending transfinite chain {Da,ct < Γ} of nonempty members of A(M) such that

f]Da = 0.

Proof. Since A(M) is not compact there exists a descending chain {Dj}je/ of nonempty sets in A{M) for which f] D{ = 0. Choose an ordinal 7 whose

iei cardinality exceeds the cardinality of the index set / . Select ίχ € I and set D\ = Dix. Proceed by transfinite induction. Let ß < 7. Having defined Da = Dia for a < ß (where ia € / ) , consider two cases.

(1) P| Da = 0. In this case there is nothing to prove—take Γ = ß. a<0

(2) f] Da φ 0. In this case there must exist i € / for which D{ is a proper

subset of f) Da. (Such a set must exist because f~) Di = 0.) Define Dp — A · a<ß ig/

Since the cardinality of 7 exceeds that of / case (2) cannot prevail throughout the induction process, so case (1) must hold for some ß < 7. Also ß must be uncountable because A{M) is countably compact. The set of all ordinals for which the conclusion of Lemma 5.3 holds is nonempty and thus by well-ordering there is a smallest such ordinal, which we again call Γ. ■

A chain {Da} that satisfies the conclusions of Lemma 5.3 for Γ will be called a Γ- chain.

Given a Γ-chain {Da}, for each a 6 Γ let

da — diam(£)a).

5.3. UNIFORM NORMAL STRUCTURE 109

Since {dQ}Q<r is nonincreasing, limdQ = d (in fact, for a sufficiently large, a

da = d). We call d the diameter of {Da} and write

d = diam({Da}).

Finally, we say that a Γ-chain {Ja} is a refinement of a Γ-chain {Da} if Ja Ç Da for each α < Γ. We now have the following.

Lemma 5.4 Under the hypothesis of Lemma 5.3, there exists a T-chain {DQ} with the property: If{Ja} is a refinement of{Da} then diam({Ja}) = diam({DQ}).

Proof. We now know there exists at least one Γ-chain {Da}· Define a sequence {Jn} of Γ-chains inductively as follows. Let J\ — {Da} and suppose Jj has been defined. Let

dj — inf {δ : there exists a refinement of Jj with diameter 6}.

Now let Jj+i = {Da+l} be a refinement of Jj = {Da} for which

di&m(Jj+i) <dj + -—-.

oo For each a < Γ define Da = f] Da. By countable compactness of A(M) each

i= i set Da is nonempty (and in A{M)). Therefore {Da} is a Γ-chain.

Suppose J is refinement of {DQ}. Then J is a refinement of Jj for each j as well, so

dj < diam(J) < diam({Z?;}) < diam(J i + 1 ) < dj + -.—-.

Since this holds for each j = 1,2, · · · , we conclude d i a m ^ ) = diam({D*}). ■

Theorem 5.5 Suppose (M,d) is a bounded metric space for which A(M) is countably compact and normal. Then A(M) is compact.

Proof. Suppose A(M) is not compact. Then by Lemma 5.4 there exists an uncountable Γ-chain I — {Da} in A(M) whose diameter d is minimal with respect to refinement. Since lim diam(Da) = d and Γ is uncountable, there

α<Γ exists β € Γ such that for all a > β, diam(Z)a) = d. We may assume d > 0 since otherwise each of the sets of the chain {Da} would eventually consist of the same point, implying Ç\Da φ 0 .

α Now let a > β. Since A(M) is normal, for each such a there exists ra <

diam(£>Q) = d and pa € Da such that Da Ç B(pa; rQ). We now come to a subtle part of the argument. Let

ln=[d(l-±),d(l-^)),n = l,2,....


Note that for each ß < a < Γ, ra G In for some n. To see this suppose for each n there exists ßn such that a > ßn => ra £ In. Since Γ is uncountable and {/?„} is countable, there exists μ < Γ such that μ > βη for each n. But this would imply that τμ for such μ could not lie in any of the sets In- a contradiction. We therefore conclude that for some m the set S = {a > β : ra G Im} is cofinal in

Γ. Let r = d ( 1 j and let J = n{B(pa;r) : a € S}. Now let μ,ν G S,

and say μ > v. Then Du Ç Dv. Also r„ G 7m so r„ < d I 1 1 = r. \ m + 1

/ Therefore,

P^ G £>μ Ç £>„ Ç B(P„ ;TV) Ç 5 ( Ρ Ι / ; Γ ) .

We conclude from this that

{Pa ■ ex G S} Ç J.

Now let K = n{B(x;r) : x G J}. Since {pa : a G 5} C J we have K Ç J. Suppose y,z e K. Then z € J and y G B(z;J") so d(y,z) < r. This implies diam(ii) < r. Also, if x G J and a G 5 then x G ß ( p a ; r) , so pQ G if, that is,

{pQ :aeS}ÇK.

Define the chain J = {JQ} by setting J„ = Da for a < ß and J a = £>α Π Ä" for /3 < a < Γ. Suppose β < μ < Γ. Then since 5 is cofinal in Γ there exists a Ç S such that μ < a. Hence pa G Da Π K Ç £)μ Π Ä" and it follows that JQ φ 0 for each a (and clearly JQ G -4(M)). Since diam(v7) < diam(K) < d and J is a refinement of J we have a contradiction. ■

5.4 Uniform relative normal structure

This is a weaker variant of the concept of uniform normal structure that does not seem to imply normal structure. Yet it does assure the existence of fixed points for nonexpansive mappings. This concept was introduced by Soardi in [149].

The admissible sets A(M) in M are said to be uniformly relatively normal if there exists c G (0,1) such that for each D G A(M) there exists zp G M such that:

(a) D Ç B(zD;c diam(D)); (b) if D C B{y\c diam(£>)) for y G M, then d(zD,y) < c diam(D).

Theorem 5.6 Let M be a nonempty bounded metric space for which A(M) is compact and uniformly relatively normal. Then every nonexpansive T : M —* M has at least one fixed point.

5.4. UNIFORM RELATIVE NORMAL STRUCTURE 111

Proof. By the compactness of A(M) and Zorn's Lemma there exists AQ G A(M) such that

Ao = cov(T(A)))·

Let r = diam(i4o). Then the set

A={zeM :A0Ç B(z; cr)} = f] B(x; cr) xGAo

is nonempty (ZA„ G A). Moreover if z G A and u G AQ, then

d(T(z),T(u)) <d(z,u) <cr;

hence T(A0) Ç B(T(z);cr), from which

Ao = cov(T(A0)) Ç B(T(z);cr).

Thus T : A -+ A Now let

tf=(p|£(2;cr))p|>l,

and let

A\ = n{D G A{M) : T(D) C D a n d D D H } .

Clearly, T : A[ —» -A'j. We show next that d ia rn^ i ) < cr. To see this, let

Since i î Ç ^Ί , then we must have F D H. Indeed, let x G H. Then x € Ay Since x G if, we have

xG p | £ ( z ; c r ) Ç f | B(z;cr),

z£A z£A\

where the second assertion is true because A\ Ç A. Now let z G F and set

G = ß(T(z);cr)nyl ' i .

Then if w € H, d(w,u) < cr for each u G A\\ hence d(u;,T(z)) < cr. Therefore, H ÇG. Also, if x G G then d(T(x),T{z)) < d{x,z) < cr and thus T(x) G G, that is, T : G -* G. It follows that A\ Ç G; hence A\ = G. Therefore B(T(z);cr) D A\ proving T(z) G F and, in turn, proving F — A\. Hence diam(Ai) = diam(F) < cr. Notice also that since A\ Ç A, AQ Ç B(x;cr) for each x G A\.


Now apply Zorn's Lemma again to obtain a set Αχ e A(M) such that A\ Ç A\ and such that cov(T(Ai)) — A\. Proceeding in this way it is possible to construct a sequence {An} Ç A(M) such that

(i) T:An^ An; (ii) diam(.An) < rcn\ (Hi) d(u, v) < rcn if u e An, v E An-i.

The argument is completed by selecting un 6 An for each n and observing that the sequence {un} is a Cauchy sequence which converges, necessarily, to a fixed point of T. ■

5.5 Quasi-normal structure

We now turn to the question of whether the assumption that A(M) is normal in Theorem 5.1 can be weakened. There is an obvious candidate for such a weakening. As before, we assume M is a bounded metric space.

Definition 5.4 The admissible structure A(M) of M is said to be quasi-normal if for each D € A(M) with diam(Z?) > 0 there exists a point p € D such that for each x € D :

d(p,x) < diam(D).

Obviously, the following implications hold:

uniformly normal => normal => quasi-normal.

The metric definition of quasi-normal was inspired by its Banach space coun-terpart which we take up later. While the concept is well understood in Banach spaces its significance in metric spaces is less clear. However the question of whether the condition is strong enough to imply the existence of fixed points for nonexpansive mappings can be quickly settled in the negative. (The answer is unknown for Banach spaces.)

Example 5.1 Let R+ denote the nonnegative reaL·, and for x, y € R+ introduce the metric

d(x,y)= , *> y I -

l + | x - y | Then if M = (R+,d), A(M) is both compact and quasi-normal. However, the mapping T : M —» M defined by T(x) = x+1 is nonexpansive (in fact, isomet-ric) and fixed point free.

Proof. We observed in Chapter 2 that d is a metric. Now suppose $ is a family of balls in M with the finite intersection property. If B(x; r) € ΰ => r > 1 then Π3 D M φ 0. Otherwise there exists B(x;r) e 3 with r < 1. In this case B(x;r) is a compact subset of R+ and {B Γ\ B(x\r) : β 6 3} has the finite

5.5. QUASI-NORMAL STRUCTURE 113

intersection property, from which ΓΊΟ φ 0 . It follows that A{M) is compact. To see that A(M) is quasi-normal note that the closed balls in M (and hence their intersections) correspond to closed intervals in R + . If such an interval is bounded then its midpoint will be a nondiametral point of the interval relative to the metric d. On the other hand, if D is unbounded in K+ then relative to d, diam(D) = 1 while d(x, y) < 1 for each x,y Ç. D. ■

The concept of quasi-normality does have fixed point theoretic implications for a special class of mappings.

Definition 5.5 A mapping T : M —* M is said to be a Kannan nonexpansive mapping if for each x,y € M,

d(T(x),T(y)) < ±[d(x,T(x)) + d{y,T(y))}.

Note that mappings of the above type may or may not be nonexpansive in the usual sense. In fact, the Kannan condition does not even imply continuity of the mapping.

Kannan nonexpansive mappings are notable for two reasons. On the one hand they (and their contractive analogs) have inspired a branch of fixed point theory devoted exclusively to the study of increasingly complex generalizations of contractive type conditions. This is an esoteric branch of the theory which seems to be of limited interest. However, it is the case that Kannan mappings are inextricably tied to the concept of quasi-normality. This fact, observed in [127], may be of wider interest.

Theorem 5.7 Suppose (M,d) is a nonempty bounded metric space. IfA(M) is (countably) compact and quasi-normal, then any Kannan nonexpansive mapping T : M —+ M has a unique fixed point. Conversely, if A(M) is not quasi-normal, then there exists a nonempty D G A{M) and a Kannan nonexpansive T : D —» D which is fixed point free.

The easy part of the above theorem is the converse. If A(M) is not quasi-normal then there exists D € A(M) for which diam(D) > 0 but which has the property that for each x 6 D there exists T(x) € D such that d(x, T(x)) — diam(£)). From this we have for each x,y G D,

d(T(x),T(y)) < diam(D) = l-[d{x,T{x)) +d{y,T{y))

We now turn to the main part of the theorem, beginning with the following lemma.

Lemma 5.5 For r > 0, assume Ar = {x 6 M : d(x,T(x)) < r} φ 0, and let Cr = cov(T(,4r)). Then Cr Ç Ar and diam(Cr) < r.


Proof. Fix x 6 Cr. If d(x,T(x)) = 0 the theorem follows, so we suppose d(x,T(x)) > 0. Let s = sup{d(T(z),T(x)) : z e Ar}. Then B(T(x);s) D T(Ar), where B(T(x); s) D cov(T(Ar)) = Cr. Since x e Cr we have d{x, Γ(χ)) < s. By the definition of s, for each ε > 0 there exists z 6 Ar such that d(T(x), T(z)) > s — ε. We now have

d(x,T(x)) - ε < s - ε < d(T(x),T(z)) < ~ d{x,T(x)) + d{z,T(z))

This immediately implies d(x,T(x)) < ά(ζ,Τ(ζ)) + 2ε < r+Ίε, that is, Cr C Ar, since ε > 0 was arbitrary.

To complete the proof of the lemma, observe that

diam(CV) = dia.m (cov(T(Ar))) = diam (τ(ΑΓ)Υ

But diam(r(Ar)) = sup{d(T(i) ,r(y)) : x,y 6 Ar) < hr + r]=r. m

Proof of Theorem 5.7. Set

r0 = i n f { d ( x , T ( x ) ) : x 6 M } .

To see that Aro ψ 0 notice that if {rn} is a sequence strictly decreasing to TQ then each of the sets

G-, D G-, D · · O Cr„ D · · ·

is nonempty and in ^4(M). Thus

> i r „ = n A^ ^ πσ»·» ^ 0 · n=l n=l

Now suppose TO > 0. Since A(M) is quasi normal there exists z € CTo such that d(z,y) < diam(Cro) < r0 for each y G Cro. But T(^) € T(Aro) Ç C*,-0, so this implies d(z, T(z)) < r0 and this contradicts the definition of ro. ■

The foregoing discussion raises a basic question about nonexpansive map-pings.

Definition 5.6 We say that a metric space M has the FPP for nonexpansive mappings if for any nonempty D € A{M) and any nonexpansive T : D —♦ D it is always the case that T has at least one fixed point.

Problem. Is there a geometric condition which lies between normal and quasi-normal which is both necessary and sufficient for the FPP?

Related questions are abundant. For example:

Problem. If M has the FPP, is A(M) necessarily compact? Is A(M) necessarily quasi-normal?

5.6. STABILITY AND NORMAL STRUCTURE 115

5.6 Stability and normal structure In this section we look at mappings which do not increase 'large' distances. Such mappings do not need to be continuous, and our objective is not to seek fixed points but rather to determine what can be said about minimal displacement for such mappings.

Definition 5.7 Let M be a metric space. A mapping T : M —* M is said to be h-nonexpansive for h > 0 if d(T(x),T(y)) < max{d(x, y), h} for x,y € M.

Our principal result is the following: The surprising aspect of the conclu-sion of this theorem is the fact that /i-nonexpansive mappings have minimal displacement strictly less than h in the presence of normality.

Theorem 5.8 Let (M,d) be a metric space for which A(M) is compact and normal, and suppose T : M —> M is h-nonexpansive for h > 0. Then there exists z S M such that d(z, T(z)) < h.

Proof. By a standard Zorn's Lemma argument there exists D € A{M) which is minimal with respect to being nonempty and invariant under T. Then clearly D = cov(T(£>)). Let r = r(D) and let C = C(D). If diam(Z?) = 0 then D consists of a single point which is fixed under T and there is nothing to prove. So assume diam(D) > 0 and let z € C. Observe that if r < h, then d(z, T(z)) <r<h and again we are finished. So we assume h < r and show that this leads to a contradiction. Let x 6 D. If d(z,x) > h then d(T(z),T(x)) < d(z,x) < r. On the other hand, if d(z,x) < h, then d(T(z),T(x)) < h < r. So in either case T(x) E B(T(z);r). Thus T(D) Ç B(T(z);r), from which D = cov(r(£>)) C B(T(z);r), and this, in turn, implies T(z) € C. Therefore, T : C —> C. But since C G A(M) this contradicts the minimality of D. ■

If A{M) is uniformly normal, it is possible to say a little more. The proof is only a slight modification of the one just given.

Theorem 5.9 Let (M, d) be a metric space for which A{M) is compact and uniformly normal with normality constant c € (0,1), and suppose T : M —> M is h-nonexpansive. Then there exists z 6 M such that d(z,T(z)) <ch.

Proof. Again assume D 6 A(M) is minimal with respect to being nonempty and invariant under T. Then clearly D = cav(T(D)). Let r = r(D) and let C = C{D). If diam(D) = 0 then D consists of a single point which is fixed under T and there is nothing to prove. So assume diam(£>) > 0 and let z € C. Next observe that if diam(Z)) < h, then d{z,T{z)) <r = c diam(JD) < ch and again we are finished. So we assume h < diam(£>) and show that this leads to a contradiction. To this end let h! = max{/i, r} and let

C = {ztD:DÇB{z\h')}.


Then C" € A(M) and since C Ç C, C Φ 0 . Let z € C and let x e D. If d(z,x) > h', then </(z,x) > /i and d(T(z),T(x)) < d(z,x) < h' so in this case T(x) e B(T(z);h'). On the other hand, if d(z,x) < h', then d(T(z),T(x)) < h < h' and again Γ(χ) e B{T{z);h'). Thus T(D) Ç B(T(z);h'), from which D = cov(T(D)) Ç B(T(z); h'), and this, in turn, implies T{z) 6 C. Therefore, T : C —* C. But since C" e A(M) is nonempty and since C" is a proper subset of D (because diam(C) < h' < diam(D)), this contradicts the minimality of D.

m

Corollary 5.1 Let (M,d) be a hyperconvex metric space and suppose T : M —» M is h-nonexpansive for h > 0. Then there exists z E M such that

d{z,T{z))<!±.

Proof. As we saw in Chapter 4, for a hyperconvex space M the admissible sets A(M) are compact and uniformly normal for c = 1/2. ■

5.7 Ultrametric spaces

Ultrametric spaces and hyperconvex metric spaces share many common prop-erties, yet they are quite different in very distinctive ways. The most striking similarity has to do with the injective extension property; the most striking difference is likely the fact that while hyperconvex metric spaces are always metrically convex, ultrametric spaces never are.

Ultrametric spaces are metric spaces for which a much stronger version of the triangle inequality holds.

Definition 5.8 A metric space (M, d) is an ultrametric space if, in addition to the usual metric axioms, the following property holds for each x,y,z € M :

d(x,z) < max{d(x,y),d(y,z)}.

It is immediate from the definition that if d(x, y) φ d(y, z) then, in fact, d(x, z) — max{d(x,2/),<i(i/,z)}. Therefore, each three points of an ultrametric space represent vertices of an isosceles triangle. This leads to another anomaly.

(l) If B(a; r\) and B(b; Γ2) are two closed balls in an ultrametric space, with ri < r 2 , then either B(a;r-i)(~)B(b;r2) = 0 or B(a;ri) Ç B{b;r2). In particular, if a e S(6;r2) , then ß (a ; r i ) Ç B(b;r2).

Ultrametrics arise naturally in the study of non-archimedean analysis; in particular, in the study of normed vector spaces over non-archimedean valuation fields ([122], [126], [139]). Here we discuss a few results which hold in the most general setting. Most of these results are taken from [131] and [130].

We begin with a characterization of ultrametric spaces in terms of nonexpan-sive extensions found in [132]. (Actually, a more general version of this theorem

5.7. ULTRAMETRIC SPACES 117

is proved in [132] in which the 'distance' values lie in linearly ordered sets with smallest elements rather than in R.)

An ultrametric space M is said to be spherically complete if every chain of closed balls in M has nonempty intersection. As a consequence of (1) the admissable sets A(M) of M coincide with the closed balls of M. Therefore, the assertion that M is spherically complete implies A(M) is compact in previous terminology.

The following is now an immediate consequence of (1).

(2) If JF is a family of closed balls in a spherically complete ultrametric space and if each two members of T intersect, then C\F φ 0.

Another consequence of (1).

(3) If B(a;ri), B(b;r2) are closed balls in an ultrametric space, if B(a;ri) Ç B(b;r2), and if 6 ^ B(a;ri), then d(b,a) — d{b,z) for each z G B{a\r\).

Proof. Since 6 $ B(a;r{) and z G B{a;r{), d(b,a) > d(z,a). Hence d(b,a) = d{b,z). m

Since the diameter of a closed ball in an ultrametric space can never exceed its radius, A(M) is never normal for an ultrametric space M.

An ultrametric space (M, d) is said to have the extension property (EP) if given any ultrametric space (X, p) and any subspace Y of X, every nonexpansive mapping / :Y —» M has a nonexpansive extension F : X -* M.

Theorem 5.10 An ultrametric space is spherically complete if and only if it /ios the extension property.

Proof. (=>) Suppose (M, d) is spherically complete, let (X, p) be an ultramet-ric space, let Y be a subspace of X, and suppose / : Y —♦ M is nonexpansive. Let 2 € X with z <£. Y and let Y' = Y U {z}. We first show that / has a nonexpansive extension F : Y' —* M.

Let T = {B(f(y);p(y,z)) : y 6 Y}- We assert that each two members of T intersect. Indeed, suppose yi,t/2 G Y with p{y\,z) < p(y2,z). Then z e B(yi;p(yi, z)) n B(y2;p(y2,z)) so by (1)

B(yi;p{yi,z)) Ç B{y2;p(y2,z)).

Therefore, d{f(y1),f(y2)) < p{y\,y2) < p(j(2,z), so f(yx) e B{f{y2);p(y2,z)). By (2) (XF φ 0 , so let p € C\F and define F(z) = p. Then if F(y) = f(y) for each y 6 Y, d(F(z),F(y)) < p(z,y) and F is a nonexpansive extension of / to Y'. The proof of this implication is now completed by using Zorn's Lemma as in the extension theorem of Aronszajn and Panitchpakdi (Theorem 4.7 of Chapter 4)·


(<=) Now assume (M,d) has the extension property but is not spherically complete. Then there exists a decreasing family {ß(xi;rj)}tg/ of closed balls in M for which (~) B(xi; rj) = 0 . Let M' = M U {p} where p £ M and define

«€/ a metric p on M' as follows. Set p(x, y) = d(x, y) if x, y 6 M; otherwise for x e M set p(p,x) = ti(x,Xj), where x £ B(XJ\TJ). By assumption, such j 6 J must exist. To see that p is well defined, notice that if x ^ B{xk\rk) then, since these balls are nested, d(xj,Xk) < d{x,Xj). Thus d(x,Xj) = d(x,Xk).

By the extension property, the identity mapping on M has an extension F : M' —» M. Also, if x, ^ B{xj\rj) it must be the case that B{XJ;TJ) Ç B{xi\rî). Thus

d{F[p),Xi) = d(F(p),F(xi)) < p(p,Xi) = ά{χ{,χά) < ru

and F(p) e f] B{xi\ r^). This contradicts the original assumption and completes

the proof. ■

The following 'fixed point result' is attributed to Petalas and Vidalis [130].

Theorem 5.11 Suppose (M,d) is a spherically complete ultrametric space and suppose T : M —> M is nonexpansive. Then either T has a fixed point, or there exists a ball B = B(x; r) in M such that (i) T : B —» B and (ii) d(z, T(z)) = r for each z 6 B.

Proof. Consider the family T — {B(x;d(x,T(x))) : x G M} ordered by set inclusion and let A be a descending chain in T. Since M is spherically complete, Π.4 = B φ 0 . Let b € B and B(a;d(a,T{a))) e A. Then if x € B{b;d(b,T(b)),

d{x,b) < d(b,T{b)) < max{d{a,b),d{a,T{a)),d(T(a),T(b))} =d(a,T(a)).

Thus

d(x,a) < max{d(a,b),d(x,b)} < d(a,T(a)).

This proves that B(b;d(b,T(b))) Ç B(a;d(a,T(a))); hence each descending chain in J- is bounded below by a member of T. It follows from Zorn's Lemma that T has a minimal element, say B(z;d(z,T(z))). For any x € B(z;d(z,T(z))), we have

d(x,T{x)) < max{d(x,z),d(z,T{z)),d{T{z),T{x))} = d(z,T(z)).

Since B(x;d(x,T{x))) C\B{z;d{z,T{z))) φ 0 , it follows (from (1)) that

B{x;d(x,T(x))) Ç B(z;d(z,T(z)))

5.7. ULTRAMETRIC SPACES 119

so by minimality, B(x;d(x,T(x))) = B(z;d(z,T(z))). If d(x,T(x)) < d{z,T(z)) then

d(x, z) = d(z, T(z)) > d(x, T(x)).

However, this would imply z φ B(x; d(x, T(x))), contradicting B(x; d(x, T(x))) = B(z\d(z,T(z))). Therefore, it must be the case that

d(x,T(x)) = d(z,T(z)) for each x € B(x;d(x,T(x))).

Corollary 5.2 Under the assumptions of the theorem, ifT:M—>Mis con-tractive then T has a fixed point.

Proof. If B = B(z;d(z,T(z))) satisfies alternative (ii), then

T(z) € B(z;d(z,T(z)))

and d(T(z),T2(z)) — d(z,T(z)). Since T is contractive this, in turn, implies z = T(z). m

Remarks. Ultrametrics arise in the study of non-archimedean analysis and in particular in the study of Banach spaces over non-archimedean valuation fields.

Definition 5.9 A (real) valuation on a field K is a mapping | · | : K —> K which satisfies for each a,b G K : (i) \a\ > 0;

(ii) \a\ = 0 O· a = 0; (Hi) \ab\ = \a\ ■ \b\; (iv) \a + b\ < | a | + |ft|.

In addition, | 1 | — 1; |— a\ — \a\; and | a - 1 | = \o.\~ ifaÔ.

The most obvious valuation is just 'absolute value' on the field of reals R. If K is the field of complex numbers the usual valuation on K is given by \z\ — γ/χ2 + y2 for z = x + iy € K.

A less obvious example arises as follows: Let Q be the field of rational num-bers and let p be a fixed prime number. Each a eQ has a unique factorization of the form a = (α/β)ρη, where a and β are relatively prime to p. The p-adic valuation on Q is given by setting \a\ = p~n, |0| = 0.

A valuation | · | is said to be archimedean if there exists an integer n in the underlying field K for which \n\ > 1. If \n\ < 1 for each integer n in K, then the valuation is said to be non-archimedean. It can be shown that a valuation | | is non-archimedean if and only if the inequality in (iv) is strengthened to

a + b\ < max{|a|, |6|}.


It is also known that a valuation on Q is non-archimedean if and only if it is equivalent to p-adic valuation for some prime p. (Two valuations | · | χ and | · | 2 on K are said to be equivalent if and only if for each a € K, \a\x < 1 4* \a\2 < 1.)

Given a valuation field K the function d(a,b) = \a — b\, a, b € K, always defines a metric on K. If the valuation is non-archimedean then the metric is an ultrametric. The p-adic numbers arise as the completion of the space (Q, | | ) , where | | is the p-adic valuation on Q.

If a norm on a normed space X over a valuation field K satisfies the inequality

||x + 2/11 < max { | |χ| | , | |ζ/ | |}

for each x,y € X, then it can be shown that, in fact, the valuation on K is non-archimedean.

5.8 Fixed point set structure—separable case

This chapter concludes with a rather deep theorem due to R. E. Brück [29]. We have already seen that if M is bounded and hyperconvex then the fixed point set F(T) of a nonexpansive mapping T : M —♦ M is always hyperconvex and thus a nonexpansive retract of M. We have also seen that if A(M) is merely assumed to be compact and normal, then F(T) is only known to be a 1-local retract of M. This raises the question of whether hyperconvexity is necessary for the stronger conclusion. A complete answer to this question is unknown, but it follows from the result proved below that in many instances F(T) is indeed a nonexpansive retract of the domain of T.

Properties of A(M) do not play a direct role in the development of this section, and to be fully appreciated the result proved here should be recast in a Banach space framework. We shall do this later. However, the proof given below illustrates the generality of the metric approach.

Some ideas will be needed here which have not yet been discussed. We consider a metric space M and the family MM of all mappings of M into M. One can think of MM as the product space

ΠΜ*. where each of the spaces Mx coincides with M itself. The usual product topology on MM then coincides with the topology of pointwise convergence on M. To be precise, a net {/a} Ç MM converges t o / e MM if and only if

lim/Q(x) = f(x) for each x € M. a

In particular, if S Ç MM, then / € S (the closure of S in the product topology) if and only if / = lim fa for some net {/Ql Ç S. Hoping to avoid confusion, we

a

shall reserve this notation to denote closure in MM, and we shall use clK to denote the closure of a subset K C M in the metric of M.

5.8. FIXED POINT SET STRUCTURE—SEPARABLE CASE 121

In the following theorem we consider a semigroup S of (nonexpansive) map-pings in MM. To say that 5 is a semigroup simply means that the composition fogeS whenever / , g 6 5. A left ideal J of S is a subset of S with the property s o j e J for each j & J and s e S. We use F(5) to denote the common fixed point set of S, that is,

F(S) - {x € M : s(x) = i for each s 6 5 } .

Also, for x 6 M,

S(x) = (s( i ) : s G 5 } .

Recall that a subset A Ç M is said to be dense in M if cl(.A) = M, and that M is separable if it contains a countable dense subset. Separability is essential to the validity of the approach given here.

Theorem 5.12 Let M be a separable complete metric space and let S be a semigroup of nonexpansive self-mappings of M. Then there exists in S a (non-expansive) retraction r of M onto F(S) if and only if one of the following two equivalent conditions holds:

(i) Each nonempty closed S-invariant subset of M contains a fixed point ofS.

(ii) Whenever x e M then clS(x) Π F(S) ψ 0 .

Proof. We begin with the easy implication. Suppose r is any retraction of M onto F(S), suppose H is a nonempty closed 5-invariant subset of M and let f e S. Then for some net {/„} Ç 5, l im/a(x) = f(x) for each x € M. Since H a is closed, f(x) € H for each x 6 H, so H is also invariant under S. Since r € S, r(H) C H. But r is a retraction of M onto F(S), so this implies r(H) Ç F(S). This proves F(S) ΠΗ 2 r(H) Φ 0 , so (i) holds. _

Now observe that in order to prove there exists in 5 a retraction r of M onto F(S) (which is necessarily nonexpansive because 5 is a nonexpansive semigroup) it suffices to show that S contains a one-element left ideal {e}. From this the conclusion is easy because then one has s o e = e for each s € S. This means that s o e(x) — e(x) for each x E M and s Ç 5 . In particular, the range of e is contained in F(S) (and, of course, F(S) = FjS))._A.t the same time e(x) = x=> s(x) = x so F(e) Ç F(S). But since e € 5 , F(S) = F(S) Ç F(e). This proves that the range of e and F(S) coincide with F(e), proving e is a retraction of M onto F(S).

We now embark on the task of proving that (ii) implies 5 contains a one-element left ideal. (We shall prove the equivalence of (i) and (ii) later.)

Note that since S is a semigroup of nonexpansive mappings it must also be the case that 5 is as well. We begin by introducing a metric p on S relative to which S is complete. A one-element left ideal of S is then constructed via Cantor's theorem as the intersection of a descending sequence of closed left ideals whose diameters converge to 0.


The metric p is defined as follows: Let {pi,P2, ■··} be a countable dense subset of M and for s, t e S set

To see that p is complete, let {sn} be a Cauchy sequence in (5, p). Then

<*(*n(Pi),am(Pi)) < 2 j r -,

ι + Φ»(ρ«),^))-Λ,α

so for each i, isn(Pi)} is a Cauchy sequence in M. Let s(pi) = lim sn(pi). (M n—»oo

OO £

is complete.) Now let ε > 0 and choose N 6 N so that ]T 2 _ t < - . Then ί=ΛΜ-1 2

nie -^ - ^ 1 ( d(S"(P')'S(P»)) \ . £

But for each i — 1, · · · , ΛΓ, there exists n^ such that n > rij =>· d(sn(pi), s(p,)) <

——. It follows that if n > max{ni : i = 1, · ■ · , ΛΓ}, then / J (S„ ,S) < e, proving <£i V

that p is complete.

Now for u e F(S) and Â: > 0, let Nfc(u) = B ( u\- ) . Then obviously

a : Nk(u) —► JVfc(u) for each s e S.

Assertion. / / 7 is any closed left ideal of S, if x € M, and if k is any positive integer, then there exists a closed left ideal J' Ç J and an element u € F(S) such that J'{x) Ç Nk(u).

Proof of the Assertion. Since J is a left ideal of 5, s o j(x) e J{x), so J(x) is 5-invariant. By continuity cl J(x) is 5-invariant as well. By (ii) there exists u € clJ(x) Π F(S). In particular, Nk(u) Π J(x) φ 0 ; hence J ' φ 0 , where

J ' = { j e J : j ( : r )eJV f c (u)} .

Also, if s G 5 and j € J', then s o j(x) e J(x) and because

rf(soj(i),u) =d(soj(x),s(u)) < d(j(x),u),

s o j(x) G Nk(u). This proves that s o j e J ' ; hence that J' is a left ideal of 5. Also, J ' is closed since J is closed and Nk(u) is closed. (Suppose {ja} is a net in J' which converges to j ' . Then j ' € 7 and lim ja{x) = j ' (x) ■ But ja(x) € Nk(u)

a

for each a and since Nk(u) is closed this implies j'(x) € Nk(u). Hence j ' € J'.) This proves the assertion.

EXERCISES 123

Returning to the proof of the theorem, let {n i j ^ j be a sequence of positive integers which has the property that every positive integer appears in the se-quence infinitely often. Define a sequence {Jk} of left ideals of S as follows. Let Jo = S and proceed by induction. Having defined Jk-\, invoke the Assertion to choose a closed left ideal Jk of S with Jk Ç Jk-i and an element Uk € F{S) such that

Jk(Pnk) Ç Nk(uk).

Now fix i 6 N. Then for infinitely many k, i —rik, and for such k

JkiPi) Ç Nk(uk).

2 It follows that diam(Jrfe(pi)) < — for infinitely many k. Since the ideals {Jn}

are descending, for each fixed i the sequence {diam(Jn(pj))} is nonincreasing; consequently, lim diam( Jn{Pi)) = 0 for each i. Since for s,t G J„,

n—»oo

this, in turn, implies that lim diam(J„) = 0 in (S,p). Since (5,p) is complete n—»oo

oo

we invoke Cantor's theorem to conclude that Ç\ Jn = {e}, where e is a left _ n=i

ideal of S. As we saw at the outset e must be a retraction of M onto F(S).

It remains only to show that (i) ■*=> (ii). To see that (ii) =*> (i), let H be a nonempty closed 5-invariant subset of M. Then if i e i i , clS(x) Ç H. But c l5(a:)nF(5) φ 0 . Hence HC\F(S) Φ 0 . To see that (i) => (ii), let x e M and note that c l5 ( i ) is closed and 5-invariant; hence by (i) S(x) Π F(S) φ 0. ■

Exercises

Exercise 5.1 Show that if M is compact then {K.(M) ,H) is compact.

Exercise 5.2 If M is compact, is the metric space (Λ (M) , H) a compact met-ric space {where here A(M) denotes the collection of all nonempty admissible subsets of M)?

Exercise 5.3 If M is a compact metric space, is A {A (M)) compact in the sense of Definition 5.1 {relative to the Hausdorff metric on A (M)) ?


Exercise 5.4 Show that if M is compact and metrically convex, then (IC (M), H) is metrically convex. Is (A (M) , H) metrically convex?

Exercise 5.5 Show that (/C(M), H) in the previous exercise need not be com-pact if M is not compact.

Hint: Take M to be the subspace of t^ consisting of all sequences {xn} which converge to 0. Consider

A = {{xn} 6 M : xn = 1 + 1/n for an odd number of n and 0 elsewhere} ; B = {{xn} € M : xn = 1 + 1/n for an even number of n and 0 elsewhere}.

We now introduce a new definition. Let M be a metric space, and for e > 0 let

ke = sup {r (D) I diam (D) : D e A (M) and diam (D) > e) .

We say that A (M) is quasi-uniformly normal if ke < 1 for each e > 0.

Exercise 5.6 Show that if M is complete and A(M) quasi-uniformly normal, then A (M) is compact.

Hint: Use the ideas developed in the proof of Theorem 5.4.

Part II

Banach Spaces



Chapter 6

Banach Spaces: Introduction

6.1 The definition

In this chapter we collect information that will be needed for an understanding of the remainder of the text. Although details are included in some cases, many of the fundamental principles of functional analysis are merely stated without proof.

The spaces considered henceforth arise when an algebraic structure is com-bined in a special way with a metric structure. Many (indeed, most) of the examples of metric spaces we have already given have an inherent algebraic structure; coordinatewise addition and scalar multiplication in the cases of R" and too, functional addition and scalar multiplication in the case of C[0,1], B[Q, 1], and so on. An abstraction of these ideas gives rise to the body of spaces, both abstract and explicit, that invariably arise in the study of analysis. Many of the fixed point results in subsequent chapters will rely heavily on this added algebraic structure.

Let S denote any nonempty set that contains with each of its elements x and each real number λ a unique element Xx, written Xx, called a scalar multiple of x. (One could also include complex numbers λ as well, but we restrict ourselves here to the real case.) Also assume that for each two elements x,y £ S there exists a unique element x+y £ S called the sum of x and y. The system {S, ·, +} is called a linear space or a vector space (over R) if the following conditions are satisfied. Here, x,y,z € S; Χ,μ, 1 6 R.

(1) x + y-y + x;

(2) x+(y + z) = (x + y) + z\

127



128 CHAPTER 6. BANACH SPACES: INTRODUCTION

(3) X(x + y) = Xx + Xy;

(4) x + y = x + z implies y = z;

(5) (λ + μ)χ = Xx + μχ;

(6) λ(μχ) = (λμ)χ;

(7) 1* = x.

Axioms (4) and (7) immediately give rise to a unique element Θ € S for which x + Θ = x for each x e 5. Following custom we denote 0 = 0, that is, the zero element of the linear space shares its notation with O s R . (Of course, R is itself a linear space over R.)

Also for x,y E S, we write x — y = x + (—l)y.

A norm on a linear space 5 is a mapping | | · | | : S —» R+ which satisfies for each x, y e S; X € R :

(8) ||x|| = 0 if and only if x = 0;

(9)| |λχ| | = |λ|||χ||;

(10)||x + y||<||x|| + |M|.

A linear space with a norm (called a normed linear space) carries a natural metric, namely, the distance d: S x S —> R+ defined by taking

d{x,y) = \\x-y\\.

Taking λ = — 1 in (9) and combining this with (10) immediately gives

I I * - y | | < | |* | |+ ||y||,

and the triangle inequality for d quickly follows from this.

Definition 6.1 A Banach space is a normed linear space (S, ||-||) which is com-plete relative to the metric d defined above.

A linear space X has many norms. In general the space may be complete relative to one norm and not complete relative to another unless the two norms

6.1. THE DEFINITION 129

are equivalent. Two norms ||.||j and ||.||2 on X are said to be equivalent if there exist a and 6 (with 0 < o < b) such that

α | | * Ι Ι ι < Ι Μ Ι 2 < * Ι Μ Ι χ

for any vector x Ç. X.

Many of the metric spaces we have already discussed are, in fact, examples of Banach spaces. These include the spaces Kn, Rp (1 < p < oo), lp (1 < p < oo), C[0,1], and B[0,1]. In each instance a norm is denned by taking ||x|| = d(x,0).

Here are three additional examples.

Example 6.1 (co) This is the space of all sequences x = {xi,X2, ■ ■ ·} of real numbers for which lim x„ = 0, with

n—»oo

||x|| = sup |x„ | . l<n<oo

Example 6.2 (c) This is the space of all convergent sequences x = {χι,χ^, ·■■} of real numbers, with

\\x\\ = sup \xn]. \<n<oo

Example 6.3 (oo(M) This is the space of all bounded real-valued functions f : M —+ R where M is a complete metric space and

| | / | | = sup | / ( s ) | . xeM

The completeness of £oo(M) follows from the fact that if {/„} is a Cauchy sequence in £<χ>(Λί) then {fn(

x)} is a Cauchy sequence in M for each x 6 M. The function denned by f(x) — lim /„(#), x € M, exists since M is complete.

n—»oo

It is quite easy to show that lim | | / n - / | | = 0. n—*oo

We will make use of the following embedding result later.

Proposition 6.1 Any complete metric space M is isometric to a closed subset of a Banach space.

Proof. Fix a point xo 6 M and for arbitrary x 6 M consider the function fx 6 lJ{M) denned by

fx(y)=d(y,x)-d(y,x0), y 6 M.

130 CHAPTER 6. BAN ACH SPACES: INTRODUCTION

First observe that since \fx(y)\ < d(x,xo), fx is bounded. We claim that the mapping F : M —♦ £00(M) defined by taking F(x) = fx is an isometry. To see this take x\ and x2 in M and consider ||/Xl - /x.2||. For y € M we have

| / i , ( y ) - / x , ( y ) | = \d(y,xl)-d(y,x0)-d(y,x2)+d(y,x0)\ = Κΐ / ,^ ι ) -d(y,X2)\ < d(X\,X2).

This proves \\fXi — fx.2\\ < ά(χχ, x2). Conversely, upon taking y — X2 we see that

l/x,(y) - Λ 2 ( ΐ / ) | = \d(x2,x\) -d(x2,xo) -d(x2,x2) + d(x2,x0)\ = d(xux2),

so ||/Xl - / X . J > d( i i , i2) -Finally notice that F(M) is a closed subset of £QO(M). To see this sup-

pose that lim | |/x — g\\ = 0. Then {/x } is a Cauchy sequence in ^οο(Λί)

and it follows that {xn} is a Cauchy sequence in M. (Notice as above that | | /i„ _ /x,„ll = d(xn,xm).) Since M is complete lim xn = x € M, from which

n—>oo

lim /x„ = fx=g. u

Remark 6.1 Since the functions fx defined above are also continuous, it follows that the isometry F maps M onto a subset of C(M), the closed subspace of (.oo(M) consisting of all bounded real-valued continuous functions f : M —» R. This fact has further implications since the space C{M) is always complete regardless of whether or not M is complete. Since F{M) is closed in C(M), F(M) is itself complete. This proves that every metric space is isometric with a dense subspace of a complete metric space. This space, which is unique up to isometries, is called the completion of M. (See the discussion in Chapter 2, Section 2.3.) In particular, the completion of any normed linear space is a Banach space.

We conclude with an intricate example due to R. C. James. We will allude to many interesting properties of this space later (but without giving details).

Example 6.4 (The James space J) Let X denote the collection of all x = (xi, x2, X3, ■ ■ ■ ) € Co for which

sup / AXPi XPi+\) ~^~ (XPn + l XV\)

-11/2

i = l

< CO,

where the supremum is taken over all positive integers n and all finite increasing sequences (pi, ■ · ■ ,pn+i) . The James space J is the completion of X with respect to the norm \\x\\j .

6.2. CONVEXITY 131

6.2 Convexity Definition 6.2 A subset K of a normed linear space is said to be convex if Xx + (1 — X)y 6 K for each x,y € K and each scalar X € [0,1].

The next fact follows from a routine induction argument.

Proposition 6.2 A subset K of a normed linear space is convex if and only if n Σ îxi 6 K for any finite set {x\,X2, · · · , xn} Q K and anV scalars Xi > 0 for i= l

n which Σ Xi — 1.

t = l

The family of all convex subsets of a Banach space forms a convexity struc-ture, in the sense that the family of such sets is closed under arbitrary inter-sections, and under unions of ascending chains. The family of all closed convex subsets of a Banach space is also closed under arbitrary intersections, but it fails the latter property. However, this class is more relevant to the study of fixed point theory. Since the assumption of completeness will be essential to much of what we do, we shall accordingly confine our attention to Banach spaces (as opposed to normed linear spaces).

We shall use (X, | | | |) to denote a Banach space, and simply write X if only one norm on X is under consideration. In keeping with previous notation, we shall use B{x\ r) to denote the closed ball centered at x 6 X of radius r > 0:

B{x;r) = {y€X:\\x-y\\<r}.

The ball B(0; 1) is called the unit ball in X and is often denoted Βχ. The boundary dB(Q;r) of B(0;r) is the sphere

Sx=S(0;r) = {xeX:\\x\\=r}.

For a subset A of a metric space X the concept

cov(i4) = n{B : B is a closed ball in X and B D A}

has been already introduced. Related to this is another important concept.

Definition 6.3 For a subset A of a Banach space, the set

conv(yl) = Π{Κ Ç X : K is convex and K ^ A}

is called the convex hull of A. The set

conv(A) = Γ\{Κ Ç X : K is closed, convex and K 3 A}

is called the convex closure (or closed convex hull) of A.


Proposition 6.3 For any set AC. X

conv(Ä) = conv(i4).

In general, cöäv(A) φ conv(.A). However, the following is true.

Proposition 6.4 If AC. X is compact, then conv(A) is compact.

A proof of the above fact will be given later in the section on condensing mappings (Chapter 7, Section 7.8).

Finally, since A Ç conv(yl) Ç conv(.A) Ç cov(A), and since we have already shown that in an arbitrary bounded metric space diam(cov(i4)) = diam(.4), we have the following.

Proposition 6.5 If AC X is bounded, then

diam(A) = diam(conv(.A)) — diam(conv(A)).

6.3 £2 revisited There is some euclidean geometry inherited by £2 that will be needed in the next section. Let x = (χχ, χ?., ■ ■ ■ ), y = (2/1,2/2.·) € £2, and let

00

ix,y) = ^2xiVi-i= l

As in Rn, the number (x, y) is called the inner product of x and y, and the fact that it is well defined follows from the Schwarz inequality, which now takes the form:

(x,y)<\\x\\\\y\\.

Properties of the Inner Product: The following algebraic properties of the inner product follow instantly from the definition. For x,y,z e £2 and λ € R :

(i) (x,x) = | | a : | | 2 >0 ,

(ii) (x, x) = 0 <=> x — 0,

(iii) (x,y) = (y,x),

(iv) (Xx,y) =X(x,y),

6.3. £2 REVISITED 133

(v) (x,y + z) = (x,y) + (x,z).

Although there is nothing that obviously suggests this in the definition, a direct calculation shows that the inner product in l<i can be expressed explicitly in terms of the norm via the Polarization Identity:

(vi) (x, y) = - II* + » I I 2 - I k - » I I 2

A similar direct computation also establishes the Parallelogram Law :

(vii)||x + y||2 + ||a;-2/||2 = 2[||x||2 + ||3/||2

Keep in mind that only the real case is being considered here. If the scalar field is taken to be the complex numbers, then the inner product takes the form

k.y) = 5Ζχ»»ί. i = l

where y{ denotes the complex conjugate of j/j. In this case (iii) becomes (x, y) =

The procedure of deriving an inner product from a norm is reversible. A linear space X in which an inner product, that is, a mapping (·, ·) : X x X —► R, can be defined which satisfies properties (i)-(v) is called an inner product space. Any inner product space X can be assigned a norm by taking

||ι|| = y/(x,x), x€X,

and moreover it can be seen, again by a direct computation, that this norm satisfies the Parallelogram Law. The converse is true as well. If a normed linear space X satisfies the Parallelogram Law, then there exists an inner product on X which ||x|| = yj{x,x), x € X. The Polarization Identity suggests how to define this inner product. Thus if a normed linear space satisfies either the Polarization Identity or the Parallelogram Law, then it necessarily is an inner product space.

If an inner product space (X, | | | |) is complete relative to its norm metric, then X is called a Hilbert space. In particular, £2 is a Hilbert space. There are many others, the most well known among these being the space £2(0,1] of Lebesgue measurable functions on [0,1]. (See the comments at the end of Section 6.5.)

In the study of calculus in Rn the inner product is (x, y) is called the dot product and denoted x ■ y. Consider the well-known formula from calculus:

C0S* = PÏTÏMÏ'


where Θ denotes the angle between the vectors x and y. By the Law of Cosines,

Therefore,

| |x- l / | | 2 = N | 2 + ||l/||2-2||x||||y||ooe(9.

2(z,y) = ||x||2 + | | y | | 2 - | | z - y | | 2 .

Replacing y with — y and multiplying both sides by —1 gives

2(ar,y> = - W 2 - | | y | | 2 + ||a: + 2/||2.

Note that upon adding the two preceding equations one obtains the Polarization Identity. This illustrates that the Polarization Identity is merely the Law of Cosines in another guise.

One useful consequence of the Parallelogram Law is the fact that if K is a closed convex subset of a Hilbert space H and if z e H\K, then there is a unique point Pz of K which is nearest to z. To see this, suppose, without loss of generality, that z — 0 ^ K, let r = inf{||x|| : x 6 K}, and suppose {xn} C K satisfies lim ||xn|| = τ. Because

T < Xn ~r Xn

< *n +1*..

it must be the case that lim η,τη—»οο

Law,

Xn ι Xf] = r. Thus by the Parallelogram

\\Xn - Xmf = 2[\\Xnf + | | X m | | 2 ] - | | *„ + Xmf ■

Since the right-hand side of the above equality tends to zero as m, n —> oo, {xn} is a Cauchy sequence, and since K is closed there exists x € K such that lim x„ = x. Clearly ||x|| = r. If there were two such points, say x,y e K, such

n—»oo

that ||x|| = \\y\\ = r with x φ y, then since K is convex -(x + y) € K. Again,

by the Parallelogram Law,

r2 = x + y

5[ lN2 + lly||2 -x-y

2

'2

= r2-x-y

2

which is a contradiction. Therefore, the point of K having smallest norm is unique.

We now take a closer look at the mapping P which associates with each point z € H the point P(z) of K which is nearest to z. The following result holds only in a Hilbert space.

6.3. ί2 REVISITED 135

Theorem 6.1 Let K be a closed convex set in a Hubert space H. Let x,y € H, and suppose Px and Py are points of K which are, respectively, nearest to x and y. Then

\\Px-Py\\<\\x-y\\.

The following will facilitate the proof.

Lemma 6.1 Let x € K with K as above in Theorem 6.1, and letpEH,x^ p. Then x is the point of K nearest to p if and only if for all z € K,

ξ = (z - x,x -p) >0.

Proof. In view of the Law of Cosines this is intuitively clear. The lemma just asserts that the angle between the vector from p to x and the vector from

7Γ x to z is at least —. A detailed proof using the Polarization Identity follows.

it

Suppose x is the point of K is to nearest p and let z 6 K. Since K is convex,

Xz + {1- X)x G K for all 0 < λ < 1.

Now

λ2 Ik ζ|Γ + 2Χξ = (X(x - z), X(x - z)) - 2(X(x -z),x-p) = (X(x~z),[X(x-z)-2(x-p)})

"2 l |2 (*-p) | | 2

||2

\\2X(x-z)-2(x-p)\f

| | ρ - ( λ * + ( 1 - λ ) * ) | | ' - | | ( * - ρ ) | | '

>0 .

Thus λ ||x - zf + 2ξ > 0 for λ 6 (0,1] and upon letting λ —» 0 + we conclude ξ > 0 .

For the converse, suppose ξ > 0. Then

\z-pf-\\p-x\\ = (z - P. 2 - p) - (p - x,p - x) = (z, z) - 2{z,p) + 2(p, x) - (x, x) = (z, z) - 2(z, x) + (x, x) + 2(2, i )

-2(p,2> + 2(p,a:)-2(x,a;)

= (z - x, z - x) + 2 (p, x) + (z, x) - (p, z) - {x, x)

>0 . \z-xf + 2(z-x,x-p)

Proof of Theorem 6.1. Let A = (Px-Py, Py-y) and B = (Py-Px, Px-x). By Lemma 6.1, A > 0 and B > 0. Write B = (Px - Py, x-Px). Then

A + B = ( 2 ( P x - P y ) , x - y + Py-Px) = 2{(Px-Py,x-y)-\\Px-Py\\2}.


Since A + B > 0, this gives

\\Px-Pyf<(Px-Py,x-y).

But the Schwarz inequality implies

(Px -Py,x-y)< \\Px - Py\\ \\x - y\\.

Combining the two previous inequalities leads to the desired conclusion:

\\Px-Py\\<\\x-y\\.

■

6.4 The modulus of convexity

Interest in the modulus of convexity arose from a careful study of geometric properties of the £2 space. Indeed, let x and y be any vectors in £2- The Parallelogram Law implies

:|* + y | | = 2 N I 2 + II2/II2 - IF -2 / I I

So if we assume that x and y are in the unit ball and bounded away from each other, meaning that | | i — y\ > ε > 0, then

which implies

Thus the midpoint of the segment joining x and y is uniformly bounded away from the surface of the unit ball by a factor depending on ε. This property is known as uniform convexity.

\\x + y

x + y

2

1 | 2 <4 -ε 2 ,

W'-î

Definition 6.4 The modulus of convexity of a Banach space X is the function δχ : [0,2] — [0,1] defined by

5x(e) = inf j 1 x + y

: INI < 1, IMI< 1,11*-2/11 > ε (6.1)

In working within a specific space we generally drop the subscript X from δχ and simply write <5. Note that for a given space the number (5(e) is the largest number for which the following inequality holds:

x + y <1-δ(ε). (6.2)

6.4. THE MODULUS OF CONVEXITY 137

One very useful observation is the rather trivial fact that the above inequality is equivalent to the following: If x,y,p € X, R > 0, and r e [0,2R], then

||a= - p|| < Ä I I W - P I I < Ä II*-2/11 >r

x + y\ < ( . - « ( * ) ) *

Related to the modulus of convexity is another important function.

Definition 6.5 The characteristic (or coefficient) of convexity of a Banach space X is the number

ε0 = ε0(Χ) = sup{£ > 0 : δ(ε) = 0}.

The following is very important, but we will not give the proof here. (For a proof see, e.g., [67].)

Proposition 6.6 Let X be a Banach space with modulus of convexity δ and characteristic of convexity CQ. Then ê is continuous on [0,2) and strictly in-creasing on [εο, 2].

The function δ can actually be discontinuous at ε = 2. However, it is always the case that

lim δ(ε) = l - f . ε->2- I

In general, it is quite difficult to calculate the modulus of convexity of a Banach space, but for l 1 x + y

> l - \ / l (5) · A straightforward calculation shows that the reverse inequality holds (see

the proof of Theorem 6.2 in the next section), so for £2 w e n a v e

M=1 - \A-(!)2· An analogous formula, which we take up in more detail in the next section, holds for the modulus of convexity δρ for the spaces X = ip (or Lp) (2 < p < 00) :

£\P\ l/P

«.)-.-(I-®-) The Lp spaces are a natural prototype of spaces which satisfy the following

more general condition.


Definition 6.6 A Banach space X with modulus of convexity δ is said to be uniformly convex if 6(e) > 0 for each e 6 (0,2], or equivalently, if εο(Χ) = 0. If εο(Χ) < 2, then X is said to be uniformly nonsquare.

Every nonempty closed convex subset of a uniformly convex space contains a unique element of minimal norm (see Exercise 6.10). In fact, a Banach space may satisfy this conclusion yet fail to be uniformly convex.

Definition 6.7 A Banach space X is said to be strictly convex {or have strictly convex norm) if for x,y,z € X

\\X-z\\ = \\y-z\\ = l\\X-y\\=>Z = ?±V.

The properties in this section are purely geometric. Later we shall combine some of these ideas with (weak) topological properties of the space to classify various categories of Banach spaces.

6.5 Uniform convexity of the tv spaces

We begin with the following elementary lemma. The key ingredient is a fun-damental inequality from analysis known as Holder's inequality, which asserts that if p, q € K+ and if 1/p + \/q = 1, then for Xj, y* € K, i = 1, · · · , n,

Y^XiVi < [Σ\Χί

t = l α=ι ''f(f>r 1/9

Lemma 6.2 If 2 2; (ii) (a2 + b2Y'2 < 2 ( P - 2 ) / 2 K + &")■

Proof. Both inequalities hold if either a or b is 0, so we may assume a φ 0 ψ b. Since

a2 + b2 < 1 and b2

a2 + b2 < 1

and p/2 > 1, we have

+ V

(a2+b2)P/2 (a2 + 62)P/2

-2 1 P / 2

a2 + b2

„2

+

+ b2

a2 + b2 a2 + b2

a2+b2

= 1.

P/2

6.5. UNIFORM CONVEXITY OF THE lP SPACES 139

Thus (i) holds. Note that part (ii) is trivial if p = 2, so assume p > 2, and set p ' = p/2 > 1 and q' = p'{p' - 1) = p/(p - 2). Then 1/p' + 1/q' = 1, so by Holder's inequality,

a2 + b2 < [(a2)"' + (b2)"']1/P' [l«' + 1 " ' ] W

= l

ap + fep]2/p [2(P-2>/P .

This leads to

(a2 + b2)p'2<2W2(a» + bp),

Theorem 6.2 77ie modulus of convexity δρ of lp, (2 >}.

Consequently,

\u + v\p + \u - v\p < \\u + v\2 + \u - v\2

= [ 2 | u | 2 + 2 M 2 j

< 2 " - 1 [ Η Ρ + |υ |ρ]

Therefore, if u = (ui, u2, ■ ■ · ), v = (i>i, V2, ■ ■ ■ ) G lv

P/2

IP/2

£>l' + £>l: i = l t = l H=l t = l

In other words,

| | u + v| |P + | | u _ v | | P < 2 P - i [ | | u | r + | | v |n

for each u, v G lp. Now let e G (0,2) and suppose u, v G ip satisfy ||u|| < l , | |v | | < 1, and

flu - vll > ε. Then

u + v < 2p

< 1

- 1 ui iP

+ PI

(§)'·

U — V


Prom this it follows that

w * i-(i-G)·) P \ l / P

«(2) For the reverse inequality, let ε 6 (0,2), let u = (υ.ι,υ.2) 6 £p satisfy

V P \ 1 / P ui={l~(W) and u 2 = i and take v = (ui, -U2). Then ||u|| = ||v|| = 1. Moreover, it is easy to see that llu - vll = ε, while

u + v £ \ P \ 1 / P

- (' - (1) )

An implicit formula for δρ is also known for the spaces X — lp (or Lp) (1 < p < 2). In 1956 Hanner showed that for 1 < p < 2 the following inequality holds:

x,y€(P \\Μ-\\ν\\\ρ + \\\*\\ + \\ν\\\ρ<\\χ-ν\\ρ + \\* + ν\\'

and from this he obtained the precise implicit formula

l - M e ) + | | P + | l - M e ) - | 2.

Therefore, for all the spaces lv (1 0 whenever e > 0.

So far we have concentrated on the sequential ip and said nothing about their functional analogs, the important class of Lp spaces.

Let (Χ,β,μ) be a finite measure space. This means that X is a set, S is a rr-algebra of subsets of X, and μ : S -* R is a finite measure. (We consider only the real case.) In particular X G S, the complement of any set in S is also in S, and S is closed under countable unions. The measure μ is a real-valued non-negative function defined on S for which μ(0) = 0 and

00 00

n = l n=J

for every disjoint countable family {An} in 5 . A mapping / : X —* R is said to be μ-measurable if / - 1 ( i / ) € 5 for every open set U Ç R.

For each p, 1 < p < 00, the space Lp(ß) — LP(X,S^) is the space of equivalence classes of μ-measurable functions / for which Jx \/\ράμ < oo. Two

6.5. UNIFORM CONVEXITY OF THE lP SPACES 141

elements f,g € 1<ρ(μ) are said to be equivalent if fx \f — g\p άμ = 0, and the norm of an element / G Lpfa) is given by

ll/ll,= (/xl/l'V)"'.

The norm for Loo(/x) is a little more technical. £οο(μ) consists of those μ-measurable functions / for which

H/IL·» = ess SUP{I/HI : ω e x) = jnL SUP l/l < oo.

where TV = {£ 6 S : μ{Ε) = 0}.

It is possible to prove that the spaces £ρ(μ) are uniformly convex for 1 ρ{μ) then the line of argument used in Theorem 6.2 leads to the fact that for almost all t 6 X

|u(t) + v(t)\* + \u(t) - v(t)\> < 2»-1 ( \u(t)\" + \v(t)\").

On integrating both sides of the above we obtain

||u + t;|r + ||«-υ||ρ<2"-1(|Η|ρ + |Η|ρ)

for each u, v € Lp^). The argument can now be completed as before. Thus we have the following.

T h e o r e m 6.3 The spaces £Ρ(μ), 1 < p < oo, are uniformly convex.

Of course, the lp spaces can be thought of as a special subclass of the LP(X,S, μ) spaces with X = N and μ the counting measure on bounded subsets 5 of N.

Among the Lp spaces, the case p — 2 is of particular interest. If μ is taken to be Lesbesgue measure on [0,1] then Z fO, 1] is the space consisting of all measurable functions / with | / | summable, identifying those functions which differ only on a set of measure zero. This space, is in, fact metrically identical to 1%. The correspondence is obtained by assigning to each / G la the sequence (αι,α2, ■ · · ) of its Fourier coefficients. Then the corresponding series oo oo 53 a\ converges. On the other hand, if the series ^Z an converges, then it can

n=l n=l

be shown that a.\,a,2,··· , are the Fourier coefficients of some / € Li. It follows that the spaces Li and li are isometric and share the same metric properties.


6.6 The dual space: Hahn-Banach Theorem If X is a Banach space, then a mapping / : X —» K is called a linear functional if for each x,y € X and a,/? € R,

/ ( a x + /?y) = <*/(*)+/?/(»).

The space of all continuous linear functionals on X is denoted X*. Since lim xn — x if and only if lim (xn — x) = 0, if a linear functional is con-

n—*oo n—*oo

tinuous at 0 then it is continuous on the whole space. Also, if / is a linear functional and if / is bounded, in the sense that there exists a constant c such that for each x £ X

\f(x)\<c\\x\\,

then clearly / is continuous at 0. The converse is true as well.

Proposition 6.7 A continuous linear functional f on a Banach (or normed) space X is always bounded.

Proof. Assume / is not bounded. Then for each n € N there exists xn € X such that

/ ( i „ ) > n | | x „ | | n = l , 2 , · · · .

Since /(0) = 0 this, in turn, implies xn φ 0. Let yn = — n , n = 1,2, ■ · · . n | |x n | |

Then lim yn = 0. On the other hand, f(yn) — f I ., " ,, I > 1, and this n - o o \ « l F n | | /

implies / is not continuous at 0. ■

In view of Proposition 6.7, for each / 6 X* there is a number ||/||„ 6 R+

such that

U/H. = s u p { | / ( x ) | : * e S x } ,

where, as usual, Sx = {x 6 X : \\x\\ = 1}. It is a fairly straightforward matter to verify that the space (X", | | | |»), called the dual (or conjugate) space of X, is, in fact, a normed space. Although it is a little less obvious, (X*, ||-||») is also complete.

Proposition 6.8 If X is a normed space, then its dual space (X*, ||-||„) is al-ways a Banach space.

Proof. Let {/„} Ç X* be a Cauchy sequence. Then given ε > 0 there exists N € N such that if m,n > N, ||/„ - / m | | , < e. Thus for m,n>N

ll/n - /m||. = SUP{|(/n - fm) (x)\ : X € SX},

6.6. THE DUAL SPACE: HAHN-BANACH THEOREM 143

from which, for each x 6 X

| / n ( x ) - fm(x)\ = | ( / „ - / m ) (X)\ < l l /n - fm\\. 11*11 < C 11*11 ·

So for each such x, {fn{x)} is a Cauchy sequence in R and has a limit which we denote f{x). Also if x,y € Sx,

f(ax + ßy) = lim /„ (ax + ßy) = a lim /„(a;) + ß lim /n(2/), n-—»oo n—»oo n—*oo

which implies / ( a i + /3y) = af(x) + ßf(y). Therefore, / is a linear functional. To see that / € X* we only need to show that / is bounded. We have already observed that every Cauchy sequence is bounded. Therefore, there exists M € R such that

Af = 8 u p { | | / n | | . : n € N } .

Let ε > 0 and choose N as above. Then for each x 6 X and m,n> N,

\fn(x)\ < \fn(x) - fm(x)\ + \fm(x)\ < (ε + M) ||x||.

Letting n —» oo gives | / (x) | < (ε + M) \\x\\, so / is bounded, hence continuous. Finally, for n> N,

11/ - /nil» = sup{|/(x) - fn(x)\ :xeSx}<e.

This proves lim /„ = / in (X*, ||·||„). ■ n—»oo

Proposition 6.8 offers an easy way to see that certain normed spaces are complete. For example, once it is known that t\ is complete and that (i\)* = oo (Section 6.8), then completeness of t^ follows.

We now state, without proof, the Hahn-Banach Theorem and one of its many useful consequences. The Hahn-Banach Theorem is usually formulated in a slightly more abstract setting and the standard proof, which utilizes Zorn's Lemma, can be found in many texts.

Theorem 6.4 (Hahn-Banach) Let X be a Banach space and let H be a linear subspace of X. Then given any continuous linear functional f £ H* there is a continuous linear functional f € X* such that

(i) f(x)

(«)

— f(x) for each x 6 H; and

< 11/11.·

It is important to note that the norm ||/||% in the above statement is relative to the dual space H*.

Corollary 6.1 Let X be a normed space and let x G X, x Φ 0. Then there exists fx 6 X* such that \\fx\\ = 1 and for which fx(x) = ||χ||.


6.7 The weak and weak* topologies In our discussion of Tychonoff 's Theorem in the Appendix we describe the product topology in terms of 'net' convergence. This is also a convenient way to describe the weak and weak* topologies. Let X be a Banach space. We shall say that a net {xa} converges to x € X in the weak topology (or converges weakly to x) if lim/(a:Q) = f(x) for each / 6 X*. When this is true we write

w- lim xa = x. a

A subset K of X is weakly closed if it is closed in the weak topology, that is, if it contains the weak limit of each of its weakly convergent nets. The weakly open sets are now taken as those sets whose complements are weakly closed. The resulting topology on X is called the weak topology on X. Sets which are compact in this topology are said to be weakly compact. It can be shown that the weak topology is the weakest topology for which all the functionals / € X* are, in fact, continuous.

It is important to know that the weak topology on a Banach space is a Haus-dorff topology, and that weak limits are unique. This is because the functionals in X* separate points in X, that is, given any two points x, y € X with x φ y there exists an / E X* such that f(x) φ f{y)- This is another consequence of the Hahn-Banach Theorem.

For x e X and / € X* define i(x)(f) — f(x)- It is easy to see that i(x) € X** and that, in fact, the mapping i : X —» X** is an isometric isomorphism, called the canonical embedding of X into X**. If i(X) = X" then X is said to be reflexive .

Now observe that if X* is the dual of a Banach space then it too has a weak topology determined by its dual space X**. Thus {fa} Ç X* converges weakly to / € X* if l imF( / a ) = F(f) for each F € X". However, as we have seen

a above, there is a natural isomorphism between X and a subspace, say i(X), of X". If it is the case that limx**(/a) = x"(f) for each x" € i(X) then {fa}

a is said to converge to / in the weak* topology. When this is the case we write

w*-\imfa = / . OL

Note that one always has the following implication:

w- lim fa — f => w'- lim fa — f. a a

However, the converse only holds if X (hence X*) is reflexive. A subset K of X* is said to be weak* closed if it contains the weak* limit of

each of its weak* converging sequences. The weak* open sets are now taken as those sets whose complements are weak* closed. The resulting topology on X* is called the weak* topology on X*. It is important to notice that this topology depends on X as well as X*. It is possible for two different Banach spaces to have the same dual space X*; consequently X* could have two different weak*

6.7. THE WEAK AND WEAK* TOPOLOGIES 145

topologies, one generated by each of its preduals.

Another concept which will be useful for us in the subsequent sections is the adjoint of operators.

Definition 6.8 Let X and Y be two Banach spaces. For any operator (contin-uous linear mapping) T : X —> Y, define the adjoint operator T* : Y* —* X* defined by

T*(y') = y'oT

or

T(y*)(x) = y'(T(x))

for any x € X.

The adjoint operator has some fundamental properties which play a crucial role in the study of Banach spaces. In particular, we have

ΡΊΙ = imi, where ||T|| is defined by ||T|| = sup{||T(x)|| : x € Sx).

Another common notation for the weak topology is σ(Χ, X*). This suggests the topology on X induced by X*. In this notation σ(Χ*,Χ**) denotes the weak topology on X*, whereas σ(Χ*,Χ) denotes the weak* topology on X*.

We conclude this section with another consequence of the Hahn-Banach Theorem. It will be used later, for example, in the proof of Theorem 9.12, Chapter 9.

Theorem 6.5 (Basic Separation Theorem) Let A and B be disjoint convex subsets of a Banach space X.

(a) If A is open, then there exists x* € X* and a number r ç R such that

x*(x) > r if x e A and x*(x) <r if x € B.

(b) If B is compact (or, more generally, weakly compact) and A is closed, then there exist x* € X* and z € B such that

x*(z) — ||2|| = max{a;*(x) : i € ß } < inf{x*(x) : x e A}.


6.8 The spaces c, CQ, i\ and i^

The space c is the Banach space of all sequences of real numbers which converge and Co is the subspace of c consisting of those sequences converging to 0. In each case 11 i n || = sup{|xn| : n = 1,2, · · ·} and thus the following subspace inclusions hold: Co Ç c Ç ^ .

A sequence {xn} in a (real) Banach space is called a (Schauder) basis for X if for each x 6 X there is a unique sequence {an} of real numbers such that

n oo

x = lim } aiXi = y üiXi. n—»oo ^—' *—*

ί = 1 t = l

The first three classical sequence spaces listed in the section heading all have bases while ^ , since it is not separable, does not. In fact, the classical unit vectors {e*}, where

e ^ i O . - . - . l . O , · · · )

and 1 appears in the ith position and 0's appear elsewhere can be shown to be a basis for both Co and i\.

Among other things we show in this section that the dual spaces of both c and co are isometrically isomorphic to ίλ. Moreover (ί\)* = i^. These facts lead to three interesting questions. If {uk}'£L1 is a bounded sequence in i\ when does {uk} converge weakly; when does {uk} converge in the weak* topology regarding i\ as the dual of c; when does {uk} converge in the weak* topology regarding l\ as the dual of Co?

The answer to these questions should help to sort out subtle differences in the concepts, and the answer to the first question is a little surprising. A bounded sequence {uk} in i\ converges weakly to u € êi if and only if lim ||u ' — u|| = 0.

fc—>oo

Spaces which have this property are said to have the Schur property. Thus the weakly compact subsets of such spaces coincide with the norm compact subsets. (Note, however, that i\ is not finite-dimensional.) We discuss the Schur property of l\ in detail later in Section 6.10.

The latter two questions are a little less complicated. First we show that (up to isomorphism) c* = cj = l\.

Let y = (2/1,3/2,···) € ^1 and ( i i , i 2 r · ·) e °ο> anc* define

0 0

y(x) = ΣΧίνί· i = l

This defines a linear functional. Also,

0 0 0 0

\y(x)\ < Σ M ^ .ML Σ M = INI«. Nli. i = l i = l

6.8. THE SPACES C, C0, ix AND £, 147

so y is a bounded (continuous) linear functional with ||y||„ < \\y\\v Now let ε > 0 and choose N so that

ΛΓ

E l w l > H » l l i - ' 1 = 1

Let

x = (sgn 2/i,sgn3/2,·· ,sgn yN,0, ■ ■ ■ ) e c0,

where "sgn" denotes the function which assigns to a number 1, —1, or 0 accord-ing to whether the number, respectively, is positive, negative, or zero. Then

N

ΐ/(*) = ΣΜ>ΙΜΙι -ε i = l

with Hill^ = 1. It follows that \\y\\m > \\y\l, ; thus ||y||, = \\y\l,. On the other hand, if / € cj$ and x = (x\, X2, ■ ■ ■ ) € CQ, then since the vectors

n

{e1} form a basis for Co we have x = lim J2 x*e%· This leads to

n n oo

t = l i = l i = l

oo In particular, £) |x i / (e l ) | exists, that is,

i=l

y = {f{el),f{e*),-.-)&tu

and moreover \\y\\x < | | / | |» . At the same time, ||/||„ > ||y||i because

l/(*)l Y^xifiJ) i = l

^ Μ Ο Ο Σ Ι / ^ Η Ι Ν Ι Ο Ο Ι Μ Ι Ι -t = l

Therefore, the mapping ψ : CQ —» i\ defined by

V(/) = ( / ( e 1 ) , / ( e 2 ) , · · · )

is an isometry of cj onto t\.

A subtlety arises in showing that l\ is also isomorphic to c* because the vectors {e'} do not form a basis for c. Suppose x — ( x 1 , X 2 , · ) € c. Then lim xn — L G R and

n—»oo

n oo

lim y^(xi - L)el — "Sîxi - L)el = (χχ - L, x2 - L, ■ ■ ■ ) € c0. n—>oo i—J ^—' t = l i = l


oo

Moreover, Σ (xi ~ L)el — x — L ■ e, where e = (1,1, · · ■ ). Thus x = L ■ e + i-l

oo 53 (xi — L)el. It follows that the sequence {e1} along with the vector e provides i= l

a basis for c. Moreover it can be shown that the mapping Φ : c* —» i\ defined by

oo

Hî) = U{e)-Yjf{e%f{e'),f{e2),-.-)

oo

is an isometry of c* onto ίχ. [Note that the convergence of Σ fie1) is assured

because / 6 c* implies / 6 c*,; hence y>(/) = (/(e1), / (e 2 ) , · ■ · ) € £\.\

Since the basis vectors e' € Co C c, z = 1,2, · ■ · , if i = (xi, X2> · · · ) € i> then the function defined by el(x) = Xj can be thought of as an element of i\ which is identified under the natural isomorphism with an element of CQ (or c). Thus a necessary condition for a bounded sequence in i\ to converge in the weak* topology induced on i\ by either CQ or c is that it converge coordinatewise. In the case of l\ = c*,, however, coordinatewise convergence is also a sufficient condition for weak* convergence of bounded sequences in t\. This is because {e1} is a basis for CQ.

We conclude by remarking that reasoning similar to the above shows that ^Ϊ = οο and that a bounded sequence in ί^ converges in the weak* topology induced by i\ if and only if it converges coordinatewise to an element of ί^. Similarly, weak convergence of a bounded sequence in £\ implies coordinatewise convergence. Also a bounded sequence {xn} in CQ converges weakly if and only if it converges coordinatewise t o i € CQ, while a bounded sequence {xn} in c converges weakly to x G c if and only if it converges coordinatewise to i and lim lim x" = lim Xj.

n—»oo i—»oo i—»oo

6.9 Some more general facts

We now collect for later reference some well-known properties of the weak and weak* topologies. Despite the fact that proofs of these results can be found in any standard functional analysis text we include selected details.

We begin with yet another consequence of the Hahn-Banach Theorem.

Propos i t ion 6.9 A convex subset K of a Banach space is weakly closed if and only if it is closed.

The above leads to the following.

Proposition 6.10 / / o subset K of a Banach space is weakly compact, then conv(K) is also weakly compact.

6.9. SOME MORE GENERAL FACTS 149

Neither of the above facts holds for the weak* topology. However the fol-lowing holds for both topologies.

Proposition 6.11 If K C. X is weakly compact (or weak* compact if X is a dual space), then K is bounded.

This is a consequence of the following classical result, which is also known as the uniform boundedness principle.

Proposition 6.12 (Banach Steinhaus Theorem) Let X be a Banach space. Then a subset S of X* is bounded in norm if and only if {/(x) : x E X} is bounded for each f 6 S.

The Banach Steinhaus Theorem can be used to prove Proposition 6.11 by identifying K with its natural imbedding in X** and invoking the fact that a continuous real-valued function defined on a compact set is bounded.

The following fact holds only for the weak* topology (except, of course, in reflexive spaces).

Proposition 6.13 (Alaoglu's Theorem) The unit ball Βχ* (hence any ball) in a dual space X* is always compact in the weak* topology.

Proof. This follows nicely from Tychonoff 's Theorem. Let

BX. = { / € * ' : U/H. < 1 }

and recall that for a net {/„} C Βχ>, weak* l im/ a = 0 ·£> lim/Q(z) = 0 for a a

all x 6 Βχ. Consider the product space Γ] Ix, where Ix — [—1,1] for each x€Bx

x e X. Then for each / € Bx-, f(x) € Ix for each x 6 Bx. Thus / € \[ Ix, *€Bx

that is,

Βχ- C H Ix. x€Bx

Since the weak* topology on Βχ· is just the topology of coordinatewise conver-gence, it coincides with the product topology on f] /X) and since Π Ix is

χζΒχ χ€Βχ

compact in the product topology by Tychonoff's Theorem (each of the sets Ix

is compact), it only needs to be shown that Βχ· is a closed subset of Γ] Ix. xeBx

To see this, let {fa} be a net in Βχ· which converges in the product topology t o / e Π Ιχ· We need to show that / G Βχ-. However, since {/„} con-

xeBx verges to / in the product topology, lim/a(a;) = f(x) for each x G Βχ. Clearly a

this implies / is linear. Moreover, since \fa(x)\ < \fa\ < 1, it follows that lim | /o ( i ) | < 1, from which \f(x)\ < 1 for each x e Bx. Thus / e Βχ·. ■


Proposition 6.14 A Banach space X is reflexive if and only if its unit ball Βχ is compact in the weak topology.

The following is a very deep fact. It says that in the weak topology com-pactness is equivalent to sequential compactness. This fact holds also for the weak* topology on X* if X is separable, because in this case the weak* topology is metrizable, but it does not hold in general for the weak* topology.

Proposition 6.15 (Eberlein-Smulian Theorem) For any weakly closed subset A of a Banach space the following are equivalent:

(a) Each sequence {xn} in A has a subsequence which converges weakly to a point of A.

(b) Each net {xa} in A has a subnet which converges weakly to a point of A.

(c) A is weakly compact.

Finally, we list several properties which characterize reflexivity. Each is at times useful. For example, property (g) offers a quick way to confirm the fact that uniformly convex Banach spaces are always reflexive.

Proposition 6.16 For a Banach space X the following are equivalent.

(a) X is reflexive.

(b) X* is reflexive.

(c) Βχ is weakly compact in X.

(d) Any bounded sequence in X has a weakly convergent subsequence.

(e) For any f € X* there exists x 6 Βχ such that f(x) = \\f\\.

(/) For any bounded closed convex subset K of X and any f E X* there exists x G K such that f(x) — sup{/(y) : y 6 K}.

(g) If {Kn} is any descending sequence of nonempty bounded closed convex oo

subsets of X, then f-) Kn φ 0 . n=\

Condition (b) is immediate from the definition, (c) is a consequence of the Hahn-Banach Theorem, and (d) follows from the Eberlein-Smulian Theorem. The characterizations (e) and (f) are due to R. C. James, and (g) is due to Smulian.

6.10. THE SCHUR PROPERTY AND Ιλ 151

6.10 The Schur property and i\ We return to the Schur property for i\, and here we include the details because the argument we give has an entirely metric flavor. Aside from an application of Alaoglu's Theorem, the key ingredients are the Baire Category Theorem and the fact that the weak* topology on X* is metrizable if X is separable.

To establish the Baire Category Theorem we need some definitions.

Definition 6.9 If S is dense in some open subset U of a topological space T (that is, S D U), then S is said to be somewhere dense in T. Otherwise, S is said to be nowhere dense in T.

Definition 6.10 A subset S of a topological space T is said to be of the first category in T if it is the union of a countable family of nowhere dense sets in T. Otherwise S is said to be of the second category in T.

Theorem 6.6 (Baire Category Theorem) Any complete metric space M is of the second category in itself.

It is the following reformulation of the Baire Category Theorem that we will actually use later.

oo Corollary 6.2 If M is a complete metric space and if M = (J Ai, then for

_ _ i = l some n, An contains an open set (i.e., An has nonempty interior).

Proof. By the Baire Category Theorem some An is not nowhere dense. This means that An contains an open set. ■

The Baire Category Theorem is an easy consequence of the following lemma. Here we use U(a; r) to denote the open ball centered at a € M with radius r > 0; as usual, B(a; r) will denote the corresponding closed ball.

Lemma 6.3 Let {Gn} be a sequence of dense open sets in a complete metric oo

space M. Then f] Gn is also dense in M. n=l

Proof. Let U be an open ball in M. Because G\ = M, G\ Π U is a nonempty open set, there exist a\ € G\ Π U and r\ > 0 such that B(a\;ri) Ç G\ Π U. Similarly, G2C\U(a\\r\) is nonempty and open, so there exist a^ 6 £?2Πί/(αι;Γι)

and r2 > 0 with r2 < -r\ such that B(a2\ τ^) Ç G2 ΠU(ai;r\). Continue in this

way to obtain

B(an;rn) Ç Gn Π t / (a„_i ; r„_i) , n = 1 ,2 , · · ,


with r n < — rn-i —> 0 as n —» oo. Observe that m > n =>

Bia^r^ D · · 2 B(an;rn) 2 · · · 2 B(am;rm).

Thus d(an,am) < r„ and it follows that {a„} is a Cauchy sequence. Since M is complete there exists a € M such that lim an — a. But for any given n,

n—oo am € B(an;rn) for all n < m. Hence a = lim αη G 5 ( a n ; r n ) . Consequently,

oo

a 6 P|£(an;rn)Ç f| G„. n = l r»=l

(oo ■. oo

Π Gn)(\U φ 0\ hence f) Gn is dense in n = l ' n = l

M. ■ Pnoo/ o/ the Baire Category Theorem. Let {An} be a sequence of nowhere

oo

dense sets in M with M — (J An- Because An is nowhere dense, An has no n = l _

interior points, so any open ball in M must intersect Gn = M\An, n = 1,2, · · · . oo

Therefore, {Gn} is a family of dense open sets in M. By Lemma 6.3, P) Gn φ 0 · n = l

oo This shows that (J i4n φ M, which is a contradiction. ■

n = l We need one other fact before proving Schur's Lemma. Suppose X is a

separable Banach space with unit ball B and let B* denote the unit ball in X*. It is easy to see that if S is a dense subset of B and if {fa} is a net in B*, then

weak* lim fa = f e B* <$ lim }a(x) = / (x) for each x € B a ot

·£> lim fa{x) = f (x) for each x e S. a

Now suppose S = {aa.aî · · · }, and for each f,geB* define

p(/.5) = f;èl/(a:i)-ff(xOI· t = l Z

Then p is a metric on B* and moreover the metric topology on {B*,p) coincides with the relative weak* topology on B* (see Exercise 6.14), that is,

weak* lim/Q = / € ß ' o l imp(/ a , / ) = 0.

L e m m a 6.4 (Schur's Lemma). Let {xn} = {(χ^,χζ,·· · )} be a bounded se-quence in êi which converges weakly to 0. Then lim ||χη||! = 0.

6.10. THE SCHUR PROPERTY AND ίλ 153

Proof. Let B* denote the unit ball in ôo· Given ε > 0, define

Ffc = {/ e B* : | / ( x n ) | < e/3 for each n > k}.

The sets Fk are weak* closed and, since lim f(xn) — 0 for each f € B*, n—*oo oo

B' C U Fk. fc=l

Since i\ is separable, 5 * is metrizable in its relative weak* topology. Also, by Alaoglu's Theorem B* is compact, hence complete, in this topology. This means we may apply the Baire Category Theorem to conclude that some Fk must have nonempty weak* interior. Fix such a k and suppose / G weak'int(Ffc). Now suppose that for each n > 0 and for each N > 0 there exists hn G B*\Fk such that |Λ" - fi\ < \jn for each 1 < i < N. By letting N —* oo it follows that

lim |/ι? - Λ| = 0.

n—oo

Since weak* convergence in B* coincides with coordinatewise convergence, we conclude that weak* lim hn = / . This contradicts / € weak*int(Fjfc).

n—»oo So, there must exist 6 > 0 and N 6 N such that

{h e B* : \hi -fi\<6, l<i<N}c Fk.

Also, since weak um xn = 0, there exists p € N such that for each n > p, n—+oo

J>?|<e/3. i = l

(Here it is necessary to use the fact that weak convergence implies coordinate-wise convergence in i\ and apply this fact N times.)

Now fix n > max{fc,p}, and define g G ß* by

_ / fi if î<i<N, 9i ~ \ sgn x? if i> N.

Then p G Ffc. Therefore by the definition of Fk,

I TV o o

Ιΐ(χη)Ι = Ε » Ι " + Σ l*"l ^£/3· | t = l i=N+l

from which (because HPH^ < 1)

f; ixri<^/3+f;kri<2e/3. i=N+l i = l

It follows that if n > max{fc,p},

oo N oo

ιι*Ίΐι = Σι*?ι = Σι*?ι+ Σ ι<ι< ε· i = l i = l i=7V+l


Schur 's Lemma was instrumental in resolving an early fundamental question in functional analysis, namely, whether every infinite-dimensional Banach space must necessarily contain an infinite-dimensional reflexive subspace. The answer is negative because Schur's Lemma implies that every reflexive subspace of £χ must be finite-dimensional.

We conclude this section with another famous result which is a consequence of the Baire Category Theorem (Theorem 6.6).

Theorem 6.7 (Closed Graph Theorem) Let X and Y be two Banach spaces. Let T : X —> Y be a linear mapping. Then T is continuous (or bounded) if and only if the graph of T, that is, the set {(x,T(x));x G X}, is closed in X x Y (endowed with the product topology).

6.11 More on Schauder bases in Banach spaces

It is certainly true that the concepts of algebraic bases and orthonormal bases are crucial to the study of finite-dimensional vector spaces and Hubert spaces, respectively. That is why one tries to find a corresponding concept in the study of Banach spaces. Although, using Zorn's Lemma, one can always establish the existence of an algebraic basis of a given Banach space, this is not really sufficient because such a basis does not provide information about the topology and the geometry of the space.

Recall that a sequence {xn} in a Banach space X is called a Schauder basis (or just a basis) for X if for each x G X, there exists a unique sequence of scalars {an} such that

n oo

x = lim > afcXfc — V^ anxn . n—»oo *—' *—'

fc=l n = l

The span of a subset S of X is the collection of all linear combinations {au + βν : u, v G S} . If the sequence {xn} does not span the entire space X and at the same time is a Schauder basis for its closed linear span, span{a;n}, then it is called a basic sequence.

We will always assume that basic sequence {xn} is normalized, that is, that | | i n | | = 1 for a l l n > 1.

Once it is known that a Banach space has a Schauder basis it is natural to raise the question of its uniqueness. In order to study this question properly, the concept of equivalence of bases is needed.

6.11. MORE ON SCHA UDER BASES IN BAN ACH SPACES 155

Definition 6.11 Two bases {xn} and {yn} are called equivalent, provided Σαηχη n

converges if and only if Σ anyn converges. n

Using the Closed Graph Theorem (Theorem 6.7), one can easily show that {xn} and {yn} are equivalent if and only if there exists an isomorphism T from span{xn} onto span{yn} for which T(xn) = yn, for all n.

A very useful method of building equivalent bases is based on the notion of block bases.

Definition 6.12 Let {xn} be a basic sequence in a Banach space X. Any non-zero vector u E X of the form

u ~ y ^ û n x n , n = p + l

where {an} are scalars andp < q, is called a block vector. A block basis of {xn} is a sequence of block vectors {um} such that

Pm+l

u m — y ^ anxn, n = p m + l

where P\ < P2 <

It is natural to associate to any vector x — £ α„ ι η , a set of integers, called

the support of x, denned by

supp(x) = {n; an φ 0} .

The basicity of a sequence {xn} can be characterized by a condition on the projections onto finite subsets of the coordinates.

Definition 6.13 Let {xn} be a basic sequence in X. Set XQ — span{x„}. A natural projection on {xn} is a mapping Pj : XQ —+ XQ defined by

/ °° Pi ί Σ a"x" I = Σ α"χ" '

\ n= l / nel

where I is a subset of I

There is a simple and useful criterion for checking whether a given sequence is a basic sequence.


Theorem 6.8 A sequence {xn} is basic if and only if there exists a number c > 0 such that for any positive integers n and p and any sequence of scalars

n

2jûfcZfc fe=l

< c n+p

y]o-kXk fc=l

Proof. Suppose that {x„} is a basic sequence for its closed linear span Xo = span{xn}. Consider the natural projections {Pk}, defined by

Pk ί Σ a"Xn = Σ an3

W = i n = l

Since the range of each Pk is a finite-dimensional space, they are bounded linear operators. Our assumption implies that for every i 6 Xo, we have

a: — lim Pfc(x) · fc—»oo

It follows from Banach-Steinhaus theorem (see Proposition 6.12) that sup | |Pn | | < oo. Thus, we should have

2_,afcZk * = i

<sup| |Pn | k=n+p

2 j akXk fc=l

for any positive integers n and p and any sequence of scalars {an}. Therefore, the constant c may be taken equal to sup | |Pn| | · Conversely, suppose that {a;n} is a sequence such that for any positive integers n and p and any sequence of scalars {an},

n

/]akXk fe=l

< c n+p

/Ô-kXk fe=l

This condition ensures that linear operators P n from XQ into Xo are uniformly bounded and their norms are less than the constant c. It is easy to check that the vectors {xn} are linearly independent. Let x e XQ and e > 0 be given.

η(ε)

Then there exists y £ spanjzn} such that ||x — y\\ < ε. We have y = Σ αηχη, n=l

for some n(e). But now, if n > n(e), we have

\\X-Pn{x)\\ < \\x-V\\ + \\V-Pn{y)\\ + \\Pn{v)-Pn{x)\\

< ε + ||P„||e < (1 + c)e.

It follows that x = lim Pn{x)· Using the properties of the operators P n , namely, n—»oo

PnPm = PmPn — Pmin(n,m), w e deduce that the first n coordinates of Pm(x)

6.11. MORE ON SCHAUDER BASES IN BAN ACH SPACES 157

is

are independent of m. This clearly implies the existence of a unique sequence oo

of scalars {an} such that x = Σ anxn. ■ n=l

The least number c which satisfies the above inequality is called the oasis

constant of {xn}. If the basis constant is 1, then the sequence < ^Z ak^k \

monotone increasing. In this case, the basis {xn} is called monotone . Looking at the proof of Theorem 6.8 closely, we see that the basis constant

of {xn} is given by

c = sup| |P„| | . n

We will say that the basic sequence {xn} is bimonotone if

| |P n | | = | | / - P „ | | = l f o r a n y n e N .

Remark 6.2 One may renorm the space X with an equivalent norm for which the basis constant is 1. Indeed, set

| x | = s u P | | P n ( x ) | | n

for x € X. Then we have

| | x | | < | x | < c | | x | |

for any x € X. Using the basic properties of the natural projections, we can easily show that

\Pm(*)\ < W

for any x € X and m > 1, which means that the basis constant of {x„} with respect to the new norm | · | is 1.

Next we discuss the relationship between basicity and duality.

Definition 6.14 Let {xn} be a Schauder basis for X. For any n G N, define the linear functional x* on X by

x*n I YâkXk ] =a„.

These functionah {x^} are called the biorthogonal functionah associated with basis {x„}. In particular, we have

Χ = Σ<(Φη n=l

for any x G X.


It is easy to check that for any n,

IKII < 2c, where c is the basis constant of {x„}. In fact, let {Pn} be the sequence of natural projections associated with {xn}. Then for any integers n < m, we have

( m \ n

fc=l / fc=l

Hence, by Theorem 6.8, the sequence {x„} ' s a basic sequence in A"* whose basis constant is also c. But in general {x*} is not a Schauder basis of X'.

Example 6.6 Consider the Banach spaces CQ, l\, and Zoo- We have seen that CQ = Ζχ and l^ = loo- The canonical basis {en} of CQ is a Schauder basis. The associated biorthogonal basis {e^} is the canonical basis of l\, which is also a Schauder basis. But the biorthogonal basis {e**} of the basis {e*} is not a Schauder basis of l^, since the Banach space Zoo îS not separable (see Exercise 6.8).

Definition 6.15 A Schauder basis {x„} of X is called shrinking if the sequence {x* } is a Schauder basis of X*.

The canonical basis of l\ has a surprising property which suggests the fol-lowing.

Definition 6.16 A Schauder basis {xn} of X is called boundedly complete if oo r n N

Σ, 0-n.Xn converges whenever the sequence < Σ akXk \ is bounded. n = l *· fc=l J

The natural basis of CQ is not boundedly complete. A careful study of the spaces CQ and l\ suggests a kind of duality between shrinkingness and boundedly completeness.

Theorem 6.9 / / the Schauder basis {xn} of X is shrinking, then {x*} is a boundedly complete basis of X*.

Proof. Assume that {xn} is a shrinking basis of X. Let {an} be a sequence of scalars such that

n

n " f c = i

< oo

We need to prove that £ α/tx* is convergent. Consider the sequence of elements n = l

{y^}oiX* defined by

n

yÛ = Yâkx'k

J f c = l

6.11. MORE ON SCHA UDER BASES IN BANACH SPACES 159

Our assumption implies that {y* } is bounded. Since X is separable, the weak*-topology on X* is sequentially compact. Hence there exists a subsequence {y*lk} which weak*-converges to an element y* in X*. Since {xn} is a shrinking, the sequence {x* } is a Schauder basis of X*. Therefore we have

oo

y* = Σν*(χη)χ*η-7 1 = 1

Since {Vnk} weak*-converges to y*, we must have

«m ynk{xi) = y*{xi) nfc—>oo

for any i > 1. By the definition of the vectors y*, we have

y*nk{xi)=ai

whenever nk > i. So it must be the case that y*(xi) = a;, for any i > 1. Therefore,

oo

n=l

which completes the proof. ■

In the next theorem, we give a nice and useful description of X" when X has a shrinking basis.

Theorem 6.10 Let {xn} be a shrinking basis ofX. ThenX** may be identified n

with the space of all sequences of scalars {an} such that sup 2~]akXl' < °°-n "fc=i

The correspondence is given by

x « ^ ( i " ( x ; ) , * * · ( * ; ) , χ * * { χ * 3 ) , ■■■).

The norm of x** is equivalent (and, if the basis constant is 1, even equal) to n

sup Σ0*1* The proof is left as an exercise.

These two dual concepts are crucial in understanding the structure of the dual space. In particular, we have the following characterization of reflexivity.

Theorem 6.11 Assume that X has a Schauder basis {xn}- Then X is reflexive if and only if {xn} *s shrinking and boundedly complete.


In order to prove this theorem, we need the following technical lemma.

Lemma 6.5 Assume that X has a Schauder basis {xn}- Then {xn} is shrinking if and only if for every x* € X', the norm of χΤ>, (which is the restriction of x* to the span of {xi}n<i) tends to 0 as n —* oo.

Proof. If {x„} is a basis of X* then, for every x* e X*, we have

lim \\P*(x') - x'\\ =0. n—>oo

Since Pn-i(x\[x,],^,) = °> w e Se t

l i m IK[*,u,H = 0· n—>oo I I * » J > » S «

Conversely, assume that \\xTtx , || —> 0 as n —» oo. Let x € X be such that

||x|| = 1. Then, by definition of the adjoint operator, we have

(x*-P:(x*))(x) = x*(x-Pn(x)) < I K [ I I 1 B + I S ( | | ( * + 1) ,

where K is the basis constant of {xn}. So we must have

lim | | ι · - Ρ η · ( ι ' ) | | = 0 . n—*oo

Back to the proof of Theorem 6.11. Assume that X is reflexive and that {xn} is not shrinking. Then by the technical lemma, there exist ε > 0, y* and a sequence {yn}, with \\yn\\ = 1 for any n > 1, such that

ll/*(yn)| > e

and yn € span{xi, n < i}. Since X is reflexive, there exists a subsequence {ynk} of {yn} which weakly converges to a point y. Since {xn} is a Schauder basis of X, we can easily see that all the coordinates of y are 0. In other words, {ynk} weakly converges to 0. So we must have

lim \y*(yn)\ = 0 > e, nk—.oo

which is a contradiction. Therefore, {xn} is shrinking. Moreover, a similar argument used in the proof of Theorem 6.9 will show that {xn} is boundedly complete.

Conversely, assume that {xn} is shrinking and boundedly complete. In order to show that X is reflexive we will show that any bounded sequence has a weakly convergent subsequence. Let {yn} be such that ||yn|| < 1, for any n > 1. Since {xn} is a Schauder basis of X,

oo

Vn = ^2ai(n)xi i= l

6.11. MORE ON SCHA UDER BASES IN BANACH SPACES 161

for any n > 1. We have

Mn) | < 2K,

where K is the basis constant of {xn} (once we assume that ||xn|| = 1). A diagonal argument will help construct a subsequence {ynk} such that {oj(nfc)} converges to Oj, as n^ —* oo. Since

i=N

y ai(n)o i = l

< K y ai(n)xj

for any N > 1, we get

i=N

22aixi i=\

i=l

< K .

<K,

Because {a;n} is boundedly complete, there exists y £ X such that

oo

y = Σα{Χ' ■ t = l

In order to complete the proof we will show that y is the weak limit of {yni}-Since {xn} is shrinking, it is enough to show that for any m > 1 we have

x'm(y)= lim x'm{ynJ, nk—oo

but this is exactly the assumption on the sequences {am(nk)}. This completes the proof of Theorem 6.11. ■

Note that the existence of a Schauder basis does not give us very much in-formation about the structure of the space. So if we wish to use bases to study the structure of a Banach space in any detail, we are led to consider bases with various special properties.

Definition 6.17 A Schauder basis {xn} of X is called unconditional if when-oo oo

ever ^ o,nxn converges, it converges unconditionally, that is, Σ απ(η)3;π(η) n = l n = l

converges for any permutation π o/N.

The following properties more fully describe the unconditional behavior of basic sequences.

Proposition 6.17 Let {xn} be a Schauder basis of X. The following are equiv-alent:

(i) {xn} is an unconditional basis.


(it) / / 53 αηχη converges, then for any sequence {εη}, where εη = ±1 for all 7 1 = 1

n, the series Σ εηαηχη converges. n>l

oo (ni) / / Σ αηχη converges, then for any sequence {bn}, where \bn\ < \an\ for

n=l oo

all n, the series ^ bnxn converges. n=l

(iv) There exists c > 1 such that if A and B are finite subsets ofN with A C B, then for any sequence {an} we have

Σ anxn < c ^ anxn

ηζΑ nÇB

The smallest constant c which satisfies the above inequality is called the unconditional basis constant of {x„}. / / the unconditional constant is 1, the basis is called unconditionally monotone.

oo (v) There exists λ > 1 such that if Σ αηχη converges, then for any sequence

n=l {εη}, where εη — ±1 for all n, we have

oo

n=l η>1

The smallest constant X which satisfies the above inequality is called the unconditional constant of {xn}. Moreover, we have

c < X < 2c,

where c is the unconditional basis constant.


Remark 6.3 Let {xn} be an unconditional basis of X. For any x € X, set

\x\ =s\ipl \\γ^χ*η(χ)εηχη

n=\

εη = ±1 , for any n > 1

It is easy to check that \ ■ \ is a norm equivalent to || · || for which the unconditional constant of {xn} is 1.

The conclusion of Corollary 6.11 can be strengthened when unconditional bases are involved.

Theorem 6.12 Let X be a subspace of a space with an unconditional Schauder basis. Then X is reflexive unless it contains a subspace isomorphic to either CQ or l\.

6.12. UNIFORM CONVEXITY AND REFLEXIVITY 163

Proof. Taking into account the conclusion of Theorem 6.11, it is enough to establish the following two propositions.

Proposition 6.18 Assume X has an unconditional Schauder basis. Then the basis is boundedly complete if and only if X does not contain a subspace iso-morphic to CQ.

Proposition 6.19 Assume X has an unconditional Schauder basis. Then the basis is shrinking if and only if X does not contain a subspace isomorphic to Ιχ.

The proofs of both propositions are left as an exercise.

6.12 Uniform convexity and reflexivity

In Chapter 9 we will see that the following is actually a special case of a much more general result. However, the proof below illustrates some nice geometric properties of uniformly convex spaces.

Theorem 6.13 If X is a uniformly convex Banach space, then X is reflexive.

Proof. We show that Smulian's condition (g) of Proposition 6.16 is satisfied. Suppose {Kn} is a descending sequence of nonempty bounded closed convex subsets of X. We rely on exercises at the end of this chapter. For each n there is a unique point xn € Kn such that

| | * n | |= inf{ | | ï | | : * € # „ } .

Since {Kn} is descending the sequence {||xn||} is nondecreasing. Moreover, for each n, | | i n | | < sup{||x|| : x € K\) and, since K\ is bounded, the sequence {llxJI} is bounded as well. Therefore lim ||xn|| = r > 0. Now let m>n. Then

n—>oo Xn T Xm r ,

€ Λ „ , SO

IWI < # n r %τη < I M ± j W I < r

Thus it must be the case that lim τη,η—*οο

Xn ~T XT\ = r. On the other hand,

Xn i %r\ < (...(fczsd)).

From this we conclude that

Um *(ΐ!*ζ£=η = o, ι,η—»oo V r I


and since X is uniformly convex it follows that lim \\xn — xm\\ = 0. This τη,η—·οο

oo proves that {xn} is a Cauchy sequence. Therefore, lim xn = x with x 6 Π Kn. n - ° ° n=l

■ Another way to see that Lp(/i) is reflexive for 1 < p < oo is to show directly

that ( £ ρ ( μ ) ) " = Lp(ß). We will not discuss the details but this follows from the

fact that (Ζ,ρ(μ))* = Lq(ß), where - + - = 1. This instantly gives (Ζ,9(μ))* = P Q

( £ Ρ ( μ ) ) " = Ζ,ρ(μ). It is also known that (Ζ,ι(μ))* = £οο(μ) and that (Ζ,οο(μ))* φ ·£Ί(Μ)· Thus

Li(/i) is not reflexive. (Similarly LOO(M) is not reflexive.) Since c* and CQ are isometrically isomorphic to t\, neither of these spaces is reflexive.

Another thing to keep in mind: It is immediate from the definition that reflexivity of X implies reflexivity of X*. However, uniform convexity of X does not necessarily imply that X* is uniformly convex.

A more general result, which we will not prove here, is the following.

Theorem 6.14 If X is a uniformly nonsquare Banach space, then X is reflex-ive.

A useful property of uniformly convex spaces which does not generally hold in reflexive spaces is the following.

Proposition 6.20 Let X be uniformly convex and let {xa} be a net in X. If weak -lim xQ = x and t/lim \\xa\\ = \\x\\, then l imxa = x.

a a a

Thus in a uniformly convex space weak convergence and convergence in norm imply strong convergence. Spaces which have this property are said to have Kadec-Klee (KK) norm.

There is another fact which distinguishes reflexive spaces from uniformly convex spaces. If (X, | | | |) is a Banach space, and if | | | | j is another norm on X which is equivalent to the norm | | · | | , then it readily follows that (X, ||·||) is reflexive if and only if (X, | | | | j ) is reflexive. Thus reflexivity is invariant under equivalent renormings. To see that this is not true of uniform convexity one need look no further than the finite-dimensional spaces Kg a n d K^,.

We conclude with one final comment for those interested in specific cases. Let 1 < p < oo and consider the spaces Lp[0,1] where Lesbesgue measure is understood. In this case a sequence {/„} in Lp\0,1] converges weakly to / € Lp[0,1] if and only if {/„} is bounded and for each x € [0,1],

pX fX

lim / fn(t)dt= / f(t)dt. n ^ ° ° Jo Jo

6.13. BANACH LATTICES 165

6.13 Banach lattices Let X be a Banach space and suppose < is a partial order on X. Recall that this means that:

(i) x < x for all x e X;

(ii) x < y and y < x => x = y;

(iii) x <y and y < z => x < z.

As usual we adopt the convention x > y ·£> y < x. Elements x € X for which x > 0 are said to be positive.

The linear structure of X is assumed to be compatible with the order struc-ture in the following sense:

(iv) x < y => x + z < y + z for all x, y, z € X\ and

(v) x < y => ax < ay for all x, y E X and a G K+.

Now suppose S C X and suppose x € X. If x satisfies the following two conditions:

(a) y < x for all y e S, and

(t>) y < z for all y € S => x < z,

then x is called the supremum of 5, written

x = sup(S) = \J y. y€S

The infimum is defined dually, that is, by replacing < with >, and writing

x = inf(S) = f\y-y€S

Definition 6.18 If x\/ y and x Ay exist for every pair of elements x,y € X then X is called a linear lattice (or Riesz space).

Remark 6.4 A good nontrivial example of a linear lattice to keep in mind as we run through the ensuing properties and definitions is the space B[0,1] of bounded real-valued functions on [0,1], taking f < g O f(x) < g(x) for all x € [0,1]. Another good example is the space CQ of sequences of real numbers which converge


to 0. In this case if x — {xn},y = {yn} € Co, take x < y o · xn < yn for all n € N. The spaces lv (1 < p < oo) with the same order relation provide additional examples. Of course, R with its usual order relation provides an example as well.

In a linear lattice it is possible to describe the positive and negative parts of elements as follows:

x+ - xVO

x~ = ( -x)VO.

Also,

\x\ = x\/ (—x).

This leads to the identities x = x+ - x~ and \x\ = x+ + x~ = x+ V x~. If x = u — v, then u A v = 0 if and only if u = x+ and v = x~.

Two elements x,y S X are said to be disjoint if |x| Λ \y\ = 0. In particular, x+ and x~ are disjoint (since |x + | = x+, \x~\ = x~, and x+ Λ x~ — 0). In particular, if x and y are disjoint, then

\x + y\ = \x-y\ = Wv|y| = M + M·

Four additional identities are obvious if one thinks about the example spaces. These also link the linear and order structure together.

x + y = xVy + xAy;

\x — y\ — x V y — x Λ y;

2(|a:|V|y|) = | i + y| + | i - y | ;

2 (|x| Λ \y\) = \x + y\ - \x - y\ .

The following familiar identities hold as well.

\\x\ - \y\ < \χ-y\ < \χ\ + \y\;

|a: + »|< W + M-

6.13. BANACH LATTICES 167

Finally, suppose u, v, w > 0 and suppose u and v are disjoint. Then u + v = uVv and, in particular,

(u Λ tu) V (v Λ w) = (u V ti) Λ ω

and

uAw + vAw = (u + v)Aw.

Definition 6.19 If X is a linear lattice which is also a Banach space, then the norm of X is called monotone if

0 | | i | | < ||y||.

If in addition, \\\x\\\ — \\x\\ for all x € X, then X is called a Banach lattice .

A Banach lattice is called Dedekind complete if every nonempty order bounded set in X has a supremum and an infimum. If every order bounded sequence has a supremum and an infimum in X then X is said to be σ-Dedekind complete.

If X is σ-Dedekind complete then for each x 6 X it is possible to define the principal band projection, Px, by

oo oo

p*iv) = V [(n Ιχΐ)Λ y+] - V [(n W ) Λ y~ ■ n=l n=l

Then Px (x) = x and both Px and I — Px have norm one in the operator norm sense. (The norm of an operator T : X —» X is given by ||T|| = sup{ \\T(x) ||; x e Bx}.)

It might be instructive to look closely at the projection Px in particular spaces. Suppose x,y & B\0,1]. Then Px(y) € S[0,1] is the function defined as follows:

P(v)(t)-f 0 w h e n i W = 0;

From this one sees that y — Px(y) € J5[0,1] is given by

v(t) - P (v)(t) = ί 2 / ( i ) w h e n a ; ( i ) = 0 ; VW rx{V)W j 0 w h e n χφ φ o

Similarly, if x = {xn},y - {yn} 6 Co, then Px(y) - {c„}, where

_ J 0 when xn — 0; °" _ \ 2/n when xn φ 0.


Finally, the norm of a Banach lattice X is said to be order continuous if inf{||a;|| : x e A} = 0 for every set A Ç X which has the property that (A, >) is a directed set with inf(.A) = 0.

Exercises

Exercise 6.1 Show that too(I) is complete, for any index set I.

Exercise 6.2 Show that c is complete and that CQ is a closed subspace of c.

Exercise 6.3 Prove Propositions 6.2 and 6.3.

Exercise 6.4 Show that, in general, conv(.A) Φ conv(.A).

Exercise 6.5 Let K be a compact subset of a Banach space X.

1. Show that conv(K) is also compact. Do we have a similar conclusion when K is weakly compact ?

2. Show that there exists a sequence {xn} in X such that lim | |xn | | = 0, and n—*oo

K C convNx n } j .

Exercise 6.6 {Ascoli-Arzelà theorem) Let (K,d) be a compact metric space, and denote byC(K) the Banach space of continuous real-valued functions defined on K.

1. Show that any totally bounded subset K. of C(K) is equicontinuous, that is, for any e > 0 there exists δ > 0 such that

d(x,y) <δ=> \f(x) - f(y)\ < ε for any f e fC.

2. Show that if K. is a bounded subset of C{K) and D is a countable dense subset of K, then any sequence of elements of /C has a subsequence which converges pointwise on D.

3. Show that any equicontinuous sequence which converges pointwise on the set S C K, converges uniformly on S.

Deduce then that a bounded subset K. of C(K) is relatively compact if and only if K, is equicontinuous.

EXERCISES 169

Exercise 6.7 For any p, 1 < p < oo, show that any bounded subset K of lp is relatively compact if and only if

■•oo ' i=n

lim y^\xi\p = 0

uniformly for x = {x;} £ K.

Exercise 6.8 Show that any Banach space with Schauder basis is separable. Deduce from this that Ι,χ, does not have a Schauder basis. Hint: Show that l^ is not separable.

Exercise 6.9 / / {xn} ond {yn} are sequences in a uniformly convex space X, show that the condition

lim ||xn|| = lim ||y„|| = lim - \\xn + yn\\ = r > 0 n—»oo n—»oo n—»oo £

implies lim ||xn — yn\\ = 0. In particular, if x,y, z € X, then n—»oo

\\X-z\\ = \\y-z\\ = l\\X-y\\*Z = Z±!L.

Exercise 6.10 Show that every nonempty closed convex subset of a uniformly convex space contains a unique element of minimal norm. Deduce from this that for any nonempty closed convex subset C of a uniformly convex space X, and for any x e X, there exists a unique point a € C such that

\\x — a\\ = dist(x,C) = inf{||x - y||; y 6 C).

Exercise 6.11 Show that any finite-dimensional space X is strictly convex if and only if it is uniformly convex.

Exercise 6.12 Show that a Banach space X with modulus of convexity δ is strictly convex if and only if 6(2) — 1.

Exercise 6.13 Show explicitly that ί,χ, and l\ are not strictly convex (thus not uniformly convex).


Exercise 6.14 Let X be a separable Banach space with unit ball B and let B* denote the unit ball in X*. Let S = {χχ, X2, ■ · ·} be a dense subset of B. For each f,g&B* define

oo 1

Show that p is a metric on B* and that the metric topology on (B*,p) coincides with the relative weak* topology on B*, that is,

weak* lim fa = feB*<& limp(fa, f) = 0. a a

Exercise 6.15 Let X be a Banach space. Define a type-function by

τ(χ) — limsup | |xn — x||, n—*oo

where {xn} is a bounded sequence. Show that r is weak lower semi-continuous.

Exercise 6.16 Let X be a Banach space. Let {x*,} be a bounded sequence in X* the dual space of X. Show that, in general, the type-function

r(x*) = limsup ||x* — x*|| n—»oo

is not weak*-lower semi-continuous.

Exercise 6.17 Let X be a Banach space with a Schauder basis {e„}. Assume that {en} is bimonotone. Show that {e*} is also bimonotone. Assume that {en} is shrinking and bimonotone. Show that any type-function

r(x* ) = lim sup | |x* — x* 11 n—»oo

is weak*-lower semi-continuous.

Exercise 6.18 Let X be a Banach space with a Schauder basis {en} with asso-ciated biorthogonal system {e*}. Let {x/t} be a bounded sequence in X such that lim e*(xfc) = 0, that is, the coefficients of {xjt} converge pointwise to 0. Show

k-~»oo

that there exists a subsequence {xia} of {x^} and a sequence {ui} of successive blocks of {e„} such that

lim \\xki -Ui\\ = 0 .

Exercise 6.19 Show that B\0,1] is a Dedekind complete Banach lattice.

Exercise 6.20 Show that CQ is a Dedekind complete Banach lattice.

Chapter 7

Continuous Mappings in Banach Spaces

7.1 Introduction

In this chapter we take up Brouwer's Fixed Point Theorem (Theorem 1.4 of Chapter 1) and some of the many results that have been inspired by this pro-found result. We have chosen what seems to us to be one of the most accessible approaches to Brouwer's theorem, taking as our of departure the following sim-ple fact about the real line.

Proposi t ion 7.1 Let V : (0 = P0 < Ρχ < ■ ■ ■ < Pk = 1) be a ■partition of the interval [0,1] with each of the points Pi, 0 < i < k, 'labeled' either 0 or 1. Then the cardinality of the set N-p = {i € {1, · · · ,k} : Pi-\ and Pi are labeled differently) is an odd integer.

Proof. This proposition has a straightforward induction proof, which the reader might want to try. However, a 'counting' argument is perhaps more illuminating for what will follow. Begin by labeling 0 = P0 and 1 = Pi, and label points Pi € (0,1) arbitrarily with O's and l's. Now count the number of 'endpoints' of subintervals defined by the partition which are labeled 0, and do this in two different ways. First, if an endpoint labeled 0 is in (0,1) then it must be counted twice since it is an endpoint of two different subintervals of the partition. By adding PQ to this total we see that there are an odd number of such endpoints, say k. Now divide the subintervals of the partition into two different subclasses: (i) Those which have only 0 as endpoints, and (ii) those which have a 0 and a 1 as endpoints. If there are m intervals in class (i) then these m intervals contribute 2m endpoints to the total k. If there are n subintervals in class (ii) then, since each such interval contributes only one endpoint to the total, we conclude that 2m + n = k. This proves that n must be an odd integer. Since n is the cardinality of N-p we are done. ■

171



172 CHAPTER 7. CONTINUOUS MAPPINGS IN BAN ACE SPACES

Note in particular that the conclusion of Proposition 7.1 implies N-p φ 0 .

We first show how to use Proposition 7.1 to give another proof of Brouwer's Theorem in R1. While the proof given here is no simpler than the one given in Chapter 1, we shall see that this same line of argument leads to a simple proof of Brouwer's Theorem in R2 and, in turn, it points the way to a proof of Brouwer's theorem in its full generality. The reader might think of quicker ways to proceed in R1, but please keep in mind that it is the method of proof that we are interested in here.

Proposition 7.2 Let Co and Cx be two closed set in R1 with 0 6 Co and 1 6 C\, and suppose [0,1] C C0 U Cx. Then C0C\CX ^ 0.

Proof. For each k € N let Vk be a partition of [0,1] into subintervals of equal length. Assign the label 0 to P0 and 1 to ?n. For i € {1,· · · ,fc - 1} assign labels as follows: If Pi € Co assign the label 0 to P»; otherwise assign the label 1 to P{. Since [0,1] C Co U Cx, this results in a labeling of Vk as in Proposition 7.1. Consequently, for each k there exists an interval of length l/k whose endpoints {Pk,Qk} are labeled differently, say Pk is labeled 0 and Qk is labeled 1. Thus for each fc, Pk € Co and Qk 6 C\. It is possible to choose convergent subsequences {/V} and {QkA with, say, lim Pfc, = P and

j—·οο

lim Qk, = Q- Since lim \Ρ^ — Qkt I = lim 1/fc, = 0, P — Q, and since both j—ΌΟ j — OO j—OO

Co and C\ are closed this point lies in Co Π C\. ■

It is somewhat remarkable that the above ideas provide the basis for the general proof of Brouwer's theorem. Proposition 7.1 is a special case of a cele-brated result known as Sperner's Lemma, whereas Proposition 7.2 is a special case of a result due to Knaster, Kuratowski and Mazurkiewicz, known as the KKM Theorem. We shall discuss more general cases below. First, however, we show how the above results lead another proof of the fact that any continuous mapping / : [0,1] —» [0,1] has a fixed point. Note that every point in P € [0,1] can be written in the form

Ρ = λ 0 ( Ρ ) 0 + λ 1 ( Ρ ) · 1 ,

where λ0(Ρ) + XX(P) = 1, by simply taking λχ(Ρ) = P and λ0(Ρ) = 1 - P. Set

Co = { Ρ € [ 0 , 1 ] : λ ο ( / ( Ρ ) ) < λ 0 ( Ρ ) } ; d = { Ρ € [ 0 , 1 ] : λ 1 ( / ( Ρ ) ) < λ 1 ( Ρ ) } .

Thus

Co = { P e [ 0 , l ] : P < / ( P ) } ; d = { P S [ 0 , 1 ] : / ( P ) P and / ( P ) < P , which is absurd, so it must be the case that [0,1] C Co U Cx. We now invoke Proposition 7.2 to conclude Co Γ) Cx Φ 0 . But if P G Co Π Ci, then clearly / ( P ) = P.

7.2. BROUWER'S THEOREM 173

7.2 Brouwer's Theorem We now give a detailed proof of Brouwer's Theorem in the case n ~ 2 and we also indicate how to prove the theorem for arbitrary n. The material in this section is inspired by a discussion found in Zeidler [163, p. 798]. Our starting point is Sperner's Lemma in R2. Let M be a closed triangle in R2 with vertices Po, Pi,P2· In order to understand the general case it is important to be precise about some definitions. The r-dimensional sides of M are:

The vertices Po, Pi, P2 for r = 1. The sides PoPÎ", P1P2, Ρ2Ρ) for r = 2. The triangle itself for r — 3. We define the base of a point P in M to be the side of M of smallest

dimension which contains P. An integer i (i = 0,1,2) is said to be a Spemer label for P if Pi belongs to the base of P in M. It is important to understand exactly what this means. The points Po, P\, P2 are labeled 0,1,2, respectively. Any other point on the side P0P1 may be labeled either 0 or 1, regardless of how any other such points are labeled. The same holds for the remaining sides. Any point strictly inside M may be labeled 0,1 or 2.

The following is Sperner's Lemma [151] for the case n = 2.

Lemma 7.1 Suppose the triangle M of R2 with vertices Po,P\,P2 *s divided into subtriangles, and suppose each vertex of each subtriangle is assigned a Sperner label. Then the cardinality of the set of subtriangles whose vertices have distinct labels is odd.

Proof. Call a subtriangle of the triangulation of M a Sperner simplex if its vertices have distinct labels, and call a side of a subtriangle distinguished if its vertices carry the labels {0,1}.

Note that if a distinguished side of T has a vertex which lies in the interior of M then this side is also a distinguished side of precisely one other subtriangle in the triangulation of M. Thus there are an even number of distinguished sides having at least one vertex lying strictly inside M. On the other hand, by Proposition 7.1, there are an odd number of distinguished sides lying on P0P1 and there none lying on either P1P2 or P2P0· Therefore, the total number of distinguished sides is an odd integer, say k. At the same time, the vertices of any subtriangle which has distinguished sides and which is not a Sperner triangle must be labeled either {0,1,1} or {0,0,1}. Each such triangle has precisely two distinguished sides. Suppose there are m subtriangles of this type and suppose there are n Sperner triangles. Then 2m + n = k and we conclude that n is odd.

We now prove the KKM Theorem for the case n = 2. Note that here we use M to denote, inclusively, the vertices of the triangle along with its sides and interior. The only new idea to enter at this point in the development is that of a 'triangulation' of a given triangle. It is easy to see how to do this in an orderly way by beginning with an equilateral triangle, partitioning each of

174 CHAPTER 7. CONTINUOUS MAPPINGS IN BAN ACH SPACES

the sides into n subintervals, and joining points of the subintervals with lines which are parallel to the sides of the given triangle. In this way it is possible to achieve a triangulation in which each of the subtriangles is also equilateral and has diameter \/n times the diameter of the original triangle.

Lemma 7.2 Suppose M is a triangle in R2 with vertices Po, Ρχ, P2, and suppose Co,Ci,C2 are closed subsets ofR2 which satisfy

(i) Pi € d, i = 0,1,2; (it) PiPjCduCj, i,j = 0,1,2; (ni) M C Co U C\ U C2.

77ien Co Π Ci Π C2 φΰ>.

Proof. For £ = 1,2,··· consider a sequence of triangulations Vk of M where the diameter of the largest triangle in Vk is ejt with ejt —» 0 as k —» oo. Assign a Sperner labeling to each triangulation with the added provision that each vertex Q is assigned a label i consistent with Q € C;. This is possible by (ii) and (iii). Then by Lemma 7.2 each triangulation Vk contains a Sperner simplex with vertices {P^k) ,P^h) ,P^k)}, where P$k) G Cu i = 0,1,2. Now choose respective

ik Λ convergent subsequences {P} }j>\ (i — 0,1,2) with

lim Pl{k')=Qi(i = 0,1,2).

j—+oo

Since the diameters of the subtriangles tend to zero as j —> oo it must be the case that Qo = Q\ = Q2 = Q, and since each of the sets d is closed, Q € Ci, for i = 1,2,3. ■

We are now in a position to prove Brouwer's Theorem for the case n = 2.

Theorem 7.1 Suppose M is a triangle in R2 with vertices Po,P\,P2, and sup-pose f : M —> M is continuous. Then f has a fixed point.

Proof. Every point P in M can be written in the form

P = λ0(Ρ)Ρθ + Xl(P)Pl + \2(P)P2,

where 0 < A4(P) < 1 (< = 0,1,2) and λ0(Ρ) + λι (Ρ) + λ2(Ρ) = 1. Set d = {Pe M : Xi(f(P)) < Xi(P)}, i = 0,1,2. Since / is continuous each

of the sets Cj is closed. Also, since Xi(Pi) = 1 for each i, Pi € Cj for i = 0,1,2. If Xi(f(P)) > \i(P) for each i = 0,1,2, then it would follow that

λο(/(Ρ)) + λ 1 ( / (Ρ)) + λ 2 ( / ( Ρ ) ) > 1 .

Since this contradicts / : M —» M it must be the case that for each P G M there exists i € {0,1,2} for which Xi(f(P)) < Aj(P), that is, M C CoUCj UC2

and (iii) of Lemma 7.2 holds. Finally, if P e P0P1 then λ2(Ρ) = 0. However, this fact along with the assumption P £ Co U C\ leads to the contradiction

λ 0 ( / (Ρ) ) + M / ( P ) ) + λ 2 ( / (Ρ) ) > λ0(Ρ) + λχ(Ρ) = 1.

7.2. BROUWER'S THEOREM 175

Similar reasoning applies to the remaining sides, so we conclude that (ii) of Lemma 7.2 also holds. Therefore, we may apply Lemma 7.2 and conclude that there exists P e Co Π C\ Π C<i. This implies

Ai(/(P)) < Ai(P), t = 0,1,2,

2 2 which, in turn, implies 1 = Σ λ ί ( / (Ρ ) ) < Σ xi(p) = l- Therefore, Xi(f(P)) =

i=0 t=0 λ;(Ρ), z = 0,1,2, and thus f{P) = P. u

We conclude this section with a few remarks on how the above theorem is extended to higher dimensions. The difficulties largely involve terminology. An n-dimensional closed simplex in K" is the convex hull of n +1 points PQ, ■ ■ ■ , Pn

in R" which do not lie in an n — 1-dimensional subspace of Rn. As before, the base of a point in M is the subsimplex of lowest dimension which contains the point. To see how the induction goes we illustrate how to pass from n = 2 to n = 3.

Let M be a closed three-dimensional simplex (i.e., a tetrahedron) in K3 with vertices labeled {0,1,2,3}, subdivide M into three-dimensional subsimplexes, and assign to each vertex of the resulting partition a number that belongs to its base in M. Now call a triangular face of a subsimplex distinguished if it carries the labels {0,1,2}, and observe that if a vertex of a distinguished triangle lies in interior of M then it is a face of precisely two distinct subsimplexes of the partition. By the case n — 2 there are an odd number of distinguished triangles on the boundary of M. (Here it is important to observe that only a face initially labeled {0,1,2} can have distinguished triangles.) Therefore, the total number of distinguished triangles is odd. On the other hand, the vertices of a simplex which is not a Sperner simplex (i.e., one that does not have distinguished labels) and which does have a distinguished triangle must have one of the following labelings: {0,1,2,0}, {0,1,2,1}, or {0,1,2,2}. Each such simplex has precisely two distinguished triangular faces. If there are m such simplexes and if n are the number of Sperner simplexes, then 2m + n must equal an odd number; thus n is odd.

In order to state Sperner's Lemma in its full generality we need a little more explanation. Let Mn = [PQ, ■ ■ ■ , Pn\ be a closed n-dimensional simplex in Rn. By a subdivision S of M " we mean a decomposition of M " into finitely many non-overlapping n-simplexes s\ , · · ·■ , Sk such that (1) the intersection of any two simplexes in S is either empty or a common face of each, and (2) each n — 1-simplex in S that is not on the boundary of M n is a common face of exactly two n-simplexes of S. The base of a vertex P in S the face of M n of lowest dimension which contains P.

Lemma 7.3 [Sperner] Let S be a subdivision ofMn and label each vertex P € S with one of the numbers {io,- ■ ■ ,i3} whenever [Pj0, ■ · · , PjJ is the base of P in Mn. Then the number of simplexes in S which are labeled {0, · · · , n} is odd.


The proof is by induction with the inductive step applied on the boundary of Mn.

Sperner's Lemma can be used to extend the KKM theorem to higher dimen-sions, and this, in turn, leads to Brouwer's Theorem. The general version of the KKM theorem needed to carry this out is the following. The proof follows closely that given in the case n = 2.

Theorem 7.2 Suppose {PQ, · ■ · , Pn} are the vertices of an n-simplex in Rn and suppose Co, · · · , Cn are closed subsets of Rn for which

k

c o n v { P i o , - - - , P i t } c I J C i ,

for every subset {Pio, ■ ■■ ,Pik) of {PQ, · · · , P n } . Then

n

t=0

7.3 Further comments on Brouwer's Theorem

If X is a topological space and M C I , then a continuous mapping r : X —> M is called a retraction if r(x) = x for all x € M. When this occurs, M is said to be a retract of X.

We begin with the following simple fact.

Proposition 7.3 Every closed convex subset K in R" is a retract o/K".

Proof. This is a special case of our earlier observations about l<i. Let p G R" and set

r = inf{| |p-a; | | : x 6 K}.

Then it is possible to choose a sequence {xn} Ç K such that

lim | | i n - p\\ = r. n—»oo

Since the sequence {xn} is bounded it has a convergent subsequence. (An exten-sion of the Bolzano-Weierstrass Theorem assures that K is compact.) So assume that lim xnk — x. Then x E K because K is closed, and clearly ||x — p\\ — r.

k—»oo

Now suppose there are two points x,y S K, such that

\\x-p\\ = \\y-p\\ = r ·

Suppose x Φ y and consider the point m = - ( x + y). Then by the Pythagorean

Theorem,

| | p - m | | + | | m - x | | = r2.

7.3. FURTHER COMMENTS ON BROUWER'S THEOREM 177

If m φ x this would imply \\p — m\\ < r. This is a contradiction since convexity of K implies m € K. Therefore m — x = y. This assures that for each point p € Rn there is a unique point p(p) € K which is nearest to p.

To conclude that p is the desired retraction we need only to show that p is continuous. However, in view of Theorem 6.1 we know even more, namely, that p is nonexpansive, that is, for each u , D 6 R n

| | p ( t t ) - p ( t i ) | | < | | « - V | | .

■ The above fact permits an extension of Brouwer's Theorem to arbitrary

bounded closed convex subsets of Rn. For such a set K select a simplex S sufficiently large that K Ç S. By Proposition 7.3 there exists a retraction r : S —* K. The composition mapping / o r is a continuous mapping of 5 into itself and therefore must have a fixed point x. Moreover, x must he in the range oi f or, which is K. Since r(x) = x for points in K,

x = f or(x) = f(x).

Therefore, we have the following.

Theo rem 7.3 Let K be a bounded closed convex subset o/Kn and let f : K —> K be continuous. Then f has a fixed point.

One very interesting consequence of Brouwer's Theorem is the following fact, which seems intuitively clear.

Proposition 7.4 The surface, S, of a nontrivial closed ball B in Rn is not a retract of B.

Proof. Suppose r is a retraction of B onto S. Then — r would map each point of S to its antipodal point. Thus — r would be a continuous mapping of B into B which is fixed point free, and this contradicts Brouwer's Theorem. ■

In fact, Proposition 7.4 is equivalent to Brouwer's Theorem. To see this, suppose / : B —* B is continuous. If f(x) φ x for each x 6 B then one could construct a retraction r : B —» S as follows: For each x follow the directed line segment from / (x) through x and find its intersection with S. Call this point r(x). It is easy to see that the mapping x ι—► r(x) is continuous, and the existence of such a retraction contradicts Proposition 7.4. Therefore, / must have a fixed point.

The fact that the above proposition fails in an infinite-dimensional Banach space answers a question posed in the famous collection of mathematical prob-lems known as The Scottish Book (see [119]). This is a book named after a cafe, in what was at that time Lvov, Poland, where Banach and his collaborators frequently gathered to discuss mathematics. Around 1935 S. Ulam raised the


following question, which appears as Problem 36 in The Scottish Book: "Can one transform continuously the solid sphere of a Hubert space into its bound-ary such that the transformation should be the identity on the boundary of the ball?" An addendum indicates that Tychonoff had provided an affirmative answer. Subsequently S. Kakutani gave several examples of fixed point free continuous mappings of the unit ball of Hubert space into itself, any of which yield an affirmative answer to Ulam's question.

Another nice solution to Ulam's problem, and one that holds in an arbitrary Banach space, was published by Victor Klee in 1955. In [101] Klee proved that any infinite-dimensional Banach space X is homeomorphic with the 'punctured' space X\{0}. Let h : X —> A"\{0} be such a homeomorphism. (Thus h is continuous, one-to-one, and surjective (onto), and h~l is also continuous.) Now assume (as one may) that h(x) = x if x 6 X and \\x\\ > 1. The required retraction is now given by the mapping

\\h(x)W

Brouwer's Theorem assures that if A" is a closed convex set in R" with its usual norm, | | | | 2 , then any continuous mapping / : K —» K has at least one fixed point. However, this fundamental result extends to Rn with any other norm. In fact, any two norms on Rn are equivalent. This is a consequence of the of the Closed Graph Theorem, which assures that the identity mapping I : (R",||-||2) —♦ (Rn,||-||) is continuous. (It is also a consequence of a more elementary fact which we prove in Section 9.1 of Chapter 9. Any linear mapping T : E —> F where E and F are Banach spaces with E finite-dimensional is necessarily continuous.)

A consequence of the preceding is the fact that any closed convex set K\ in (R n , | | | | ) is homeomorphic to a closed convex set K2 in (R n , | | | | 2 ) via the identity mapping I : K2 —» K\. Let / : K\ —» K\ be continuous. Now consider the mapping I~l o / o I : K2 —> K2- This mapping is continuous and has a fixed point x G B2 via Brouwer's Theorem in Rn. But

I~l o / o I(x) - x => / o I(x) = I(x) = x,

so x is also a fixed point of / in K\. Since any finite-dimensional Banach space can be viewed as Rn with an

equivalent norm simply by identifying the respective basis vectors, the following is the most useful formulation of Brouwer's Theorem.

Theorem 7.4 Suppose K is a nonempty bounded closed convex subset of a finite-dimensional Banach space X, and suppose f : K —> K is continuous. Then f has at least one fixed point.

The following facts will be needed in the next section. If (X, ||·||) is an infinite-dimensional Banach space and if {x\, x2, ■ ■ ■ , xn} Ç X, then the smallest

7.4. SCHAUDER'S THEOREM 179

subspace Xm of X which contains {x!, χ^, ■ ■ ■ , xn} is, in fact, finite-dimensional (having dimension at most n). (This is because all linear combinations of the

n form Σ αίχί form a subspace of X so some subset of {x\,X2, ■ ■ ■ ,xn} serves

i = l

as a basis for this subspace.) Therefore (Xm,||-| |) has the same topological properties as Rm. Consequently, if K = conv({xi,X2i · · · i^n}), then A" is a bounded closed convex subset of Xm and therefore K is compact.

7.4 Schauder's Theorem

We begin by observing that Brouwer's Theorem cannot be extended to any Banach space.

Example 7.1 Consider the unit ball B in the Banach space CQ. For x — {x\,X2, ■ ■ ■ ) € B define

T(x) = (1 - |a;1|,a;1,X2,···)·

It is easy to check that T : B —> B. Also, T is continuous; in fact,

| |Γ(χ)-Γ(2/) | | = | | χ -2 / | | for x, y 6 B.

However, T(x) = x implies xl = x2 = ■ ■ ■ = 0 which, in turn, implies 1 — |xi| = 0—a contradiction.

This shows that Brouwer's Theorem fails in some infinite-dimensional spaces and, in fact, the classical result of Dugundji assures that it fails in any infinite-dimensional space. The discussion about retractions in the previous section leads to this conclusion as well. On the other hand, bounded closed convex subsets of E n are compact. It turns out that this is precisely the assumption needed to assure the validity of an infinite-dimensional version of Brouwer's Theorem.

Theorem 7.5 (Schauder's Theorem) Let K be a nonempty compact convex subset of a Banach space X, and suppose f : K —» K is continuous. Then f has at least one fixed point.

Proof. As we have seen before f(K) is precompact, hence totally bounded. Therefore, for each n G N there exists {yi,j/2i · · · ,ï//v„} Q f{K) such that for any ι ί ί ί

, m]n | | / ( x ) - y i || < - . ι<ι<Νη η

For each such i define

ai(x) = max \ | | / ( i ) - y<||, 0 \ ;

180 CHAPTER 7. CONTINUOUS MAPPINGS IN BANACH SPACES

thus for each x E K there must exist at least one i E {1,2, · · ■ Nn} for which aj(x) φ 0. Now define the so-called Schauder operator Pn : K —► K as follows:

Nn

^2ai(x)yi

Pn(x) = i=1 N„

Σ<α{χ)

Note that Pn(x) E K because Pn(x) is a convex combination of the elements {yi 12/2, · ' ' i VNn} ■ Also, since / is continuous, each of the functions aj is as well.

Now let Kn — conv({t/!, j/2, · ·· , VNn })· Then Kn is a bounded closed convex subset of the finite-dimensional Banach space which is spanned by the vectors {î/i> 2/2. · · ■ i2//v„}, and moreover P n : Kn —» K„. Brouwer's Theorem applies. Each of the mappings Pn has a fixed point xn € Kn Ç Α\ Since Ä" is compact the sequence {xn} has a subsequence {xn/fc} which converges, say lim xnk — x E K.

k—>oo Now observe that for any n,

| | P „ ( i ) - / ( x ) | | = \ t=l i = l ;

i N,i 1 /N" i

It follows that x is a fixed point of / since

||*n4. - / ( l ) | | < I! ) - / ( X n J | | + | | / ( X n J - / ( x ) | |

and the right-hand side tends to 0 as k —» oo while the left-hand side tends to | | x - / ( x ) | | . ■

7.5 Stability of Schauder 's Theorem

In some sense continuity is an 'ideal' assumption, often merely added to make otherwise difficult problems manageable. This section deals with situations in which the function under consideration cannot be shown to be continuous, but at the same time can be shown to be 'almost continuous'. For such functions an extension of Schauder's Theorem is useful.

Definition 7.1 Let X be a topological space and M a metric space. A map-ping f : X —» M is said to be ε-continuous if for every x E X there exists a neighborhood Ux of x such that d iam/( t / x ) < e.

We shall apply some basic topological facts to give a detailed proof of the following classical result of V. Klee [102].

7.5. STABILITY OF SCHAUDER'S THEOREM 181

Theorem 7.6 Suppose K is a compact convex subset of a normed linear space, and suppose f : K —► K is ε-continuous. Then there exists a point xo 6 K such that ||/(xo) - zo|| < e.

Our proof uses a standard 'partition of unity' argument to show how / may be approximated by a suitable continuous function g to which Schauder's Theorem may be applied. (Klee's original approach involved approximating the domain of / with a polyhedral set, using triangulation to approximate / with a piecewise linear functional, and then applying Brouwer's theorem.)

The proof of Theorem 7.6 is a direct consequence of the following extension theorem.

Theorem 7.7 Suppose K is a compact convex subset of a normed linear space X, and suppose f : K —> K is ε-continuous. Then there exists a continuous mapping f' : K —♦ K such that | | /(x) — f(x)\\ < ε for each x € K.

Before proving Theorem 7.7 we need some terminology. If X is a topological space and φ : X —» R, then the support of φ is defined to be the closure of the set φ~ι(Μ. — {0}). In particular, if x ^(supportφ) then there is some neighborhood of x on which φ vanishes.

Definition 7.2 Let {U\,· ■ ■ ,Un} be an open covering of the topological space X. A family of continuous functions

φί : X-* [0,1], ί = 1 , · · · , η ,

is said to be a partition of unity of X dominated by the family {Ui} if (i) (support 0j) Ç Ui for each i; and

n (ii) Σ Φί(χ) = 1 for each x 6 X.

t=l

The topological fact we need is the following. Its proof can be found in any standard topology text (e.g., [49], [89], [125]).

Theorem 7.8 (Partitions of unity) Let {U\,· ■ · , Un} be a finite open covering of the metric (or, more generally, normal topological) space M. Then there exists a partition of unity of M dominated by the family {Ui}.

Proof of Theorem 7.7. Since / is e-continuous, for each x € K there exists δχ > 0 such that dia.m(f(U(x;6x)) < ε. (Recall that U(x;6x) denotes the open ball centered at x with radius δχ.) Since K is compact there exists a finite set

{ i i , · · · ,xj} Ç K such that K C (J U(XÏ,6XJ2). Let 6 = mî{èXi : 1 < i < j}. i= l

Note that we now have the following: If x, y 6 K and \\x - y\\ < 5/2 then there exists an i such that i , t / 6 U(xi\ 6Xi); hence | |/(x) - f(y)\\ < ε. Now again use

n the fact that K is compact to select A — {αχ, · · · , αη} Ç K so that K Ç (J Ui,

i = l where Ui = U(üi;6/2), i = 1, · ■ · ,n. Then the family {Ui} is a finite open


covering of K so there exists a partition of unity {< }™=1 of A" dominated by the family {Ui}. Define the function / : K —> X as follows:

n

f(x) = 2^'φί(χ)/(αί) for each x G K. %=\

n Since f(A) C K and £ 0{(χ) = 1 we see that, in fact, f{K) Ç cönv(/(A)) Ç K.

i=\

Furthermore, since each of the functions <j>i is continuous, / is continuous. Finally, observe that for x G K,

| | / ( x ) - / (x ) | | = f(x)-^l(x)f(ai)

n

X;^(x)[/(x)-/(ai)i

i= l ι=1

Since </>i(x) = 0 iî x £ Ui while \\f(x) - /(ui) | | < ε if x e ί/i, we conclude that

| | / ( x ) - / ( x ) | | < £ .

By Schauder's Theorem / has a fixed point Xoi where ||/(xo) — XQ\\ < £· ■

7.6 Banach algebras: Stone Weierstrass Theorem Certain Banach spaces, especially those whose elements are themselves functions (e.g., the space C[0,1]), possess additional algebraic properties. We pause here to discuss a fundamental fact from real analysis that will be referred to in the next section.

Definition 7.3 A Banach space (X, | | | |) is said to be a Banach algebra if, in addition to addition and scalar multiplication, there is a product operator xy defined for x, y G X which satisfies for each x,y,z £ X, and a G K,

(i) x(yz) = (xy)z; (ii) x(y + z) = xy + xz\ (iii) (y + z)x = yx + zx; (iv) (ax)y = a(xy) = x(ay); (v) llxyll < ||z|| |M|.

If / , g G C[0,1] and one defines fg as usual by setting

(fg)(t) = f(t)g(t),

7.7. LERAY-SCHA UDER DEGREE 183

then all the above properties hold. In particular,

II/5Ü = sup | / ( t )f l ( i ) l< sup | / ( i ) | sup \g(t)\ = | | / | | \\g\\. «€[0,1] t€[0,l] t6[0,l)

The following well-known theorem is one of the most useful in all of anasysis. A proof based implicitly on Banach's contraction mapping principle can be found in [67].

T h e o r e m 7.9 (Stone- Weierstrass Theorem) Let S be a compact Hausdorff topo-logical space and let C(S) be the Banach algebra of all bounded, continuous, real-valued functions on S with the usual norm: | | /[ | = sup | / (s ) | . Suppose X is

«es a subalgebra of C(S) which contains the constant functions and which separates points of S. Then X is dense in C(S).

We should be clear about what the theorem says. To say that X is a sub-algebra of C(S) means that X is a subspace of C(S) which has the property that f,g e X and a e R implies fg £ X and af G X. A constant function is a function with the property f(s) = a for s € S. The condition that X separates points of S means that if u, v € S and u^ v then there exists / € X such that f(u) Φ f{v)- The conclusion is that the topological closure X of X in C(S) coincides with C(S).

The following is the familiar special case of the Stone-Weierstrass Theorem we allude to in the next section.

Corollary 7.1 The space P[0,1] of polynomial functions is dense in the space C[0,1].

To see that the corollary is a special case of the general theorem one need only to observe that P[0,1] is a subalgebra of C[0,1] and that the identity function separates points in [0,1].

In view of the corollary we know that if / € C[0,1] and e > 0, then there is a polynomial J e P[0,1] such that | | / - J\\ < z.

7.7 Leray-Schauder degree

One of the principal tools in applications of nonlinear functional analysis is the concept of degree of a mapping. We give an outline of the development of this theory here.

Classical degree theory, in its most general sense, is the study of mapping degree for various classes of continuous mappings with domains contained in one Banach space X and which take values in a (possibly different) Banach space Y. Such a theory consists of the algebraic count of the number of solutions of the equation f(x) = i/o, where / is defined on the closure G of an open set G in X and i/o is a given point in Y which is assumed to lie in the complement of the image of the boundary dG of G in y (i.e., i/o ^ f(dG)). The value of the degree function for any such count is an ordinary integer, either positive or negative.


Degree theory in Kn was introduced by L. E. J. Brouwer in 1912. In this section we seek to make the above observations more intuitively clear in Rn by beginning with the case n = 1 and G = (0,1). (There is nothing special about the interval (0,1)—we could just as well consider G = (a, b).)

Suppose / 6 C[0,1] and /(0) Φ 0 Φ / ( l ) . Then by Weierstrass's approxima-tion theorem if e > 0 there is a polynomial function / such that | | / — / | | < e. In particular, any such / has at most a finite number of zeros, χχ, ■ ■ ■ ,xm in [0,1]. Also, since / has at most a finite number of critical points (n - 1 if its degree is n), by adding an arbitrarily small constant to / if necessary, we can retain the condition | | / — / | | < ε and at the same time insure that f (xi) Φ 0 for all i. Having done all this, let G = (0,1) and define deg(/,<3) — 0 if / has no zeros in (0,1); otherwise set

m

deg(7,G) = £ s g n ( 7 ' ( x ; ) ) , i = l

where sgn is the function

sgn(x)

Now define deg(/,G) = deg(7,G).

In order to see that the function deg is well defined, it is necessary to show that it is independent of the choice of / . In other words, it needs to be shown that if e > 0 is sufficiently small and if / and ~g are polynomials for which | | / - / | | < e and | | / - Tj\\ < ε then deg(/, G) = deg(g, G). However, this is easy to see if one thinks about how polynomials behave. Note that since we are assuming /(0) φθφ / ( l ) , we can take ε = min{|/(0)j_,|/(l)|}. Then / and / will have the same respective signs at 0 and 1 if | | / — / | | < ε. There are three cases.

Case 1: /(0) · / ( l ) > 0. In this case either / will have no zeros, or / will have an even number of zeros, say X\ < xgn(7 ' (x i ) ) = [/'(xi) + / ' (x2)] + ■ ■ ■ + [/'(m-i) + /'(*„.)] = 0. i= l

Case 2: /(0) < 0 and / ( l ) > 0. In this case / will have an odd number of zeros, say x\ < x 0 and / (xm) > 0. Thus

m

deg(7,G) = £sgn(7'(i i)) = l· i = l

" { 1 if x > 0, - 1 if x < 0 .

7.7. LERAY-SCHA UDER DEGREE 185

Case 3: /(O) > 0 and / ( l ) < 0. This is the same as case (2), but with

7'(xi) < 0 and / ( x m ) < 0. Thus

m

deg(7,G) = 53sgn(7 ' (n ) ) = - l . t = l

Since the above analysis holds for any polynomial function / for which | | / - / | | < £) deg(/, G) is well defined and takes on the values 0, 1, or - 1 according to the cases prescribed.

Mapping Degree in R".

In passing to Rn the situation is less obvious, but the principle is the same. We only present an outline of the procedure here.

Let G be a bounded open subset of Rn with / : G —► Rn continuous, let y € Rn, and assume f(x) φ y for x e dG. As in the case n — 1 it is possible to approximate / with a function / : G —» Rn, which is continuously differentiable on G and which satisfies

s u p | | / ( i ) - 7 ( * ) | | < inf■Jf(x)-y\\,

and for which the equation f(x) = y either has no solutions in G or has at most finitely many solutions, x1,X2,· · · , x m , in G. For each such solution Xfc the derivative / (xfc) is given by the linear transformation represented by the n x n matrix

/ <**) = Ä(*<> It may be further assumed (this is a much deeper fact) that each of the solutions is regular, that is,

det(7'(xfc)) φ 0, k = l,--,m.

Note that the boundary condition assures that f(x) ψ y for each x 6 dG. If f(x) = y has no solutions, set deg(/, G, y) = 0. Otherwise set

m

deg(f,G,y) = ^sgn(det (7 ' (x f e ) ) . fc=l

It can be shown that the number deg(/, G, y) is independent of the approximat-ing function / and that the function deg is the unique function defined for all continuous / : G —* Rn which satisfies the following properties.

Property 1: If y £ dG, then deg(/, G, y) is an integer, and if deg(/, G, y) Φ 0, then the equation f(x) = y has a solution.


Property 2: If / = I (the identity), then deg(I,G,y) = 1 if y 6 G and deg(/,G,3/) = 0 i f y £ G .

Property 3 (Additivity): deg(/, uGt, y) = £ d e S ( / > Φ , y) if φ , i = 1, · · · , m i

are disjoint open sets with y £ dGi for each i.

Property 4 (Homotopy): If h : [0,1] x G —» i?" is continuous, and if ht{x) = h(t, x) φ y for all x € dG, then

deg(/i0, G, y) = deg(/ij, G, y).

Property 5 (Excision): If K is a closed subset of G and if y $. f(K), then

deg(f,G,y) = deg(f,G\K,y),

where / denotes the restriction of / to G\K = {x E G : x $ K}.

To illustrate the usefulness of the above theory, we use it to prove a simple extension of the Brouwer fixed point theorem. Properties (1), (2), and (4) play the key roles.

Theo rem 7.10 Suppose B = 0(0; r) is a closed ball in Rn and suppose T : B —> Rn is a continuous mapping which satisfies

(LS) T(x) φ \x for all x € dB and X > 1.

Then T has a fixed point in B.

Proof. For t £ [0,1] let ht be the mapping defined by taking ht{x) = x -tT(x), x e B, that is, ht = I - tT. First note that ht(x) φ 0 for x e dB and t e [0,1) since ht(x) = 0 implies tT(x) = x, from which T(x) = t~lx, contradicting (LS). If /ii(x) = 0 for some x € dB, then f(x) = x and we are finished. Otherwise 0 $. ht(dB) for all t e [0,1], so by (1), deg(/i t ,5,0) is defined. By (4) and (2)

l = deg(/iO)ß,0) = deg(/i1 ,ß)0),

so by (1) again, 0 G h\(B). Clearly this implies T has a fixed point in B. ■ Note that in the above theorem one could replace B with the ball B(xo;r)

and replace (LS) with the condition

T(x) - XQ φ \(x - xo) for all x ε dB and λ > 1.

7.8. CONDENSING MAPPINGS 187

A final comment about Brouwer's Theorem. In 1904, Bohl proved a theorem in [14] which can be formulated as follows: Suppose B — B(0; r) is a closed ball in E " and suppose / : B —> Rn is a continuous mapping for which f(x) φ 0 for each x € B. Then f(x) = μχ for some x € dB and μ < 0. Taking T = I — f and λ = 1 — μ this becomes the contrapositive formulation of Theorem 7.10.

Mapping Degree in Infinite Dimensions

We now turn briefly to the theory developed by Leray and Schauder (and for whom the boundary condition (LS) is named). Let X be an arbitrary Banach space. A mapping T defined on a subset D of X and taking values in X is said to be compact if T is continuous and if T maps bounded sets into sets whose closures are compact. Using approximations similar to the one used in the proof of the Schauder Fixed Point Theorem it is possible to extend degree theory to mappings of the form I — T where T is a compact mapping. This theory of degree is characterized by the same properties that characterize the degree function in Rn and leads to the following theorem.

Theorem 7.11 {Leray-Schauder Fixed Point Theorem) Let G be a nonempty bounded open subset of a Banach space X and let T : G —» X be a compact mapping which satisfies for some XQ € G :

(LS) T(x) -χ^φ \[x - xQ) for all x e dB and λ > 1.

Then T has a fixed point in G.

Subsequently degree theories have been established for many classes of map-pings, including those we take up in the next section.

7.8 Condensing mappings

A well-known generalization of Schauder's theorem which bridges the gap be-tween the metric and topological theories involves the notion of a 'measure of noncompactness'.

Definition 7.4 Let D be a bounded subset of a metric space (M,d). If D — n U Pi, then we say that P — {P\,- ■■ ,Pn} is a finite partition of D, and we set

t = l

| |P| | = sup{diam(Pi) :i = 1,··- ,n} .

The Kuratowski measure of noncompactness x(D) of D is defined as follows:

x(D) = inf{||P|| : P is a finite partition of D).


Certain fundamental properties of χ are easy to see. For example:

(i) X(D) = x(D), and

(ii) x(D) = 0 if and only if D is compact,

(iii) AÇB^ χ(Α) < χ(Β).

(iv) χ ( Λ υ Β ) =max{x(A),x(B)}.

The above properties all follow immediately from the definition, although for (ii) one should remember that a set is compact if and only if it is closed and totally bounded.

There are other important properties of the noncompactness measure χ that hold for bounded subsets A, B of a Banach space.

(ν)χ(Α + Β)<χ(Α) + χ(Β).

(νϊ)χ(ΧΑ) = \\\χ(Α), Λ 6 Κ .

(vii) x(conv(A)) = χ(Α).

Properties (v) and (vi) are easy and left to the reader. Property (vii) is less obvious, and since its validity is a critical step in the proof of the main result of this section we give a detailed proof.

Step 1. For any bounded set A, χ(ΝΕ(Α)) < χ(Α) + 2ε, where

Ne(A)= \JU(x;e).

Proof. This follows from the trivial fact that for any bounded set S in a metric space, diam(iV£(S)) < diam(S) + 2ε.

Step 2. Suppose C ~ A U B where A and B are bounded and convex. Then

x(conv(C)) < max{X(A),x(B)}.

Proof. Let x € conv(C) and suppose x £ AuB. Then there exist Xi e C and n n

λ» > 0, i — 1, · · · ,n, 52 λί = 1, such that x — Σ XiX{. By regrouping we may i= l i = l

7.8. CONDENSING MAPPINGS 189

suppose that Xj € A for 1 0 and choose TV e N so that — < ε.

For each i = 0,1, · · · , N, let

d = < x € conv(C) : x = — u + I 1 - — I v for some u G .A, V G 5 >.

Next observe that if a € (0,1), then a G z i + 1

for some 0 l) ,x(ß) max v(/Ve(Ci))}

< max{x(A),>:(B), imax^[x(Ci)+2e]}.

But by (v) and (vi), for each i,

m) = x fa+(. - 1 ) B) < ±xW + (. - 1 ) ,(B), which implies

x (Ci )<max{x(A) ,x (ß)} .

Therefore,

x(conv(C)) < max{x(yl),x(ß)} + 2ε.

Step 2 now follows from the fact that ε > 0 is arbitrary.

Step 3. Suppose C = | J Ai where each Ai is bounded and convex. Then t = l

x(conv(C)) < max χ(Α). Κ ι < η


Proof. This follows from Step 2 and an obvious induction argument.

Proof of (vii). Let a > χ(Α). Then A can be written as a finite union

A Ç (J Ai, where diam(.Aj) < a and since diam(conv(.Ai)) = diam(Ai) we may

assume that each of the sets Ai is convex. By Step 3,

X(conv(i4)) < max x(Ai) < max diam(.Ai) < a. \<i<n !<»<"

Since this is true for all α > χ{Α) the proof of (vii) is complete. ■

Definition 7.5 Let K be a subset of a metric space M. A mapping T : K —> M is said to be a k-set contraction, k E (0,1), ifT is bounded and continuous, and if for all bounded subsets D of M

X(T(D)) < kX(D).

T is said to be condensing ifT is bounded and continuous, and if

X(T(D)) < x(D)

for all bounded subsets D of M for which x(D) > 0.

Example 7.2 Let K be subset of a Banach space X. Suppose T\ : K —> X is a contraction mapping with Lipschitz constant k < 1, and suppose T2 : K —» X is a compact mapping (T is continuous and T(D) is compact for every bounded D Ç X). Then it is easy to see that the mapping T = 7\ + T? is a k-set contraction.

The following theorem was proved by Sadovskii in 1976. In 1965 Darbo proved the same theorem for fe-set contractions, k < 1. Such mappings are obviously condensing.

Theorem 7.12 Suppose K is a nonempty bounded closed and convex subset of a Banach space and suppose T : K —> K is condensing. Then T has a fixed point.

Proof. Fix x € K and let Σ denote the family of all closed convex subsets D of K for which x E D and T : D -> D. Now set

B=f)D,

and let

C = cönv{T(ß)U{a:}}.

7.9. CONTINUOUS MAPPINGS IN HYPERCONVEX SPACES 191

Since x e B and T : B —> B it must be the case that C Ç B. This, in turn, implies that T(C) Ç T{B) Ç C. Since x € C it follows that C € Σ. Therefore, B Ç C, from which we conclude B = C.

We now have T(C) = Γ ( β ) C C and

X(C) = x(ccmv{T(B) U {*}}) = χ({Γ(Β) U {x}} = χ(Τ(Β)) = x(T(Q).

Since T is condensing this can only happen if x(C) — 0, that is, if C is compact. Therefore T is a continuous mapping of the compact convex set C into itself, so by Schauder's theorem T must have a fixed point. ■

The above theorem has applications in abstract hyperconvex spaces as well.

Theorem 7.13 Let (M,d) be a bounded hyperconvex metric space, and suppose T : M —> M is condensing. Then T has a fixed point.

Proof. Since M is a complete metric space, M can be isometrically embedded in a Banach space X. If we use M to denote its image in X as well, then M as a hyperconvex subspace of X, and as such it is a nonexpansive retract of X. In particular, there exists a nonexpansive retraction R which maps conv(M) onto M. Note that now we have

ToR-.œnv(M) -» M.

Now let D Ç cönv(M) and suppose \(D) > 0. If x(R{D)) = 0 then since T is continuous, T maps compact sets into compact sets and so χ(Τ ο R(D)) = 0 < x(D). On the other hand, if x(R(D)) > 0, then since R is nonexpansive, x(R(D)) < x{D), and since T is condensing,

X(ToR(D))<x(R(D))<x(D).

We therefore conclude that T o R is a condensing mapping of the closed convex set conv(M) into itself. Hence by Sadovskii's theorem, there exists x € conv(M) such that ToR(x) = x. This implies that x € M from which R(x) = x = T(x).

7.9 Continuous mappings in hyperconvex spaces In this section we look more closely at compact hyperconvex spaces. For this we need to recall some notation of Chapter 4. Let (M, d) be a metric space and let D Ç M be nonempty and bounded. Set:

cov(D) = (1{B : B is a closed ball and DC B}.

As before let A(M) = {A Ç M : A = cov(i4)} denote the class of admissible sets in M.

We begin with the following consequence of Theorem 7.13.


Theorem 7.14 Let M be a compact hyperconvex metric space and let f : M —► M be continuous. Then f has a fixed point.

Our next result specifically concerns the class 3 of nonempty admissible sets of M endowed with the Hausdorff metric.

Theorem 7.15 Let M be a compact hyperconvex metric space and let f : M —> M be continuous. Define the mapping f : S —♦ 3 by setting

f{D) = cov(/(JD)).

_ °° Set DQ = M, let Dn = / ( /?„_i) = fn(M), and suppose D = f) Dn. Then

n = l f(D) = D, and D = lim Dn, where the limit is taken relative to the Hausdorff

n—oo metric H on G. In particular, if f(x) = x then x € D.

Remark 7.1 The existence of a largest {relative to set inclusion) D 6 3 such that f{D) — D {even without assuming continuity of f) follows easily from the fact that f is monotone {D\ Ç D<j. => f{D\) Ç /(D2)) and every descending chain in $ has an infimum, namely, the intersection of its members. From this the existence of such a D is immediate from a fixed point theorem of Amman {see Theorem 11.D of [162]). It is the fact that this largest fixed point can be obtained by iterating f that is of interest here.

Example 7.3 Theorem 7.15 can be thought of as an abstract formulation of a well-known fact in interval analysis, the simplest case of which is the following. Let [a, b] be an interval on the real line and let f : [a, b] —» [a, b] be a continuous function. If M{a,b) denotes the set of all nonempty closed subintervals of[a,b], then M{a, 6) corresponds to the set 3 of all nonempty admissible sets described above, and the Hausdorff metric H is given by

H{[x,y], [u,v\) = max{|x -u\,\y- v\}.

In this case one can define f : M {a, b) —♦ M {a, b) by taking f{J) = f{J) for J € M(a,6), since by continuity of f the latter set is always in M{a,b). Then the mapping f has a greatest fixed point J 6 M{a, b) (relative to set inclusion)

0 0

and for this fixed point J one has J — Γ) Jn and J — lim Jn, where JQ = [a, b] n=l n ^ ° °

and Jn — /"(Jo)· {This example is taken from Chapter 11, Section 10 of [162].)

We now turn to the proof of Theorem 7.15. Suppose M is hyperconvex and p > 0, and for D € A{M) set

NP{D) = {zeD: dist(z, D) < p).

7.9. CONTINUOUS MAPPINGS IN HYPERCONVEX SPACES 193

Since the hyperconvexity of M implies that each point z 6 M has at least one 'nearest' point in D it follows that

NP(D) = U{B(x; p):x£ D).

The Hausdorff metric H on 3 may be described as follows: For U, V G 9 ,

H(U, V) = inf{p > 0 : U C NP(V) and F Ç NP(U)}.

The proof is based on the fact that a hyperconvex space is complete and on several lemmas, the first of which is a restatement of Lemma 4.2 of Chapter 4 (Section 4.6).

Lemma 7.4 If M is hyperconvex and if D — f) ß(zjjrj) 6 A{M), then for 16/

P > 0 ,

Np(D) = f]B(zi;ri + p).

The following lemma may be of independent interest since it does not require compactness of the space.

Lemma 7.5 Let M be a bounded hyperconvex metric space and 3 the collection of all nonempty admissible sets of M endowed with the Hausdorff metric H. Then Q is hyperconvex.

Proof. Let {Da}aeA Q $> and suppose for pa > 0 the family {B(Da;pQ)} satisfies

H{Da,D0)<pa+p0.

We need to show that there exists £ ) g 9 such that

D€ f]B(Da;pa). aÇA

Since each of the sets Da is admissible, for each a 6 A there exists an index set Ia and a family of closed balls {B(xai;rai)}içia in M such that

Da = P | B(xai;rai).

Moreover, if i 6 Ia and j € Iß, since H(DQ, Dß) < ρα+ Ρβ it follows that

d(xai, Xßj )<ra,+pa+ rß. + p0.

Since M is hyperconvex the family

{B(xai;rai + pa) : i € Ia; a€ A}


has a nonempty intersection D, and clearly D € 0. In view of Lemma 7.4,

D=f)NpJDa). (*) αζΑ

In order to show that D € (~) B(Da; pa) it suffices to show that for fixed β £ A, aeA

D Ç NPß(Dß) and Dp C NP0(D) (thus H(D,D0) < p0). The first inclusion is obvious from (*). For the second, since H(Da,Dß) < pa + p0, Lemma 7.4 implies that for each a 6 A, a φ β,

D0 C NPn+Pß(Da) = niaaB(xai;rai +pa + pß).

Therefore, since Dp = P) B(xß.;rß_), again using Lemma 7.4

Dß Ç Np0(D) = n{B{xai;rax + pa + p0) : i G / α ; a € Λ}.

Lemma 7.6 Let M be a bounded hyperconvex metric space, let B be the collec-tion of all nonempty closed subsets of M endowed with the Hausdorff metric H, and let A,BeB. Then H(cov(A),cov(B)) < H(A, B).

Proof. Suppose H(A,B) = p > 0. Then if B(x;r) D A it follows that B(x; r + p)D NP{A) 3 B. Therefore, if cov(,4) = f| B{xi; rj, by Lemma 7.4

i€l

Np(cow(A)) = p | B{xi-n + p)2 Np{A) D B. i€l

From this it follows that cov(ß) Ç jV^cov^)) . Similarly, οον(Λ) Ç iVp(cov(ß)), from which H(cav(A), cov(B)) < p. ■

We also need the following elementary fact.

Lemma 7.7 Suppose {Dn} is a sequence of nonempty closed subsets of a com-pact metric space M and suppose D\ D D 0. Clearly D Ç Νε(ΰη) for each

n. Suppose for each n there exists xn € Dn such that xn £ Ne(D). Since M is compact {xn} has a subsequence which converges to a point x which is necessarily in D. However, dist(xn,Z?) > ε implies dist(x,D) > e. Since this is a contradiction, it follows that D„0 Ç Ne(D) for some integer no; hence H{Dn,D) <ε if n > n0. ■

EXERCISES 195

Proof of Theorem 7.15. Let B denote the family of all nonempty closed subsets of M. Since / : M —* M is (uniformly) continuous, it is easy to see that the induced mapping D ι—» f(D) is a continuous mapping from $ to B. Since Lemma 7.6 shows that the mapping A i—> cov(.A) from B to 3: is also continuous, it follows that / is a continuous mapping from 0 —> 3 . Also, since M is compact, the space B with the Hausdorff metric is also compact [107, p. 38]. Since Ö is hyperconvex (Lemma 7.5), $ is a complete (hence closed) subspace of B. Thus Q itself is compact. Let DQ = M and let Dn = / ( D n _ i ) , n = 1,2, · · · . Then Dx = cov(f(D0)) <Ξ -Do, so it follows that f(Dx) Ç f(D0); hence D2 = cov(/(r>!)) Ç cov(/(Z?0)) = D\. Continuing, D0 2 A D · · · D

CO

A i 2 · · ■ . By Lemma 7.7, lim Dn = D = f] Dn, and since / is continuous

f(D) = D. m

Remark 7.2 / / / is assumed to be a compact mapping, then Theorem 7.14 remains true without the compactness assumption on M, provided M has the property that for any compact subset S of M it is the case that cov(S) is compact as well. We mention abo that this problem has recently been studied by R. Espinola and G. Lopez (Houston J. Math. 25 (1999), 371-378) using the Isbell hull discussed in Chapter 4, Section 4-7.

Exercises

Exercise 7.1 Prove that if M is a compact metric space, then the metric space M consisting of all nonempty closed subsets of M endowed with the Hausdorff metric is abo compact.

Exercise 7.2 Prove the opening statement in the proof of Theorem 7.1; namely, that each point of a triangular region has a unique expression as a convex com-bination of the vertices of the triangle.

Exercise 7.3 Let B — {x 6 R2 : ||x|| < 1} denote the closed unit ball in R2, and let S = {x G B : \\x\\ = 1}. Suppose f : B —> B is a continuous mapping which has the property f (x) = x for all x € S. Show that f is surjective.

Exercise 7.4 With B and S as in the previous exercise, suppose g : B —> B is a continuous mapping satisfying g (x) = —x for each x € S. Show that there exists x € B such that g (x) = 0. [Hint: Use the previous exercise]


Exercise 7.5 Show that if a subset X of Rn has the fixed point property (for continuous mappings) and if Y is a retract of X, then Y also has the fixed point property.

Exercise 7.6 Consider whether the following subsets o/R2 have the fixed point property:

(a) A figure eight (two circles with exactly one point in common);

(b) A solid figure eight (two discs with exactly one point in common);

(c) A cross ({(x,y)eR2:\x\< l,\y\< 1, and x ■ y = θ}).

Exercise 7.7 Can you think of a subset o/R2 which is not a retract o/R2 but which does have the fixed point property?

Exercise 7.8 Let B be the closed unit ball in any infinite-dimensional Banach space. Show that given any a € [0,1) there exists a continuous mapping f : B —» B which has the property: \\x — f {x)\\ = a. [You may use Dugundji's result that Brouwer's Theorem fails in any infinite-dimensional space.]

Exercise 7.9 Prove properties (v) and (vi) of the measure of noncompactness χ (Section 7.8).

Chapter 8

Metric Fixed Point Theory

8.1 Contraction mappings

In this chapter we examine metric fixed point theory in a Banach space setting, taking an application of Banach's contraction mapping principle as a point of departure.

Since Banach's Theorem is couched in a complete metric space setting, one might wonder what more can be said if the underlying space is richer. The following is a fundamental domain invariance result which is an easy consequence of Banach's Theorem, yet it requires an underlying linear structure for its very formulation.

T h e o r e m 8.1 Let G be an open subset of a Banach space X and letT : G —> X be a contraction mapping. Then (I — T)(G) is an open subset of X.

Proof. Let f = I — T and let XQ € G. What we must show is that for p > 0 sufficiently small, the ball B(f(xo);p) Ç f(G). The approach is classical. Choose r > 0 so that B(xo;r) Ç G, and choose p = (1 — k)r, where k is the Lipschitz constant of T. Now fix z € B(f(xo);p)- We wish to show that there exists w 6 B{XQ\ r) such that f(w) = w — T(w) = z, from which

B(f(xo);p)Çf(G).

Define the mapping Tz : G —* X by setting

Tz(x)=T(x) + z, x&G.

Note that the problem has now been reduced to showing that the (contraction) mapping Tz has a fixed point which lies in B{XQ\T).

We assert that Tz : B(xo;r) —» B(xo\r), from which the conclusion follows

197



198 CHAPTER 8. METRIC FIXED POINT THEORY

via Banach's Theorem. To see this, let x 6 B(xo;r). Then

| | Γ , ( χ ) - χ 0 | | = ||Γ(*) + 2 - * ο | | = ||T(x) + Γ(χ0) - Τ(χο) + ζ - soll < | |Γ(χ)-Γ(χ0) | | + | | ζ - ( χο -Γ(χο) | | < fc||x-x0|| + | | z - / (xo) | | < kr + ρ = r.

We now conclude that for some w e B(xo',r), Tz(w) = w and the proof is complete. ■

The above argument provides an elementary and elegant example of how it is possible to reduce a mapping theorem to a fixed point theorem.

We look at another result in which an underlying linear structure is required. Recall that we use the symbol dK to denote the boundary of K. (A point x € K is in the boundary of K C X if each ball centered at x contains both points of K and points of X\K.)

T h e o r e m 8.2 Let K be a closed convex subset of a Banach space X with int(Ä") φ 0 , and let T : K —» X be a contraction mapping which satisfies T : dK —* K. Then T has a unique fixed point in K.

Results more general than the above are known. In fact, the theorem holds for more general mappings (e.g., the condensing mappings) under even weaker boundary assumptions. However, we state Theorem 8.2 to motivate the fol-lowing observation due to Felix Browder, from which it follows immediately. Browder made this observation several years ago yet it appears be not so widely known.

Theorem 8.3 Let K be a bounded closed convex subset of a Banach space X with 'mt(K) φ 0, and letT : K —* X be a lipschitzian mapping which satisfies T : dK —» K. Then for λ > 0 sufficiently small, the mapping

Τλ(χ) = ( 1 - λ ) χ + λΤ(χ), xeK,

maps K into K (and has the same fixed points as T).

Proof. Without loss of generality we may suppose 0 € 'mt(K). Let p denote the Minkowski functional of K associated with 0, that is,

p(x) = inf {μ > 0 : x € μΚ}, x € X.

Suppose B(0; r) Ç K. Then since K is bounded there exists t > 0 such that K Ç B(0; tr). Consequently, since p (x) - 1 x € dK for each x φ 0,

r <p (x ) _ 1 | | x | | <tr

8.2. BASIC THEOREMS FOR NONEXPANSIVE MAPPINGS 199

and thus there exist constants c\ and c2 such that

ci ||x|| < P(x) < c2 \\x\\

for each x € X. Now let k be the Lipschitz constant of T and let x be any point of K. If

y = p(x)~lx then y 6 ÖÄ- and y is also on the ray joining 0 to x. Therefore,

p(y -X) = P ( ( P ( Z ) _ 1 - i)x) = ( P ( X ) " 1 - l)p(x) = l - ρ(χ).

Now let A e (0,1). Then

p(Tx(x)) = p((l - X)x + XT(x)) < (1 - A)p(ar) + λρ(Γ(χ)).

On the other hand, since T(y) G K, it follows that p(T(j/)) < 1 and

P(T(x)) = p(T(y) + T(x)-T(y)) < p(T(y))+p(T(x)-T(y)) < l + c2\\T(y)-T(x)\\ < l + c2fc | |y-x | | < 1 + c2kcïlp(y - x) < \-Ykx(\-p{x%

where k\ = c2kc\x. FVom this we have

ρ(Τχ(χ)) < (1-Χ)ρ(χ) + Χ(1 + ^(\-ρ(χ))) = ( 1 - λ ( 1 + *!))?(*)+ A(l + fc!).

Now observe that if p{x) < 1 (i.e., if x 6 K) and if λ is chosen so small that λ(1 + ki) < 1, then p{T\{x)) < 1 and we conclude Τχ : K -> K. ■

Now suppose G is a subset of a Banach space X with 0 € int(G), suppose T : G —> X, and consider the condition

T(x) φ μχ for x € dG and μ > 1. (8.1)

Theorem 8.4 Let G be a subset of a Banach space X with 0 € int(G), and let T : G —+ X be a contraction mapping which satisfies (8.1) on dG. Then T has a fixed point in G.

See Exercise 8.2.

8.2 Basic theorems for nonexpansive mappings

In this section we discuss applications of the results and techniques of Chapters 4 and 5 to Banach spaces. The following definition goes back to a 1948 paper of Brodskii and Milman [20]. We specialize the definition here to allow for the fact that only weakly compact sets or, if the underlying space is a conjugate space, the weak* compact sets may be of interest.


Definition 8.1 A convex subset K of a (possibly conjugate) Banach space is said to have normal structure [resp., weak, weak* normal structure) if any bounded closed [resp., weak, weak* closed] convex subset H of K for which diam(H) > 0 contains a point XQ for which

sup{||zo - x\\ : x € H} < diam(ii).

Such a point XQ is called a nondiametral point of H.

We adopt notation consistent with that used previously in metric spaces. For a bounded closed convex subset K of a Banach space X, let

r x ( / 0 = s u p { | | a : - y | | : y e A " } .

let

r(K) = inf{rx( /0 : x € K},

and let

C{K) = {x€K:rx(K)=r{K)}.

Thus the assertion that X has normal structure means that for any bounded closed convex subset K of X it is the case that r(K) < diam(.K') whenever diam(A') > 0. If K is weakly [resp., weak*] compact, and if X has weak [resp., weak*] normal structure then C(K) is a nonempty proper weakly [resp., weak*] compact convex subset of K whenever diam(Ä') > 0. This latter assertion follows from the fact that C(K) is admissable in the sense that it can be expressed as an intersection of closed balls centered at points of K, and closed balls are both weak and weak* closed. Specifically:

C(K) = ne>oC£(K),

where

C€ (K) = (nx€K (B (x; r (K) + e))) Π K.

Many spaces which have normal structure satisfy an even stronger condition.

Definition 8.2 A convex subset K of a Banach space is said to have uniform normal structure if there exists a constant c < 1 such that any bounded closed convex subset H of K for which diam(/f ) > 0 contains a point XQ for which

sup{||a;o - x\\ : x € H} < c diam(J/).

It should be noted that the above definition is more restrictive than its metric counterpart of Chapter 5 since here it is required that the condition apply to all convex subsets of K—not just the admissible subsets of K.


We begin with one of the first observations to evolve from the introduction of the concept of normal structure. A bounded sequence {xn} in a Banach space is said to be a diametral sequence if

lim dist(x„+i,cönv{xi,X2i · · · ,χη}) = diam{xi,X2, · · · } . n—»oo

In the paper [20] in which the notion of normal structure was introduced, Brod-skii and Milman proved the following useful fact.

Proposition 8.1 A bounded convex subset K of a Banach space has normal structure if and only if it does not contain a diametral sequence.

Proof. If K contains a diametral sequence {xn}, then

H = conv{x!,X2! · ■ ·}

is a convex subset of K each point of which is a diametral point; hence K fails to have normal structure.

Now suppose K contains a convex subset H which contains more than one point and each point of which is diametral, and let ε G (0, d) where d = diam(/f). Select χχ e H, let yo = χι, and assume xn e H has been defined. Set

n

Σ Χΐ — ·

i= i n

By assumption, yn-i is a diametral point, so there exists a point xn+i € H such that

lkn+1 - î / n - l | | > d T.

n n

Let x 6 convfx!, · · · , x n } , say x = ^Z ajxj> where ctj > 1 and 53 <*j = 1· Now

take

a = max{ai, · · · , a n } .

Since a > 0, it is possible to rewrite yn-\ as follows:

1 A / 1 α Λ 2/n-i = — x + V -)xi-

na *—' \n not I


( 1 ct \ 1 n / 1 α · \ 1 > 0 and that 1- Γ 3- ) = 1.

n na) na jî \n na J Consequently,

d Ö < \\χη+\ - y n - i |

< — ||ι„+1 - i|| + Y" ( 3- ) \\xn+i - ij\ na s—1 \n na I

3 = 1 N '

na ^-^ \n na I 7 = 1 v '

— ||a:n+1 - x|| + (1 ) d. na V na '

Therefore,

n2 \_ na) ) P n + i - z | | > na

d ε = na |

na n* = d-™>dJi.

n n

Thus dist(xn+i,conv{x1, · · ■ ,xn}) > d with ε arbitrary, so {xn\ is a di-n

ametral sequence. ■

Corollary 8.1 Every compact convex subset K of a Banach space X has nor-mal structure. In particular, every finite-dimensional Banach space has normal structure.

Proof. Let K be a compact and convex subset of X. If K does not have nor-mal structure then by Proposition 8.1 K contains a diametral sequence {xn}, and clearly {xn} can have no convergent subsequence. This contradicts com-pactness of K. ■

Proposition 8.2 Every uniformly convex Banach space X has uniform normal structure.

Proof. Let if be a bounded closed convex subset of X for which d =

diam(/i) > 0. Then there must exist u,v € H such that \\u — v\\ > d/2. Let

m = - (u 4- v), and let x G H. Then (see Chapter 6, Section 6.4)

\u — i | | < d \\v — x|| < d II« - v\\ > d/2 v-x\\<d \=ï\\x-m\\< (l - δ (^λλ d= (l - δ (^\λ d.


Since X is uniformly convex, 6 I - ) > 0, so we conclude that

sup \\m -x\\<(\-6 ( | J J d<d.

îetral point of B

-O-'G)) Thus m is a nondiametral point of H which satisfies the uniform normal struc-

ture condition for c

An examination of the above proof reveals even more. Let ε > 0 and choose

u,v € H such that \\u — v\\ > d(l - ε). Then if m = - ( u + v), one obtains

v — x\\ < d > =» llx — ml ί(ι-«(*^))<-('-«'-'>)*

Note that m will be a nondiametral point of H if it is the case that 6(1 — ε) > 0, where ε can be chosen arbitrarily small. Recall that the characteristic of convexity εο(Χ) of a Banach space is the number

e o P O = s u p { i : i 5 ( i ) = 0 } .

Therefore, if εο(Χ) < 1, then it is always possible to choose ε > 0 small enough that «5(1 — e) > 0. This gives the following extension of the previous proposition.

Proposition 8.3 Suppose X is a Banach space for which εο(Χ) < 1. Then X has uniform normal structure.

Now suppose K is either a weakly compact convex subset of a Banach space X or a weak* compact convex subset of a dual space X*. In either case K is bounded. Let A(K) denote the set of all admissible subsets of K. Thus D € A{K) if D = K Π (C\B(xi;ri)), where {Bfa;^)} is a family of closed balls centered at points of K. Each such set D is itself closed and convex, hence weakly compact if K is weakly compact. If K is weak* compact then, since closed balls in X" are themselves weak* closed (indeed by the Alaoglu theorem they are weak* compact) each D € A{K) is weak* compact. In either case A(K) is compact in the sense of Section 5.1, Chapter 5. If X has normal structure (or more particularly, if K has normal structure), then it follows that A(K) is normal as well. Therefore the following are immediate consequences of Theorem 5.1 of Chapter 5.

T h e o r e m 8.5 Let K be a weakly compact convex subset of a Banach space, and suppose K has normal structure. Then every nonexpansive mapping T : K —* K has at least one fixed point.


Theorem 8.6 Let K be a weak* compact convex subset of a dual Banach space, and suppose K has weak* normal structure. Then every nonexpansive mapping T : K —* K has at least one fixed point.

A Banach space in which every bounded closed convex subset has the fixed point property for nonexpansive self-mappings is said to have the FPP. (In met-ric spaces the FPP designation was reserved for spaces in which the admissible sets have the fixed point property.) If we restrict the convex subsets to be compact for the weak or weak* topologies, we refer to the weak- or weak*-FPP.

As a consequence of Theorem 8.5 (which was proved in 1965) the search for spaces having the FPP centered on identifying reflexive spaces which have nor-mal structure. However, it was soon discovered (in 1974) that normal structure is not an essential condition for the FPP. (Whether or not reflexivity is essential remains unknown.) Later (in 1981) it was shown that the classical space L\ [0,1] fails to have the weak-FPP (Alspach), while reflexive subspaces of L\ [0,1] do have the FPP (Maurey). Also it was shown that CQ has the weak-FPP (Mau-rey) but, of course, fails the FPP. As we shall see momentarily, i\ provides an example of a space which fails the FPP yet has the weak*-FPP.

Some of the history of this development is discussed, for example, in [1] and [67].

We proceed with another consequence of the metric development. It is known that a Banach space X is hyperconvex if and only if it is isometrically isomorphic to a space C(K) of continuous scalar valued functions where A" is a compact extremally disconnected (the closure of each open set is itself open) Hausdorff space. Such spaces are of interest in a classical sense because they include the £00 (^00) spaces. From Theorem 4.8 (Chapter 4) we have the following.

Theorem 8.7 Let B — B(0; 1) be the closed unit ball (more generally, an ad-missible set) in a hyperconvex Banach space (e.g., L^) o,nd letT.B-^Bbe nonexpansive. Then the fixed point set of T is a nonempty nonexpansive retract ofB.

Remark 8.1 It is worth noting that the argument of Lemma 5.2 (Chapter 5) can be modified to prove the following.

Theorem 8.8 If X is a Banach space which has uniform normal structure, then X is reflexive.

The proof involves replacing "cov" with "conv" in the proof of Lemma 5.2 and then applying property (g) of Proposition 6.16 (Chapter 6).

Corollary 8.2 Suppose X is a Banach space for which εο(Χ) < 1. Then X is reflexive.

Corollary 8.3 Every uniformly convex space X is reflexive.

8.3. A CLOSER LOOK AT *, 205

8.3 A closer look at t\

Using the fact that a bounded sequence in i\ is weak* convergent relative to l\ = CQ if and only if it converges coordinate-wise (Section 6.8, Chapter 6) we immediately have the following.

Lemma 8.1 If {xk}k*Ll is a bounded sequence in t\ which is weak* convergent to x relative to ί\ = c*,, then for any y E ί\,

r{y) = r(x) + \\x - y\\,

where r(y) = lim sup ||x — y\\. fc—«oo

Proof. As usual, let {e1 ,e2, ■ ■ ■ } be the standard unit basis for l\ and for each i let P; denote the natural projection of i\ onto its subspace spanned by {e1, · · ■ , e '}; specifically,

P i ( x 1 , x 2 , · · · ) = (xi,X2,··· ,Χί,Ο,Ο,-··).

Also let Qi = I -Pi. Then

Qi(xi,x2,-) = (0,0, · · · , 0 , x i + 1 , x i + 2 , · · ) .

For any i = 1,2, · · · ,

r(x) = lim sup ||xfc - x|| = lim sup ||Qi(xfc - x) | | . k—*oo k—*oo

Since

\\Pi(xk-y)\\ = J2\xn-y»\> n = l

then we have

H* - y || = lim ( l i m ||Pi(xfc - y)\\) . i—«oo yfc-»oo ' " /

Also,

oo

lim ||Qi(x - y)\\ = lim V " |xn - yn\ = 0. 1—»OO I—»OO *—·

n=i

Now

||^(xfe -y)\\ + ||Qi(xfc - χ)\\ - \\Qi(x - v)\\ < \\Pitf -v)\\ + \\Qi(xk - v)|| = \\*k-y\\


and letting k —» oo and i —► oo, we get

\\x-y\\+r{x) <r(y).

On the other hand,

\\xk-y\\<\\xk-x\\ + \\x-yh

so r(y) <r(x) + \\x-y\\. ■

Theorem 8.9 Weak* compact convex subsets of ί\ viewed as the dual of CQ have weak* normal structure. Hence l\ has the weak* -FPP relative to this weak* topology.

Proof. Let K Ç ίγ be weak* compact and convex and let H be a weak* closed and convex subset of K. We need to show that if άΪΆχη(Η) > 0 then H contains a nondiametral point. We have already seen that this is the case if H is compact. If H is not compact, then H contains a sequence {xk} such that

weak* lim xk = XQ E H k—»oo

but for which {xk} does not converge in norm to XQ. For each y & H, let

r(y) = limsup||ar*: - y\\. k—»oo

By the previous lemma

r(y) = r(x0) + \\x0 - y||,

where r( io) > 0. Thus

\\xo - 2/11 = r(y) - r(x0) < diam(ii) - r(x0).

This proves that x0 is a nondiametral point of H. ■

Since the admissible subsets of l\ are weak* compact relative to any weak* topology on ^ι, we have the following.

Corollary 8.4 Let D be an admissible subset of t\ and let T : D —» D be nonexpansive. Then T has a fixed point in D.

At this point it might be interesting to note that there exist weak*-compact convex subsets of t\ viewed as the dual of c which fail to have the fixed point property. An example can be found in [67].

8.4. STABILITY RESULTS IN ARBITRARY SPACES 207

8.4 Stability results in arbitrary spaces This topic is a little specialized but it highlights a 'metric' approach to the stability result discussed in connection with Brouwer's Theorem.

Let if be a bounded closed convex subset of a Banach space X and let T : K —» K be a mapping which satisfies, for some h > 0 and p > 0, the condition

| | Τ ( χ ) - Γ ( ^ ) | | < / ι | | χ - ι / | | ρ , (x,yeK). (8.2)

These mappings are well understood for various choices of h and p. If p = 1 and h < 1, then such mappings always have a fixed point because of the Banach contraction mapping principle, and if p — 1 and h = 1 these are the nonexpansive mappings. In the case p — 1 and h > 1 (the usual Lipschitz condition), a fair amount is known about minimal displacement properties of such mappings (e.g., see [67], [69]) although a number of questions remain open. Since mappings satisfying (8.2) are degenerate if p > 1 regardless of the value of h, what remains is the case p < 1. This is the case we take up here. Central to our observations is the fact that mappings which satisfy the Holder condition (8.2) for 0 < h,p < 1, also satisfy the following more general condition:

\\T(x)-T(y)\\<maX{h,\\x-y\\}, (x,y 6 K). (8.3)

Thus such mappings are /i-nonexpansive in the terminology of Section 5.6, Chap-ter 5. As was the case there, our objective is not to assure that such mappings have fixed points (indeed, mappings satisfying (8.3) may not even be continu-ous), but rather to find conditions under which there always exists at least one x £ K for which ||x - T(z)| | < h.

This section is motivated also by the results of Chapter 7. It was proved in Section 7.5 of Chapter 7 that if K is a compact convex subset of a normed linear space and if ψ : K —► K is e-continuous in the sense that each point x G K has a neighborhood Ux for which diam(<^(t/x)) < ε, then for every ε' > ε there exists z € K such that \z — i£>(.z)|| < ε'. It is our purpose here to show that by adding the assumption ||<^(") — "^(^ll 5· | |u — v\\ f°r u>v $■ Ux the same conclusion holds in noncompact settings, and in many cases (including the compact case) a slightly stronger conclusion holds.

For convenience we take as our point of departure the following variant of Klee's definition introduced in Bula [34]. Let (M,d) be a metric space, D Ç M, and / : D —> M. Then / is said to be w-continuous at xo € D (for w > 0) if for each ε > 0 there exists δ = δ(ε) > 0 such that whenever x e D and d(xo, x) < δ then d(f(xo), f(x)) < ε + w. f is said to be w-continuous (on D) if / is w-continuous at each XQ € D. (The analog of Klee's result in Bula's terminology is that if ψ : K —> K is uniformly ω-continuous, then for every w' > w there exists x0 ζ K such that ||a;o ~ "^(ô)!! < w'■)

We begin by considering an analog of «»-continuity for contractions.


Definition 8.3 Let (M, d) be a metric space. A mapping T : M —» M is said to be a w-contraction, w > 0, if there exists k € (0,1) such that for each x, y 6 M, d(T(x),T(y))<kd(x,y) + w.

Note that if k 6 (0,1) and if I 6 (k, 1) then for x,y £ M,

kd(x, y) + w < (d(x, y) Φ> d(x, y) > w t-k

This fact leads to the following related definition.

Definition 8.4 A mapping T : M —» M if said to be a he-contractive if for h>0 and te (0,1),

d(T{x),T{y))<emax{d{x,y),h} x,y E M. (8.4)

Two facts are immediate.

Proposition 8.4 IfT:M-*M is a w-contraction with contractive constant w

k 6 (0,1), then T is a he-contractive for h = -—- and i Ç. (k, 1). c K

Proof. Iîd{x,y) > h then d(T(x),T(y)) < kd(x,y) + w < £d(x,y). Otherwise d(T(x),T(y)) < kd(x,y) +w < kh + w = th. ■

Proposition 8.5 IfT:M—>Misa w-contraction with contractive constant k, then

w mi{d(x,T(x)) : x £ M} < -—r·

1 K

Proof. Let ί e (k,l), let h(£) = -—-, and let x £ M. Suppose I — k

d(Tn(x),Tn+l{x)) > h{t),n = 1,2,· · ■ .

Then Definition 8.4 in conjunction with the geometric series argument for prov-ing Banach's contraction principle implies {Tn(x)} is a Cauchy sequence, and clearly this leads to a contradiction. Therefore, for some n it must be the case that d(Tn{x),Tn+1(x)) < h{l). Since I € (fc,l) is arbitrary, the conclusion follows. ■

In the limiting case k = 1, the above result becomes meaningless. However, in the limiting case in Definition 8.4 it seems that a great deal can be said. For this case we return to the definition introduced in Chapter 5.

Definition 8.5 If T : M —* M is h\-contractive then we say that T is h-nonexpansive.

8.4. STABILITY RESULTS IN ARBITRARY SPACES 209

Now let K be a nonempty bounded closed convex subset of a Banach space X and suppose T : K —* K is ft-nonexpansive. For t € (0,1) and fixed z & K, define Tt : K —» K by setting

Tt(x) = (1 - t)z + ίΓ( ι ) , x 6 /f.

Then it follows that

in f{ | |x -T(x) | | :x£K}<h.

To see this, note that for each x,y € K,

\\Tt(x) - Tt(y)\\ - t \\T(x) - T(y)\\ < t max{||x - y||, h}.

Therefore, for any x € K and n > 1,

||Tt"(x)-Tf"+1(x)|| < tmax{\\T»-i(x)-Tn(x)\\,h}

< i m a x i i m a x i j l T 7 1 - 2 ^ ) - ^ - 1 ^ ) ! ! ^ } ^ } = max{t2 | | r n - i (x)-rn-1(a!) | | , i / i} <

< max{tn\\x-T(x)\\,th}.

Since K is bounded and lim tn — 0, this proves n—»CXJ

inf{ | |x-T ( (x) | | : x 6 K) < th.

But

||z - T(x)|| < t - 1 ||x - Tt(x)|| + i - ' ( l - t) \\x - z\\.

Therefore, inf {\\x - T(x)\\ : x € K) < ft + i_ 1

( l — t) ||x - z\\, so upon letting

t —> l - , we have the following.

Theorem 8.10 If K is a nonempty bounded closed convex subset of a Banach space X and ifT:K—>K is h-nonexpansive, then

iuf{\\x-T(x)\\:x€K} <ft.

If T satisfies a Holder condition, the conclusion is sharper.

Theorem 8.11 Suppose K is a nonempty bounded closed convex subset of a Banach space X, and suppose T : K —> K satisfies

\\T{x)-T{y)\\<h\\x-y\\v, (x,y e K),

for some ft,p € (0,1). Then

mi{\\x-T{x)\\:xeK}<hl^l-pl


Proof. Since h < 1, Theorem 8.10 implies there exists x e K such that | | x - T ( x ) | | < 1. Thus

\\T(x)-T2(x)\\<h\\x-T(x)\\p<h,

from which

\\T\x) - T\x)\\ < h \\T(x) - THx)\\P < / i 1 + p ,

and by induction

\\Tn+\x) -Tn+2(x)\\ < h^+p+-+p"\ n = l , 2 , · ■■

Therefore,

lim | | Τ η + 1 ( χ ) - Τ " + 2 ( χ ) | | < lim / ι ( 1 + ρ + ' + p" ) = h^l~p\ n—*oo " " n—*oo

and the conclusion follows. ■

Since / i1^1 -?) < h for h,p 6 (0,1), the minimal displacement estimate of Theorem 8.11 is better than that of Theorem 8.10.

As we have seen, if K is a closed convex subset of a Banach space X and if K is weakly compact and has normal structure, then the family of all admissible subsets of K is compact (via weak compactness of K) and normal. Therefore the following is a direct consequence of the principal result of Section 5.6, Chapter 5.

Theorem 8.12 Suppose K is a weakly compact convex subset of a Banach space, suppose K also has normal structure, and suppose T : K —> K is h-nonexpansive. Then there exists a point z € K such that \\z — T(z)\\ < h.

Corollary 8.5 Suppose K is a compact convex subset of a Banach space and suppose T : K —► K is h-nonexpansive. Then there exists a point z € K such that \\z-T(z)\\ < h.

Proof. As we have seen, compact convex sets have normal structure. ■

Theorem 8.13 Let X be a Banach space which has uniform normal structure with coefficient c, let K be a bounded closed convex subset of X, and suppose T : K —► K is h-nonexpansive. Then there exists z € K such that \\z — T(z)\\ < ch.

Again, if T satisfies a Holder condition a sharper conclusion holds.

Theorem 8.14 Let X be a Banach space which has uniform normal structure with coefficient c, let K be a bounded closed convex subset of X, and suppose T: K -» K satisfies

| | Τ ( χ ) - Γ ( ΐ , ) | | < Λ | | ι - ν | | ρ , (x,v£K),

where h,p € (0,1). Then if p is sufficiently near 1,

ΐ η ΐ { | | χ - Γ ( χ ) | | : χ € / ί ' } < ch.

8.5. THE GOEBEL-KARLOVITZ LEMMA 211

Proof. By Theorem 8.13 there exists z€K such that \\z - T(z)\\ < ch. Thus

\\T(z)-T2(z)\\<h\\z-T(z)\\p <cPh1+".

Since c < 1, cph1+p < ch for all p sufficiently near 1. ■

The above estimate invites a comparison with the displacement estimate of Theorem 8.11. Clearly for certain values of p, h and c it will be the case that ch < / i 1 ^ 1 - ^ ; in particular, this is always true if c < h and p < 1/2, and also for all h < 1 if p is sufficiently small.

Since it is known that £<» is hyperconvex, the following is an immediate consequence of Corollary 5.1 (Section 5.6, Chapter 5).

Theo rem 8.15 Let K be an admissible subset of £&, and let T : K —+ K be h-nonexpansive. Then there exists z € K such that \\z — T(z)|| < hfl.

As a very special case of the above we have:

Corollary 8.6 Suppose K — [a, 6] C K1 and suppose T : K —> K is h-nonexpansive. Then there exists z € K such that \z — T(z)\ < h/2.

The results of this section are qualitative in that no effort has been made to determine optimal minimal displacement for each of the classes of mappings and spaces considered. A number of questions in the spirit of those raised in [69] remain open.

Also there is another question which the above analysis leaves open:

Quest ion. Let K be a bounded closed convex subset of a Banach space, and suppose T : K —> K is /i-nonexpansive for h > 0. Under what conditions can one conclude that there exists z € K such that \\z — T(z)\\ < hi

As we have seen, if K is weakly compact and has normal structure, then one obtains even more, namely, the existence of z 6 K for which ||z — X"( r)j| < h. This leaves the tempting suggestion that weak compactness alone suffices for an affirmative answer to the above question. The limiting case of the above (when h = 0) involves a question about fixed points of nonexpansive mappings which remains unresolved, and it is plausible that the two problems may be related.

8.5 The Goebel-Karlovitz Lemma

Any nonempty set K of a Banach space which is minimal with respect to being closed, convex and invariant under any mapping T : K —» K must satisfy cönv(T(K)) = K. This has nothing to do with properties of the mapping T. However, if T is nonexpansive and if K is also bounded, then much more can be said.


Lemma 8.2 Each point of K is a diametral point.

Proof. Suppose there exists a point z 6 K such that r = rz(K) < diam(A'), and let

Cr(K) = LeK:uef] B(x;r)\ = K f] ( f | B(x,rj). { x€K ) xeK

Then obviously Cr(K) is nonempty (z 6 Cr(K)), closed and convex. At the same time note that u 6 Cr{K) if and only if u € K and K Ç B(u\r). In particular, if u e Cr(K) then by nonexpansiveness of T, T(K) Ç B(T(u);r). But this, in turn, implies that

K=mSv(T(K)) Ç B(T(u)-r).

Therefore, T(u) € Cr(K), that is, T : Cr(K) -* Cr(K). However, this contra-dicts the minimality of K. u

Since a nonexpansive mapping is /i-nonexpansive for any h > 0, the following is a consequence of Theorem 8.14.

L e m m a 8.3 If K is a bounded closed and convex subset of a Banach space, and ifT:K—*Kis nonexpansive, then there exists a sequence {xn} Ç K such that lim ||xn - T ( i n ) | | = 0.

n—>oo

There is a fundamental fact which, in conjunction with the previous lemma, has proved extremely useful in extending fixed point results for nonexpansive mappings beyond those spaces which possess normal structure. It appears to have been discovered independently and at about the same time by Goebel [66] and Karlovitz [88].

L e m m a 8.4 {Goebel-Karlovitz) Let K be a subset of a Banach space X which is minimal with respect to being nonempty, weakly compact, convex, and T-invariant for some nonexpansive mapping T, and suppose {xn} Q K satisfies

lim | | ι „ - Γ ( χ η ) | | = 0 . n—>oo

Then for each x € K, lim \\x — xn\\ = di&m(K). n—>oo

Proof. As we have just seen, if diam(.ft') > 0 then each point of K is a diame-tral point. Now suppose there exists {xn} Ç K for which lim ||xn — T(x„)|| =

n—>oo

0, but for which lim ||x - xn | | = r < diam(Ä') for some x € K. Let η—Όο

C = {z e K : lim sup ||z - xn | | < r} . n—»oo

It is easy to see that C is convex and not difficult to see that C is closed. Indeed, if {un} Q C and if lim Uj = u, then for each i = 1,2, · · · ,

i—»oo

limsup||u — xn| | < ||u — u H + limsup||ttj - xn | | < ||u — Uj|| +r.

8.6. ORTHOGONAL CONVEXITY 213

Letting i —> oo gives lim sup ||u — xn| | < r. Thus u € C and C is closed. Also if n—»oo

U 6 if,

U m s u p | | r ( u ) - x „ | | < limsup (| |T(u) - T(xn) | | + ||T(xn) - xn | | ) n—»oo n—»oo * '

- l imsup | |T(u) -T(x„) | | n—»oo

< limsup||u - xn | | < r. n—»oo

This proves that T : C —* C. Since any closed convex subset of a weakly compact set is itself weakly compact, we conclude that C = K by minimality of K. However, if e > 0 is chosen so that r+ε < diam(Ä"), then the family {B(x\ r+ε) : x £ K} has the finite intersection property since each such ball contains all but a finite number of terms of the sequence {xn}- Since these balls are weakly compact,

Kf]l f] ß(x;r + £)) / 0 . \x€K )

But this implies r(K) < r + ε < diam(K'), contradicting the fact that K is a diametral set (all its points are diametral). ■

8.6 Orthogonal convexity

The Goebel-Karlovitz Lemma has been applied extensively in the study of fixed point theory for nonexpansive mappings, both as formulated in Lemma 8.4 and as stated in the language of ultraproducts (see [1] and references cited therein). This section is devoted to a typical application of the Goebel-Karlovitz Lemma. The approach is a recent one due to A. Jiménez-Melado and E. Lloréns-Fuster in [85]. It arises in connection with their introduction and study of a generalization of uniform convexity called orthogonal convexity.

We begin by describing the concept of orthogonal convexity. For points x, y of a Banach space X and λ > 0, let

Mx(x,y) = LeX: max {\\z - x\\, \\z - y\\} 0, let

D({xn}) = limsup(limsup ||XJ - Xj||); I—»OO j—»OO

^λ({χη}) = limsup(limsup|MA(xj,Xj)|). i—ΌΟ j — OO


Definition 8.6 A Banach space is said to be orthogonally convex (OC) if for each sequence {xn} in X which converges weakly to 0 and for which D({xn}) > 0, there exists λ > 0 such that A\({xn}) < D({xn}).

It is shown in [86] that every uniformly convex Banach space is OC. Other examples given in [86] include Banach spaces with the Schur property (hence ίι, Co, c, and James's space J, described in Section 6.1 of Chapter 6.

Theorem 8.16 Let K be a nonempty weakly compact convex subset of a Ba-nach space X, and suppose K is orthogonally convex. Then every nonexpansive mapping T : K —> K has a fixed point.

Proof. Invoking Zorn's Lemma we may assume K satisfies the assumptions of Lemma 8.4. We assume that di&m(K) > 0 and show that this leads to a contradiction. Let {xn} c K satisfy lim ||a;„ — T"(a;n)|| = 0. By a suitable

n—»oo

translation and passing to a subsequence, we may assume 0 € K, with {xn} converging weakly to 0. Also, for convenience, we assume that diam(Ü') = 1. The argument in the general case merely requires multiplying everything by d~l, where d is the diameter of K.

Now observe that the Goebel-Karlovitz Lemma implies that D({xn}) = 1. Hence by orthogonal convexity there exist positive real numbers λ and b such that .A.\({xn}) < b < 1. Thus there exists a subsequence of {xn}, which we again label {xn}, such that \Mx(xn)xm)\ < b for all m φ η, and again by passing to a subsequence we may assume

lim ||a:„ n—*oo

t n + l | = 1; lim n \\χ„ n—»oo

• Γ ( χ η ) | | = 0 . (*)

Apply Banach's contraction mapping principle to obtain the (unique) solution zn to the equation:

where m„ = X-n ~r X-n+l

1 - -) T(zn) + -mn, n I n

For i — 0,1, we have

ln+i - Zn 1 - - T{xn+i) + n

Using the definition of zn, we get

zn - xn+i =h-lA (T(zn) - Τ{χη+ίή

I 1 \Xn^-i) Xn+i n

+

+

1 j \T(xn+i) - Xn+i j

-hnn - xn+i),

from which (using the fact that T is nonexpansive)

\Zn - * n + i | | < I 1 Zn X n-\-i + I 1 -1

+ -n

T{Xn+i) ~ xn+i\

mn — x n+i


Multiplying by n and using the definition of mn, we have (again for z = 0,1)

\\Zn - Xn+i\\ <(n-l) \\T{xn+i) - Xn+i\\ + - \\xn+i - Xn\\ ■

In view of (*), we have for n sufficiently large:

max I \\zn - xn\\, \\zn - xn+i\\ } < — ~ — H 2 ^ 1 ~ X"H ·

This proves that {zn} is eventually in M$xn,xn+$; hence

\\Zn\\ < AX({xn}) oo

The resulting contradiction establishes the theorem. ■

8.7 Structure of the fixed point set

It is useful to know that the fixed point set of a nonexpansive mapping is a nonexpansive retract of its domain. The following simple facts serve to illustrate this. Throughout this section we use F(T) to denote the fixed point set of a mapping T.

Theorem 8.17 Suppose M is a metric space which has the property that the fixed point set of every nonexpansive mapping of M into M is a nonempty nonexpansive retract of M, and suppose T and G are commuting nonexpansive mappings of M into M. Then F(T) Π F(G) φ 0.

Proof. First observe that if x € F(T) then ToG(x) = GoT(x) = G(x). This proves G : F(T) —» F(T). By assumption there is a nonexpansive retraction r of M onto the nonempty set F(T). Thus G o r : M —> F(T) is nonexpansive and by assumption has a nonempty fixed point set. However, since r is a retraction onto F{T), it must be the case that F{G o r) = F(G) Π F{T). This proves that F(T) Π F(G) Φ 0 (and at the same time that F(T) Π F(G) is itself a nonexpansive retract of M). ■

A mapping T : M —♦ M is said to be eventually nonexpansive if there exists an integer N such that for each n> N,Tn is nonexpansive.

Theorem 8.18 Suppose M is a metric space which has the property that the fixed point set of every nonexpansive mapping of M into M is a nonexpansive retract of M, and suppose T : M —* M is eventually nonexpansive. Then F(T) φ 0 .


Proof. For TV sufficiently large the mappings TN and TN+Ï are commut-ing nonexpansive mappings of M into M. By the previous theorem F(TN) Π F(TN+1) φ 0. But it is trivial to check that

F(T) = F(TN)C)F(TN+1).

m

Of course, the above results are not really about nonexpansive mappings. All one needs to know is that there is a retraction from the domain of the mapping onto its fixed point set which can be drawn from a class of mappings for which the original domain has the fixed point property. It turns out, however, that the class of nonexpansive mappings often works—a fact we have already noted in hyperconvex spaces. We now take up this question in a Banach space setting, beginning with the separable case since most of the difficulties in this case have been disposed of in Section 5.8 of Chapter 5. It is perhaps surprising that in any separable space the following simple assumption is enough to assure the desired result.

Definition 8.7 A bounded closed convex subset K of a Banach space is said to have the generic fixed point property for nonexpansive mappings ( GFPP) if every nonexpansive mapping f : K —» K has a fixed point set in every nonempty bounded closed convex f-invariant subset of K.

By what we have already seen, any bounded closed convex subset of a re-flexive Banach space which has normal structure satisfies (GFPP), as does any weakly compact convex subset of an orthogonally convex Banach space. Other examples will be discussed later.

Our first result is the following.

Theorem 8.19 Suppose K is a bounded closed convex subset of a separable Banach space, suppose K has (GFPP), and let T : K —> K be nonexpansive. Then the fixed point set F(T) of T is a (nonempty) nonexpansive retract of K.

The above follows quickly from Theorem 5.12 of Chapter 5 which, for con-venience, we restate here.

Theorem 8.20 (5.12) Let M be a separable complete metric space and let S be a semigroup of nonexpansive self-mappings of M. Then there exists in S a nonexpansive retraction r of M onto F(S) if and only if one of the following two (equivalent) conditions holds:

(i) Each nonempty closed S-invariant subset of M contains a fixed point of S.

(it) Whenever x € M then c\S(x) Π F(S) φ 0 .

S. 7. STRUCTURE OF THE FIXED POINT SET 217

Proof of Theorem 8.19. Let KK denote the family of all mappings of K into K and put

S = {s e KK : s is nonexpansive and F(s) D F(T)}.

Obviously, 5 is a semigroup on K and F(S) 2 F(T). Since T 6 S we conclude that F(S) = F(T). Iîsus2eS and λ 6 (0,1) then Xsx + (1 - A)s2 € 5. Thus for each x 6 K the set 5(x) = {s(x) : s G S} is clearly nonempty, convex, and T-invariant. Since T is continuous, clS(x) is a nonempty bounded closed convex T-invariant subset of K, so by (GFPP) cl5(x) Π F(T) φ 0. This shows that S satisfies (ii). The conclusion now follows from Theorem 5.12. ■

The following is now a direct consequence of the above and the theorem stated at the beginning of the section.

Corollary 8.7 Suppose K is a bounded closed convex subset of a separable Banach space, suppose that K has (GFPP), and let T be a finite family of commuting nonexpansive mappings of K into K. Then the common fixed point set F(J-) of T is a (nonempty) nonexpansive retract of K.

The above corollary can be extended to arbitrary families T. Rather than pursue this further, however, we take up the question of what can be said in the nonseparable case. The central observation is that the separability assump-tion on the space can be replaced with reflexivity or, more generally, with the assumption that the domain is weakly compact. The ideas are due to Brück [29].

Theorem 8.21 Let K be a nonempty weakly compact convex subset of a Banach space X and suppose T : K —» K is a family of nonexpansive mappings with a nonempty common fixed point set A. Suppose further that

A intersects every nonempty T-invariant closed convex subset of K. (8.5)

Then A is a nonexpansive retract of K.

Proof. As before, let KK denote the family of all mappings of K into K and let

N = {/ € KK : f is nonexpansive and F(f) D A},

where F(f) denotes the fixed point set of / . Obviously, T Ç N so N ψ 0 . Notice also that by Tychonoff's Theorem KK is compact in the topology of weak pointwise convergence, since this is the product topology on the space n x e / f K induced by the weak topology on K.

Now for / , / ' e N, we say / ~ / ' if ||p - / (x) | | = ||p - / ' ( i ) | | for each p e A and x e K, and for / G N let

[/] = { / ' € # : / ' - / } .


Let [TV] = { [ / ] : / e TV} and introduce the partial order -< on [TV] by defining [/] ^ [9} if and only if ||p — /(x) | | < ||p - g(x) || for eachp E A and x € K.To see that ([TV], ) has a minimal element, let {[/α],α € / } be a descending chain in ([N], -<). Since KK is compact in the topology of weak pointwise convergence the n e t {/<*} has a subnet {/a(} which converges in this topology to some / E KK. Thus for each x E K,

weak - lim fa( (x) = f(x).

For x,y E K, {fa^{x) - faîv)} converges weakly to f(x) - f(y) (algebraic operations are always weakly continuous) so

| |/(*) - /(v) | | < Inn | | / „ t ( i ) - fQ((y)\\ < Nor - y\\.

Also, if x E A then fa(x) = x for each a € I so it follows that f(x) = x. Thus F(f) 2 A and this proves that / e TV.

Now let x 6 K and p E A. Since {[/α],α Ε 1} is descending in ■< it is the case that the limit limç ||p — /Qf (x)|| exists, and since norm-closed balls in X are weakly closed it must be the case that

||p - /COU < lim \\p - /Q <(i) | | = lim ||p - /„WH.

Since lim||p - / a (x ) | | = lim||p - /Q'(OII i f /« ~ /α ' . {[/α]>ΰ Ε 1} is bounded a a'

below by [/] in ([TV],X). We may now invoke Zorn's lemma to conclude that ([■N],;^) contains a minimal element [r]. We show that r E \r) is the desired

retraction. Now suppose there exists a point z E K such that r(z) $. A. Since

| | p - r o r ( i ) | | < | | p - r ( i ) | |

for each p E A and x E K, [r or] ■< [r]. But since [r] is minimal, \r or] — [r]. In particular, if ZQ — r{z) then for all p E A,

\\p-r(z0)\\ = \\p-z0\\>Q.

Let

C={for(zQ):fEN\.

Since f,g Έ TV implies Xf + (1 — λ)ρ € TV for any λ E [0,1], C is convex. To see that C is weakly compact we only need to show that C is closed. Let {xa} be a net in C for which limx« = x. Then for each a there exists /„ 6 TV such that

a xa = fa ° r(zo)- But since KK is compact in the topology of weak pointwise convergence {/„} has a subnet {fat} which converges to some f E KK. Arguing as before, / is nonexpansive and F(f) D A. Hence / E TV, so / o r(z0) E C. But

x = weak-lim/c{ or(zo) = / ° r(zo)-

8.8. ASYMPTOTICALLY REGULAR MAPPINGS 219

This proves that x £ C; hence C is closed. Let f e N. Then F(f) D A so for each p € A,

\\p-f°r{zo)\\ = \\f(p)-for(zo)\\<\\p-r(zo)\\.

Thus [for] ;< [r]. But since [r] is minimal this implies \\p — f ° r(zo)\\ — \\p - r(z0)\\ > 0 for each p e A. Therefore, A Π C = 0 . On the other hand, we have already observed that T Ç iV, so s : C —> C for each s € T. Thus .A Π C ^ 0 by our basic hypothesis. This is a contradiction, so we conclude that r(z) e A for each z 6 K. m

The following is an obvious special case of the preceding theorem.

Corollary 8.8 Suppose K is a weakly compact convex subset of a Banach space, suppose K has (GFPP), and let T : K —> K be nonexpansive. Then the fixed point set F(T) of T is a (nonempty) nonexpansive retract of K.

8.8 Asymptotically regular mappings

In this section we show that in terms of fixed point properties of nonexpansive mappings defined on a convex subset of a Banach space, it always suffices to study only a special class of mappings, the so-called asymptotically regular mappings. Let D be a subset of a Banach space X.

Definition 8.8 A mapping f : D —» D is said to be asymptotically regular if for each x € D it is the case that lim | | / η (χ) - / n + 1 (a ; ) | | = 0.

n—»oo " "

The key observation of this section is the following: If D is bounded and convex, and if T : D —» D is nonexpansive, then the mapping / — (I + T)/2 has the same fixed point set as T and moreover, / is asymptotically regular. This very useful fact was discovered by Ishikawa in 1976 [76]. It is an easy consequence of the following lemma.

Lemma 8.5 Suppose {xn} o,nd {yn} are sequences in a Banach space X which satisfy for all n g N:

(i) xn+i = ^(xn+yn)

(«) l|yn+i-yn|| < ||ζ„+ι-z„||.

Then for o ! / t , n € N ,

( l + - J \\yi - Xi\\ < 2" \\yi - Xi\\ - \\yi+n - xi+n\\ + \\yi+n - Xi\\.


Proof. The proof is by induction on n. Observe that the lemma is trivially true for all i if n — 0, and make the inductive assumption that the theorem holds for a given n E N and all i. This gives (upon replacing i with i + 1 in the inequality)

( l + 7jj | | j / i+i-s»+i | | < 2 n Hî/i+i-Xi+i | | - \\yi+n+i - *i+n+A j

+ ||2/i+n+l - Z i + l | | ·

Next observe that by (i), the triangle inequality, and (ii), respectively:

\yi+n+\ - xi+i\\ < (1/2) Vi+n+l - Xi\\ + WVi+n+l - V;

< (1/2) lbt+n+1 - X.ll + Σ WVi+k+i - î/t+fc|| k=0

n

< (1/2) WVi+n+1 - Xi\\ + 5 Z \\Xi+k+l - Xi+k\ k=0

Combined with the previous inequality, this gives

( l + £ ) l l l / i + i - * i + i | | < 2" WVi+i - Xi+\\\ - \\yi+n+i - art+n+il

+(1/2) WVi+n+i - Xi\\ + Σ H^+fc+i - Xi+k\\ k=0

Thus

2(l + £) | |y i + 1 -* i + 1 | | < 2"+1

+

2/i+i - z»+i| | - WVi+n+i - Xi+n+i

n

Vi+n+l - Xi\\ + Σ HX»+fc+l ~ Xi+k\\ fc=0

Next we observe that (i) and (ii) imply

WVn+l - ^n+ll l < ||2/n+l ~ Vn\\ + ||î/n ~ Xn+l\\

< ]\xn+i - x„|| + \\yn - xn+i\\ = \\Vn - Xn\\ ■

This means that the sequence {\\yn - xn\\) is monotone nonincreasing. Thus

Σ WXi+k+l - Xi+k\\ = 9 Σ 11^+* ~ Xi+fcH - " ^ 9 ^ ~ XiH ' fc=0 fc=0

8.8. ASYMPTOTICALLY REGULAR MAPPINGS 221

and we have

2 ( l + ^ ) \\yi+i ~ xi ->n+l +111 < 2'

+

= 2 n + 1

Vi+l - Xi+l || - | | î / i+n+l - Xi+n+l |

tt+n+i - n | | + ((n + l)/2) ||yi - i i | |

2/i - * t | | - | | î / i+n+l - 2Γχ+η+ΐ||

+ 2 n + 1 | | 3 / i + i - X i + i | |

+[((n + l)/2) - 2"+1] ||tf< - n u + | |y i + n + 1 - n | |

Since 2 ( l + | ) - 2 n + 1 < 0,

[2(l + )-2"+1]||2/i+1-n+1|| +

> [2(1 + f)-2n+1] H* -xi|| +

2 " + 1 - ( ( n + l ) / 2 ) j | | y i - n |

2 " + 1 - ( ( n + l ) / 2 ) ] | | y i - n |

l + ( n + l ) / 2 l | | t t - n | |

and we have

l + ( n + l ) / 2 J | | t t - X i | | < 2 n + 1 [ | | t t + i - a ; i + i | | - | | 2 / i + n + i - n + n + i |

+ ||yi+n+i - Zill ■

This completes the induction. We now prove the main result of this section.

Theo rem 8.22 Let K be a bounded convex subset of a Banach space and sup-pose T : K —> K is nonexpansive. Then the mapping f : K —> K defined by setting f(x) = (x + T(x))/2 is asymptotically regular.

Proof. Select x0 G K, let t/o = /(xo), and having defined xn take yn = / (x„) and set xn+i = ( i n + yn)/2,n = 1,2, ·■ · . It is easy to see that the resulting sequences {xn}, {yn} satisfy the assumptions of the lemma. Thus for each i, n €

(1 + | ) lb, -nil < 2" W ~Xi\\- WVi+n ~Xi+n\

+ WVi+n - I t | | ■

Since {||ni — Vn\\} = {\\xn — f(xn)\\} is monotone nonincreasing, there exists a number r > 0 such that lim ||xn+i - /(xn+i) | | = r- Now let i —» 00 in the

n—»oo above inequality. Then

(1 + 77V <diam(K).

Clearly this implies r = 0.


Two remarks are in order. First, a slight modification of the preceding argument yields the fact that each of the mappings fa = (1 - a) I + aT are asymptotically regular for each a 6 (0,1). Moreover, it can be shown that the convergence in Theorem 8.22 is uniform in the sense that the rate of convergence does not depend on the initial point xo, nor indeed on the mapping T. Details may be found in [67].

8.9 Set-valued mappings

We conclude this chapter with some observations about set-valued nonexpansive mappings. For this we need some preliminary facts. For a bounded sequence {xn} in a Banach space X and K a bounded subset of X we associate the number

r = rK({xn}) = inf{limsup | | i„ - y\\ : y G K} n—»oo

and the set

A = AK{{xn}) = {y € K : limsup||x„ - y|j =rK{{xn})}. n—»oo

r and A are called, respectively, the asymptotic radius of {xn} and asymptotic center of {xn} relative to K.

If X is reflexive and K is closed and convex, then Ακ{{χη}) is always a nonempty closed convex subset of K for any bounded sequence {xn} in X. To see this observe that for each ε > 0 the set

Ce = {y e K : limsup \\xn - y\\ < r + ε} n—>oo

is nonempty by definition of r, and straightforward arguments show that each of the sets Ce is closed and convex. Hence

Ακ({Χη})=Ρ\Οε,

and the latter set is nonempty by weak compactness of K.

Definition 8.9 A bounded sequence {xn} is said to be regular with respect to a bounded subset K of X ifrx({xn}) = rj({{xnk }) for every subsequence {xnk } of {xn}, and {xn} is said to be asymptotically uniform if Ακ({χη}) — Ακ({χηί}) for every subsequence (i,,,.} of {xn}.

Lemma 8.6 Let K be a bounded subset of X and {xn} a bounded sequence in X. Then {xn} has a subsequence which is regular with respect to K.

8.9. SET-VALUED MAPPINGS 223

Proof. To avoid double subscripts we use the notation {vn} -< {#„} to indicate that {vn} is a subsequence of {x„}. We begin by setting

r0 = inf{rfc({u„}) : {vn} -< {xn}}.

Obviously, there exists {v„} ~< {x„} such that

rK({vln})<r0 + l.

Set

r , = inf{rfc({î;n}) : {vn} -< {i£}}

and proceed inductively. Having defined {v*n} X {i£ - 1} , set

r i = inf{r j b ({« n }) : {«„}-<{<}}

and choose {t/J,+1} ~< {v'n} so that

rK(K+1})<n+ l

i + l

Note that the sequence i n } is nondecreasing and bounded above, so r = lim ri i—>oo

exists. Since rt < r/f({ujl+1}) < 7\ + - — - it follows that

i + 1

lim ΓΚ({ν*η}) = r. 1—»OO

Now consider the diagonal sequence {vn} — {v\,v%,-· ,v™,···} and let p — ΐ'κ{{νη})- Since {vn} -< {vl

n} for each i, p > rt. On the other hand, since K > n M { < + 1 } ,

P<rK({vil+1})<ri + j l - .

i + l

Therefore, u < p < ri + \/{i + 1) for each i, and letting i —► oo, we conclude that p = r. Also, if {un} -< {ϋη} then {un} -< {v^} and {u„} -< { < + 1 } . This implies r/<-({un}) = r, completing the proof. ■

It is worth noting that if in addition X is uniformly convex and K is nonempty closed and convex, then Ακ({χη}) consists of exactly one point. Thus every regular sequence in X is asymptotically uniform. However, much more can be said.

Lemma 8.7 In addition to the assumptions of the previous lemma, assume K is separable. Then any bounded sequence {xn} in X has an asymptotically uniform subsequence.


Proof. In view of the preceding lemma we may assume at the outset that {xn} is regular. Let S = {2/1,2/2, · · } D e a countable dense subset of K. There exists a subsequence {i„ } of { i n } such that lim 2/1 — xn^ exists. Sim-

n—»oo

ilarly there exists a subsequence {i„ } of {i„ } such that lim 2/2 — xn n—»oo

exists. Proceed in this way to obtain a subsequence {x„ } of {xn } such

that lim yk+i - xlfc+1' exists. Then the diagonal sequence {vn} = {i„ } is 71—»OO

a subsequence of {xn} for which lim \\yk+i ~ vn\\ exists for each k = 1,2, · · · , n—»00

and since S is dense in K, it follows that lim \\y — i>„|| exists for each y € K. n—OO

Now let {un} be any subsequence of {vn}. Since {xn} is regular, rK({un}) — rK({xn}) and since {un} is a subsequence

of {xn} it must be the case that

AK({xn}) Ç AK({un}).

On the other hand, for each y e Ax({un})

lim | | y - u „ | | =rK({un}) = rK{{xn}). n—*oo

This proves that Ακ{{ηη}) Ç Αχ({χη}). ■

Our principal result in this section is the following.

Theorem 8.23 Let X be a uniformly convex Banach space, let K be a bounded closed convex subset of X, and let T be a nonexpansive mapping of K into the nonempty compact subsets of K endowed with the Hausdorfj metric H. Then there exists x € K such that x € T(x).

Proof, lite (0,1) and if z € K is fixed, the mapping Tt defined by

Tt{x) = (l-t)z-rtT(x)

is a set-valued contraction mapping defined on K (with closed values in K), so by Theorem 3.20 of Chapter 3, for each such t there exists xt € K such that

xt € T(xt).

Therefore, there exists x't 6 T(xt) such that

xt = (l-t)z + tx't.

Thus ||xt - x't\\ = (1 - i) \\z - x't\\ —» 0 as t —> 1. It follows that there exists a sequence {xn} in K such that

lim dist(xn,T(xn)) = 0.

8.10. FIXED POINT THEORY IN BANACH LATTICES 225

Moreover, by passing to a subsequence, we may assume {xn} is regular. Since X is uniformly convex, Aj<({xn}) is a singleton, say {v}. Let r = τκ({χη}). For each n select yn € T(xn) so that lim ||xn - yn|| = 0, and for each n select

n—>oo

zn e T{v) so that \\yn - zn\\ = àxst{yn,T{v)). Then

\\yn - zn\\ < H(T(xn),T(v)) < \\xn - v\\.

Since T(v) is compact, {zn} has a convergent subsequence, say {znk}, with lim znk = w 6 T(v). On the other hand,

k—>oo

l k nt

- H I < l k nk

- y n j | + | | î / „t

- Z „ J | + \\znk

- ^ | |

and lbn f c -Zn„\\ < \\Xnk ~ v\\ .

From this we have

limsupHw - x„k\\ < lim sup \\v - xnk\\ = r. fc—>oo k—»oo

Since {xn} is regular this proves v = w G Τ(υ). ■

In the above theorem we only used Lemma 8.6. Lemma 8.7 can be used in conjunction with a set-valued extension of Schauder's Theorem to extend The-orem 8.23 to spaces which have the property that Ακ{{χη}) is always compact for bounded sequences { i n } , Details may be found in [67].

8.10 Fixed point theory in Banach lattices

In this section we illustrate briefly how the lattice structure of a Banach lattice can be exploited to prove fixed point theorems for nonexpansive mappings de-fined on closed convex subsets of such spaces. This section has no bearing on material covered in Chapter 9 and may be omitted. However, the techniques are interesting in that they require intertwining the geometric and order theoretic properties of such spaces in order to attain the goal. Since such classical spaces as co,c, dp, and Lp (1 < p < oo) are all Banach lattices, it is not surprising that such an approach would yield positive results, if only to show how similar results can be achieved by diverse methods.

Many of the results discussed in this section were first recorded in the 1997 University of Newcastle (Australia) thesis of T. Dalby [39].

8.10.1 Weak orthogonality

The concept of weak orthogonality in Banach lattices dates back to Borwein and Sims [16] who, in turn, derived it from the known fact mentioned earlier that weakly compact convex subsets of CQ always have the fixed point property for nonexpansive mappings.


Definition 8.10 A Banach lattice X (dual Banach lattice X*) is said to be weakly orthogonal (weak!' orthogonal) if whenever {xn} converges weakly (weak*) to 0, it follows that

lim n—»oo

I n Λ X 0

for all x € X.

Note that in the weak* case it may be necessary to view {xn} as a net. In this discussion we confine our remarks only to the weakly orthogonal case.

Borwein and Sims [16] used a slightly weaker definition of weak orthogonality, namely, that whenever {xn} converges weakly to 0, then

liminf liminf |xn | Λ |x m | = 0 . n—*oo m—*oo

By embedding the problem in the quotient space X = i°°(X)/co(X) they showed that a weakly orthogonal Banach lattice with Riesz angle ct(X) < 2 has the weak fixed point property. The Riesz angle of X is the number

a ( * ) = s u p { | | x | v | y | | | : i , y € f l ( 0 ; l ) } .

It was later observed by Sims [51] that the requirement a(X) < 2 is superfluous if the stronger definition of weak orthogonality is used, and he also obtained stability results in the sense that Banach spaces sufficiently near X in the sense of Banach-Mazur distance (Chapter 9, Definition 9.2) also have the weak fixed point property. Similar results were obtained by Khamsi and Turpin in [94] by different methods. It is these methods we employ below.

The two main ideas in [94] are the definition of a mapping that is not linear but mimics to some extent the principal band projections in Banach lattices, and the use of an auxiliary nonexpansive mapping F\ (discovered by K. Goebel; see [67]) which is created from T.

Suppose X is a Banach lattice, and for u,v € X, let

Su(x) = x+ Λ |u| — x~ Λ |u|.

We pause to look at Su in CQ. If u = {un},x = {xn} € CQ then Su(x) = c„, where

0 if ii„ = 0; min{|u„| ,x„} if x„ > 0; max{— |ωη| , χ η } if xn < 0.

Now some general properties of 5U: Since (x+ Λ |u|) Λ (χ~ Λ \u\) = 0,

Su(x)+ = x + Λ |u| and ,i>u(x)- = x~ Λ |u|.

8.10. FIXED POINT THEORY IN BAN ACH LATTICES

Thus

\Su(x)\ - (x+Λ H ) + ( x -Λ H ) = (x + Λ jttj) V (x~ Λ |u|) = |x| Λ |rt|,

and in particular this implies

| | 5 u (x) | |< inf{ | |x | | , |H |} .

Now we turn to the mapping I - Su. If x € X then

x - Su(x) = (x+ - x+ A |u|) — (x~ - x~ A \u\),

where

(x + - x + Λ |u|) Λ (x~ - x " Λ \u\) = 0.

Therefore,

( x - 5 u ( x ) ) + = x + - x + A | u |

and

(x - Su(x))_ — x~ - x~ A \u\.

This leads to the following calculation:

| x - 5 u ( x ) | = x++ x~ - (x+A \u\ + x~ A \u\) = x| — |x| Λ \u\ = x\ + \x\ V |uj - |x| - \u\ = x| V |u| — |u| < x| V |u| - |u |A |x |

= 11 - MI < lx — u\.

By monotonicity of the norm

i | x - 5 „ ( i ) | | < | | i - t t | | .

Also,

\Su(x) - Su(y)\ = |x+ Λ \u\ - χ - Λ \u\ - y+ A \u\ + y~ A

< |x+ - y + | Λ \u\ + \y~ — x~\ A \u\

< 2 ( | x - y | A | « | ) .

Hence

| | 5 „ ( i ) - 5 „ ( t f ) | | < 2 \x-y\A\u\


Now

so

Su(x) + Sv(x) = 5 | „ | Λ | „ | ( ΐ ) + S | u | v | v | ( x ) ,

\\Su(x) + 5„(χ)|| < ||SMAM(*)|| + ||5|u|vN(x)||.

Using (8.6),

| | 5 U ( I ) + 5 „ ( I ) | | < i n f { | | i |

< ΙΙχΙΙ + inf ·

= \\x + « Λ v

u| Λ |υ|

lui Λ Ιι»

+ inf I ||x

lui V Ivl

\u\ V \v\

(8.9)

We are now ready to prove a key lemma.

Lemma 8.8 Let X be a weakly orthogonal Banach lattice and let {u„}, {vn} be sequences in X which converge weakly to some c € X. Suppose

lim n—>oo

Un - C Λ \Vn - C\ 0.

Then for every sequence {wn} in X and for every x S X,

21imsup||w„ - c\\ < lim sup ||u>n — x|| + limsup||u;n - un | | n—>oo n—>oo n—»oo

+ limsup||u)n - vn | | . n—»oo

Proof. Without loss of generality assume c = 0. Since

2u)„ =wn- SUn{wn) + Su„{wn) + wn- SVti{wn) + SVn(wn),

by the triangle inequality and (8.7)

2 | K | | < \\SUn{^n) + SVtl(wn)\\ + \\wn-SUn(wn)\\ + \\wn-SVn(wn)\\ < \\Sun(Vn) + S«Ju>n) | | + I K ~ «n | | + I K - 1/„|| ·

Using (8.8) and weak orthogonality,

lim sup \\SUn(wn) - SUll(wn - x)\\ < 2 lim sup |x| A |u„| —0. n—»oo n—»oo '

Similarly,

lim sup ||S„„(tt>„) - SVn(wn - x)|| = 0.


Therefore, using (8.9),

lim sup \\SU„ (wn) + SVu (wn) < lim sup \\SUn(wn) - SVn(wn - x)\\ n—*oo

+ lim sup \\SUn (wn -x) + Sv„ {wn - x) n—oo

+ lim sup ||5„„ (wn - x) - SVn (wn)\\ n—»oo

= \imsup\\SUn(wn-x) + SVn(wn - x)\\ n—*oo

< limsup \\wn - x\\ + limsup |un | Λ \vn\ n—Όθ n—>oo I

= limsup \\wn - x\\,

which completes the proof. ■

We now prove a technical lemma which invokes the Goebel-Karlovitz Lemma.

Lemma 8.9 Let X be a weakly orthogonal Banach lattice with C a nonempty weakly compact convex subset of X and suppose T : C —* C nonexpansive. Assume C is a minimal invariant set for T with diam(C) = 1, and assume that {wn} is an approximate fixed point sequence for T which converges weakly to 0. Then there exists a subsequence {un} of {wn} for which

lim k—ΌΟ

= 0 and lim \\uk - Ufc+i|| = 1. k—»oo

\Uk\ Λ |Ufc+1|

Proof. For any x 6 C, by weak orthogonality and minimal invariance,

lim \wn\ A\x\\ = 0

and

In particular, for fixed m €

lim ||x - wn\\ = 1.

lim n—oo

\wn\ Λ \w„

and

lim \\w„ n—*oo

= 0

1.

(8.10)

(8.11)

(8.12)

(8.13)

Let {£*} Ç 1 + satisfy lim ek = 0. By (8.12) and (8.13) there exists m e k—oo

such that for all n > n-i,

\wn\ A\wi\ < ei and \\wi - wn\\ - 1 < εχ


Also there exists n2 > n\ such that for all n > n2,

\wn\ Λ |u>„, | < e2 and \\wni - wn\\ - 1 < e2.

Having defined η* choose η ^ + 1 > nyt so that for all n > rik+\

\wn\ Λ |w n J < efc+i and ||iw„fc - w„|| - 1 < ek+\-

Since

m "fc + 11 Λ \wnk | < £fc+i and < Cfc+l,

the sequence {tunfc} clearly has the desired properties.

We now digress to make some general observations. Let C be a bounded closed and convex subset of a Banach space X, and let T : C —> C be non-expansive. Fix x e C and λ G (0,1), and define the mapping T\ : C —> C by

Tx{y) = (1 - \)x + λΤ(2/) y 6 C.

As we have seen, T\ is a contraction mapping and hence there exists a unique point F\{x) for which T\{F\{x)) = F\{x). This, in turn, induces a mapping F\ : C —* C, which has a number of interesting properties. In particular,

(a) F\ is nonexpansive.

(b) T(Fx(x)) - x = A - X ( F A ( I ) - x).

(c)\\Fx(x)-x\\<X(l-X)-1\\T(x)-x\\.

Among other things, an approximate fixed point sequence for T is also an approximate fixed point sequence for F\ and the fixed point set of T coincides with the fixed point set of F\. However,

(d) \\Fx(x) - T(Fx(x))\\ = (1 - λ μ - 1 ||x - Fx(x)\\.

So it is also the case that if {xn} is an approximate fixed point sequence for F\, then {F\(xn)} is an approximate fixed point sequence for T.

Finally, if C is a minimal invariant weakly compact convex subset for T then, by the Goebel-Karlovitz Lemma, for all x,y E C,

( e ) ü m | | F A ( x ) - i / | | = d i a m ( C ) .

This brings us to the main result of this section.


Theorem 8.24 Let X be a weakly orthogonal Banach lattice. Then X has the weak fixed point property.

Proof. Assume the theorem is false. Then there exists a nonempty weakly compact convex subset C of X and a nonexpansive mapping T : C —» C which has no fixed point. We may further assume that C is a minimal such set with 0 E C and diam(C) = 1.

Let {an} be an approximate fixed point sequence for T and assume weak lim an = 0. Fix λ € (0,1) and set un = Fx{an). Then if x* 6 X*,

n—*oo

|x*(un)| = \x*(Fx(an))\ = |(1 - λ)χ*(α„) + λχ* (T(Fx(an)))\ < |χ*(αη)| + λ |x* (T(Fx(an))) - χ*(αη)| < |χ*(αη)| + λ | | χ 1 | | Γ ( ί \ ( « „ ) ) - α η | | = |χ-(αη) | + | | χ · | | | | ^ λ ( α η ) - Ω η | | .

Also, by (c) and (d) above, {un} is an approximate fixed point sequence for T. Since the last term tends to 0 as n —> oo we conclude that weak- lim un = 0.

n—»oo

By Lemma 8.9 {un} has a subsequence, which we again label {un}, for which

lim ||un - w„+i| | = 1 and lim |un | Λ |un+i | = 0.

Now set wn = - (u„ + u„+i) . Then

lim \\wn - un\\ = lim - ||u„ - u n + 1 | | = - . n—>οο n—*oo / £

Similarly, lim \\wn — un+i | | = 7:· Also,

\\Fx(wn) - un)\ < \\Fx(wn) - Fx(un)\\ + \\Fx(un) - «n | | < \\Wn - Un\\ + $l - $~l \\T(un) - un\\

from which lim sup ||^λ(ιοη) — u„\\ < - . Similarly, n—>oo ^

lim sup ||Fx(tu„) - u„+i|| < - . n—*oo ^

Since 2 (|ttn| Λ |un+i|) = \un + u„+i| - |u„ - u„+i| (lattice identity) and

lim ( |u„ |A |u n + i | ) = 0 n—»oo

we conclude

lim n—»oo

\un + un+i || - | |un - u n + 1 1 | < lim |u„ + un+x \ - \un - un + 11 = 0. I n—*oo | M


Therefore lim ||ωη + ωη + 1 | | = lim ||u„ — u n + 1 | | = 1, from which

lim | K || = - . n—>oo ^

By Lemma 8.8

21imsup||FA(u;„)|| < limsup||FA(w„) - FA(0) | | + limsup||FA(u> n) un n—»oo n—»oo n—»oo

+ Um sup \\F\(wn) - u n + 1 | | n—»oo

< limsup||u>n|| + - + -n—»oo ~ ^

3

This, in turn, implies that

3 limsup||FA(w„)|| < - .

n—>oo 4

Now, let λη € (0,1) with lim λη = 1. Since the above inequality holds for n—»oo

all λ € (0,1), for each n there exists zn € C such that

\\Fxn(Zn)\\<J + - .

4 n On the other hand, by (d),

| | F A B ( Z „ ) - T(FxJzn))\\ = (1- Xn)X~l \\zn - Fx(zn)\\ - 0

as n —♦ oo. Therefore, {F\u (zn)} is an approximate fixed point sequence for T so by the Goebel-Karlovitz Lemma lim | | ί \ „ (ζ η ) | | = 1. This obviously contradicts

n—»oo

limsup||FA..(2™)||<|. ■ n—KX) 4

Having proved the above theorem, where does it apply? It is known ([39]) that the class of Banach lattices whose lattice operations (Λ and V) are weakly sequentially continuous and which do not contain subspaces isomorphic to CQ are weakly orthogonal. This fact follows rather directly from results which can be found, for example, in [122]. The most familiar spaces with this property are the £p spaces, 1 < p < oo.

8.10.2 Uniformly monotone Banach lattices

The first recognition that uniform monotonicity plays a role in fixed point the-ory seems to occur in [54], where it is shown that if a Banach lattice X has a uniformly monotone norm and if l\ is not finitely representable in X (see Chap-ter 9) then X has the weak fixed point property. At the same time it is known that L\ has uniformly monotone norm, while Alspach's example [2] shows that L\ does not have the weak fixed point property. Thus the following condition alone is not sufficient for the weak fixed point property.


Definition 8.11 The norm of a Banach lattice X is said to be uniformly mono-tone if there exists a strictly increasing continuous modulus δ : [0,1] —» R with 6(0) = 0 such thatifx,y>0 satisfy 1 = \\y\\ > \\x\\, then ||x + y\\ > l+<5(||x||).

In this section we show that with an additional assumption there is a connec-tion between uniform monotonicity of a Banach lattice and normal structure; hence the weak fixed point property. In proving this result we need to recall (Proposition 8.1 of Chapter 8) that a bounded convex subset K of a Banach space has normal structure if and only if it does not contain a diametral se-quence, that is, a bounded sequence {x„} for which

lim dist(a;n +i )cönv{ii ,X2,· · ' ,^η}) = diam{xi,X2, · · · }■

We also need another direct consequence of weak orthogonality. If {xn} converges weakly to 0 in a weakly orthogonal Banach lattice then by definition for each x Ç X,

lim n—>oo

= 0. | χ η | Λ | χ |

As noted in the previous argument, in conjunction with the lattice identity

2 (|x| Λ |y|) = | |x + y\ - \x - y\

this gives

lim n—»oo

IX ~t~ X j i | |X Χγ\ = 0.

This is precisely the observation that leads to the definition of the weak orthog-onality condition labeled WORTH, which we take up in the next section.

Theo rem 8.25 Suppose X is a Banach lattice with uniformly monotone norm and suppose X is weakly orthogonal. Then weakly compact convex subsets of X have normal structure.

Proof. Let K be a weakly compact convex subset of X with diam(Ä') > 0 and assume K does not have normal structure. Then by Proposition 8.1 K contains a diametral sequence {xn}i and by weak compactness we may assume that {xn} converges weakly to p Ç. H — œîïv{x1,X2, · · ■}. By translating H we may assume p = 0 and by normalization we may assume diam(i7) = 1. Thus \\xn\\ S 1 f°r e a c n n- Since

lim dist(x„+i,cönv{xi,X2,· · · ,χη}) = 1 n—*oo

and 0 € cönvjxx, X2, · · · } , we also have

lim \\x„ n—oo

1.


Now let x € A" with x φ 0 and assume liminf ||xn — x|| < 1. Uniform mono-n—>oo

tonicity of the norm of X implies that for all n,

«(!) + !< 'Ml + 11*11 Thus (using weak orthogonality in the second step and using \a\ — \b\ < \a - b\ in the third step)

l|i| |(«(l) + l) < liminf 11*11

= lim inf n—>oo

< lim inf n—*oo

< lim inf n—*oo

l - I H I

Xn + \X\

\Xn\ - X

Ikll

l*»ll + 1.

r X n Xn + lim inf ||x„ - x||

Since {x„ — x} converges weakly to x, ||x|| < lim inf ||x„ — x|| < 1. If ||x|| = 1 n—>oo

then <5(1) < 0 which is a contradiction. So ||x|| < 1 and

||*|| (5(1)+ 1 ) < 2 - | | * | | .

Solving for ||x|| we obtain

\X < (5(1)+ 2

< 1 .

Taking x = xn for n sufficiently large, we see that this leads to a contradiction.

8.10.3 WORTH In this section we study a weak orthogonality condition labeled WORTH, intro-duced independently by H. Rosenthal in 1983 [?] and B. Sims in 1988 [?]. As we just observed, this condition arises as an outgrowth of the concept of weak orthogonality in a Banach lattice.

Definition 8.12 A Banach space X is said to have WORTH if whenever {xn} is a sequence in X which converges weakly to 0, then for any x G X

limsup ||x„ — x|| = limsup ||xn + x| | .

/ / weak convergence is replaced with weak* convergence in a dual space X* then X* is said to have WORTH*.

8.10. FIXED POINT THEORY IN BAN ACH LATTICES 235

We begin with the following.

Propos i t ion 8.6 If X is a separable Banach space and if X* has WORTH*, then X has WORTH.

Proof. Assume X* has WORTH* while X fails WORTH. Then there exists a sequence {xn} in X and an element u & X, for which weak- lim xn = 0 while

n—»oo

limsup||x„ + u\\ > limsup||xn — u\\. n—»oo n—»oo

Let {xnk} be a subsequence of {xn} for which

Then

limsup| |xn + u\\ = lim | |xn f c+u| | n—»oo fc—»oo

l imsup | | x n + u|| = lim | |xn ] t+zi | | n—>oo fc—»oo

> limsup||x„ — u\\ n—»oo

> limsup||x„A. — u\\, k—OO

so by again passing to a subsequence we may assume

lim ||x„ + u\\ > Um ||xn - u | | . n—»oo n—»oo

For each n e N choose x* € 5 χ · such that i * ( i „ + u) = ||x„ + u | | . Since (by separability of X) X* is metrizable in the weak* topology, Βχ- is weak* sequentially compact so by yet again passing to a subsequence we may suppose that

weak* - lim x* = x* 6 Βχ>. n—*oo

Therefore, recalling that weak lim xn = 0, and then using WORTH*,

lim ||xn + u|| = lim x*,(xn + u) n—»oo n—*oo

= lim i ' ( i n ) + lim Xniu) ~ 2 u m x*(xn) n—*oo n—*oo n—»oo

= lim (x* - x*) (x„) + Um x* (u - x„) n—»oo n—»oo

= Um ((x* - x * ) - x * ) ( x „ - u ) n—*oo

< Umsup\\(x*l-x*)-x*\\\\xn-u\\ n—»oo

= l imsup| | (x*l-x*) + x * | | | | x n - u | | n—»oo

= ümsupIKHIIzn-u l l n—»oo = limsup| |xn — u\\. n—»oo

This contradicts our original assumption.

We now use the preceding result to establish a connection between WORTH* and normal structure.


Theorem 8.26 Suppose X is a separable Banach space for which X* is uni-formly nonsquare and has WORTH". Then X has normal structure.

Proof. By assumption, €Q(X*) < 2 and X* (hence X) is super-reflexive. Moreover, by Proposition 8.6 X has WORTH. It can be shown that there exists a sequence {xn} in Βχ such that x\ = 0, weak- lim x„ — 0, lim ||x„|| = 1,

n—*oo n—*oo

diam({xi, £2 · ■ ■ }) = 1, and for each n

dist(a:n,conv{xi,··· ,x„_i}) > 1 - 1/n.

(This is a refinement of Proposition 8.1 due to B. Turett, which is left as an exercise; alternatively, see Step 1 in the proof of Proposition 3 of [159].)

Since convjxx, · · · ,x„_i} is compact there exists m n e convjxj, · · · ,x„_i} such that

dis^Xn.convfx!,··· ,x„_i}) = | | x n - m n | | .

By the Hahn-Banach Theorem, there exists x* G 5 χ · such that

x* (xn - mn) = | |xn - mn\\.

Thus for j = 1,··· ,n,

1 < inf{||x„ - x||;x G conv{xi,· ■ · ,xn}} n

= x*n{xn - m„) = x* (x„) - x ; (m n ) < x ; ( x „ ) - X*n{Xj) < 1 - X*n{Xj).

Since x\ = 0, we conclude that for each n > 2,

1 > Χη(χη) > 1 - 1/n.

Also, since Βχ· is weak* sequentially compact we may assume

weak*- lim x*—x*£ Βχ·. n—»00

Now fix ε > 0 and select jo € N so that jo > 1, 1/jo < e/2 and so that |x*(xJO)| < ε/4. Then for i > jo sufficiently large

| ( x * - x * ) ( x j o ) | < e / 4 , | χ* ο ( χ , ) |<ε /2 ,

and, since X has WORTH, ||XJ0 — Xi|| + ε > ||XJ0 + Xi\\. In particular,

\ζΧ*3ο)\-\χ·(χ»)\<\Μ-χ·){χ*,)\>

so |x* (xj0)| < ε/2. Hence (for ε < 1)

> - e / 2 + l - l / J o > 1 - e .

8.10. FIXED POINT THEORY IN BAN ACH LATTICES 237

Also,

. / xi xjo \ 4 \\\*i-*Jo\\)

> X± [Xi Xjo)

= x*{xi) ~ x'Àxjo) > 1 - 1/i - ε/2 > 1 - ε .

Now for μ e (0,1) and x Ç. Sx, let

S(BX.,χ,μ) = {x* € Bx. : x(x*) > 1 - μ}

denote the χ,μ-slice of Βχ-. Then in view of the above inequalities,

Moreover, since

and

we have (for ε < 1)

ll*î-(-**,)H

-x;a,x;çs\Bx. Xi X Jo

\Xi - X ,e

Jo I

Xi + Xjo\\ < Ι|*ι-*<οΙΙ + ε < 1 + ε

Κ+^^+*)(^)

> (l + eîx'i+xîxi + Xjo) = (1 + ε ) " 1 (xf (ii) + x*(xjo) + x*jo(xi) + x%{xjo)) > (1 + ε ) - 1 (1 - 1/i - ε/2 - ε/2 + 1 - l / j 0 ) > (1 + ε ) - 1 ( 2 - 2 ε ) = 2 ( 1 - ε ) / ( 1 + ε ) .

If μ > ε then diam S ( Βχ 0 ' 'Jo

\Xi - X -, μ 1 > diam 5 I > diam 5 ( Βχ·,

Xi — X Jo ,e

With the above, this implies that for all μ € (0,2),

Now let ε0(Χ*) < ε' < 2. Then tf(e') > 0. On the other hand, for any μ > 0,

there exist u*,u* e 5 \BX., ,.Xi ~ Xio„ ,μ ) such that ||u* - υ * | | > ε'. At the V iFi-Zjoll /

same time,

u* + v*

*jo\

(u* +V*\ ( Xj-X

Xj Xjo \ , * / Xj Xjo \

ll*i-*joll/ \\\xi-xJo\\J > - [ Ι - μ + 1-μ]


This proves that δ(ε') < μ, which is obviously a contradiction if μ < δ(ε'). Therefore, it must be the case that εο(Χ') = 2, contradicting the original assumption that X* is uniformly nonsquare. ■

Exercises

Exercise 8.1 Show that if G is convex and T : dG —► G, then T satisfies condition (8.1).

Exercise 8.2 Prove Theorem 8.4. [Hint: Let f = (I-T), letGt = (I-tT)(G) fort e [0,1]. Let

H={te[0,l}:0eGt}.

Observe that H φ 0. Use Theorem 8.1 to show that H is open in [0,1]. Then show that H is closed in [0,1]. From this it follows that H = [0,1] and therefore

Exercise 8.3 Let K be a convex subset of a Banach space andp > 1. Show that any mapping T : K —> K which satisfies | |T(x) — T(y)| | < \\x — y\\p , x,y € K, is a constant mapping.

Exercise 8.4 Let X be a Banach space. Define the modulus of uniform con-vexity in the direction z € X (||ζ|| = 1) by

S(z ,ε) = inf 11 - \x + y : x,y £ Βχ, x — y = az, and 1*-»11>ε}

for any ε 6 [0,2). X is said to be uniformly convex in every direction ( U. C.E.D.) if δ(ζ,ε) > 0, for any z and ε > 0. Show that if X is V. C.E.D., then X has normal structure.

Exercise 8.5 Let K be a weakly compact convex subset of a Banach space which is U.C.E.D. Show that the asymptotic center Ak{{xn}) of any sequence {xn} in K is a singleton.

Exercise 8.6 Let X be a Banach space. Define the modulus of smoothness px

ofXby

p(v) = s u p | x + f]y

2 + x-r)y

2 1:11*11 < 1, llvll - } for any η > 0. X is said to be uniformly smooth (U.S.) if

l i m ^ M = 0 . »7—0 η

EXERCISES 239

Show that

r) -0 η 2

where εο(Χ*) is the characteristic of convexity of X*.

Exercise 8.7 Use the previous exercises to show that if

,. ΡχΜ 1 7j—o η 2

then X and X* have uniform normal structure property. In particular, uni-formly smooth Banach spaces have uniform normal structure property.

Exercise 8.8 Let X be a Banach space. X is said to be nearly uniformly convex if for every ε > 0, there exists δ(ε) > 0 such that if \\xn\\ < 1 and

inf | | | i n - i m | | : n Φ m\ > ε

then

cönv{a :„}f |ß (0 ; l - Ä(e)) ψ 0 .

Show that if X is nearly uniformly convex, then X has normal structure.

Exercise 8.9 Let X be a Banach space. X is said to satisfy Opial's condition if for every sequence {xn} which converges weakly to u>,

lim inf | | i„ - ω|| < lim inf ||xn - x|| 7t—»oo n—>oo

holds for any x φ ω. Show that if X satisfies Opial's condition, then X has weak normal structure.

Exercise 8.10 Let X be a Banach space. For any nonempty bounded subset A of X, define the Kuratowski's measure of noncompactness a of A by

a(A) = inf Ir > 0, A C [ J Ai with diam(i4i) < r | . l < t < n

The modulus of noncompact convexity of X is defined by

Δχ(ε) = inf < 1 — inf ||χ|| : A is convex subset of the unit ball with a(A) > ε >

for any ε € (0,2). The characteristic of noncompact convexity ε\ of X is given by

ει(Χ) = 3ηρ{ε:Δχ(ε)=0}.

Show that if ε\{Χ) < 1, then X is reflexive and has normal structure.


Exercise 8.11 Let X be a Banach space with a Schauder basis {xn}. Define ßp(X), for p > 1, to be the infimum of the set of numbers λ such that

( N l p + IHIp)1/P<A||x + y||

for every x,y G X which satisfy supp(x) + 1 < supp(y), that is, for every i € supp(x) and any j 6 supp(y), we have i + 1 < j . Show that X has weak normal structure provided βρ(Χ) < 21/p.

Exercise 8.12 (James space) Let CQ denote the space of all real sequences that converge to 0. The James space J (resp. J\) consists of all sequences (a„) 6 CQ for which | | (a„) | | j < oo (resp. \\(an)\\Jl < oo), where

(an)\\j =supl J2 K - a p i + . ) 2 + K n + i - Qp.)2 \ -l < i < n

resp.

(«n)| | j , = sup < Σ (aPi - a P i + 1 ) 2 + o ^ + 1 \ , l < i < n

the supremum being taken over all finite increasing sequences of positive numbers Pl,P2,--,Pn+l-

1. Show that /32(Ji) = 1. Deduce from it that J\ has weak normal structure.

2. Show that for any sequence {xn} in J which converges weakly to 0, there exists a subsequence {xn·} of {xn} such that

sup^ limsup||xn< — χς|| \ > liminf ||x„'||. g *■ η'—oo -» η'—οο 9

Deduce that J has weak normal structure.

Exercise 8.13 Show that any dual Banach space, with a shrinking uncondi-tional basis with the unconditional constant equal to 1, has the weak"-fixed point property. Hint: Use the conclusion of Exercise 6.17.

Exercise 8.14 Let X be a Banach space. Let C be a nonempty closed convex subset of X and T : C —> C be nonexpansive.

1. Show that T has a bounded orbit if and only if any orbit ofT is bounded.

EXERCISES 241

2. Assume that T has bounded orbits. Show that there exists a bounded closed convex subset K of C which is T-invariant, that is, T(K) C K.

3. Show that if X has the fixed point property and T is a nonexpansive map-ping defined on a closed convex set, then T has a fixed point if and only if it has bounded orbits.

Exercise 8.15 Show that the mapping F\ defined in Section 8.10.1 satisfies properties (a)-(d).

Chapter 9

Banach Space Ultrapowers

9.1 Finite representability

The central concept in this chapter derives from the fundamental notion of finite representability introduced by R. C. James [78] in 1972.

Definition 9.1 A Banach space Y is said to be finitely representable in a Ba-nach space X if for every ξ > 0 and every finite-dimensional subspace YQ of Y there is a finite-dimensional subspace XQ of X (with dimXo = dimYo) and an isomorphism (one-to-one linear mapping) T : Yo —* XQ for which

(i-OIMI<IIT(y)ll<(i + Oll!/ll

for all y 6 Vo-

lt is possible to formulate the above definition in a slightly different way.

Recall that, in general, if X and Y are Banach (or normed) spaces and if T :

Y —> X is a linear mapping, then T is continuous if and only if ||T|| < oo, where

| | r | | = s u p { | | r ( » ) | | : y e y a n d ||y|| = 1}.

Now suppose Y is finitely representable in X and let η > 0. Choose ξ > 0 so

that — j | = 1 + 7?. Then with T : Y0 -> X0 as in Definition 9.1, ||Γ|| < 1 + ξ

and HT"11| < (1 - 0 ~ \ so ||Γ|| ||3—x || 0 and every finite-dimensional subspace YQ of Y there is a finite-dimensional subspace Xo of X and an isomorphism T of Y0 onto Xo with

imiHT-1!! < ! + ,,.

243



244 CHAPTER 9. BANACH SPACE ULTRAPOWERS

This version of the definition is sometimes more useful, and even it can be refined. Having such an isomorphism T let F = | | Τ - 1 | | Γ . Then F~l = T-VI lT" 1 ! ! and thus

| | F | | < 1 + 7, and H ^ h l .

This rather minor observation will be useful later in this section.

These facts can be expressed in terms of the Banach-Mazur distance.

Definition 9.2 Let X and Y be Banach spaces. The Banach-Mazur distance between X and Y is denoted by d(X, Y) and is defined as

d(X,Y) = inf{||T|| | | Γ _ 1 | | :T is an isomorphic from X onto Y}.

When X and Y are not isomorphic we simply set d(X, Y) — oo.

Therefore, Y is finitely representable in X if for any 0 < e < 1 and for any finite-dimensional subspace YQ of Y, there exists a subspace XQ of X such that dimXo = dim Vo and d(Xo, ô) < 1 + ε.

It is immediate from the definition that finite representability is transitive: If Z is finitely representable in Y and if Y is finite representable in X, then Z is finitely representable in X. Also, if Y is finitely representable in X then Y possesses any Banach space properties which are 'locally' determined, that is, determined by uniform behavior on finite-dimensional subspaces of X.

Definition 9.3 Let X be a Banach space and let P be a Banach space property. Then X is said to be super-P if every Banach space which is finitely representable in X is P.

For example, we have the following.

Definition 9.4 A Banach space X is super-reflexive if it is the case that any Banach space Y which is finitely representable in X is necessarily reflexive.

Since a Banach space is always finitely representable in itself, if X is super-P for any P then X is P. Thus any super-reflexive space is necessarily reflexive.

Properties which are preserved under finite representability are called 'su-per' properties. Finite dimensionality is a trivial super property. Also a cele-brated theorem of Dvoretzky assures that ii is finitely representable in every infinite-dimensional Banach space. On the other hand, reflexivity is not a su-per property. In particular, there exist reflexive Banach spaces in which l\ is finitely representable; hence there exist reflexive Banach spaces which are not super-reflexive. An example of such a space is the space

*-(n<f)

9.1. FINITE REPRESENTABILITY 245

The elements of X are sequences x — (xi,X2,· ■ ■ ), where xn 6 v™ (the n-dimensional i\, i.e., the space R") and

/ oo \ !/2

H= Σ>»ιι? <0°· < n = l

It is not difficult to see that X is reflexive. In fact, X is isomorphic to X*, where

\n€N ) 2

To see that ίι is finitely representable in X we begin with an elementary fact about finite-dimensional spaces mentioned in Chapter 7.

Lemma 9.1 Suppose E and F are Banach spaces with E finite-dimensional, and suppose T : E —> F is linear. Then T is continuous.

Proof. Suppose E is m-dimensional and let {ei,· ·· , e m } be an algebraic

basis for E. The unit sphere S it™ ) of r™' is compact and the mapping Φ :

S I £j ) —> R that sends (u\, · · · , um) € S ( 1 \ ' 1 into the number Σ ujej

continuous and nonzero (since the vectors {ei, ■ · -, em} are linearly independent).

Thus there exists a number 6 > 0 such that for all such (ui, · · ■ , um) G S ί ί™ ) ,

is

XSeJ J = l

>δ.

From this it follows that for every sequence (αχ, · · · , a m ) € £j ,

Therefore,

imi = sup

= sup

T Σ α ^ kj=l

/

/ Σα3εό J'=l

(ai, · · · , am) 6 l\

: ( a 1 , - - - , a m ) e 4 m )

(m)

< βιιρ{||Τ(β,·)||/ί: j = l , · · · , m } < o o .

246 CHAPTER 9. BAN ACH SPACE ULTRAPOWERS

Now let £ > 0 and let YQ be a subspace of t\, say of dimension m, with basis {yi,··· i2/m}· Further assume ||j/j|| = 1, i = 1,■■ · , m. As in the above proof there exists 6 > 0 such that for every sequence (αχ, · · · , a m ) € q m ,

Choose μ > 0 so that

m

*ΣΜ< j = l

m

Σα^ j = l

i l l < 1 + ε,

and for i = 1, ■ · · , m, choose ^ e ii so that y\ has only finitely many nonzero coordinates, and so that ||t/i — yill < A4· Then

Yâiivi-yl) i = l

< (X>iU, v i = l

from which

Also,

hence

m

< τη

i = l

771

+ μ £ | α ί | < ( ΐ + £) i = l

X]ai2/i i=\

J2aiV'i i=l

> ( ' - ! ) Σαίνί

i = l

( ' " ! )

m

i = l

< m

Σα ί 2 / ί i = l

('+!) Σαί^ t = l

This means that the operator T defined from YQ onto span{yi, · · · ,y'm}, denned by

T(y1) = y[,--,T{ym)=y'm

(and extended by taking linear combinations), satisfies

||T||<l + f andl l r - l^ l -^ ) - 1 . Let rifc be the index of the last nonzero term in y'k, k = 1, ·■ · , m, and

let n = max{rifc : 1 < k < m}. Then span{yi,· · · ,y'm} is isometric to an m-dimensional subspace of q" ' and £j is a subspace of X. Therefore, T composed with this embedding is the desired isomorphism.

Uniform convexity provides a nontrivial example of a super-property. This is a consequence of the following result, which illustrates the 'two-dimensional' character of the modulus of convexity.

9.1. FINITE REPRESENTABILITY 247

Theorem 9.1 Let X and Y be Banach spaces with respective moduli of con-vexity δχ and δγ, and suppose Y is finitely representable in X. Then for each εΕ[0,2),δγ(ε)>δχ(ε).

Proof. Let e > 0 and let x,y € Y with ||a;||y = \\y\\Y — 1 and ||a; — y\\Y > e. Suppose η > 0 and let F be an isomorphism of span{x, y) onto some two-dimensional subspace Xo of X which satisfies | |F| | < 1 + η and | | -F - 1 | | = 1-If

x' = F(x) and y' = F(y),

then \\x'\\x < 1 + η and \\y'\\x <1+η, while

\\x'-y'\\x = \\F(x-y)\\x>\\x-y\\Y>e.

According to the definition of δχ,

2\l+v^l+vJ x - Χ\ί + η]

Using the fact that δχ is continuous1 and letting η —» 0, we see that

2 ^ + w') < 1 - δχ (e).

But since | | F _ 1 | | HI -

2 (x + ) -( Ϊ^+Λ) < 5(*' + y')

The definition of δγ now gives iy(e) > δχ(ε).

Corollary 9.1 If X is a uniformly convex Banach space and if Y is finitely representable in X, then Y is uniformly convex.

Corollary 9.2 If X is a Banach space and if Y is finitely representable in X, then ε0(Υ) < ε0{Χ).

A Banach space X is said to be uniformly nonsquare if eo(-X") < 2. Thus we conclude that this is also a super-property:

Corollary 9.3 If X is a uniformly nonsquare Banach space and if Y is finitely representable in X, then Y is uniformly nonsquare.

We conclude this section with a collection of important facts which follow from results proved elsewhere.

First, since uniformly convex spaces are reflexive it follows from Corollary 9.1 that uniformly convex spaces are super-reflexive. In fact, since it is known that uniformly nonsquare spaces are reflexive (R. C. James [81]), the following is true.

' A proof of this fact can be found in [67].


Theorem 9.2 Every uniformly nonsquare Banach space is super-reflexive.

Dvoretsky's result gives another corollary to Theorem 9.1.

Corollary 9.4 If X is any infinite-dimensional Banach space, then δχ < 6e2.

The following deep result of P. Enflo [55] reveals a surprising connection between super-reflexivity and uniform convexity.

Theorem 9.3 A Banach space is super-reflexive if and only if it can be given an equivalent uniformly convex norm.

There have been a number of more technical characterizations of super-reflexivity, many of which are due to R. C. James. One such characterization is the following, which we shall use later.

Theorem 9.4 (James) A Banach space X is not super-reflexive if and only if the following condition holds: If0<6< 1 and 0 < ε < 1, then for every positive integer n there exist subsets {zi,··· ,Zn} Ç Βχ and {gi,··- ,gn} Ç Bx. such that

gi(zj) ζ=θί}ι< j and gi(zj) =0ifi>j,

and, for every sequence of numbers {a;} and every k <n,

n

YÔ-iZj >-¥ k

i = l

9.2 Convergence of ultranets The purpose of this section is to lay the groundwork for how to construct for any Banach space X a much larger Banach space X which has the property that X both contains an isometric copy of X and yet is always finitely representable in X. The fact that X is 'much larger' than X is important; otherwise X itself would do. The usefulness of this particular type of construction will be made apparent in subsequent sections.

Recall (see Appendix) that a net {xa}açA in a set X is a mapping φ : A —♦ X, where A is a directed set and xa = ¥>(<*)> oc E A. A net is an ultranet if given any subset G Ç X either {xa} is eventually in G or {xQ} is eventually in X\G (the complement of G). Thus given any subset G of X one of the two following mutually exclusive alternatives must hold.

(a) There exists c*o such that a > cto implies xa € G.

(b) There exists ctQ such that a > c*o implies xa $ G.

One nice property of ultranets is that they automatically pass from one set to another.

9.3. THE BAN ACH SPACE ULTRAPOWER X 249

Proposition 9.1 Let X and Y be sets and let f : X —» Y. Suppose {xa} is an ultranet in X. Then {/(xa)} is an ultranet in Y.

Proof. Let H be any subset of Y and let G = {x € X : f(x) 6 H). Since {xa} is an ultranet in X, {xa} is eventually in G or in X\G. If {xa} is eventually in G, then {f(xa)} is eventually in H; otherwise {f(xa)} is eventually in Y\H.

M

Proposition 9.2 If{xa}aeA is an ultranet in a topological space X and if some subnet {xa,}ξ^Ε of {xa} converges to x G X, then limxQ = x.

Proof. Let U be any open set in X which contains x. Since {xa(} converges to x there exists ξ0 € E such that ξ > ξ0 implies xa( G U. Also, since {xa£} is a subnet of {xa} , given any a £ A there exists ξχ € E such that if ξ > ξ1, then aç > a. Since E is directed there exists ξ € E such that ξ > ζ0 and ξ > ξ1. For such ξ, we have QÇ > a and xa( € C/. This proves that {xQ} is not eventually in X\U; hence {xa} is eventually in U, proving limxQ = x. ■

a Our next fact relies on the following characterization: A subset K of a

topological space is compact if and only if every net {xa} in K has a subnet which converges to a point of K.

Proposition 9.3 Let K be a compact subset of a topological space and let {xa} be an ultranet in K. Then {xQ} converges (to a point of K).

Proof. Since K is compact {xa} has a subnet which converges to a point x 6 K. In view of the previous proposition, this implies that {xQ} itself converges to x. ■

We now bring to bear a fundamental fact discussed in the Appendix. We may take this as an axiom since, indeed, it is known to be equivalent to the Axiom of Choice.

Lemma 9.2 Every net {xa} in a set S has a subnet {xa{} which is an ultranet.

9.3 The Banach space ultrapower X

It is not our intention here to study Banach space ultrapowers in general. This theory has a rich history and elegant treatments may be found elsewhere, for example, in [73], [147]. Also, an excellent exposition which deals with fixed point theory of nonexpansive mappings is found in [1]. However, a few facts have emerged more recently.

For our immediate purpose we shall only need to have at hand one Banach space ultrapower associated with each Banach space, and the one described


below will be adequate. It is common practice to construct Banach space ul-trapowers using ultrafilters but we use ultranets instead. The end result is the same.

It suffices to begin with the smallest infinite ordinal ω. As a well-ordered set this set is linearly ordered, therefore a net. By Lemma 9.2 it has a subnet {φ, E) which is an ultranet. Thus E is a directed set and φ : E —> ω satisfies the following condition.

(i) Given any n Εω there exists ξ0 G E such that if ξ > ξ0 then φ(ξ) > φ(ξο).

We follow the usual convention and denote φ(ξ) — τΐξ, ξ e E. In particular, we use {ηξ} to denote {φ,Ε}. Throughout this section the ultranet subnet {ης} of ω will remain fixed.

Now let X be a Banach space and let

£οο(Χ) = {* = {xn) <Ξ X ■■ sup | | χ { | | <οο} . 1 < « ο ο

It is known (and easily checked) that ioo(X) is a Banach space with the norm defined by

for {xn} € 4o(X). Set

M={x

The Banach space ultrapower X of X (relative to the ultranet {nç}) is the quotient space £oo(X)/N. Thus the elements of X are equivalence classes \(xn)] of bounded sequences (xn) C X, where one agrees that two such sequences (xn) and (yn) are equivalent if and only if

lim ||χ„ξ - yni \\ = 0.

The norm | | |L in X is the usual quotient norm. Thus for x = [{xn)] 6 X,

||*|i£=inf{||(*„ + yn)| |TO:(yn)eA0.

There is another approach which leads to the same thing. Note that since {||χη||} is bounded it lies in a compact subset of R. Therefore lim ||χηί || always

exists. Thus one could just as well define for x = [{xn)] G X,

| |ί | |ξ = 1ϊπι||χη{||.

We check that the second definition coincides with the first.

Propos i t ion 9.4 Let \\χ\\ξ = inf{||(xn + y « ) ^ : (j/„) e TV}. Then

\\χ\\ξ = lim ||a:nt | | .

ll(*n)lloo = S U P ll^ll l<t<oo

= (in) € ίοο(Λ-): lim ||χηί | | = 0 } .

9.3. THE BANACH SPACE ULTRAPOWER X 251

Proof. Let x = \(xn)] € X. Then x - {(xn + yn) : (yn) e Af}. Therefore, if (yn) e N,

lim ||x„£|| = lim | | i „ t || - \\ynt \\ < lim \\xn( +ytti\\ < ||(x„ + ynJII«, ·

This proves that lim ||χηξ || < | |i |L·

For the reverse implication, let ε > 0 and let

/ = { n e N : | | * „ | | < l i m | | : r n J + e } .

Clearly {n^} is eventually in / . Define (yn) 6 ôo(^) by setting

_ j —xn if n ^ / , Vn ~ \ 0 otherwise.

Then lim y„£ = 0 and it follows that

| | ΐ | | ί < IK^n + yn)\\oo = sup ||(x„ + y„)|| = sup ||x„|| < lim | | i n || + ε. n n€I ί

Since ε > 0 is arbitrary, this completes the proof. ■

Proposi t ion 9.5 If X is a Banach space, then X is a Banach space.

Proof. There are two ways to see this. One way is to show that M is a closed subspace of ôo(^) and invoke the fact that the norm on X is the quotient norm on the quotient space £00(Χ)/Λί. (It is well known that the quotient space of a Banach space over one of its closed subspaces is itself a Banach space.) However, we proceed with a direct proof.

Let { î fc}^! be a Cauchy sequence in X. Since a Cauchy sequence converges if it has a convergent subsequence, it suffices to show that such is the case for {ifc}·

We pass to a subsequence and assume at the outset that

\xk -ifc-ιΙΙξ < -£ , k= 1,2,·

Now set

U\ — X\, U2 = X2 ~ Xl, ,Ufc = Xk - X f c - l .

k i Then îk — Σ üj and ||£tfe|) < — for all k = 1,2, · ■ · .

For each k we can think of ük as an equivalence class of the form [(u„ )].

Moreover, since Hüfcl = lim {u^) < —ç we can select (u„ ') € [(ul )] by

setting, for each n, υ„ — u„ if ,(«=) 2*

< -£·, and Vn = 0 otherwise. (Note


u(fc> - tAfc> = 0.) The sequence thus obtained satis-that one then has lim

ί

fies k>n < ■%£ for all n > 1. It now follows that for each n the sequence

Σ vn \ is a Cauchy sequence in X and hence it has a limit un e X.

„0") t _ v< f c + 1 ) < 1

2fc+i ; hence (Let wk = Σ vn, k = 1,2, · · · . Then ||wfc+i - wfc|| =

oo

53 ll^fc+i - tffell < oo.) Let û = [(un)]. We complete the proof by showing that fc=l

lim ||îfc - ù|L = 0. :—»pin ' fc—»OO

Let k < m. Then

k

i=fe+i < Σ IKj)

j=k+l

1

Letting TU —> oo gives

<-h But

Thus

i= i i=\ j=\

lim ί "« = l l * * - û | | € < 2*-

Therefore, {xjt} converges to ù.

9.4 Some properties of X

We continue with the notation of the previous section, with X a given Banach space.

Proposition 9.6 X contains a subspace isometrically isomorphic to X.

Proof. For each x 6 X let (xn) denote the sequence for which xn = x, and let x = [{xn)} G X. Then the subspace

X = {x:xeX}

9.4. SOME PROPERTIES OF X 253

is isometric to X via the mapping x —► x since ||x||, = lim ||x„£ || = lim ||x|| =

χ ■

The following is less obvious. However, it ties the observations of this section to those of the first section.

T h e o r e m 9.5 X is finitely representable in X.

Proof. Let Xo be any finite-dimensional subspace of X, say of dimension ro, and let {x1, · ■ · , x m } Ç Xo be an algebraic basis of Xo for which ||xfe||f = 1, k = 1, · · · , m. For each k = 1, · · · , m, let xfc = [(x£)] and for each n € N let X n

be the subspace of X spanned by {xj,, · · · , x™}. Now define Tn : Xo —* Xn by setting Tn (xJ) = xJ

n, j — 1, · · · , m and extending Tn linearly to all of XQ. Thus for

m

x = yjajX·* € ΛΌ i= i

set

in m m

Tn{x) = £ {Τηαά&) = ΣαάΤη(χή = £ > Χ 6 Xn. j=i i=\ j = i

What we propose to show is that given e > 0 there exists an n € N such that Tn is an isomorphism of Xo onto Xn for which

(1 - ε) \\χ\\ξ < | |Γη(χ)|| < (1 + e) | | î | | ç (x e X0).

m

To this end, let ε > 0 and let x = V ] α,χ7 € XQ be arbitrary. Since 3=1

\\Tn(i)\\ = Σ α ^η

we have

U m | | r n t ( f ) | | = l i m Σα^χ: 3 = 1

= x le-

in particular, for each x & Xo there exists an index £ä (in some directed set) such that if £ > ξΐ,

( ι - | ) | | ί | | ξ < | | Γ η ί ( χ ) | | < ( ι + | ) | | χ | | ξ .


Moreover, since T„£ is (uniformly) continuous (Lemma 9.1), each x in the unit sphere So of XQ has a neighborhood Vj with the property that y € Vf and ξ > ξχ imply

( l - e ) | | » | | € < | | r n t ( y ) | | < ( H - e ) | | j ? | | € .

Since SO is compact there exists a finite set {5i, · · ■ , zp) Ç S0 such that

~s0c(jv-Zi.

Now, if n — τΐζ where ξ is chosen so that ξ > ξ£ for all i = 1, · ■ · ,p, then for

each y € So,

(l-s)\\y\\c<\\Tn(y)\\<(l + e)\\y\\r

The preceding fact is important because it assures that X possesses all the super-properties that X possesses. The following is a special case.

Corollary 9.5 If X is uniformly convex then X is uniformly convex.

In fact, one can say more.

Theorem 9.6 For any Banach space X, δχ = δχ; hence εο(Χ) = £o{X)·

Proof, δχ (ε) > δχ(ε) by Theorem 9.1. The reverse inequality follows from the fact that the spaces X and X Ç X are isometric. ■

We also have the following.

Proposition 9.7 If X is super-reflexive, then X is also super-reflexive.

Proof. X is reflexive because X is finitely representable in X. If Y is finitely representable in X, then since finite representabihty is transitive and X is finitely representable in X it follows that Y is finitely representable in X. Since X is super-reflexive it must be the case that Y is reflexive. Hence X is also super-reflexive. ■

In fact, the above proof suggests a more general result. Indeed, if X is super-P, then any Banach space Y which is finitely representable in X is also super-P. In particular, X is also super-P.

Another important fact about super-reflexive spaces is the following result of J. Stern [155]. We will use this fact later.

Theorem 9.7 If X is super-reflexive, then (X)* = X*.

9.5. EXTENDING MAPPINGS TO X 255

What this means precisely is the following: Each functional / 6 (X)* is the form / = [(/n)] € X", and f(x) = lim/n< (x„{ ) for each x € X.

So far we have been talking about a very special type of ultrapower X be-cause this is all that is needed for the specific results we have in mind. However, in order to consider the class of all Banach spaces which are finitely repre-sentable in a given Banach space, a more general definition is needed; namely, one in which the ultrapower is built by using_arbitrary ultranets rather than just those which are subnets of ω. If (X)* = X* holds for any such Banach space ultrapower of X, then it can be shown that X is necessarily super-reflexive. So with this extended definition the converse of Theorem 9.7 is also true. In fact, it can be shown that any space Y which is finitely representable in X is a subspace of some ultrapower of X. Another fact is the following (cf. [9, pp. 228-232]).

Theorem 9.8 If a Banach space ultrapower X of X is reflexive, then X is super-reflexive.

9.5 Extending mappings to X

Now let K Ç X be bounded closed and convex, and suppose T : K —> K. Letting

K = {x = \(xn)] € X : xn € K for each n},

there is a canonical way to extend T t o a mapping T : K —* K by setting for x = [(*„)] e K,

f(x) = [(T(x„))].

It is immediate that A" is a bounded closed and convex subset of X. Now assume that T is nonexpansive. Then for x,y € K,

f(x) - f(y)\\^ = lim | |T(xn t) - T(yni)\\ < lim ||χηξ - yn< || = ||£ - y\\( ,

so T is nonexpansive as well. Moreover, since K is bounded and T is nonexpan-sive there always exist sequences (xn) C K such that

lim Κ - Γ ( χ η ) | | = 0. n—»oo

From this it follows that T{x) = i , where x — [(x„)], that is, Fix(T) Φ 0 .

Not only does the mapping T always have fixed points, but in fact Fix(T) also has a nice structure.


Theorem 9.9 Fix(T) is metrically convex.

Proof. All we need to do is show that for any x,y € Fix(jT) there exists z e Fix(f) such that

ιι* - *i i { = iiy - *ιι€ = 5 I F - i/ii€ · Suppose x = [(i„)], y = \(yn)] and fix n e N. Set

f <5n = max{\\xn-T{xn)\\,\\yn-T{yn)\\};

\ dn = 2 l l^n-ynll-

Now choose {εη} with e„ 6 (0,1) so that

lim en = lim — = 0, η—Όθ η—Όθ £η

and set

1 ~εη, Vn = ——<5n.

For each n let

Kn = B(xn; dn + ηη) Π B(yn; dn + ηη) n K.

Then obviously Kn is closed and convex, and since ηη > 0, Kn is also nonempty

(since, in particular, - ( x n + yn) € Kn).

Next define for each n a mapping Tn : Kn —> K by setting for each z e Kn>

Tn(z) = (1 - e„)T(z) + εη Q ( i „ + yn)

It is easy to see that Tn is a strict contraction with Lipschitz constant 1 — ε„. Less obvious is the fact that Tn maps Kn into Kn. To see this write

Xrx -Tn(z) = (1 -εη)(χη -T(z)j +εη i ^ n ~yn)j ■

Thus (using in the last step the fact that (1 — en)<5n = £nVn)

\\xn-Tn(z)\\ < ( l - e „ ) ( | | i „ - r ( x n ) | | + | | T ( i n ) - T ( 2 ) | | ) + ^ | | i „ - y „ | |

< (1-εη)(<5„ + dn +ηη) + = dn + (l- εη)δη + (1 - εη)ηη

= άη+ηη.

9.6. SOME FIXED POINT THEOREMS 257

Replacing xn with yn in the above gives \\yn - Tn(z)|| < dn + ηη as well. There-fore, Tn : Kn —* Kn and so there exists a unique zn e Kn such that Tn(zn) — zn. In particular,

\\Xn - Zn\\ < dn + Vn a n d \\Vn ~ *n | | < dn + ηη.

Let z — [{zn)\- Since limci„£ = - ||x — y\\^ and limTj = 0, the above gives

P - «ll€ = i™ ll^nt - 2n£ || = 2I I* - 11« ;

||y - 5||€ = lim \\ynt - znt || - - | | i - y\\^ .

We now only need to show that z 6 Fix(T). However, this is a direct consequence of the fact that lim εη = 0 and the fact that

n—>oo

zn = (1 - εη)Γ(ζ„) + ε„ ί - ( ι „ + y„) J ,

from which

\zn - T ( z n ) | | = ε η Î2(x'»+2/"-) - Γ ( Ζ η ) diam(Ji').

9.6 Some fixed point theorems

It was Maurey's surprising discovery [121] that all bounded closed convex sub-sets of reflexive subspaces of Ll have the fixed point property for nonexpansive mappings that set the stage to some amazing fixed point theorems. These re-sults are deep and nonconstructive in the sense that they involve ultrapowers of Banach spaces. One of the ingredients behind most of these new results is a property satisfied by the ultraproduct of the minimal convex set.

Indeed, let X be a Banach space and C be a nonempty weakly compact convex subset of X. Assume that there exists a nonexpansive mapping T : C —» C whose fixed point set Fix(T) is empty. Since C is weakly compact, there exists K a closed convex subset of C, which is T-invariant and minimal. In Chapter 8, we have seen some of the properties of this minimal set. Define K as usual to be

K — <x = [(xn)\ € X : xn € K for each n\.

We have the following result:


Lemma 9.3 Let {wn} be an approximate fixed point sequence (a.f.p.s.) forT in K, that is, \\T(wn) — û)n||ç —♦ 0 as n goes to oo. Then for any x 6 K, we have

lim \\wn — x\\e = diam(.ft'). n—»oo

Proof. Let {wn} be an a.f.p.s. for T in K and x € K. We will show that diam(fi) is the only possible limit for {||wn — i | | i}· Without loss of generality, we can assume that lim | |ώη — x\\e = d. Write <5„ — ||Τ(ώ„) - wn\\e and

n—»oo

ώη = [(wn(m.)} with wn(m) G K. Let {en}i with ε > 0, such that lim en = 0. n—»oo

Set

4fc = {TO e N; ||w„(m) - x|| < d + 2efc and \\wn(m) - T(wn(m))\\ < ek + <5„} ·

FVom the properties of the ultranet ξ, we can construct increasing sequences of integers {n^} and {m*} such that

\\wnk(mk) - x\\ < d + 2ek and \\w„K(mk) - T(wnk(mk))\\ < ek + 6Uk

hold for any k € N. The sequence {wnk(mk)} is an a.f.p.s. for T. Using the Goebel-Karlovitz Lemma (Lemma 8.4) we get

diam(fi') = lim \\wnk(mk) — x\\ < d-fc—»oo

By definition of K, we obviously have d < diam(A") which completes the proof of Lemma 9.3. ■

A direct consequence of this lemma is the following result:

Corollary 9.6 Let W be any nonempty closed convex subset of K which is invariant under T, that is, T(W) C W. Then

sup j ||ώ - x\\ç; w eW> = diam(Ä")

holds for any x G K.


The first of these new results was discovered by Lin [113] in Banach spaces with unconditional bases.

Theorem 9.10 Let X be a Banach space with an unconditional basis {en}. Assume that the constant of unconditionality is 1. Then for any weakly compact convex subset C of X and any nonexpansive mapping T : C —» C, the fixed point set Fix(T) is not empty.


Proof. Assume that there exists a nonexpansive mapping T : C —» C such that Fix(T) = 0 . Since C is weakly compact, there exists a closed nonempty convex subset K of C which is minimal invariant under T. Without loss of generality, we may assume that diam(A') — 1. By approximation, there exists an approximate fixed point sequence {x„}. We may assume that {xn} weakly converges to 0 6 K. In view of Goebel-Karlovitz Lemma we have

lim ||x„|| = 1 n—>oo

and by passing to a subsequence we may further assume

lim | | i n -xn+i\\ = 1 n—*oo

or, even stronger,

lim dist (xn,{xn+i,xn+2,-■■}) = 1. 71—»OO V /

Let Pn denote the natural projection of X onto the subspace spanned by {ei, ·· · , e n } . Since {xn} weakly converges to 0, then we must have

lim | | f t ( i „ ) | | = 0 n—·οο

for any k > 1. Also, for each n > 1,

lim | | ( J -P f c ) (*n) | | = 0. K—»OO

By passing to a subsequence, we may assume that there exists a sequence {Fn} of finite subsets of N such that maxFj < minFi+i and satisfy

(i) lim \\PFn(xn)\\= um ||*„|| = 1; n—»oo n—»oo

(ii) lim | | ( / - P F „ ) ( x n ) | | = 0; n—»oo

and obviously we have Pfn o PFm = 0, for ηφτη.

Now we turn our attention to the ultrapower of X. First we translate all of the above into X. Indeed, consider the set K, the mapping T : K —> K. Set x = {{xn)\ and y = [(xn +i)]. From the above properties, we have

f(x)=x, f(y) = y, and | | i - » | | € = l.

Define the two projections (in X) by

P = [{PFJ] and P = [(PFn+l)}-

We have P o Q = 0, and

P{x) = x, Q{y) = y, and Q{i) = P(y) = 0.


Finally, for any x 6 K C K, we have

P(x) = Q{x) = 0.

At this stage, we will make use of the constant of unconditionality of {en} being equal to 1. Indeed, we have

\\PFn(Xn) - Pf„+l(Xn+l)|| = \\ΡΡΛ*η) + Ρρη+ι(*η+ΐ)\\

for any n > 1. This clearly implies that

\\P(ï)-Q(y)\k = \\P(î) + Q(y)k-

So we have

\\i-yh = \\P{i)-Q{y)h = i

and

||£ + oil« = \\P{£) + Q(y)k = \\P&) - Q(y)\k = i ·

Thus, finally,

Ρ + ίΙΐ£ = Ι|ί-ίΙΙ{ = ΐ·

Next we will show that the above fact leads to a contradiction. Indeed, consider the set

W — < w € K\ max{||tt) - i | |ç, ||û) — ί | |ξ, |{τΣ> — y||ç} < - for some x € K >.

x ~\~ v " Clearly, —-— 6 W (where x = 0). Also since x and y are fixed point for T, we

have

f | | f(«;)-r( i) | | c<|| t i i - i | | c

\\Τ(ύ)-χ\\ξ = \\Τ(ύ)-Τ(χ)\\ζ<\\*-ϊ\\ξ

{ \\f(w)-y\k = \\f(w)-f(y)\k<\\ü-v\k

showing that W is invariant under T. Let w € W. Then there exists x G K such that

max{||ù) - i lk , ||ώ - i | | € , \\w - y\\^} < -.

On the other hand, we have

2ώ = (P + Q)(w) + ( / - P ) ( t i ) + ( / - Q)(Û>),


which implies

11«% < \ (\\(P + 0)(ώ)11ί + II (i - P)Wk + HU - Q)Wk).

But

(P + Q)(X) = ( / - P ) ( i ) = ( / - « ) ( » ) = 0,

so

INIi < | ( | | ( P + Q)(û - i ) | | ç + ||(J - P)(w - χ)\\ξ + | | ( / - Q)(w - y)\k)·

Since the constant of unconditionality of {en} is 1, then we must have

| |P + Q | | < 1 , | | / - P | | < 1 , | | / - Q | | < 1 .

Thus all of the above yield

\\w\k < g (||ώ - ill« + II* - ill« + ||ώ - Olli) < ^

This contradicts the conclusion of Corollary 9.6, which gives

sup j lHIç , w e W\ =diam(Ä") = 1.

The proof of Theorem 9.10 is therefore complete. ■

Example 9.1 Consider the Hubert space I2 with its usual norm. For any ß > 1 and x 6 I2, set

||x||/J = max(|H|/a;jS|M|/ee).

The functional \\.\\p is a norm which is equivalent to \\.\\t2. Indeed, we have

ΙΙ·ΙΙ<2<ΙΙ·ΙΙ0<Μ,·

Write XQ = [I2, ||.||/j). These spaces have been studied extensively. First it is easy to see that Χ^ / α ^ to have normal structure. This answers the question whether any reflexive Banach space has normal structure. In fact, we can show that Χβ has normal structure property if and only if β < \ /2. It was then natural to ask whether Χβ has the fixed point property. This question was open until Lin proved Theorem 9.10. Indeed, the constant of unconditionality of the canonical basis of I2 for \\.\\p is 1.

Remark 9.1 In fact, Lin has improved the conclusion of Theorem 9.10. Indeed, if X is a Banach space with an unconditional basis {e„} with X as its constant of unconditionality, then the conclusion of Theorem 9.10 holds, provided

V53-3 A < _ 2

It is still unknown whether the conclusion of Theorem 9.10 holds for any value of\.

262 CHAPTER 9. BAN ACE SPACE ULTRAPOWERS

9.7 Asymptotically nonexpansive mappings Asymptotic fixed point theorems are those theorems from which fixed point properties are derived from the behavior of large iterates of the mapping. For the nonexpansive mappings this assumption, in its most direct case, takes the following form. Let X be a Banach space, D Ç X, and T : D —» D. If there exists a sequence {h} C K+ with fcj —» 1 as i —» oo for which

||Γ*(χ) - r ' (y ) | | < h \\x - y\\ for all x, y € D,

then T is said to by asymptotically nonexpansive. If T is asymptotically nonexpansive then T will be asymptotically nonexpan-

sive as well, but T may not be nonexpansive. However, the mapping T defined as follows is nonexpansive. For x — [(xn)] € K set

f(x) = [{T(x1),T2(x2),T

3(x3),-..}}.

Then

f(x) - f(y)\\^ = lim \\Tn<(xn() - T"<(y„<)|| < lim/cn£ \\xnt - ynt\\ = \\x - y\\( .

We show how this observation can be used to give a very quick proof of the original fixed point theorem of Goebel-Kirk [68] about the existence of fixed points for asymptotically nonexpansive mappings.

Theorem 9.11 Suppose K is a bounded closed convex subset of a uniformly convex Banach space X and suppose T : K —> K is asymptotically nonexpansive. Then T has a fixed point.

Proof. We adopt the notation (T and K) as above. Since the modulus of convexity is a super-property, X is uniformly convex. We first show that T has a fixed point in K. Consider the mapping T : K —» K defined by

f({(xn)}) = [(Tn(xn))}, ([(xn)]€K).

Since T is nonexpansive and X is uniformly convex, T has a nonempty fixed point set F in K, and further, F itself is closed and convex. Now consider the mapping H : K -* K, where H = T o f. Then if x = [ ( i n ) ] , y = \{yn)\ 6 K,

\\H(x)-H(y)\\ = l i m ç l j T ^ + i ^ J - T ^ + H y n J H < limi (kni + i) \\xnt - 2 / n J

= P-5ll· Thus H is also nonexpansive, and further H o T = T o H. It follows that H : F —» F and therefore H has a fixed point, say x G F. We now have f o f(x) =x = t{x), from which f(x) = x.

9.8. THE DEMICLOSEDNESS PRINCIPLE 263

Now let K be the closed convex subset of K defined as follows:

K = {x = [(z„)] :xn = x e K).

As we have seen, T has a fixed point x E K. Note that if x — x € K then

x - f(x) = lim \\T(x) - x\\ = 0, and it follows that T(x) = x. On the other

hand, if x £ K, then there is a unique point i 6 K such that | | i — i|L =

dist(i,Ä'). Since T is asymptotically nonexpansive

lim n—»oo

Tnz-x = lim Ç η—ΌΟ

Tnz - Tnx < lim kn Ç n—*oo

Tnz - Tnx

and since X is uniformly convex this implies lim \(T)n(z) — z\ = 0 . There-n—>oo I I ξ

fore, T(z) = z. Hence T(z) = z and in either case we are finished. ■

9.8 The demiclosedness principle

A fundamental result in the theory of nonexpansive mappings is Browder's demiclosedness principle [24].

Definition 9.5 Let X be a Banach space and K Ç X. A mapping f : K —» X is demiclosed (at y) if the conditions {XJ} converges weakly to x and {f(xj)\ converges strongly to y then x 6 K and f(x) = y.

Browder's demiclosedness principle asserts that if K is a closed convex subset of a uniformly convex Banach space X and if T : K —» X is nonexpansive, then I — T is demiclosed.

The following fact is proved in [24] using sharp properties of the modulus of convexity and a clever subsequential thinning process. The ultrapower approach given here is direct and simple (although it does require the deep fact mentioned earlier that if X is a super-reflexive space then (X)* — X").

Theorem 9.12 Suppose K is a bounded closed convex subset of a uniformly convex Banach space X and suppose T : K —> K is asymptotically nonexpansive. Then I — T is demiclosed at 0.

Proof. We continue with the notation used previously and we use the fact that X is uniformly convex.

Let K be the closed convex subset of K defined as follows:

K = {i = [(in)] :xn = xeK).

Let {xn} be a sequence in K satisfying

weak- lim xn = x and lim | | i n — T(xn) | | = 0.


By assumption [(xn)] € Fix(T). Since T is asymptotically nonexpansive, it is easy to see that Fix(T) is closed and convex. Let ε > 0 and let / € X* ■ Since X is super-reflexive there exists (/„) C X* such that for each ü — [(un)) € X, f(ü) = l im/„{(un {) . Also, since {xn} converges weakly to x, for each n € N

there exists an integer kn such that

and clearly we may assume {kn} is increasing. Thus

i = [ ( * f c „ ) ] e F i x ( i ) .

We now have

/(£) - f{±) = lim | /η ί (xfcri{ ) - fnt (x) <e.

Thus inf | | / ( ù ) - f(x) : ü e Fix(T)} = 0. Since Fix(f) is weakly compact,

this implies f{u) = / ( i ) for some û € Fix(!T). By Theorem 6.5 (Separation Theorem) this, in turn, implies x 6 Fix(T); hence T(x) = x. ■

9.9 Uniformly non-creasy spaces

A Banach space X is said to have the FPP if every bounded closed convex subset of X has the fixed point property for nonexpansive self-mappings. The results of the previous two sections suggest that there are implications for asymptotically nonexpansive mappings in the class of spaces X for which FPP is a super-property, in particular, for which X has the FPP. Of course, the uniformly convex spaces belong to this class, as do Banach spaces X for which EQ(X) < 1, but in general, the class of spaces X for which X has the FPP is not yet well understood. In particular, it is not even known whether uniform normal structure is a super-property. Recently S. Prus [135] has introduced a class of spaces, which he calls uniformly non-creasy, which also have the super-fixed point property. We take up this fact here.

There are two stages to the approach. First it is shown that uniformly non-creasy is a super-property. Then, it is shown (via an ultrapower approach) that uniformly non-creasy spaces have the FPP.

We need some more notation. We shall use Βχ and S\ to denote, respec-tively, the unit ball and unit sphere of a Banach space X. For any f & Sx· and δ 6 [0,1] we define the slice S(f, 6) to be the set

S(f,S) = {xeBx:f{x)>l-6\.

Certain geometric properties of Banach spaces are easily thought of in terms of slices; indeed, uniform convexity can be characterized in such a way.

9.9. UNIFORMLY NON-CREASY SPACES 265

Propos i t ion 9.8 A Banach space X is uniformly convex if and only if

(a): for every e > 0 there exists δ > 0 such that diam(5(/,δ)) < ε for each f&Sx·.

Proof. Assume X is uniformly convex, let ε > 0 and let δ = δ(ε). Suppose x,y e S(f,6) for some f eSx-. Then

l-6<f(±(x + vf)<±\\x + y\\\\f\\ = \\\x + y\\.

Since X is uniformly convex it must be the case that ||x — y|| < ε, that is,

diam(S(/,«5)) < e.

Now assume (a) holds and assume X is not uniformly convex. Then δ(4ε) — 0 for some e > 0. Let δ > 0 be the number in condition (a) which corresponds to ε. Since 6(4e) = 0 there exist x, y G Βχ such that \\x — y\\ > 4e but for which

Ö llx + Î/II > 1 — δ· Choose / G Sx- so that

/ \\(x + y)J = ^\\x + y\\-

Then - (x + y) 6 S(f, δ), and it also must be the case that either x e S(f, δ) or

y e S(f, (5), from which diam (5(/,<$)) > 2e > ε. ■

Banach spaces X for which AT* is uniformly convex can be characterized by the intersection of slices. For this we need another definition.

Definition 9.6 A Banach space X is said to be uniformly smooth if

fx(h) = H"x + thf-W exists uniformly for each (x,h) € Sx x Sx. (It is easy to check that this limit defines a functional fx G X*.)

A Banach space property (P) is said to be self-dual if X has property (P) if and only if X* has property (P). It is well known that a Banach space X is reflexive if and only if X* is reflexive; that is, reflexivity is self dual. On the other hand, there exist uniformly convex spaces X for which X* is not uniformly convex. However, uniform convexity does entail another type of duality. It is known that a Banach space X is uniformly convex if and only if X* is uniformly smooth (and conversely, X is uniformly smooth if and only if X* is uniformly convex).

Now consider two functionals f,g E Sx· and a scalar δ G [0,1], and set

S(f,g,6) = S(f,6)nS(g,6).


Proposition 9.9 A Banach space X is uniformly smooth if and only if

(b): for every ε > 0 there exists δ > 0 such that if f,g E Sx· and \\f — g\ > e, thenS(f,g,6) = 0 .

In view of the comments immediately preceding its statement, the proposi-tion will follow from a conjunction of the following two facts:

(i) If X satisfies (a), then X* satisfies (b).

(ii) If X* satisfies (b), then X satisfies (a).

Proof, (i) Suppose (a) holds for X and (b) does not hold for X*. Then (since (a) implies X** = X) there exists ε > 0 such that for any 6 > 0 there exist x,y eBx for which ||x - y\\ > 2ε and S{x,y,δ) φ 0 . Since / E S(x,6)(lS(y,6) implies x,y E S{f,6) and hence diamS(/ , δ) > 2e > ε, this clearly contradicts (a) for δ sufficiently small.

(ii) Now suppose (b) holds for X* and (a) does not hold for X. Then there exists ε > 0 such that for any δ > 0, diam S(f, δ) > 2ε for some f E Sx·. Choose x,y E S(f,6) such that \\x — y\\ > e. Then / E S(x,y,6). This contradicts (b) for δ > 0 sufficiently small. ■

A Banach space is said to have a crease if there exist two distinct functionals f,g G Sx- such that diamS(/ , g,0) > 0. Since f(x) < 1 for each / € Sx· and x € Βχ, given / € Sx· the hyperplane {x E X : f{x) = 1} supports the unit ball Βχ. Thus to say that X has a crease means that the sphere Sx of X contains a segment of positive length that lies on two different hyperplanes which support the unit ball Βχ.

A space X is called noncreasy if its unit sphere does not have a crease. Any Banach space X with dim(X) < 2 is trivially noncreasy, as is any strictly convex space.

Definition 9.7 A Banach space is said to be uniformly noncreasy (UNC) if given any ε > 0 there is a δ > 0 such that if f,g E Sx· and \\f — g\\ > ε, then

diam (S(f,g,«)) < ε.

Also, Propositions 9.8 and 9.9 show that UNC is implied by both uniform convexity and uniform smoothness. As we shall soon see, UNC is a self-dual property. However, we first prove the following.

Theorem 9.13 / / a Banach space X is UNC, then X is super-reflexive.

Proof. Assume X is not super-reflexive. Then by James's characterization of Theorem 9.4 for any number δ E (0,1) there exist points x; E Sx and functionals fi E Sx· for i = 1,2,3 for which fi(xj) = 1 - δ if j > i and fi(xj) — 0 if i > j .


This implies | |/i - f2\\ > (/i - /2)(*i) = 1 - 5 . Moreover, x2,x3 € S(/i,/2,<5) so diamS(/ i , / 2 ,5) > ||x3 - X2II > /3(x3 — x2) = 1 — 5. This proves that X is not UNC. ■

In contrast to uniform convexity and uniform smoothness, UNC is a self-dual property.

Theorem 9.14 A Banach space X is UNC if and only if X* is UNC.

Proof. It suffices to show that if X is UNC then X* is as well. The converse will then follow from the fact that UNC implies reflexivity. So, suppose X is UNC and X* is not. Then (making the identification X** = X) there exists a constant ε > 0 such that for each δ & (0, —) there exist elements ii ,X2 € Sx

with ||xi — x2|| > ε and functionals / i , / 2 € S(xi,x2,6) for which | |/i — /2II > ε. In particular, \\fi\\ > frfa) > 1 - δ and ||/2| | > /2(x2) >1-δ.

Now set / = / , / ΙΙΛ || and g = f2/ | | / 2 | | . Then

II/-SII = /ll/ill + /(i-||/ill)-flll/2||-ff(i-||/2ll)

>

Thus

f\\h\\-g\\f2\\ - l l / ( i - | | / i | | ) | | - | | s ( i - | | / 2 | |

l l / - f f l l > l l / i - / 2 l | - ( i - | | / i l | ) - ( i - | | / 2 | | ) > e - 2 Ä > | .

Moreover, Xi,x2 G S(f,p,6).This contradicts the assumption that the space X is UNC. _ ■

Finally, we observe that since the ultrapower X of a Banach space inherits all finite-dimensional geometric properties of X we have the following.

Theorem 9.15 A Banach space X is UNC if and only if X is UNC.

Proof. In view of Theorem 9.13 we may assume X is reflexive. Now suppose every five-dimensional subspace of X has UNC (for the same δ corresponding to a given ε), and suppose X fails the UNC assumption. Then there exists ε > 0 such that for any δ (take δ — δ(ε)) there exist / ,g 6 5 χ · with | | / — g\\ > ε and for which

diam(s(/,5,<5)) > e.

Now pick x\, x2, £3 G X such that

= / ( * i ) , \\9\\=9{X2), | | / - f l | | = ( / - 0 ) ( * 3 ) .

Also pick Χ4, X5 G S(f, g, δ) such that ||x4 — x5|| > e. Then the five-dimensional space spanned by {χχ, · · ■ ,xs} fails the UNC assumption. This shows that the property of UNC has five-dimensional character, so any space which is finitely representable in X must also be UNC. ■

This fact can now be used to show that X (hence X) has the fixed point property for nonexpansive mappings.


Theorem 9.16 Let K be a bounded closed convex subset of a Banach space X which is UNC, and letT : K —» K be a nonexpansive mapping. Then T has a fixed point.

Proof. Assume T does not have a fixed point. Since X is reflexive, we may use Zorn's Lemma and assume that K is minimal with respect to being nonempty bounded closed convex and T-invariant. Also, for convenience, we may assume that 0 € K and that diam(A') — 1. Now let {xn} Ç K satisfy lim ||xn — T(xn) | | = 0. By passing to a subsequence and translating we may

n—»oo also assume that {xn} converges weakly to 0. Now for each n 6 N select a functional /„ E Sx- such that / „ ( i „ ) = | |xn | | . Since X is reflexive, then X* is also reflexive. Hence {/„} has a weak limit point, say / , in Βχ·. We now need some detailed calculations.

Since lim / (x n ) = 0 it is possible to choose ni so that | / ( x n ) | < - for all n—»oo 4

n >ηχ. Again use the fact that {xn} converges weakly to 0 to choose n% € N, ri2 > ni, so that | / n i ( x n ) | < - and | / ( x n ) | < for all n > Π2, and also so that 2 8

l(/n2 — /X^nJI < - for i = 1,2. Furthermore, in view of the Goebel-Karlovitz

Lemma, n 1 — ^. Note that we now have

and

2

Ι / η , Ο Ο Ι ^

| /na(*„.) | < | / (*„ , ) | + l(/n2 - / ) (*„.)! < \ + \ = \

We proceed one step further. Choose n^ € N, TI3 > 712, so that |/n2(xn)| < -r

and | / ( x n ) | < 7^ for all n > 713, and also so that | ( / n - / ) ( x n i ) | < r for 16 8

i — 1,2,3. Furthermore, in view of the Goebel-Karlovitz Lemma, 712 can also be chosen so that ||x„2 - x„3|| > 1 — - . Note that we now have

| / n , ( * n j | < \

and

I l-l 8 + 8 ~ 4

| /n s (*„J | < | / ( ϊη 2 ) | + |(/n3 ~ / ) ( ^ n j | < ö + Ö = 7"

Proceeding by induction it is possible to choose ni < τΐ2 < ri3, · · ■ so that for i = 1, · · · ,k — 1,

|/nl+,(x„J| < 2fc a n d |/nt(*nt + I ) | < ^fc


and also so that ||xnA: - χηλ+1 || > 1 — —.

Now set x = [(*„„_,)], y = [(x„,J) € X, and / = [(/„„_,)], g = [ ( / n , J ] € X*. Then x,y € Sx, and

/ ( f ) = ϋ ι η / η ^ . , ί χ η · ^ - , ) = lim = 1.

Similar calculations show that ||^||^ = g(y) = 1, /(y) = g(x) = 0, and ||x - y\\^ = 1.

Now, since Fix(T) is metrically convex (Theorem 9.9) there exists w € K such that | |ώ| | . = 1 (another consequence of the Goebel-Karlovitz Lemma) and such that

1 Ι Ρ - ζ | | ξ = l|w- ■»llc = 5 ·

Also, /(û>) = f(w-y) < \\w - y\\( = 1/2. On the other hand, 1/2 = ||x - ιΐ;||ξ > f{x - w) = 1 - f(w) so f(w) > 1/2. Therefore, f(w) = 1/2. Similarly, g(w) = 1/2.

Thus

This shows that

and similarly

/(2(x - wj) = 1 and <7(2(x - ώ)) = - 1 .

2(x-w)eS(f,-g,0)

2(w-y)eS{f,-g,0).

Moreover, / φ —g since / — (—g) f + 9 > if + g)(x) = 1. So, since X

is UNC it must be the case that x — w — w — y, that is,

- 1 , - -,

Now choose a functional hk € Sx· for each k such that

frfcV^nafc-i + xn2k) = \\xn2k-\ ~l~ xn2k\\ ·

There is a subsequence {xmk} of {x„} for which

lim /ifc(xmJ = lim / n i l . , ( i m J = 0 k—·οο fc—·οο

and

lim x„2fc_, - i m i = 1 · k—«oo


Put z — (xmk) G X and h — (hk) 6 X*. Then clearly \\h = 1; moreover,

h(x + y) = lim hk( (xn _, + xn ) = lim Xn2ki - 1 "r ^"21^ = ||ί + ί

Also, ft(z) = f{z) = 0 and ||x - z||€ = 1. Therefore,

1 > Λ(χ) = 2/ι(ώ) - h(y) > 2 -

Similarly, /i(y) = 1. Thus

llvllf = i ·

i , i - z e 5 ( / , g , 0 ) ,

where h - / > (h-f){y) — h{y) — 1. Therefore,

||z||{ = | | i - ( i - z)||ç < diam(5(/,5,0)) = 0,

from which z = 0, which, of course, is a contradiction since ||z||^ = 1 by the Goebel-Karlovitz Lemma. ■

Exercises

Exercise 9.1 Show that X is a finite-dimensional Banach space if and only ifX is a finite-dimensional Banach space. In this case, show that dim(A') = dim(X).

Exercise 9.2 Show that if X is not a finite-dimensional Banach space, then X is not separable.

Exercise 9.3 Show that X is strictly convex if and only if X is uniformly convex.

Exercise 9.4 Let {xn(*)}n>i be a Schauder basis o/C([0,1]), the space of con-tinuous functions on [0,1]. Set Xn to be the space spanned by x\,· ■ ■ ,xn. Let

(i) Show that C([0,1]) is finitely representable in X.

(ii) Show that X is separable and reflexive.

EXERCISES 271

(iii) Show that any separable Banach space is finitely representable in X.

Exercise 9.5 Let X, Y, X\, Υλ be Banach spaces, and T : X —♦ Y, 71 : X\ —» Y\ operators. We say that T\ is finitely representable in T if, for every e > 0, every finite-dimensional subspace X° of X\, one can find a finite-dimensional subspace X° of X, and (setting Yf = T(X{), Y° = T(X°)), two isomorphism U:Xf-> X" and V : Yf — Y° such that T o U = V o Tx and

/ \\U\\-\\U-'\\ < 1 + e; \\V'l\\ < 1 + e.

Show that this definition is transitive. Show that T : X —* Y is finitely repre-sentable in T.

Exercise 9.6 (Ultrapower of Banach lattices) Let X be a Banach lattice. Con-sider the ultrapower X of X. For x € X, we write x > 0, if there exists a sequence {xn} of positive elements in X such that x = [(xn)}· Show that X is a Banach lattice with

{(xn)} V [(yn)} = [(xn V yn)] and [(x„)] Λ [(yn)] = [(xn Λ y„)].

Show also that \[(xn)] — {(\xn\)\-

Exercise 9.7 Let X be a Banach space with a Schauder basis {en}. Set

• μ — sup{||u — v\\;u and v are disjoint blocks with \\u + t>|| < 1};

• c\ = sup{| |/ — Pn\\; Pn is the natural projection on the interval \l,n]};

• c2 = sup{||7 — PF\\',F is any interval in N}.

The constant c is the basis constant of {en}. Show that if

c^μ-rc-rC2 < 4,

then X has the fixed point property. Hint: Use the same ideas developed in the proof of Theorem 9.10.

For example, for the James space J, show that μ = \/2, c^ = \/2, and c — c\ — 1. In particular, we have c\ß + c + a = \/2 + \/2 + 1 < 4.

Exercise 9.8 Show that if X has normal structure, then X has uniform nor-mal structure. Deduce that if X has super-normal structure, then X is super-reflexive.

Appendix

Set Theory

A.l Mappings

Familiarity with the basic rudiments of elementary set notation, and concepts such as set inclusion, intersection, union, and so on, is assumed throughout this discussion. We pause here to look a little more closely at some of the underlying set-theoretic concepts that arise in the text. Obviously it is important in math-ematics to be precise about what is meant by a mapping (or function) from one set to another.

Definition A . l If A and B are sets, then the Cartesian product of A and B is the set A x B of all ordered pairs (a,b) with a € A and b Ç. B.

Of course, the above definition raises the question of what is meant by an 'ordered pair'. Commonly this means that one should distinguish between the 'left' element a from the 'right' element b. To make this idea mathematically rigorous, logicians define (a, 6) = {a, {a, 6}}, where the bracket notation {u, v} denotes an unordered pair. Thus one can identify a with the element a, and 6 with the element {a, b} of the unordered pair {α, {α,ί>}}. This validates the accuracy of the intuitive approach.

Definition A.2 A mapping (or function) f from A to B, denoted f : A —» B, is a subset of Ax B which satisfies:

(i) If a £ A then there exists b € B such that (a,b) e / ; and (ii) (a,b) 6 / and (a, c) 6 / => 6 = c.

/ / (a,6) € / one denotes b — / (a ) . / / only (i) is assumed then f is called a relation (with domain A).

Thus in common language, corresponding to each a € A there is exactly one element b 6. B such that b = f(a).

273



274 APPENDIX. SET THEORY

A. 2 Order relations and Zermelo's Theorem Let 71 be a relation on a set S. By this we mean that 7i is a subset of S x S. Next, say a < b for a, b € S if the pair (a, b) 6 5. The mere introduction of this notation prompts one to refer to 71 = (S, <) as an order relation. There are several types of order relations that arise in analysis; we list the most common.

Definition A.3 (S,<) is said to be a partially ordered set (POS) if for each a,b,c G S:

(i) a < a; (ii) a a < c.

Definition A.4 (S, <) is said to be a directed set (DS) if (S, <) is a (POS) and if, in addition:

(iv) a,b € S => there exists c Ç S such that a < c and b < c.

Definition A.5 (5, <) is said to be a linearly ordered set (LOS) if (S, <) is a (DS) and if, in addition:

(v) a,b Ç. S =$■ a < b or b < a.

Definition A.6 (S,<) is said to be a well-ordered set (WOS) if (S,<) is a (LOS) and if, in addition:

(vi) every nonempty subset of S has a smallest element.

Thus it is immediate from the definitions that for a given ordered set (S, <) the following implications hold:

(WOS) => (LOS) => (DS) => (POS).

The set N of natural numbers (with usual ordering) is a well-ordered set. The set R of real numbers (again with usual ordering) is a linearly ordered set which is not well ordered.

The set V(S) (or 2s) of subsets of a given set is a directed set when ordered by set inclusion (A < B o A Ç B) but it is not a linearly ordered set (if S contains at least two elements). Finally, the nonempty subsets of V(S) is a partially ordered set when ordered by reverse set inclusion (A < B ■«· B Ç A) which, in general, is not a linearly ordered set.

Each of these concepts play fundamental roles in analysis.

Linearly ordered sets are often called chains, especially when referring to families of sets ordered by set inclusion. It is possible for a partially ordered set to contain a linearly ordered subset. When this is the case, such a subset is also called a chain. The concepts of 'bounded above', 'least upper bound', and

A.3. ZORN'S LEMMA AND THE AXIOM OF CHOICE 275

so on, which were defined initially for the real numbers, carry over verbatim to the more abstract setting of either a linearly or partially ordered set.

We are now in a position to state yet another fixed point theorem. The roots of this theorem go back to a 1908 result of Zermelo [168], and proofs are readily at hand to those who have access to, for example, [52], p. 5; [166], p. 504. (In the latter citation the result is attributed to both Kneser and Bourbaki. The idea is only implicit in [168].)

A quick proof of this theorem based on Zorn's Lemma is given in the next section. At the end of this Appendix we add a 'constructive' proof.

Theorem A . l (Zermelo) Suppose (S,<) is a partially ordered set in which each chain has a least upper bound, and suppose f : S —» S is a mapping which satisfies x < f(x) for each x 6 S. Then f has a fixed point.

A.3 Zorn's Lemma and the Axiom Of Choice The abstract discussion of the previous section would have been impossible without language involving "sets". The concept is so basic that it requires only a brief explanation. A set is a collection of objects, usually called elements or points, along with the primitive concept of belonging; that is, a set is determined by the collection of objects "belonging" to it. If an object x belongs to a set A we write x S A. Given any object imaginable and any set A, either the object belongs to A or it does not belong to A—the framework of mathematical logic allows for no ambiguity; indeed the core of mathematics as an intellectual discipline rests upon this premise.

In Section A.l the cartesian product Ax B oî two sets A and B was defined as the collection of all ordered pairs (a, 6) with a € A and b € B. It is clear that A x B is itself a set since we have described the objects belonging to it (especially since we have defined Ordered pair' as well). Suppose one has a finite number of sets, say A\,· ■ ■ ,An. Surely one would want to define the cartesian product A\ x A^ x · · · x An to be the collection (ax,a2, · ■ · ,an) of all ordered n-tuples where ai € Ai, i = 1,2, ■ · · ,n. Mathematical induction can make this definition precise. Having defined A\ x A2, suppose A\ x ■ ■ ■ x ^4n-i has been defined, and define

Aix A2x ■■■ x An = (AiX ■■■ x An-i) x An.

Beginning with an infinite number of sets, say A\, A2, ■ ■ ■ , then surely it would make sense to define the cartesian product A\ x Ai x ■ ■ ■ to be the collection of all sequences (αχ,α2,·- , α η , · · · ) where α^ Ç. Ai, i — 1,2,··· . Something subtle arises here, for in order to make such a definition it is necessary to simultaneously select an infinite number of elements ai € Ai, i = 1,2, · · · . While the basic axioms of classical Zermelo-Praenkel set theory do not allow for such a selection, most mathematicians accept the validity of such a procedure


by admitting the so-called Countable Axiom of Choice. Since the concept of 'sequence' seems so fundamental to mathematics this is usually not viewed as a serious problem. Very little mathematics as we know it could be accomplished without an assumption of this type.

The general Axiom of Choice is a different matter entirely. Suppose T = {Aa}a€i is a collection of sets with no restriction placed on the index set / . Thus I may be very large, especially uncountable (see Section A.6 below). The Axiom of Choice states that there is a function / : / " — ♦ \J Aa such that

f{Aa) e Aa for each a £ I. Such a function / is called a choice function. Upon knowing that such a function exists, it is now possible to define the cartesian product Y[ Aa to be the collection of all such choice functions. The Countable

Axiom of Choice now becomes simply the assertion that there exists a choice function for each countable family of sets.

Mathematicians study both the assumptions upon which the subject is based and the objects about which those assumptions are made. As Moore observes in [125], rarely in history has there been more controversy about one of the central premises of mathematics than the one which arose when Zermelo proposed the Axiom of Choice in 1908 [169]. The Axiom of Choice has had profound impli-cations in the recent development of mathematics and without it mathematics would be much different than we know it today. Indeed, if the most severe 'con-structivists' were to prevail, mathematics would be reduced to the study of a collection of algorithms. Consequently, most mathematicians accept the Axiom of Choice without question. Indeed, it is such a natural assumption that its use often goes unnoticed. But it is deceptive in its simplicity. Try, for example, to describe a choice function for the family of all nonempty subsets of the real numbers. (It is easy for the family of all closed subsets—take an element of each set which is nearest 0.)

On the one hand, the Axiom of Choice is different from other generally accepted assumptions in mathematics because it can be used to prove theorems which run deeply counter to intuition. For example, it can be used to show that in a euclidean space of dimension n > 3 any two arbitrary sets with nonempty interior can be decomposed into the same (finite) number of mutually disjoint subsets that are, respectively, isometric to one another. This is the astonishing Banach-Tarski paradox. (For an excellent discussion, see [84]; also see [125] for a discussion of the impact of the Axiom of Choice in mathematics and the controversy surrounding it.) At the same time, the Axiom of Choice validates many intuitively attractive propositions. Because of the efficiency it brings to many proofs we have used the Axiom of Choice in this text without reservation (although not necessarily without comment), despite the fact that many of the results we discuss can be shown to be true using weaker axioms.

One of the most useful consequences of the Axiom of Choice (in fact, it is equivalent to Axiom of Choice) is the celebrated Zorn's Lemma [170].

L e m m a A . l (Zorn) Let (S, <) be a partially ordered set in which every chain has an upper bound. Then S has a maximal element relative to <.

A.4. NETS AND SUBNETS 277

It is important to be precise. A maximal element of S is an element m € S which has the property that if x € S and if m < x, then x = m. In particular, nothing in S is larger than m; it is not asserted that every element of S is < m.

We return to Zermelo's Theorem of the previous section to illustrate the power of the Axiom of Choice. Despite the fact that it can be proved without recourse to the Axiom of Choice, we show here that it is a trivial consequence of Zorn's Lemma.

Theorem A.2 (Zermelo) Suppose (S, <) is partially ordered set in which each chain has a least upper bound, and suppose f : S —* S is a mapping which satisfies x < f(x) for each x G S. Then f has a fixed point.

Proof. By Zorn's Lemma S has a maximal element xo € S. Since xo < /(xo) it must be the case that /(xo) = xo- ■

Given that it is such an immediate consequence of Zorn's Lemma, it might seem remarkable that the above theorem can be proved without recourse to the Axiom of Choice (see the proof at the end of this Appendix).

A.4 Nets and subnets

There usually comes a time in the study of analysis when it becomes essential to confront the concept of convergence of nets (or filters). The concept of sequential convergence is all that is needed to describe continuity and various topological notions in metric spaces. It is convenient, however, to use nets to describe topo-logical concepts in those non-metric settings in which sequential convergence is not adequate. We do this in the discussion of Tychonoff's Theorem below, and we did so in Chapter 6 in discussing weak topologies.

We begin with a special kind of partial order.

Definition A.7 A directed set is a partially ordered set (I, <) which has the property

α,β € I => a <y and β < y for some 7 € / .

Obviously, any linearly ordered set is a directed set. The collection of all subsets of a given set X ordered by set inclusion (A < B O A Ç B) is directed. Also, if x € X then the family of all subsets A Ç X with the property x € A is directed when ordered either by set inclusion or reverse set inclusion (A < B <» A3 B). The family of all finite subset of a given set is also directed when ordered by set inclusion. There are many such examples.

A subset J of a directed set / is cofinal in I if for each a € I there is a β 6 J such that α < β. It is easy to see that under these circumstances J itself is directed.

We now come to the principal definition.

Definition A.8 Let S be a set. A net in S is a mapping φ : I —♦ S where I is a directed set.


Observe that every sequence is a net, a fact that gives rise to some standard notation. If a € / we usually write φ(α) = xa, and we use {xa}aei, or simply {xa} when there is no confusion, to denote the net φ.

Recall that a sequence in 5 is a mapping φ : N —> 5, where N denotes the natural numbers. Therefore, any sequence is a net, and while sequences and nets share many common properties, there is a fundamental difference between subsequences and subnets. If φ : N —» S is a sequence and if / : N —» N satisfies / ( n i ) > / ( n 2 ) whenever n\ > ri2, then φ = φο{ : N —» S is called a subsequence of φ. In the analogous definition for nets the mapping / is far less restricted.

Definition A.9 If φ : I —> S is a net in S and if f : J —* I is a mapping of a (possibly different) directed set J into I which satisfies

for each a € I there exists jo £ J such that f(j) > a for all j > jo

then the net φο f : J —> S is called a subnet of φ.

The essential difference, of course, is the fact that the domain of the subnet may differ from that of the original net. This is not the case for subsequences. Consequently, one cannot say that a subnet of a sequence is a subsequence (although a subsequence is a subnet).

There is one very surprising consequence of Zorn's Lemma which involves nets. Let S be a set and H Ç S. A net {xa}aei m S is said to be eventually in H if there exists cto € I such that xa e H for all a > ao-

Definition A.10 A net {xa} in o set S is called an ultranet (or universal net) if, given any subset G Ç S, either {xQ} is eventually in G or {xa} is eventually in S\G.

It can be shown that the following is a consequence of the Axiom of Choice via Zorn's Lemma. (Indeed it can be shown to be equivalent to the Axiom of Choice.) This observation is crucial to the development of Chapter 9.

Proposition A . l Any net {xa} in a. set S has a subnet which is an ultranet.

A. 5 Tychonoff's Theorem TychonofF's Theorem states quite simply that the cartesian product of an ar-bitrary family of compact topological spaces is itself compact in the so-called 'product topology', and it is well known that this result is equivalent to the Axiom of Choice. The concept of net convergence offers a convenient way of un-derstanding Tychonoff's Theorem and in fact the proposition about ultranets of the previous section can be used to give a quick proof of Tychonoff's Theorem.

Since Tychonoff's Theorem is a topological fact we briefly review some con-cepts. Here r denotes the collection of open sets which defines the topology on X.

A.5. TYCHONOFF'S THEOREM 279

Definition A. 11 A net {xa} in a topological space (Χ,τ) is said to converge to x € X, written l imxa = x, if x € U Ç. τ implies there exists cto such that

a

xa € U whenever a > c*o. Thus the net {xa} is eventually in U for any open set U containing x.

A topological space (X, r) is said to be Hausdorff if given any two points x, y 6 X with x ψ y there exist (open) sets U,V e r such that x 6 U, y E V and UnV = 0. All the topological spaces encountered in this text will be Hausdorff. The significance of this concept lies in the fact that in a Hausdorff space limits are unique: If l i m i a = x and l i m i a = y then x = y.

a a

Propos i t ion A.2 A subset K of a topological space X is closed if and only if for any net {xa} Ç K, lima:a = x implies x 6 K.

a

Definition A.12 A subset K of a topological space (X, r ) is compact if when-ever K is contained in the union of any family T of sets in r then the union of some finite subfamily of T also contains K.

This leads to the following result. This fact is true in metric spaces for sequences, but its validity in general topological space requires the use of nets.

Propos i t ion A.3 A subset K of a topological space is compact if and only if every net {xa} in K has a subnet which converges to a point in K.

Tychonoff's Theorem involves the product of a family of topological spaces. A convenient way to describe the product topology in such a setting is to first define its closed sets. Then one can say a set is open in the product topology if and only if its complement is closed.

Let (Xi,Ti)iej be a family of topological spaces (/ is some index set) and consider the cartesian product Γ] Xf where, as before,

Y[Xi = {/ : I - ( J X i with f(i) e X, for each i € / } . »6/ «6/

One now says that a net {ga} in Γ] Xt converges to g € Π - « ( m t n e produ ct »e/ t€/

topology) if and only if the net {ga{i)} converges to g(i) for each t e J. By letting f(i) = Xi one can think of points / e [ ] X j a s /-tuples of the form

(xi)i£i- Thought of in this way, convergence in the product topology is the same a 'coordinatewise' convergence.

In order to define what is meant by a closed set in the product topology, we take our cue from Proposition A.2.

Definition A. 13 A subset K of the product space X — Γ] Xi is said to be

closed (in the product topology) if and only if for any net {fa} Ç K, l im / a = a

f € X implies f € K. A set is open in X if and only if its complement is closed.


We now come to the point of this section.

Theorem A.3 (Tychonoff) If (Xi)i€i is any family of compact topological spaces, then X = Y\ Xi is compact in the product topology.

In his original paper Tychonoff proved only that the product of any number of copies of the closed interval [0,1] is compact, but his proof lent itself to the above generalization. Hence the theorem carries his name.

It can be shown rather easily that Tychonoff's Theorem is consequence of the following three facts.

(1) A topological space X is compact if and only if every net in X has a convergent subnet.

(2) Every net in any set S has a subnet which is an ultranet.

(3) If X is compact and if {xa} is an ultranet in X, then {xa} converges to a point of X.

A.6 Cardinal numbers

At times it is necessary to make distinctions between certain classes of infinite sets. Given two infinite sets, A and B, there are precisely three mutually ex-clusive possibilities. Either there is a one-to-one mapping from A onto B, in which case we say that A and B have the same cardinality and write \A\ = \B\, or no such mapping exists. If no such mapping exists, then either there is a one-to-one mapping from A onto a proper subset of B or there is a one-to-one mapping from B onto a proper subset of A. In the former instance we write \A\ < \B\ and in the latter \B\ < \A\. (Caution: Note that \A\ < \B\ means that there is a one-to-one mapping from A onto a proper subset of B and at the same time there does not exist a one-to-one mapping from A onto B.) For any two sets A and B the following trichotomy holds: \A\ < \B\ or \B\ < \A\ or \A\ — \B\. This is another of the many fundamental results in mathematics which is known to be equivalent to the Axiom of Choice. It is this fact that enables us in some sense to order infinite sets according to their 'size'. (The fact that for infinite sets A and B, \A\ < \B\ and \B\ < \A\ =» |J4| = \B\ is known as the Schröder-Bernstein Theorem. Its proof does not require the Axiom of Choice.)

At this point the notion of cardinality may still seem vague, so we begin with something familiar. The cardinality of any finite set is simply the number of elements in the set. This suggests the use of the term "cardinal number" to refer to the cardinality of a set. We say that a set A is countable if it is either finite or if |J4| = |N|. By the trichotomy of cardinality, any infinite set S contains an infinite countable (countably infinite) subset; thus |N| < \S\ for any infinite set S. It is traditional to denote |N| = No where the symbol N (aleph) is the first letter of the Hebrew alphabet. Thus No is the smallest infinite cardinal number. As we observed in the previous section there does not exist a one-to-one mapping from the set N onto R. Thus |N| < |R|. Often the letter c (which

A. 7. ORDINAL NUMBERS AND TRANSFINITE INDUCTION 281

stands for "continuum" ) is used to denote the cardinality of M; thus there are at least two infinite cardinal numbers. In fact, there are many. It can be easily shown (using induction) that there does not exist a one-to-one mapping from any finite set S onto the set P(5) of all subsets of S. Thus for any finite set 5, | 5 | < | P ( 5 ) | .

The above fact extends to infinite sets as well. There is a one-to-one corre-spondence between P(S) and the set 2s of all functions from S into the two-element set {0,1}, and it can be shown that there does not exist a function mapping S onto V(S). (This might be an interesting exercise for those who have not studied any set theory.) Therefore, in general, one has \S\ < \P(S)\ (or, equivalently, \S\ < |2S | ) . In terms of cardinality, there is no 'largest' set.

There is a lot more that can be said about cardinal numbers but this is about all that is needed for an understanding of anything in the present text. This is fortunate because at this point things get a bit tricky. For example, it is known that relative to the order relation defined above there is a smallest cardinal number, denoted Νχ, which is larger than No, and Cantor conjectured that c = Nj. To this day this conjecture, known as the continuum hypothesis, remains unproved. However Kurt Godel has shown that it cannot be proved to be false within the standard axioms of set theory. There is something subtle here. The fact that it cannot be proved false does not mean it is true. Another option remains, namely, that it is unprovable within the axioms of set theory.

We have been admittedly vague about the use of the term "cardinal number" except to hint that "numbers" of the form ^1,^2,^3, · · · can be assigned to classes of sets of different cardinalities. In the next section we discuss ordinal numbers and the well-ordering theorem—facts which will shed further light on the concept of cardinal numbers.

A.7 Ordinal numbers and transfinite induction

It is usually possible to avoid recourse to transfinite methods by using some other approach (e.g., Zorn's Lemma). But for some arguments, especially ones involving iterative processes, transfinite induction seems the natural avenue. The idea is based on an extension of ordinary mathematical induction and the fact that any nonempty set of positive integers has a smallest element. Recall that a linearly ordered set (5, <) is said to be well ordered if every nonempty subset of S has a smallest element.

It is important to distinguish in the above definition the notion of 'smallest element' from that of an infimum. If B is a nonempty subset of the nonnegative real numbers, then B is bounded below and hence inf B exists. However inf B is a smallest element of B only if inf B e B, a fact which is not generally true.

There is a notion analogous to cardinality for well-ordered sets. Two well-ordered sets A and B are said to have the same order type if there is a one-to-one order-preserving mapping from one onto the other. This leads to a classification of well-ordered sets in terms of special well-ordered sets called Ordinal numbers' which are obtained by extending the natural numbers.


An ordinal number is a well-ordered set a (more precisely, an equivalence class of such sets all having the same order type) which has the property that for each ξ € α, ξ = {η € a : η < ξ}. To see how the natural numbers can be viewed as ordinal numbers, begin with 0 (which satisfies the condition vacuously) and let 1 = {0}, 2 = {0,1}, · · · , n + 1 = {0,1, · · · , n). Viewed this way each n G N is an ordinal number since n has the property that for all A; G n

k = {i G n : i < k}.

Also for each n, n + 1 is the smallest element of the set {x G N : n < x}. Any finite well-ordered set has the same 'order type' as the set {1,2, · · ■ , n} for some natural number n. Now observe that ω = {0,1,2, · · · } is also an ordinal number. One can continue this procedure by defining

ω + l = {0,1,2,·· ω + 2 = {0,1,2,·· ω + n + l = {0,1,2,··

• ,<*>}, • ,ω,ω+1}, ■ ,ω,ω + Ι,- ,ω + η}.

Now let

ω2 = {0,1,2, · · · , α / , ω + Ι , ω + 2 , · · · } .

At this point it is worth observing that both ω and u>2 are countable sets and thus they have the same 'cardinality'. However, ω φ ω2 because they have different order types. There is an easy way to see this. Suppose Γ is an ordinal number and suppose a G Γ. Then by our scheme of definition if a G Γ and if the set {x G Γ : a < x} is not empty, then it has a smallest element called the successor of a, and this element is denoted a + 1. This shows that the nonzero elements of Γ fall into two categories—those which are successors of other elements of Γ and those which are not. If a nonzero element of Γ is not a successor it is called a limit ordinal. To see that ω φ ω2 note that ω does not have any limit ordinals while u>2 has exactly one, namely u>.

It is possible to actually list the first few ordinals as follows (but this is only the beginning!).

0, 1, 2, (j2, ω2 + 1, ω2 + 2,

um, ωη + l, ωη + 2,

ωη, ωη + 1, ωη + 2,

ωω, α>ω + 1, ωω + 2,

£ο, εο + 1, εο + 2,

ω + 1, « 3 + 1 ,

ω2 + 1,

ωη2, ω η 2 + 1 ,

ε0ω, ε0ω + 1,

Examples of limit ordinals in the above diagram include ω,ωη,ω2,ωω,εο, and so forth.

A.7. ORDINAL NUMBERS AND TRANSFINITE INDUCTION 283

To summarize, each well-ordered set has an order type which coincides with the order type of some generic well-ordered set called an ordinal number. Each ordinal number can be thought of as a set consisting of the union of all strictly smaller ordinal numbers. And any ordinal number is either a successor or a limit ordinal.

Here is the principle of transfinite induction. Its usefulness rests on the premise that there exist well-ordered sets of arbitrarily large cardinality.

Proposition A.4 (Principle of transfinite induction) Suppose Γ is an ordinal number and suppose S Ç Γ. Suppose for each β € Γ the following is true.

(i) IfaeS for each a < ß then ß € S. Then S = Γ.

The proof of this fact is surprisingly simple. Suppose S φ Γ. Then the set U — {7 € Γ : 7 £ 5} is nonempty and thus has a smallest element 70. Since β € U, 7 0 < β. But 7 0 < β implies j 0 G S by (i)—a contradiction. On the other hand, 7 0 = β implies β 6 S by (i)—again a contradiction.

In practice it is the verification of (i) that differs from ordinary mathematical induction. It is typical to have to consider two separate cases, (a) β = a +1 for some a € Γ, and (b) β is a limit ordinal. The verification (a) usually mimics the corresponding step in ordinary induction, and indeed, if Γ = ω this is all there is to do and transfinite induction coincides with ordinary induction. (Note that 0 € 5 by assumption since the statement "a 6 S for each a < 0" is vacuously true.) However, it may be the case, in general, that β is a limit ordinal, and the verification in this case usually involves taking intersections or passing to a limit.

The usefulness of transfinite induction stems largely from the following fa-mous theorem, which is due to Zermelo.

Theorem A.4 ( Well-ordering theorem) If S is a set, then there exists an order relation on S that is a we 11-ordering.

It is well known that the above theorem is equivalent to the Axiom of Choice. One of its consequences is that ordinal numbers exist which are larger (in the sense of cardinality) than any given set. In particular (and often this is all that is needed) there exists an uncountable ordinal, and therefore a smallest uncountable ordinal. (Curiously, this weaker fact does not require the full Axiom of Choice.)

We conclude this section with the following useful fact. Let Ω denote the smallest uncountable ordinal. In terms of cardinality discussed in the previous section, |Ω| = Ni.

Proposition A.5 If A is a countable subset ο/Ω then there exists β € Ω such that β > a for all a € Ω.


oo

Proof. Since A is countable one can write A= {αι,α^» · · · }■ Let β = (Ja*. t = l

Then β is a countable ordinal since /3 is a countable union of countable sets. Hence β € Ω, and /? > a n , n = 1,2, · ■ · . ■

The above fact is often used in conjunction with the real numbers as follows.

Proposition A.6 Suppose {xa}açQ ÇK is bounded and either nonincreasing or nondecreasing. Then there exists β such that xa = Χβ for all a > β.

Proof. Suppose {xa}açn is nonincreasing. Then there exists x eR such that

limxQ = x = inf{xQ : a € Ω}. Of

Assume xa φ x for all a 6 Ω (otherwise there is nothing to prove). Then for

each n € N there exist xQn 6 I x, x H— I . By the previous proposition there exists β e Ω such that β > an, n = 1,2, · · · . Then if α > β it must be the case that xa = Χβ = x. ■

A.8 Zermelo's Fixed Point Theorem

Most of the purely constructive aspects of metric fixed point theory (i.e., those which can be proved wholly within the basic axioms of Zermelo and Fraenkel) essentially rely on at least some version of the 1908 result of Zermelo. These results include many of those for nonexpansive mappings discussed in this text. This is an observation that seems to have originated with Fuchssteiner [58].

The following is a revised formulation of Zermelo's Theorem. We use the notation EE to denote the collection of all mappings / : E —» E, and we use Fix(/) to denote the fixed point set of / . (Thus Fix(/) = {x 6 E : f(x) = x}.) Note that implicit in the conclusion of the following theorem is the fact that Fix(/) φ 0 .

Theorem A.5 Let (E, <) be a partially ordered set in which each chain has a least upper bound, and let

ΦΕ = {/ e EE : f(x) > x for each x £ E).

Then there exists a mapping ψ : Φβ —» Φβ such that for each f G Φβ, ψ{}) is a retraction of E onto Fix(/).

Proof. Let / £ Φ#. First we claim that for each x £ E there exists wx G E such that x < wx and f{wx) = wx. To this end, with x € E fixed, we call a set B Ç E (x, /)-admissible if (i) x e B; (ii) f(B) Ç B; (iii) every least upper bound of any totally ordered subset of B lies in B. Note that E itself is (x, / ) -admissible and also that if B is (x, /)-admissible, then {u € B : x < u) is also (x, /)-admissible. Now let

A - (1{B : B is (x,/)-admissible}.

A.8. ZERMELO'S FIXED POINT THEOREM 285

Then A is also (x, /)-admissible, and x / ( u )} ·

The next step is to show that A\ is (x, /)-admissible; hence A\ = A. Obvi-ously x € i4i since x /(it) . If z > f(u) then /(z) > z > f(u). If 2 < u, either 2 / (u ) , and from this we conclude f{A\) Ç Αχ. Now let u e P and let {za} be a totally ordered subset of A\ with 2 = sup zQ. If zQ > f(u) for some a then z > za > f(u). Otherwise za < u for all a from which 2 < u. Thus : £ J 4 I and Αι = Λ.

We now know that if u 6 P and 2 g A , either 2 < u or / (u) < 2. We use this fact to show that P is (x, /)-admissible. We have already observed that x 6 P. Now let u 6 P and y & A and suppose j / < f(u). Then y /Ë/(u) so it must be the case that y < u. If y / (y) < / (u) . This proves that / ( P ) Ç P. Finally, let {ua} be a totally ordered subset of P with u — sup itQ. Then if y < it it must be the case that y < ua for some a from which /(y) /(ω) > it and this, in turn, implies that A is totally ordered. By assumption A has a least upper bound wx. Then since wx 6 A, wx < f(wx) < wx, yielding f(wx) = wx and x < wx. At this point we already have the conclusion of Zermelo's Theorem as originally stated.

Now define φ : Φβ —> Φ Ε by setting, for each / 6 Φ Ε ,

<p(/)(a;) = wx-

Then since wx > x, ip(f)(x) > x so <£>(/) € Φ^. Furthermore, if x € Fix(/), then tp(f)(x) — x. To see this observe that A in the intersection of all (x , / ) -admissible sets and for x € Fix(/), {x} itself is an (x, /)-admissible set. It follows that A = {x} and wx = x. Hence <p(f) leaves each point of Fix(/) fixed, and for arbitrary i G Ê w e have φ(/)(χ) =wxE Fix(/). ■

We have chosen to include the above argument in detail for two reasons. First, it illustrates the power of the Axiom of Choice when compared with the efficient Zorn's Lemma proof given in Section A.3. On the other hand, it illustrates that deep results can be obtained without recourse to the Axiom of Choice.


A.9 A remark about constructive mathematics Although Theorem 1.4 is generally attributed to Brouwer (1912) (the citation is [21]), his correspondence suggests that it was known to him even earlier, and it was almost surely known earlier to others as well, most notably by P. Bohl [14]. It is curious to note, however, that Brouwer later adopted an intuitionist view of logic that led him to reject the Bolzano-Weierstrass Theorem, the premise upon which the theorem bearing his name rests. Brouwer expressed his position as follows:

The belief in the universal validity of the principle of the excluded third in mathematics is considered by the intuitionists as a phe-nomenon of the history of civilization of the same kind as the former belief in the rationality of π, or in the rotation of the firmament about the earth. [22]

The constructive approach grows out of the fundamental idea of interpreting 'existence' in terms of an intuitive notion of 'computability'. We shall not be concerned with this approach here, except to note in passing that there are es-sentially three modern constructive schools of mathematics which enjoy at least a limited following. In addition to Brouwer's intuitionism (I), there is the con-structive approach developed by Erret Bishop (B), and the algorithmic approach of the Russian school founded by Markov (R). The constructive approach which seems closest to the classical approach we shall take is (B). This is because every proposition in (B) has an immediate classical interpretation, and a proof in (B) is also a classical proof, a fact which is true in neither (I) nor (R). In fact, it is possible to pass from (B) to classical mathematics by simply adopting the law of the excluded middle. (This law asserts that given any proposition P, either ' P ' holds or 'not P ' holds.)

Brouwer's Theorem in (B) is the following:

Theorem A.6 If f : B —> B (B the unit ball in R") is uniformly continuous, then for each ε > 0 there exists x € B such that

d(x,f{x))<e.

Obviously this is no longer a 'fixed point theorem'. However fixed point the-ory does exist within (B). If in addition to the above assumptions it is assumed that / is contractive (d(f(x),f(y)) < d{x, y) for each i / y ) then / has a unique fixed point (see [19]).

To illustrate how these three constructive approaches differ, consider the following assertion.

Proposition A.7 Every pointurise continuous function f : [0,1] —» R is uni-formly continuous.

This proposition is true in (I), false in (R), and unprovable in (B).

EXERCISES 287

For a detailed account of the constructive approaches, see Bridges and Rich-man [18].

Exercises

Exercise A . l If X is compact and if {xa} is an ultranet in X, then {xa} converges to a point of X.

Exercise A.2 Now use (1), {2), and (3) to prove Tychonoff's Theorem {The-orem A. 3).

Exercise A.3 Prove that if a set S has n elements, then the set of all subsets of S has 2n elements. [Try induction, and remember that 0 is a subset of S.}

Exercise A.4 Prove there is a one-to-one correspondence between V{S) and the set 2s of all functions from S into the two-element set {0,1}. Hint: For each A € ~P(S) consider the characteristic function χΑ : S —► {0,1} defined by

i s _ / lifxeA; XA{X) \ 0 otherwise.

Bibliography

[1] A. G. Aksoy and M. A. Khamsi, Nonstandard Methods in Fixed Point Theory, Springer-Verlag, New York, Berlin, 1990, 139 pp.

[2] D. E. Alspach, A fixed point free nonexpansive map, Proc. Amer. Math. Soc. 82(1981), 423-424.

[3] N. Aronszajn and P. Panitchpakdi, Extensions of uniformly continuous transformations and hyperconvex metric spaces, Pacific J. Math. 6(1956), 405-439.

[4] J. S. Bae and S. Park, Remarks on the Caristi-Kirk fixed point theorem, Bull. Korean Math. Soc. 19(1983), 57-60.

[5] J.-B. Bâillon, Nonexpansive mapping and hyperconvex spaces, in Fixed Point Theory and Its Applications (R. F. Brown, ed.), Contemporary Math. vol. 72, Amer. Math. Soc, Providence, 1988, pp. 11-19.

[6] J.-B. Bâillon and R. Brück, The rate of asymptotic regularity is 0(—), in n

Theory and Applications of Nonlinear Operators of Accretive and Mono-tone Type (A. G. Kartsatos, ed), Marcel Dekker, New York, 1996, pp. 51-81.

[7] S. Banach, Sur les opérations dans les ensembles abstraits et leurs appli-cations, Fund. Math. 3(1922), 133-181.

[8] J. Banas and K. Goebel, Measures of Noncompactness in Banach Spaces, Marcel Dekker, New York and Basel, 1980.

[9] B. Beauzamy, Introduction to Banach Spaces and Their Geometry, 2nd revised edition, North-Holland, Amsterdam, New York, Oxford, 1985.

[10] C. Bessaga and A. Pelczynski, Selected topics in infinite-dimensional topology, Monografie Matematyczne 58(1957), p. 353.

[11] L. M. Blumenthal, Distance Geometries: A Study of the Development of Abstract Metrics, Univ. of Missouri Studies, vol. XIII, 1938.

289



290 BIBLIOGRAPHY

[12] L. M. Blumenthal, Theory and Applications of Distance Geometry, Claren-don Press, Oxford, 1953.

[13] L. M. Blumenthal and K. Menger, Studies in Geometry, W. H. Freeman, San Francisco, 1970.

[14] P. Bohl, Über die Bewegung eines mechanischen Systems in der Nähe einer Gleichgewichtslage, J. Reine Angew. Math. 127(1904), 179-276.

[15] J. M. Borwein, P. B. Borwein and D. H. Bailey, Ramanujan, modular equations, and approximations to pi or how to compute one billion digits of pi, Amer. Math. Monthly 96(1989), 201-219.

[16] J. M. Borwein and B. Sims, Non-expansive mappings on Banach lattices and related topics, Houston J. Math. 10(1984), 339-356.

[17] D. W. Boyd and J. S. W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc. 20(1969), 458-464.

[18] D. Bridges and F. Richman, Varieties of Constructive Mathematics, Lon-don Math. Soc. Lecture Note Series, no. 97, Cambridge Univ. Press, Cam-bridge, 1987.

[19] D. Bridges, F. Richman, W. H. Julian, and R. Mines, Extensions and fixed points of contractive maps in Rn, J. Math. Anal. Appl. 165(1992), 438-456.

[20] M. S. Brodskii and D. P. Milman, On the center of a convex set, Dokl. Akad. Nauk SSSR 59(1948), 837-840. (Russian)

[21] L. E. J. Brouwer, Über Abbildungen von Mannigfaltigkeiten, Math. Ann. 71(1912), 97-115.

[22] L. E. J. Brouwer, Brouwer's Cambridge Lectures on Intuitionism (D. Van Dalen, ed.), Cambridge Univ. Press, Cambridge, 1981.

[23] F. E. Browder, On the convergence of successive approximations for non-linear functional equations, Nederl. Akad. Wetensch. Ser. A71 = Indag. Math. 30(1968), 27-35.

[24] F. E. Browder, Semicontractive and semiaccretive nonlinear mappings in Banach spaces, Bull. Amer. Math. Soc. 74(1968), 660-665.

[25] F. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sei. U.S.A. 54(1965), 1041-1044.

[26] F. E. Browder, Nonlinear Operators and Nonlinear Equations of Evolu-tion, Proc. Sympos. Pure Math. XVII I , Amer. Math. Soc, Providence, 1976.

BIBLIOGRAPHY 291

[27] F. E. Browder, Remarks on fixed point theorems of contractive type, Non-linear Anal. 3(1979), 657-661.

[28] F. E. Browder, Fixed point theory and nonlinear problems, Bull. Amer. Math. Soc. 9(1983), 1-39.

[29] R. E. Brück, Properties of fixed-point sets of nonexpansive mappings in Banach spaces, Trans. Amer. Math. Soc. 179(1973), 251-262.

[30] R. E. Brück, A common fixed point theorem for a commuting family of nonexpansive mappings, Pacific J. Math. 53(1974), 59-71.

[31] R. E. Brück, Structure of the approximate fixed point sets in general Banach spaces, Fixed Point Theory and Applications (J.-B. Bâillon and M. Théra, eds.), Longman Scientific & Technical, Essex, 1991, pp. 91-96.

[32] N. Brunner, Topologische maximalprinzipien, Z. Math. Logik Grundlag. Math. 33(1987), 135-139.

[33] V. Bryant, Metric Spaces, Iteration and Application, Cambridge Univ. Press, Cambridge, 1985.

[34] I. Bula, On the stability of the Bohl-Brouwer-Schauder theorem, Nonlinear Anal. 26(1996), 1859-1868.

[35] G. Buskes and A. van Rooij, Topological Spaces, Springer, New York, Berlin, 1997.

[36] J. Caristi, The Fixed Point Theory for Mappings Satisfying Inwardness Conditions, Univ. of Iowa, Ph.D. Dissertation, May 1975.

[37] J. Caristi, Fixed point theorems for mappings satisfying inwardness con-ditions, Trans. Amer. Math. Soc. 215(1976), 241-251.

[38] J. A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc. 40(1936), 396-414.

[39] T. M. Dalby, Facets of the Fixed Point Theory for Nonexpansive Mappings, Ph.D. Thesis, University of Newcastle, 1997.

[40] L. Danzer, B. Grünbaum, and V. Klee, Helly's theorem and its relatives, in Convexity (V. L. Klee, ed.), Proc. Sympos. Pure Math. VII, Amer. Math. Soc, Providence, 1963, pp. 101-177.

[41] G. Darbo, Punti uniti in trasformazioni a codominio non compatto, Rend. Sem. Mat. Univ. Padova 24(1955), 84-92.

[42] M. M. Day, R. C. James, and S. Swaminathan, Normed linear spaces that are uniformly convex in every direction, Canad. J. Math. 23(1971), 1051-1059.

292 BIBLIOGRAPHY

[43] R. Deville, G. Godefroy and V. Zizler, Smoothness and Renormings in Banach Spaces, Pitman Monographs Pure Appl. Math. 63, Longman Sci-entific k Technical, Essex, 1993.

[44] C. L. DeVito, Functional Analysis, Academic Press, New York, 1978.

[45] J. Diestel, Geometry of Banach Spaces—Selected Topics, Springer Lecture Notes in Math. no. 485, Springer, Berlin, New York, 1957.

[46] P. N. Dowling and C. J. Lennard, Every nonreflexive subspace of Li[0,1] fails the fixed point property, Proc. Amer. Math. Soc. 125(1997), 443-446.

[47] P. N. Dowling, C. J. Lennard, and B. Turett, Reflexivity and the fixed point property for nonexpansive mappings, J. Math. Anal. Appl 200(1996), 653-662.

[48] D. Downing and W. A. Kirk, A generalization of Caristi's theorem with applications to nonlinear mapping theory, Pacific J. Math. 69(1977), 339-346.

[49] J. Dugundji and A. Granas, Fixed Point Theory, Polska Akademia Nauk, Instytut Matematyczny, PWN-Polish Scientific Publ., Warszawa, 1982.

[50] D. van Dulst, Reflexive and Super-reflexive Banach Spaces, Mathematical Centre Tracts 102, Amsterdam, 1978.

[51] D. van Dulst and B. Sims, Fixed points of nonexpansive mappings and Chebyshev centers in Banach spaces with norms of type (KK), in Lecture Notes in Mathematics 991, Springer, New York, 1983, pp. 91-95.

[52] N. Dunford and J. Schwartz, Linear Operators, Part I: General Theory, Wiley Interscience, New York, 1957.

[53] A. Dvoretzky, Some results on convex bodies and Banach spaces, Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960), Jerusalem Academic Press, Jerusalem, 1961, pp. 123-160.

[54] J. Elton, Pei-Kee Lin, E. Odell and S. Szarek, Remarks on the fixed point problem for nonexpansive maps, Fixed Points and Nonexpansive Mappings (R. Sine, ed.), Contemp. Math., vol. 18, Amer. Math. Soc, Princeton, 1983, pp. 87-120.

[55] P. Enflo, Banach spaces which can be given an equivalent uniformly convex norm, Israel J. Math. 13(1972), 281-288.

[56] R. Espinola, Darbo-Sadovski's theorem in hyperconvex metric spaces, Rend. Cire. Mat. Palermo (2) Suppl. 40(1996), 129-137.

[57] J. Franklin, Methods of Mathematical Economics, Springer-Verlag, New York, Heidelberg, Berlin, 1980.

BIBLIOGRAPHY 293

B. Fuchssteiner, Iterations and fixpoints, Pacific J. Math. 68(1977), 73-80.

J. Garcia-Falset, The fixed point property in Banach spaces whose charac-teristic of uniform convexity is less than 2, J. Austral. Math. Soc. 54(1993), 169-173.

J. Garcia-Falset, Stability and fixed points for nonexpansive mappings, Houston J. Math. 20(1994), 495-505.

J. Garcia-Falset and B. Sims, Property (M) and the weak fixed point property, Proc. Amer. Math. Soc. 125(1997), 2891-2896.

J. Garcia-Falset, B. Sims and M. Smyth, The demiclosedness principle for mappings of asymptotically nonexpansive type, Houston J. Math. 22(1996), 101-108.

M. A. Geraghty, On contractive mappings, Proc. Amer. Math. Soc. 40(1973), 604-608.

A. Gillespie and B. Williams, Fixed point theorem for nonexpansive map-pings on Banach spaces, Applicable Anal. 9(1979), 121-124.

K. Goebel, Convexity of balls and fixed point theorems for mappings with nonexpansive square, Compositio Math. 22(1970), 269-274.

K. Goebel, On the structure of minimal invariant sets for nonexpansive mappings, Ann. Univ. Mariae Curie-Sklodowska Sect. A 29(1975), 73-77.

K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cam-bridge Univ. Press, Cambridge, 1990, 244 pp.

K. Goebel and W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 35(1972), 171-174.

K. Goebel and T. Komorowski, Retracting balls onto spheres and mini-mal displacement problem, in Fixed Point Theory and Applications (J.-B. Bâillon and M. Théra, eds.), Longman Scientific, Essex, 1991, pp. 155— 172.

K. Goebel, W. A. Kirk, and R. L. Thele, Uniformly lipschitzian families of transformations in Banach spaces, Canad. J. Math. 26(1974), 1245-1256.

D. Göhde, Zum Prinzip der Kontraktiven Abbildung, Math. Nachr. 30(1965), 251-258.

O. Hanner, On the uniform convexity of Lp and £p, Ark. Mat. 3(1956), 239-244.

S. Heinrich, Ultraproducts in Banach space theory, J. Reine Angew. Math. 313(1980), 72-104.

294 BIBLIOGRAPHY

[74] R. Huff, Banach spaces which are nearly uniformly convex, Rocky Moun-tain J. Math. 10(1980), 743-749.

[75] J. R. Isbell, Six theorems about injective metric spaces, Comment. Math. Helv. 39(1964), 65-76.

[76] S. Ishikawa, Fixed points and iteration of a nonexpansive mapping in a Banach space, Proc. Amer. Math. Soc. 59 (1976), 65-71.

[77] J. R. Jachymski, An extension of A. Ostrowski's theorem on the round-off stability of iterations, Aequationes Math. 53(1997), 242-253.

[78] R. C. James, A non-reflexive Banach space isometric with its second con-jugate space, Proc. Nat. Acad. Sei. U.S.A. 37(1951), 174-177.

[79] R. C. James, Reflexivity and the supremum of linear functionals, Ann. of Math. (2) 66(1957), 159-169.

[80] R. C. James, Weak compactness and reflexivity, Israel J. Math. 2(1964), 101-119.

[81] R. C. James, Uniformly non-square Banach spaces, Ann. of Math. (2) 80(1964), 542-550.

[82] R. C. James, Super-reflexive Banach spaces, Canad. J. Math. 24(1972), 896-904.

[83] J. Jaworowski, W. A. Kirk, and S. Park, Antipodal Points and Fixed Points, Research Inst. of Math. GARC, Seoul, Korea, 1995.

[84] T. Jech, The Axiom of Choice, North-Holland, Amsterdam, London, 1973.

[85] A. Jiménez-Melado and E. Lloréns-Fuster, A sufficient condition for the fixed point property, Nonlinear Anal. 20(1993), 849-853.

[86] A. Jiménez-Melado and E. Lloréns-Fuster, The fixed point property for some uniformly nonsquare Banach spaces, Boll. Un. Mat. Ital. A (7) 10(1996), 587-595.

[87] I. Kaplansky, Set Theory and Metric Spaces, Allyn and Bacon, Boston, 1972.

[88] L. Karlovitz, Existence of fixed points for nonexpansive mappings in spaces without normal structure, Pacific J. Math. 66(1976), 153-156.

[89] J. L. Kelley, Banach spaces with the extension property, Trans. Amer. Math. Soc. 72(1952), 323-326.

[90] J. L. Kelley, General Topology, Van Nostrand, Princeton, Toronto, Lon-don, 1955.

BIBLIOGRAPHY 295

M. A. Khamsi, On metric spaces with uniform normal structure, Proc. Amer. Math. Soc. 106(1989), 723-726.

M. A. Khamsi, One-local retract and common fixed point for commuting mappings in metric spaces, Nonlinear Anal. 27(1996), 1307-1313.

M. A. Khamsi, KKM and Ky Fan theorems in hyperconvex metric spaces, J. Math. Anal. Appl. 204(1996), 298-306.

M. A. Khamsi and Ph. Turpin, Fixed points of nonexpansive mappings in Banach lattices, Proc. Amer. Math. Soc. 105(1989), 102-110.

Tae Hwa Kim and W. A. Kirk, Fixed point theorems for lipschitzian mappings in Banach spaces, Nonlinear Anal. 26(1996), 1905-1911.

W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72(1965), 1004-1006.

W. A. Kirk, Krasnosel'skiï's iteration process in hyperbolic space, Numer. Funct. Anal. Optimiz. 4(1981/82), 371-381.

W. A. Kirk, Nonexpansive mappings in separable product spaces, Boll. Un. Mat. Ital. A (7) 9(1995), 239-244.

W. A. Kirk, The fixed point property and mappings which are eventually nonexpansive, in Theory and Applications of Nonlinear Operators of Ac-cretive and Monotone Type (A. G. Kartsatos, ed.), Marcel Dekker, New York, 1996, pp. 141-147.

W. A. Kirk, Some questions in metric fixed point theory, in Recent Ad-vances on Metric Fixed Point Theory (Tomas Domfnguez Benavides, ed.), Univ. de Sevilla, 1996, pp. 73-97.

W. A. Kirk, C. Martinez Yanez, and S. S. Shin, Asymptotically nonex-pansive mappings, Nonlinear Anal. 33(1998), 1-12.

V. Klee, Some topological properties of convex sets, Trans. Amer. Math. Soc. 78(1955), 30-45.

V. Klee, Stability of the fixed point property, Colloq. Math. V I I I (1961), 43-46.

B. Knaster, C. Kuratowski, and S. Mazurkiewicz, Ein Beweis des Fix-punktsatzes für n-dimensional simplexe, Fund. Math. 14(1929), 132-137.

A. N. Kolmogorov and S. V. Fomin, Functional Analysis, Vol. I: Metric and Normed Spaces, Graylock Press, New York, 1957.

J. Kulesza and T. C. Lim, On weak compactness and countable weak compactness in fixed point theory, Proc. Amer. Math. Soc. 124(1996), 3345-3349.

296 BIBLIOGRAPHY

[107] W. Kulpa, The Poincare-Miranda Theorem, Amer. Math. Monthly 104(1997), 545-550.

108] K. Kuratowski, Topologie II, Warsaw, 1950.

109] H. E. Lacey, The Isometric Theory of Classical Banach Spaces, Springer-Verlag, Berlin, Heidelberg, New York, 1974.

110] R. Larsen, Functional Analysis, Marcel Dekker, New York, 1973.

I l l ] C. J. Lennard, A new convexity property that implies a fixed point prop-erty for Li, Studia Math. 100(1991), 95-108.

112] T. C. Lim and H. K. Xu, Fixed point theorems for mappings of asymp-totically nonexpansive type, Nonlinear Anal. 22 (1994), 1345-1355.

1131 P· K. Lin, Unconditional bases and fixed points of nonexpansive mappings, Houston J. Math. 116(1985), 69-76.

114] P. K. Lin, The Browder-Göhde property in product spaces, Pacific J. Math. 13(1987), 235-240.

115] M. G. Maia, Un'osservazione sulle contrazioni metriche, Rend. Sem. Mat. Univ. Padova 40(1968), 139-143.

1161 E. Maluta, Uniformly normal structure and related coefficients, Pacific J. Math. 111(1984), 357-369.

1171 R. Manka, Some forms of the axiom of choice, Jahrb. Kurt-Gödel-Ges.. vol. 1, 1988, pp. 24-34.

118] J. Matkowski, Integrable solutions of functional equations, Diss. Math. 127, Warsaw, 1975.

1191 J· Matkowski, Nonlinear contractions in metrically convex spaces, Publ. Math. Debrecen 45(1994), 103-114.

120] D. Mauldin (ed.), The Scottish Book: Mathematical Problems from the Scottish Cafe, Birkhauser, Boston, 1981.

1211 B. Maurey, Points fixes des contractions sur un convexe fermé de L\, Sém-inaire d'Analyse Fonctionelle, vols. 80-81, Ecole Polytechnique Palaiseau, 1981.

122] P. Meyer-Nieberg, Banach Lattices, Springer-Verlag, Berlin, Heidelberg, New York, 1991.

1231 J· Mikusinski and P. Mikusinski, An Introduction to Analysis, Wiley, New York, 1993.

124] A. F. Monna, Analyse Non-archimedienne, Springer-Verlag, Berlin, Hei-delberg, New York, 1970.

BIBLIOGRAPHY 297

125] G. H. Moore, Zermelo's Axiom of Choice; Its Origins, Development, and Influence, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1982.

126] J. R. Munkres, Topology, A First Course, Prentice-Hall, Englewood Cliffs, NJ, 1975.

1271 S. B. Nadler, Jr., Multi-valued contraction mappings, Pacific J. Math. 30(1969), 475-488.

128] L. Narici, E. Beckenstein, and G. Bachman, Functional Analysis and Val-uation Theory, Marcel Dekker, New York, 1971.

1291 W. Oettli and M. Théra, Equivalents of Ekeland's principle, Bull. Austral. Math. Soc. 48(1993), 385-392.

1301 G. Oldani and D. Roux, Convexity structures and Kannan maps, in Non-linear Functional Analysis and Its Applications (S. P. Singh, ed.), D. Rei-del, Dordrecht, Boston, Lancaster, Tokyo, 1986, pp. 327-333.

1311 A. M. Ostrowski, The round off stability of iterations, Z. Angew. Math. Mech. 47(1967), 77-81.

1321 C. Petalas and T. Vidalis, A fixed point theorem in non-archimedean vector spaces, Proc. Amer. Math. Soc. 118(1993), 819-821.

1331 S. Prieß-Crampe, Der Banachsche Fixpunktsatz für ultrametrische Räume, Results Math. 18(1990), 178-186.

1341 S. Prieß-Crampe, Some results of functional analysis for ultrametric spaces and valued vector spaces, Geom. Dedicata 58(1995), 79-90.

1351 S. Prus, Banach spaces which are uniformly noncreasy, Nonlinear Anal. 30(1997), 2317-2324.

1361 E. Rakotch, A note on contractive mappings, Proc. Amer. Math. Soc. 13(1962), 459-465.

1371 W. O. Ray, The fixed point property and unbounded sets in Hubert space, Trans. Amer. Math. Soc. 258(1980), 531-537.

138] W. O. Ray and R. Sine, Nonexpansive mappings with precompact orbits, Fixed Point Theory (E. Fadell and G. Fournier, eds.), Lecture Notes in Math., no. 886, Springer-Verlag, New York, Berlin, 1981, pp. 409-416.

139] W. O. Ray, Real Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1988.

140] S. Reich, The almost fixed point property for nonexpansive mappings, Proc. Amer. Math. Soc. 88(1983), 44-46.

[141] A. C. M. van Rooij, Non-archimedean Functional Analysis, Marcel Dekker, New York, Basel, 1978.

298 BIBLIOGRAPHY

142] H. Rosenthal, Some remarks concerning unconditional basic sequences, Longhorn Notes, Texas Functional Analysis Seminar 1982-83, The Uni-versity of Texas, Austin, pp. 15-47.

143] S. Russ, Bolzano's analytic programme, Math. Intelligencer 14, no. 3 (1992), 45-53.

144] B. Sadovskiï, On a fixed point principle, Funkcional. Anal, i Prilozen. 1(1967), 74-76 (Russian).

145] I. Shafrir, The approximate fixed point property in Banach and hyperbolic spaces, Israel J. Math. 71(1990), 211-223.

146] Yu. A. Shashkin, Fixed Points, Mathematical World, vol.2, Amer. Math. Soc, Providence, 1991.

147] B. Sims, " Ultra"-techniques in Banach Space Theory, Queens's Papers in Pure and Appl. Math., no. 160, Queen's University, Kingston, Ontario, 1982, 117 pp.

148] B. Sims, Orthogonality and fixed points of nonexpansive maps, Proc. Cen-tre Math. Anal. Austral. Nat. Univ. 20 (1988), 178-186.

149] W. Sierpinski, General Topology (2nd ed.) Univ. of Toronto, Totonto, 1952.

150] R. Sine, On nonlinear contractions in sup norm spaces, Nonlinear Anal 3(1979), 885-890.

151] R. Sine, Hyperconvexity and approximate fixed points, Nonlinear Anal 13(1989), 863-869.

152] V. Smulian, On the principle of inclusion in the space of type (B), Mat. Sb. (N.S.) 5(1939), 317-328.

153] P. Soardi, Existence of fixed points of nonexpansive mappings in certain Banach lattices, Proc. Amer. Math. Soc. 73(1979), 25-29.

[154] E. Sperner, Neuer Beweis für die Invarianz der Dimensionszahl und des Gebietes, Abh. Math. Sem. Univ. Hamburg 6(1928), 265-272.

[155] J. Stern, Propriétés locales et ultrapuissances d'espaces de Banach, Ex-posés 7 et 8, Séminaire Maurey-Schwartz 1974-75, Ecole Polytechnique, Palaiseau, France.

[156] F. Sullivan, Ordering and completeness of metric spaces, Nieuw Arch. Wisk. (3) 29(1981), 178-193.

[157] K. K. Tan and H. K. Xu, Fixed point iteration processes for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 122(1994), 733-739.

BIBLIOGRAPHY 299

1581 J· Tits, A 'Theorem of Lie-Kolchin Type' for trees, Contributions to Alge-bra: A Collection of Papers Dedicated to Ellis Kolchin, Academic Press, New York, 1977, pp. 377-388.

1591 S. Troyanski, On locally uniformly convex and differentiable norms in certain nonseparable Banach spaces, Studia Math. 37(1971), 173-180.

1601 B. Turett, A dual view of a theorem of Bâillon, Nonlinear Analysis and Applications (S. P. Singh and J. H. Burry, eds.), Lecture Notes in Pure and Appl. Math., vol. 80, Marcel Dekker, New York, 1982, pp. 279-286.

1611 A. Tychonoff, Über die topologische Erweiterung von Räumen, Math. Ann. 102(1930), 544-561.

1621 W. Walter, Remarks on a paper by F. Browder about contraction, Non-linear Anal. 5(1981), 21-25.

1631 C. S. Wong, Close-to-normal structure and its applications, J. Fund. Anal. 16(1974), 353-358.

1641 H. K. Xu, Existence and convergence for fixed points for mappings of asymptotically nonexpansive type, Nonlinear Anal. 16(1991), 1139-1146.

1651 H. K. Xu, Geometrical coefficients of Banach spaces and nonlinear mappings, in Recent Advances on Metric Fixed Point Theory (Tomas Domfnguez Benavides, ed.), Univ. de Sevilla, Seville, 1996, pp. 161-178.

1661 E. Zeidler, Nonlinear Functional Analysis and its Applications I: Fixed-Point Theorems, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1986.

1671 E. Zeidler, Nonlinear Functional Analysis and Its Applications IV: Ap-plications to Mathematical Physics, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1988.

168] E. Zermelo, Neuer Beweis für die Möglichkeit einer Wohlordung, Math. Anal. 65(1908), 107-128.

1691 E. Zermelo, Untersuchungen über die Grundlagen der Mengenlehre I, Math. Ann. 65(1908), 199-215.

1701 M. Zorn, A remark on method in transfinite algebra, Bull. Amer. Math. Soc. 41(1935), 667-670.

Index

absolute value, 7 accumulation point, 4 adjoint operator, 145 admissible set, 36, 84 archimedean valuation, 119 asymptotic center, 222 asymptotically nonexpansive map-

ping, 262 asymptotically regular mapping, 219 asynptotic radius, 222 axiom of choice, 276

Baire Category Theorem, 151 Banach lattice, 167 Banach space, 128 basic sequence, 154 basis constant, 157 bimonotone basis, 157 binary ball intersection property, 79 biorthogonal functionals, 157 Bolzano Weierstrass Theorem, 4 boundedly complete, 158 Brouwer's Fxed Point Theorem, 10

Cantor's Intersection Theorem, 27 cardinal numbers, 280 cartesian product, 273 Cauchy sequence, 26 Cauchy-Schwarz inequality, 9 characteristic of convexity, 137 Chebyshev center, 80 Closed Graph Theorem, 154 closed set, 16 compact set, 17 complete space, 26 completion, 29, 130 condensing mapping, 190

Contraction Mapping Principle, 41 convex closure, 131 convex set, 131 convexity structure, 35 countably compact structure, 106

degree of a mapping, 185 demiclosed mapping, 263 dense set, 29 diameter, 80 directed set, 274 dual space, 142

equivalent bases, 155 equivalent norms, 129 euclidean space, 19 externally hyperconvex spaces, 88

finite representability, 244 finitely representable, 243 FPP, 114

Goebel-Karlovitz Lemma, 212 greatest lower bound, 4

h-nonexpanisve mapping, 208 Hahn-Banach Theorem, 143 Hausdorff metric, 24 Helly's Theorem, 72 Hubert space, 20 hyperconvex hull, 97 hyperconvex space, 78

injective envelope, 78, 97 injective space, 77 inner product, 9, 20 inner product space, 133 Intermediate Value Theorem, 6



302 INDEX

isometry, 15

James space, 130, 240

KKM Theorem, 173

least upper bound, 3 limit point, 16 linear functional, 142

Maximum Value Theorem, 6 measure of noncompactness, 187 metric convex hull, 36 metric convexity, 35 metric segment, 35 metric space, 14 metric topology, 15 metric transform, 24 modulus of convexity, 136 monotone basis, 157 motion, 15

natural projection, 155 net, 248 nonexpansive mapping, 77 nonexpansive retraction, 74 normal structure, 102 normal structure, Banach spaces, 200 normed linear space, 128

open set, 16 ordinal number, 282 orthogonal convexity, 214

p-adic valuation, 119 parallelogram law, 133 partially ordered set, 277 partition of unity, 181 polarization identity, 133 precompact set, 18

quasi-normal structure, 112

radius, 80 reflexive, 144 regular sequence, 222 retraction, 176 Riesz angle, 226

Sadovskii's Fixed Point Theorem, 190 Schauder basis, 154 Schauder's Fixed Point Theorem, 179 Schroder-Bernstein Theorem, 280 Schur property, 146 Schur's lemma, 152 semimetric space, 13 separable space, 33 Separation Theorem, 145 shrinking basis, 158 Sperner's Lemma, 173 spherically complete space, 117 Stone-Weierstrass Theorem, 183 strictly convex norm, 138 super-reflexive space, 244 support, 155

transfinite induction, 283 triangle inequality, 7 Tychonoff's Theorem, 280

ultrametric space, 116 ultranet, 248 ultrapower (Banach space), 250 unconditional basis, 161 unconditional basis constant, 162 unconditional constant, 162 unconditionally monotone, 162 uniform boundedness principle, 149 uniform continuity, 18 uniform convexity, 138 uniform normal structure, 106 uniform relative normal structure,

110 uniformly monotone lattice, 233 uniformly noncreasy spaces, 266 uniformly nonsquare space, 138, 247 uniformly smooth space, 265 upper bound, 3

weak compactness, 144 weak normal structure, 200 weak orthogonality, 226 weak topology, 144 weak* topology, 144 well-ordered set, 274

Zermelo's Fixed Point Theorem, 275

PURE AND APPLIED MATHEMATICS A Wiley-Interscience Series of Texts, Monographs, and Tracts

Founded by RICHARD COURANT Editors: MYRON B. ALLEN III, DAVID A. COX, PETER LAX Editors Emeriti: PETER HILTON and HARRY HOCHSTADT, JOHN TOLAND

ADÂMEK, HERRLICH, and STRECKER—Abstract and Concrete Catetories ADAMOWICZ and ZBIERSKI—Logic of Mathematics AINSWORTH and ODEN—A Posteriori Error Estimation in Finite Element Analysis AKIVIS and GOLDBERG—Conformai Differential Geometry and Its Generalizations ALLEN and ISAACSON—Numerical Analysis for Applied Science

*ARTIN—Geometric Algebra AUBIN—Applied Functional Analysis, Second Edition AZIZOV and IOKH VIDOV—Linear Operators in Spaces with an Indefinite Metric BERG—The Fourier-Analytic Proof of Quadratic Reciprocity BERMAN, NEUMANN, and STERN—Nonnegative Matrices in Dynamic Systems BOYARINTSEV—Methods of Solving Singular Systems of Ordinary Differential

Equations BURK—Lebesgue Measure and Integration: An Introduction

*C ARTER—Finite Groups of Lie Type CASTILLO, COBO, JUBETE and PRUNEDA—Orthogonal Sets and Polar Methods in

Linear Algebra: Applications to Matrix Calculations, Systems of Equations, Inequalities, and Linear Programming

CHATELIN—Eigenvalues of Matrices CLARK—Mathematical Bioeconomics: The Optimal Management of Renewable

Resources, Second Edition COX—Primes of the Form x2 + ny2: Fermât, Class Field Theory, and Complex

Multiplication "CURTIS and REINER—Representation Theory of Finite Groups and Associative Algebras *CURTIS and REINER—Methods of Representation Theory: With Applications to Finite

Groups and Orders, Volume I CURTIS and REINER—Methods of Representation Theory: With Applications to Finite

Groups and Orders, Volume II DINCULEANU—Vector Integration and Stochastic Integration in Banach Spaces

*DUNFORD and SCHWARTZ—Linear Operators Part 1—General Theory Part 2—Spectral Theory, Self Adjoint Operators in

Hubert Space Part 3—Spectral Operators

FARINA and RINALDI—Positive Linear Systems: Theory and Applications FOLLAND—Real Analysis: Modern Techniques and Their Applications FRÖLICHER and KRIEGL—Linear Spaces and Differentiation Theory GARDINER—Teichmüller Theory and Quadratic Differentials GREENE and KRANTZ—Function Theory of One Complex Variable

»GRIFFITHS and HARRIS—Principles of Algebraic Geometry GRILLET—Algebra GROVE—Groups and Characters GUSTAFSSON, KREISS and ÖLIGER—Time Dependent Problems and Difference

Methods



HANNA and ROWLAND—Fourier Series, Transforms, and Boundary Value Problems, Second Edition

*HENRIC1—Applied and Computational Complex Analysis Volume 1, Power Series—Integration—Conformai Mapping—Location

of Zeros Volume 2, Special Functions—Integral Transforms—Asymptotics—

Continued Fractions Volume 3, Discrete Fourier Analysis, Cauchy Integrals, Construction

of Conformai Maps, Univalent Functions ♦HILTON and WU—A Course in Modern Algebra »HOCHSTADT—Integral Equations JOST—Two-Dimensional Geometric Variational Procedures KHAMSI and KIRK—An Introduction to Metric Spaces and Fixed Point Theory

*KOBAYASHI and NOMIZU—Foundations of Differential Geometry, Volume I ♦KOBAYASHI and NOMIZU—Foundations of Differential Geometry, Volume II LAX—Linear Algebra LOGAN—An Introduction to Nonlinear Partial Differential Equations McCONNELL and ROBSON—Noncommutative Noetherian Rings MORRISON—Functional Analysis: An Introduction to Banach Space Theory NAYFEH—Perturbation Methods NAYFEH and MOOK—Nonlinear Oscillations PANDEY—The Hubert Transform of Schwartz Distributions and Applications PETKOV—Geometry of Reflecting Rays and Inverse Spectral Problems

♦PRENTER—Splines and Variational Methods RAO—Measure Theory and Integration RASSIAS and SIMSA—Finite Sums Decompositions in Mathematical Analysis RENELT—Elliptic Systems and Quasiconformal Mappings RIVLIN—Chebyshev Polynomials: From Approximation Theory to Algebra and Number

Theory, Second Edition ROCKAFELLAR—Network Flows and Monotropic Optimization ROITMAN—Introduction to Modern Set Theory

*RUDIN—Fourier Analysis on Groups SENDOV—The Averaged Moduli of Smoothness: Applications in Numerical Methods

and Approximations SENDOV and POPOV—The Averaged Moduli of Smoothness

♦SIEGEL—Topics in Complex Function Theory Volume I—Elliptic Functions and Uniformization Theory Volume 2—Automorphic Functions and Abelian Integrals Volume 3—Abelian Functions and Modular Functions of Several Variables

SMITH and ROMANOWSKA—Post-Modern Algebra STAKGOLD—Green's Functions and Boundary Value Problems, Second Editon

♦STOKER—Differential Geometry ♦STOKER—Nonlinear Vibrations in Mechanical and Electrical Systems ♦STOKER—Water Waves: The Mathematical Theory with Applications WESSELING—An Introduction to Multigrid Methods

'WHITHAM—Linear and Nonlinear Waves fZAUDERER—Partial Differential Equations of Applied Mathematics, Second Edition

♦Now available in a lower priced paperback edition in the Wiley Classics Library. *Now available in paperback.

Documents

An Introduction to Metric Spaces and Fixed Point Theory