Applied Mathematical Sciences Volume 108

Editors J.E. Marsden L. Sirovich

Advisors s. Antman J.K. Hale P. Holmes T. Kambe J. Keller K. Kirchgiissner B.J. Matkowksy C.S. Peskin

Springer Science+ Business Media, LLC

Eberhard Zeidler

Applied Functional Analysis Applications to Mathematical Physics

With 56 Illustrations

Springer

Eberhard Zeidler Max-Planck-Institut fur Mathematik

in den Naturwissenschaften InselstraBe 22-26 D-04103 Leipzig Germany

Editors J.E. Marsden Control and

L. Sirovich Division of

Dynamical Systems, 107-81 California Institute of Technology Pasadena, CA 91125 USA

Applied Mathematics Brown U niversity Providence, RI 02912 USA

Mathematics Subject Classification (1991): 34A12, 42A16, 35J05

Library of Congress Cataloging-in-Publication Data Zeidler, Eberhard

Applied functional analysis : applications to mathematical physics / Eberhard Zeidler

p. cm. - (Applied mathematical sciences ; 108) Includes bibliographical references and index.

1. Functional analysis. 2. Mathematical physics. 1. Title. II. Series: Applied mathematical sciences (Springer-Verlag New York Inc.) ; v. 108. QA1.A647 voI. 108 [QA3201 510 s--dc20 94-43219 [515'.71

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ISBN 978-1-4612-6910-6 SPIN 10738833

ISBN 978-1-4612-6910-6 ISBN 978-1-4612-0815-0 (eBook)DOI 10.1007/978-1-4612-0815-0

To My Students

Textbooks should be attractive by showing the beauty ofthe subject.

Johann Wolfgang von Goethe (1749-1832)

I am not able to learn any mathematics unless I can see some problem I am going to solve with mathematics, and I don't understand how anyone can teach mathematics without having a battery of problems that the student is going to be inspired to want to solve and then see that he or she can use the tools for solving them.

Steven Weinberg (Winner of the Nobel Prize in physics in 1979)

The more I have learned about physics, the more convinced I am that physics provides, in a sense, the deepest applications of mathematics. The mathematical problems that have been solved, or techniques that have arisen out of physics in the past, have been the lifeblood of mathematics .... The really deep questions are still in the physical sciences. For the health of mathematics at its research level, I think it is very important to maintain that link as much as possible.

Sir Michael Atiyah (Winner of the Fields Medal in 1966)

David Hilbert (1862-1943)

John von Neumann (1903-1957)

Stefan Banach (1892-1945)

Preface

A theory is the more impressive, the simpler are its premises, the more distinct are the things it connects, and the broader is its range of applicability.

Albert Einstein

There are two different ways of teaching mathematics, namely,

(i) the systematic way, and

(ii) the application-oriented way.

More precisely, by (i), I mean a systematic presentation of the material governed by the desire for mathematical perfection and completeness of the results. In contrast to (i), approach (ii) starts out from the question "What are the most important applications?" and then tries to answer this question as quickly as possible. Here, one walks directly on the main road and does not wander into all the nice and interesting side roads.

The present book is based on the second approach. It is addressed to undergraduate and beginning graduate students of mathematics, physics, and engineering who want to learn how functional analysis elegantly solves mathematical problems that are related to our real world and that have played an important role in the history of mathematics. The reader should sense that the theory is being developed, not simply for its own sake, but for the effective solution of concrete problems.

viii Preface

This introduction to functional analysis is divided into the following two parts:

Part I: Applications to mathematical physics (the present AMS Vol. 108);

Part II: Main principles and their applications (AMS Vol. 109).

Our presentation of the material is self-contained. As prerequisites we assume only that the reader is familiar with some basic facts from calculus. One of the special features of our introduction to functional analysis is that we try to combine the following topics on a fairly elementary level:

(a) linear functional analysis;

(b) nonlinear functional analysis;

(c) numerical functional analysis; and

(d) substantial applications related to the main stream of mathematics and physics.

I think that time is ripe for such an approach. From a general point of view, functional analysis is based on an assimilation of analysis, geometry, alge­bra, and topology. The applications to be considered concern the following topics:

ordinary differential equations (initial-value problems, boundary-eigen-value problems, and bifurcation);

linear and nonlinear integral equations;

variational problems, partial differential equations, and Sobolev spaces;

optimization (e.g., Cebysev approximation, control of rockets, game the-ory, and dual problems);

Fourier series and generalized Fourier series;

the Fourier transformation;

generalized functions (distributions) and the role of the Green function;

partial differential equations of mathematical physics (e.g., the Laplace equation, the heat equation, the wave equation, and the Schrodinger equa­tion);

time evolution and semigroups;

the N-body problem in celestial mechanics;

capillary surfaces;

minimal surfaces and harmonic maps;

superfluids, superconductors, and phase transition (the Landau-Ginz­burg model);

viscous fluids (the Navier-Stokes equations);

Preface ix

boundary-value problems and obstacle problems in nonlinear elasticity;

quantum mechanics (both the Schrodinger equation approach and the Feynman path integral approach);

quantum statistics (both the Hilbert space approach and the C* -algebra approach);

quantum field theory (the Fock space);

quarks in elementary particle physics;

gauge field theory (the Yang-Mills-Dirac equations);

string theory.

We also study the following fundamental approximation methods:

iteration method via k-contractions;

iteration method via monotonicity in ordered Banach spaces;

the Ritz method and the method of finite elements;

the dual Ritz method (also called the Trefftz method);

the Galerkin method; and

cubature formulas.

We shall make no attempt to present concepts in the most general way but will rather try to expose their essential core without, on the other hand, trivializing them. In the experience of the author, it is substantially easier for the student to take a mathematical concept and extend it to a more general situation than to struggle through a theorem formulated in its broadest generality and burdened with numerous technicalities in an attempt to divine the basic concept. Here it is the teacher's duty to be helpful. To assist the reader in recognizing the central results, these propositions are denoted as "theorems." A list of the theorems along with a list of the most important definitions can be found at the end of this book. Furthermore, a number of schematic overviews should help the reader to understand the interrelations between the abstract principles and their applications.

Functional analysis is a child of the twentieth century. It provides us with a new language that allows us to formulate apparently different top­ics in a unique way. It seems that functional analysis is deeply rooted in our real world, since it is the appropriate tool for describing quantum phenomena in terms of mathematics. For example, the famous Heisenberg uncertainty principle on position and momentum of particles follows easily from the Schwarz inequality, which represents the most important inequal­ity in Hilbert space theory. In the study of many problems, the following steps are used:

(i) translating the given concrete problem into the language of functional analysis;

x Preface

(ii) applying abstract functional analytic theorems;

(iii) verifying the assumptions in step (ii), which often requires applying very specific analytical tools.

The basic idea of functional analysis is to formulate differential and in­tegral equations in terms of operator equations. For example, the operator equation

Au=j, UEX, (E)

may represent a concise formulation of the following integral equation for the unknown function u:

u(x) -lb A(x, y)¢(u(y), y)dy = j(x),

provided we introduce the operator A through

(Au)(x) := u(x) -lb A(x, y)¢(u(y), y)dy

a S x S b,

for all x E [a, b]. (Op)

From the abstract point of view, we assume that u is an element of the "space" X, where X := era, b] denotes the set of all continuous functions

u: [a,b] ~ R

More precisely, the definition of the operator A is to be understood in the following sense. To each function u E X we assign a new function Au on the interval [a, b] given by (Op). For example, if u( x) == 1, then the function Au is given by

(Au)(x) = 1 -lb A(x, y)¢(l, y)dy for all x E [a, b].

The set X is also called a function space. Typically, functional analysis employs the fact that the function spaces possess an additional structure. For example, the space era, b] can be equipped with the norm

Ilull := max lu(x)l· a~x~b

We callilull the length of the vector (function) u. This way, era, b] becomes a so-called Banach space. In the special case where ¢( u, y) == u, the integral equation (Eint ) is said to be linear. Generally, the importance of nonlinear problems stems from the fact that they describe processes in nature with interactions.

In contrast to the integral equation (Eint ), the operator equation (E) may also correspond to the following boundary-value problem for a differential equation:

u"(x) + c(x)u(x) = f(x), a S x S b,

u(a) = u(b) = 0 (boundary condition),

Preface xi

provided we define the operator A through

(Au)(x) := ul/(x) + c(x)u(x) for all x E [a, b].

Naturally enough, we now assume that u is an element of the space X, where X denotes the set of all functions u: [a, b] ---+ IR. that are twice con­tinuously differentiable on the interval [a, b] and that satisfy the boundary condition u(a) = u(b) = O.

Finally, set u := (Ul, U2), f := (il, h), where Ul, U2, il, and 12 are real numbers, i.e., u, f E IR.2 . Then, the following system of real equations

corresponds to the original operator equation (E), too. Here we define the operator A through

for all u E X,

where we set X := IR.2 . Obviously, u E X implies Au E X. Thus, the operator A: X ---+ X maps the space X into itself.

Furthermore, for example, the abstract minimum problem

F(u) = min!, u E X, (M)

corresponds to Euler's classical variational problem

lb L(x,u(x),u'(x))dx = min!, (Mvar)

u(a) = u(b) = 0 (boundary condition),

provided we set

F(u) := lb L(x, u(x), u'(x))dx for all u E X,

where X denotes an appropriate space of functions that satisfy the bound­ary condition u( a) = u(b) = O. Since F( u) is a real number for each function u EX, the operator F: X ---+ IR. from the space X to the space IR. of real numbers is called a functional. In addition, many problems in optimization and control theory can be formulated in terms of the abstract minimum problem (M). Roughly speaking:

Functional analysis provides us with existence theorems for both the op­erator equation (E) and the minimum problem (M) and with convergent approximations methods for (E) and (M).

xii Preface

Typically, the spaces X are infinite-dimensional. From the physical point of view, such spaces describe physical systems with an infinite number of degrees of freedom.

Problems of the type

minmaxL(u,p) = maxminL(u,p) = L(uo,Po) uEA pEB pEB uEA

(Minimax)

and, more generally,

inf supL(u,p) = sup inf L(u,p) UEApEB PEBUEA

represent basic problems in game theory and duality theory. This will be shown in Chapter 2 of AMS Vol. 109.

Functional analysis also establishes a calculus for linear operators. For example, let us consider the abstract differential equation

u'(t) = Au(t), t > 0, (D)

u(o) = Uo (initial condition),

where A is a linear operator. Formally, the solution of (D) is given by

u(t) = etAuo.

It is the goal of the theory of semigroups to give the formal symbol etA a rigorous meaning. Equation (D) describes many time-dependent processes in nature. It turns out that if (D) corresponds to an irreversible process in nature (e.g., diffusion or heat conduction), then the symbol etA only makes sense for time t 2: o.

Let us briefly discuss the contents of the present AMS Vol. 108 and of AMS Vol. 109.

Chapter 1 concerns Banach spaces. For the convenience of the reader, the most important notions of functional analysis are explained in terms of the simple space era, b] of continuous functions without using the Lebesgue integral. This way, the first chapter may serve as a quite elementary intro­duction to functional analysis. The applications to be studied in Chapter 1 concern existence proofs for ordinary differential equations as well as for linear and nonlinear integral equations. Here, we will use the two most im­portant fixed-point theorems due to Banach and Schauder. We also justify the following fundamental principle in mathematics:

A priori estimates yield existence.

In an abstract functional analytic setting, this principle was established by Leray and Schauder in 1934.

Riemann's famous Dirichlet principle stands at the beginning of Chapter 2, which is devoted to Hilbert space theory. We give an elegant functional

Preface xiii

analytic justification for the Dirichlet principle based on an existence theo­rem for quadratic minimum problems in Hilbert spaces. In this connection, the use of the Lebesgue integral is indispensible. Basic facts about this in­tegral are summarized in the appendix. Thus, the book is also accessible to those readers who are not familiar with the Lebesgue integral. In fact, our abstract setting for the Dirichlet principle represents one of several equiva­lent formulations of the so-called linear orthogonality principle for Hilbert spaces, which will be studied in Section 2.13. In terms of geometry, the linear orthogonality principle tells us that:

In Hilbert spaces, there exists a perpendicular from any point to any closed plane.

In other words, there exists an orthogonal projection onto closed linear subspaces of Hilbert spaces. If one tries to generalize this fundamental or­thogonality principle to nonlinear operators, then one obtains an existence theorem for so-called monotone operator equations.

Each Hilbert space is a Banach space. But Hilbert spaces possess a richer structure than Banach spaces, since the concept of orthogonality is avail­able.

In Chapter 3 we shall show that complete orthonormal systems in Hilbert spaces are the right tool for solving the convergence problem for Fourier series and more general series expansions of functions. This convergence problem was a famous open problem in the nineteenth century.

Hilbert discovered around 1900 that many eigenvalue problems of clas­sical analysis for differential and integral equations can be formulated in terms of a general theory for compact symmetric operators in Hilbert spaces. This approach, which is closely related to Chapter 3, will be stud­ied in Chapter 4. This way, it is possible to understand why the "Fourier method" of physicists works. In terms of physics, this method represents general states as superpositions of so-called eigenstates, which correspond to eigenoscillations of the system under consideration. Functional analysis rigorously establishes the old conjecture by Daniel Bernoulli (1700-1782) that physical systems with an infinite number of degrees of freedom possess an infinite number of eigenoscillations.

Around 1935 Friedrichs found out that the partial differential equations of mathematical physics can be understood best by means of the Friedrichs extension of symmetric operators. This extension procedure generates self­adjoint operators, which von Neumann introduced in connection with his mathematical foundations of quantum mechanics in 1932. From the physi­cal point of view, the Friedrichs approach is intimately related to the con­cept of energy. This will be studied in Chapter 5, where we also show that time-dependent processes in nature can be described mathematically either by semigroups (irreversible processes) or by one-parameter groups (reversible processes).

Near 1950 Kato proved that the Schrodinger equation for large classes of

XIV Preface

physical systems corresponds to a uniquely determined self-adjoint Hamil­tonian. This way Kato showed that von Neumann's abstract setting for quantum mechanics from 1932 represents the right tool for the mathemat­ical description of the behavior of atoms and molecules.

Chapter 5 represents the heart of the present book. It is devoted to the close relations between functional analysis and both classical and modern mathematical physics. For example, in Sections 5.21 through 5.24, which discuss the Dirac calculus and the Feynman path integral in quantum physics,

we try to build a bridge between the language and thoughts oj physicists and mathematicians.

The mathematician should have the following in mind. Until today, it has not been possible to develop a mathematically rigorous quantum field the­ory for describing the behavior of elementary particles. For about 40 years, however, physicists have worked with dubious mathematical methods that are in fantastic coincidence with experiment (e.g., in quantum electrody­namics).

As a typical example for the difference between the language of physicists and mathematicians, let us consider the "delta Junction" 8, which the fa­mous physicist Paul Dirac introduced around 1930. In terms of physics, the function 8 = 8(x) describes the mass density of a point of mass m = 1 at x = 0 on the real line. This physical interpretation of 8 leads us immediately to

8 ( x) = { 0 ~f x # 0 +00 If x = 0,

(I)

as well as I: 8(x)dx = total mass = 1 (II)

and I: J(x)8(x)dx = J(O) . (mass at 0) = J(O). (III)

Using the substitution x := z - y and g(z) := J(z - y), we also get

I: g(z)8(z - y)dz = g(y) for all y E lit (IV)

Set u(x) := 8(x - y). Applying (IV) to the Fourier transformation

v(k) = I: e-ikXu(x)dx for all k E JR.

and the inverse Fourier transformation

for all x E JR.,

Preface xv

we formally obtain that

e-iky = 100 e-ikX8(x - y)dx -00

for all k, y E lR (V)

and

8(x - y) = (27r)-1 I: eik(x-y)dk for all x, y E R (VI)

From a mathematical point of view, there is no classical function 8 that satisfies (I) and (II). At a first glance, it seems that (I) along with (II) is nonsense. In the introduction to his famous 1932 monograph Foundations of Quantum Mechanics, John von Neumann points that the Dirac calculus lacks a rigorous justification. Therefore, von Neumann did not use this calculus. Around 1950 Laurent Schwartz created the theory of generalized functions (distributions), which allows a rigorous definition of the delta distribution related to Dirac's "delta function." As we will show in Chapters 2 and 3, the theory of generalized functions gives formulas (III) through (VI) a precise meaning. However, physics textbooks do not use the rigorous mathematical approach to generalized functions. Physicists prefer formulas (I) through (VI) because of their mnemotechnical elegance. Experience shows that, generally, the calculi used by physicists possess the advantage of working on their own and leading very quickly to the desired results at least on a heuristic level. Therefore, it is useful to learn both the language of physicists and the language of mathematicians. The present book tries to support this.

A mathematician who teaches mathematics to physics students should try to help the students understand the differences and connections between the two different languages of mathematics and physics. In order to avoid confusion, we clearly distinguish between physical motivations and purely mathematical results. The word "proof" is always understood in the sense of a rigorous mathematical proof.

Let us now briefly discuss the contents of AMS Vol. 109. In Chapter 1 of AMS Vol. 109 we show that the Hahn-Banach theorem

allows us to solve interesting convex optimization problems. Here, in terms of geometry, we use the separation of convex sets by hyperplanes.

Chapter 2 of AMS Vol. 109 is devoted to variational principles. In partic­ular, we generalize the classical Weierstrass existence theorem for minimum problems via weak convergence. Furthermore, we consider the Ekeland vari­ational principle on the existence of quasi-minimal points. For example, combining this principle with the Palais-Smale condition, we will get the mountain pass theorem on saddle points. Functional analysis explains why the nineteenth-century mathematicians encountered many difficulties in es­tablishing existence theorems for variational problems. The reason for this is the following simple geometric fact:

The closed unit ball in an infinite-dimensional Banach space is not com­pact.

xvi Preface

At the end of the 1920s, Banach proved a number of important theorems on linear continuous operators in Banach spaces, which follow from the Baire category theorem, which, in turn, is a consequence of a straightforward generalization of Cantor's nested interval principle to Banach spaces. These so-called principles of linear functional analysis are presented in Chapter 3 of AMS Vol. 109. Applications to linear and nonlinear operator equations are studied in Chapters 4 and 5 in AMS Vol. 109. In particular, in Chapter 4 of AMS Vol. 109 we will use the implicit function theorem in order to study the local behavior of nonlinear operators (diffeomorphisms, submersions, immersions, and subimmersions). This is important for global analysis (i.e., the theory of finite-dimensional and infinite-dimensional manifolds).

Chapter 5 of AMS Vol. 109 is devoted to a study of linear and non­linear Fredholm operators along with bifurcation theory. Many differential and integral operators correspond to Fredholm operators in appropriate function spaces. The theory of Fredholm operators generalizes the classical Fredholm alternative for integral equations formulated first by Fredholm around 1900. In fact, the theory of linear and nonlinear Fredholm opera­tors represents the completely natural generalization of the classical theory for finite systems of real equations to infinite dimensions. Bifurcation the­ory mathematically models an essential change of the behavior of systems in nature (e.g., the buckling of beams, ecological catastrophes, etc.). The theory of nonlinear Fredholm operators dates back to a 1965 fundamental paper by Smale.

The creation of functional analysis by Hilbert around 1900 was strongly influenced by the theory of integral equations. Until the 1930s, partial dif­ferential equations were treated by being reduced to integral equations. The more successful modern functional analytic approach to partial differ­ential equations is based on an inspection of the operator equations that correspond directly to the differential equations (cf. (E) and (Ediff))' This approach dates back to von Neumann and Friedrichs in the 1930s. In fact, this point of view works successfully in numerical analysis, too. Note that all the basic equations of physical field theories (elasticity, hydrodynam­ics, thermodynamics, gas dynamics, electrodynamics, quantum mechanics, quantum field theory, general relativity, gauge field theory, etc.) are partial differential equations. It seems fair to say that the theory of integral equa­tions has reached a certain final shape. In contrast, there are still many deep open questions in the theory of those partial differential equations related to physics.

At the end of each chapter, the reader will find problems. Most of them are routine. I hope that such a carefully selected collection of fairly simple problems will help the student to check her or his basic understanding of the material. Some more advanced problems are marked with a star and provided with hints for further reading. For an in-depth presentation of non­linear functional analysis and its many applications to the natural sciences, the reader is referred to the five-volume treatise Nonlinear Functional Anal-

Preface xvii

ysis and Its Applications by the same author. In particular, Vols. 4 and 5 contain a detailed motivation of the basic equations in classical and modern mathematical physics along with both abstract existence proofs and inter­esting applications to concrete problems in physics, chemistry, biology, and economics.

The representation takes into account that in general no book is read completely from beginning to end. We hope that even a quick skimming of the text will suffice to grasp the essential contents. To this end, we recom­mend reading the introductions to the individual chapters, the definitions, the "theorems" (without proofs), and the examples (without proofs) as well as the motivations and comments in the text, which point out the meaning of the specific results. The proofs are worked out in great detail. Grasping the individual steps in the proofs as well as their essential ideas is made easier by the careful organization. It is a truism that only a precise study of the proofs enables one to penetrate more deeply into a mathematical theory.

Readers have the following two options:

(i) Those who want to become acquainted as quickly as possible with the Hilbert space approach to mathematical physics and numerical anal­ysis can immediately begin with Chapter 2 after glancing at the last section of Chapter 1, which summarizes important notions concerning Banach spaces.

(ii) Those interested in the main principles of functional analysis and their applications might skip to AMS Vol. 109 after reading Chap­ter 1.

The book is based on lectures I have given for students of mathematics and physics at Leipzig University. The manuscript has been finished during a stay at the "Sonderforschungsbereich 256" of Bonn University and at the Max Planck Institute for Mathematics in Bonn. I would like to thank Professors Stefan Hildebrandt and Friedrich Hirzebruch for the invitations and the kind hospitality. Finally, my special thanks are due to Springer­Verlag for the harmonious collaboration.

I hope that the reader of this book enjoys getting a feel for the unity of mathematics by discovering interrelations between apparently completely different subjects.

Leipzig Spring 1995

Eberhard Zeidler

Prologue

Each progress in mathematics is based on the discovery of stronger tools and easier methods, which at the same time makes it easier to understand earlier methods. By making these stronger tools and easier methods his own, it is possible for the individual researcher to orientate himself in the different branches of mathematics.

The organic unity of mathematics is inherent in the nature of this science, for mathematics is the foundation of all exact knowledge of natural phenomena.

David Hilbert, 1900 (Paris lecture) 1

In order to understand the great achievement of Hilbert (1862-1943) in the field of analysis, it is necessary to first comment on the state of analysis at the end of the nineteenth century. After Weierstrass (1815-1897) had made sure of the foundations of complex function theory, and it has reached an impressive level, research switched to boundary-value problems, which first arose in physics. The work of Riemann (1826-1866) on complex function theory, however, had shown that boundary-value problems have great importance for pure mathematics as well. Two problems had to be solved:

1 In this fundamental lecture, Hilbert formulated his famous 23 open prob­lems, which strongly influenced the development of mathematics in the twentieth century.

xx Prologue

(i) the problem of the existence of a potential function for given boundary values; and

(ii) the problem of eigenoscillations of elastic bodies, for example, string and membrane.

The state of the theory was bad at the end of the nineteenth century. Riemann had believed that, by using the Dirichlet princi­ple, one could deal with these problems in a simple and uniform way. After Weierstrass' substantial criticism of the Dirichlet principle in 1870, special methods had to be developed for these problems. These methods, by C. Neumann, Schwarz, and Poincare, were very elabo­rate and still have great aesthetic appeal today; but because of their variety they were confusing, although at the end of the nineteenth century, Poincare (1854-1912), in particular, endeavoured with great astuteness to standardize the theory. There was, however, a lack of "simple basic facts" from which one could easily get complete results without sophisticated investigations of limiting processes.

Hilbert first looked for these "simple basic facts" in the calculus of variations. He considered so-called regular variational problems which satisfy the Legendre condition. In 1900 he had an immediate and great success; he succeeded in justifying the Dirichlet principle.

While Hilbert used variational methods, the Swedish mathemati­cian Fredholm (1866-1927) approached the same goal by developing Poincare's work by using linear integral equations. In the winter semester 1900/01 Holmgren, who had come from Upsala (Sweden) to study under Hilbert in Gottingen, held a lecture in Hilbert's sem­inar on Fredholm's work on linear integral equations which had been published the previous year. This was a decisive day in Hilbert's life. He took up Fredholm's new discovering with great zeal, and com­bined it with his variational methods. In this way he succeeded in creating a uniform theory which solved problems (i) and (ii) above.

In 1904 Hilbert's first note on the "Foundations of a General Theory of Linear Integral Equations" was published in the Gottinger Nachrichten. These results were based on lectures which Hilbert held from the summer of 1901 onwards. Fredholm had proved the exis­tence of solutions for linear integral equations of the second kind. His result was sufficient to solve the boundary-value problems of po­tential theory. But Fredholm's theory did not include the eigenoscil­lations and the expansions of arbitrary functions with respect to eigenfunctions. Only Hilbert solved this problem by using finite­dimensional approximations and a passage to the limit. In this way he obtained a generalization of the classical principal-axis transfor­mation for symmetric matrices to infinite-dimensional matrices. The symmetry of the matrices corresponds to the symmetry of the ker­nels of integral equations, and it shows that the kernels appearing in oscillation problems are indeed symmetrical.

Prologue xxi

From our point of view today, Hilbert's paper of 1904 appears clumsy, compared to the elegance of Erhard Schmidt's method pub­lished in 1907 which he developed in his dissertation written while a student of Hilbert in Gottingen. But the first step had been made. In the same year, 1904, Hilbert, in his second note, was able to apply his theory to general Sturm-Liouville eigenvalue problems. His third note in 1905 contained a very important result. Of the great prob­lems which had Riemann posed with the complex function theory, there was still one left open; the proof of the existence of differential equations with a prescribed monodromy group. Hilbert solved this problem by reducing it to the determination of two functions which are holomorphic in both the interior and the exterior of a closed curve, and whose real and imaginary parts satisfy appropriate lin­ear combinations on the curve (the Riemann-Hilbert problem). The solution to this problem is a classic example for the axiomatics of lim­iting processes demanded by Hilbert. No concrete limiting processes are used, but everything results from the existence of the Green func­tion for the interior and the exterior of the closed curve, and from the Fredholm alternative which says that either the homogeneous inte­gral equation has a nontrivial solution or the inhomogeneous integral equation has a solution.

Hilbert soon noticed that limits are set to the method of integral equations. In order to overcome these limits he created, in his fourth and fifth notes in 1906, the general theory of quadratic forms of an infinite number of variables. Hilbert believed that with this theory he had provided analysis with a great general basis which corresponds to an axiomatics of limiting processses. The further development of mathematics has proved him to be right.

Otto Blumenthal, 1932

The perfection of mathematical beauty is such that whatsoever is most beautiful and regular is also found to be most useful and ex­cellent.

D'Arcy W. Thompson, 1917 On Growth and Form

Contents

Preface vii

Prologue xix

Contents of AMS Volume 109 xxvii

1 Banach Spaces and Fixed-Point Theorems 1 1.1 Linear Spaces and Dimension 2 1.2 Normed Spaces and Convergence 7 1.3 Banach Spaces and the Cauchy Convergence Criterion 10 1.4 Open and Closed Sets 15 1.5 Operators 16 1.6 The Banach Fixed-Point Theorem and the Iteration Method 18 1.7 Applications to Integral Equations 22 1.8 Applications to Ordinary Differential Equations. 24 1.9 Continuity . 26 1.10 Convexity 29 1.11 Compactness 33 1.12 Finite-Dimensional Banach Spaces and Equivalent Norms 42 1.13 The Minkowski Functional and Homeomorphisms . 45 1.14 The Brouwer Fixed-Point Theorem. 53 1.15 The Schauder Fixed-Point Theorem 61 1.16 Applications to Integral Equations 62 1.17 Applications to Ordinary Differential Equations . 63

XXIV Contents

1.18 The Leray-Schauder Principle and a priori Estimates. 64 1.19 Sub- and Supersolutions, and the Iteration Method in

Ordered Banach Spaces 66 1.20 Linear Operators . . . . . . . . . 70 1.21 The Dual Space. . . . . . . . . . 74 1.22 Infinite Series in Normed Spaces 76 1.23 Banach Algebras and Operator Functions 76 1.24 Applications to Linear Differential Equations in Banach

Spaces. . . . . . . . . . . . . 80 1.25 Applications to the Spectrum 82 1.26 Density and Approximation . . 84 1.27 Summary of Important Notions 88

2 Hilbert Spaces, Orthogonality, and the Dirichlet Principle 2.1 Hilbert Spaces 2.2 Standard Examples.

101 105 109

2.3 Bilinear Forms . . . 120 2.4 The Main Theorem on Quadratic Variational Problems 121 2.5 The Functional Analytic Justification of the Dirichlet

Principle. . . . . . . . . . . . . . . . . . . . . . . . . 125 2.6 The Convergence of the Ritz Method for Quadratic

Variational Problems. . . . . . . . . . . . . . . . . . 140 2.7 Applications to Boundary-Value Problems, the Method of

Finite Elements, and Elasticity . . . . . . . . 145 2.8 Generalized Functions and Linear Functionals 156 2.9 Orthogonal Projection . . . . . . . . . . . . 165 2.10 Linear Functionals and the Riesz Theorem. 167 2.11 The Duality Map. . . . . . . . . . . . . . . 169 2.12 Duality for Quadratic Variational Problems 169 2.13 The Linear Orthogonality Principle. . . . . 172 2 .14 Nonlinear Monotone Operators . . . . . . . 173 2.15 Applications to the Nonlinear Lax-Milgram Theorem and

the Nonlinear Orthogonality Principle . . . . . . . . . .. 174

3 Hilbert Spaces and Generalized Fourier Series 195 3.1 Orthonormal Series. . . . . . . . . . . . 199 3.2 Applications to Classical Fourier Series. 203 3.3 The Schmidt Orthogonalization Method 207 3.4 Applications to Polynomials 208 3.5 Unitary Operators . . . . . . . . . . . . 212 3.6 The Extension Principle . . . . . . . . . 213 3.7 Applications to the Fourier Transformation 214 3.8 The Fourier Transform of Tempered Generalized Functions 219

Contents xxv

4 Eigenvalue Problems for Linear Compact Symmetric Operators 229 4.1 Symmetric Operators. . . . . 230 4.2 The Hilbert-Schmidt Theory 232 4.3 The Fredholm Alternative . . 237 4.4 Applications to Integral Equations 240 4.5 Applications to Boundary-Eigenvalue Value Problems 245

5 Self-Adjoint Operators, the Friedrichs Extension and the Partial Differential Equations of Mathematical Physics 253 5.1 Extensions and Embeddings 260 5.2 Self-Adjoint Operators . . 263 5.3 The Energetic Space . . . . 273 5.4 The Energetic Extension. . 279 5.5 The Friedrichs Extension of Symmetric Operators. 280 5.6 Applications to Boundary-Eigenvalue Problems for the

Laplace Equation . . . . . . . . . . . . . . . . . . . . . . 285 5.7 The Poincare Inequality and Rellich's Compactness Theorem 287 5.8 Functions of Self-Adjoint Operators. . . . . . . . . . . . 293 5.9 Semigroups, One-Parameter Groups, and Their Physical

Relevance . . . . . . . . . . . . . . 298 5.10 Applications to the Heat Equation . . . . . . . . . . . .. 305 5.11 Applications to the Wave Equation. . . . . . . . . . . .. 309 5.12 Applications to the Vibrating String and the Fourier Method 315 5.13 Applications to the Schrodinger Equation 323 5.14 Applications to Quantum Mechanics 327 5.15 Generalized Eigenfunctions . . . . . 343 5.16 Trace Class Operators . . . . . . . . 347 5 .17 Applications to Quantum Statistics . 348 5.18 C*-Algebras and the Algebraic Approach to Quantum

Statistics .......................... 357 5.19 The Fock Space in Quantum Field Theory and the Pauli

Principle. . . . . . . . . . . . . . . . . . . . . . . . . . .. 363 5.20 A Look at Scattering Theory . . . . . . . . . . . . . . .. 368 5.21 The Languase of Physicists in Quantum Physics and the

Justification of the Dirac Calculus ...... 373 5.22 The Euclidean Strategy in Quantum Physics ...... 379 5.23 Applications to Feynman's Path Integral. . . . . . . . . 385 5.24 The Importance of the Propagator in Quantum Physics 394 5.25 A Look at Solitons and Inverse Scattering Theory 406

Epilogue 425

Appendix 429

xxvi Contents

References

Hints for Further Reading

List of Symbols

List of Theorems

List of the Most Important Definitions

Subject Index

443

457

461

467

469

473

Contents of AMS Volume 109

Preface

Contents of AMS Volume 108

1 The Hahn-Banach Theorem and Optimization Problems 1.1 The Hahn-Banach Theorem 1.2 Applications to the Separation of Convex Sets 1.3 The Dual Space era, b]* 1.4 Applications to the Moment Problem 1.5 Minimum Norm Problems and Duality Theory 1.6 Applications to Cebysev Approximation 1. 7 Applications to the Optimal Control of Rockets

2 Variational Principles and Weak Convergence 2.1 The nth Variation 2.2 Necessary and Sufficient Conditions for Local Extrema

and the Classi8al Calculus of Variations 2.3 The Lack of Compactness in Infinite-Dimensional Banach

Spaces 2.4 Weak Convergence 2.5 The Generalized Weierstrass Existence Theorem 2.6 Applications to the Calculus of Variations 2.7 Applications to Nonlinear Eigenvalue Problems

xxviii Contents of AMS Volume 109

2.8 Reflexive Banach Spaces 2.9 Applications to Convex Minimum Problems and

Variational Inequalities 2.10 Applications to Obstacle Problems in Elasticity 2.11 Saddle Points 2.12 Applications to Duality Theory 2.13 The von Neumann Minimax Theorem on the Existence of

Saddle Points 2.14 Applications to Game Theory 2.15 The Ekeland Principle about Quasi-Minimal Points 2.16 Applications to a General Minimum Principle via the

Palais-Smale Condition 2.17 Applications to the Mountain Pass Theorem 2.18 The Galerkin Method and Nonlinear Monotone Operators 2.19 Symmetries and Conservation Laws (The Noether

Theorem) 2.20 The Basic Ideas of Gauge Field Theory 2.21 Representations of Lie Algebras 2.22 Applications to Elementary Particles

3 Principles of Linear Functional Analysis 3.1 The Baire Theorem 3.2 Application to the Existence of Nondifferentiable

Continuous Functions 3.3 The Uniform Boundedness Theorem 3.4 Applications to Cubature Formulas 3.5 The Open Mapping Theorem 3.6 Product Spaces 3.7 The Closed Graph Theorem 3.8 Applications to Factor Spaces 3.9 Applications to Direct Sums and Projections 3.10 Dual Operators 3.11 The Exactness of the Duality Functor 3.12 Applications to the Closed Range Theorem and to

Fredholm Alternatives

4 The Implicit Function Theorem 4.1 m-Linear Bounded Operators 4.2 The Differential of Operators and the Frechet Derivative 4.3 Applications to Analytic Operators 4.4 Integration 4.5 Applications to the Taylor Theorem 4.6 Iterated Derivatives 4.7 The Chain Rule 4.8 The Implicit Function Theorem

Contents of AMS Volume 109 xxix

4.9 Applications to Differential Equations 4.10 Diffeomorphisms and the Local Inverse Mapping Theorem 4.11 Equivalent Maps and the Linearization Principle 4.12 The Local Normal Form for Nonlinear Double Splitting

Maps 4.13 The Surjective Implicit Function Theorem 4.14 Applications to the Lagrange Multiplier Rule

5 Fredholm Operators 5.1 Duality for Linear Compact Operators 5.2 The Riesz-Schauder Theory On Hilbert Spaces 5.3 Applications to Integral Equations 5.4 Linear Fredholm Operators 5.5 The Riesz-Schauder Theory on Banach Spaces 5.6 Applications to the Spectrum of Linear Compact

Operators 5.7 The Parametrix 5.8 Applications to the Perturbation of Fredholm Operators 5.9 Applications to the Product Index Theorem 5.10 Fredholm Alternatives via Dual Pairs 5.11 Applications to Integral Equations and Boundary-Value

Problems 5.12 Bifurcation Theory 5.13 Applications to Nonlinear Integral Equations 5.14 Applications to Nonlinear Boundary-Value Problems 5.15 Nonlinear Fredholm Operators 5.16 Interpolation Inequalities 5.17 Applications to the Navier-Stokes Equations

References

Subject Index