APPLICATIONS OF TOPOLOGICAL TENSOR

PRODUCTS TO INFINITE DIMENSIONAL

HOLOMORPHY

Raymond A. Ryan

( t

(

A thesis submitted for the degree of Ph.D. at Trinity

College, Dublin.

March 1980

This thesis is entirely my own work.

It has not been submitted for a degree or any other

award at any other University.

Raymond A. Ryan

- ii

TO MY WIFE, JANE, AND TO MY PARENTS .

- iii

A C K NOW LED GEM E N T S

My Supervisor, Dr. Richard M. Aron introduced me

to all the problems studied in this thesis. I am

immensely grateful to him for his patient guidance,

and for sharing with me his many insights and his deep

knowledge.

I would also like to thank Dr. Phil Boland,

Professor Joe Diestel, Dr. Paul Berner and, most

particularly, Professor Sean Dineen for many helpful

and enlightening discussions.

Finally, I would like to express my gratitude to

Professor B. H. Murdoch and Professor S. J. Tobin for

their advice and encouragement, and to the Department

of Education and the Trinity Trust for financial support.

Raymond A. Ryan

Dublin, March 1980

- iv

CON TEN T S

ACKNOWLEDGEMENTS .......................... ~V

INTRODUCTION ................. Vl

CHAPTER I LINEARIZATION OF POLYNOMIAL AND GATEAUX

HOLOMORPHIC MAPPINGS ON VECTOR SPACES ..... l

CHAPTER II : LINEARIZATION OF CONTINUOUS POLYNOMIALS

AND HOLOMORPHIC MAPPINGS ON LOCALLY

CONVEX SPACES ....................... 23

CHAPTER III : DUALITY FOR POLYNOMIALS AND HOLOMORPHIC ...

MAPPINGS ON FRECHET SPACES .......... 46

CHAPTER IV WEAKLY COMPACT POLYNOMIAL AND HOLOMORPHIC

MAPPINGS ON BANACH SPACES ............ 64

CHAPTER V BASES AND REFLEXIVITY FOR SPACES OF

POLYNOMIALS ON BANACH SPACES ..........92

REFERENCES .....108

- V

I N T ROD U C T ION

The aim of this work is to use the ideas and

methods of the theory of topological tensor products of

locally convex spaces as a starting point in tackling

some problems in Infinite Dimensional Holomorphy. The

basic idea is to use tensor products to "linearize"

non-linear problems.

In Chapters I and II we develop the basic machinery.

The problem posed here is the following: given a locally

convex space E, and a class, F, of mappings whose domains

are E, construct a locally convex space EF ' and a

mapping X: E ~ EF which belongs to the class F

so that for every locally convex space F, and every

mapping f: E ----4 F of class F, there is a continuous

linear mapping f: EF ---+ F such that foX = f Thus questions concerning mappings on E from the class F

can be reduced to questions concerning continuous linear

mappings on EF In Chapter I the algebraic side of this

problem is considered, where E is a vector space, and

continuity is not in question. Here F is taken first as

the class of homogeneous polynomials of a fixed degree,

and then as the class of Gateaux-holomorphic mappings

on E. In Chapter II E is taken to be a locally convex

space. Using Grothendieck's work on, topological tensor

products [13J , a locally convex space h~E is constructed for each natural number n, and continuous

n-homogeneous polynomials X : E ~hnE are defined n 'IT which have the effect of linearizing continuous

n-homogeneous polynomials on E. Thus for each locally

- vi

convex space F we obtain a canonical isomorphism of the

vector space L(h~EiF) of continuous linear mappings from hnE into F with the vector space p(nE;F) ofTI continuous n-homogeneous polynomials from E into F.

When E is a Complex Frechet space, we construct a

locally convex space hTIE and a holomorphic mapping

X : E~h E which yields a canonical isomorphism ofTI the vector space L(hTIE;F) of continuous linear

mappings of h E into the locally convex space F withTI the vector space H(E;F) of holomorphic mappings from

E into F.

In Chapter III we obtain the topological dual

spaces of (P( nE),T ) and (H(E) ,TO) of n-homogeneousOscalar-valued polynomials and holomorphic mappings,

respectively, on a Complex Frechet space E, where TO

denotes the topology of uniform convergence on compact

subsets of E. We define an approximation property which

we call the Strict Approximation Property and we give

some examples of spaces which have this property. We

show that every space with the strict approximation

property has the Grothendieck approximation property.

If E is a Frechet space with the strict approximation

property we show that the dual of the space (H(E),T )O

can be identified with the space of Nuclear Holomorphic

Germs at the origin in the space E~, where c denotes

the topology on E' of uniform convergence on compact

subsets of E. This space of germs can in turn be

identified with the space, which we define, of Entire

Functions on E' of Nuclear Exponential Type. This c enables us to prove an approximation theorem similar to

that of Gupta [15J for the kernel of a partial differential

operator on (H(E),T )O

- vii

Chapter IV is devoted to a study of weakly compact

polynomial and holomorphic mappings on Banach spaces.

Our inspiration in the first part of this chapter is the

work of Aron and Schottenloher [2J on compact holomorphic mappings. Motivated by this we define a holomorphic

mapping f: E ---- F to be Weakly Compact if f maps some

neighbourhood of every point of E into a relatively

weakly compact subset of F. Weakly compact polynomials

have been defined by Pelczynski [22J. We prove a sequence of results, similar to those of Aron and

Schottenloher, which show that many of the classical

theorems concerning weakly compact linear mappings

generalise to the case of weakly compact polynomial and

holomorphic mappings. We prove a generalization to

holomorphic mappings of the factorization theorem of

Davis, Figiel, Johnson and Pelczynski [5]; namely, we

show that a holomorphic mapping f: E ---- F of Banach

spaces is weakly compact if and only if there is a

reflexive Banach space, G, a bounded linear mapping

T : G - F , and a holomorphic mapping g : E----1' G

such that f = Tog

In the second part of Chapter IV we consider

Banach spaces with the Dunford-Pettis Property the

Banach space E has the Dunford-Pettis Property (DP) if

for every Banach space F and every weakly compact linear

mapping T: E ~F , T maps weak Cauchy sequences in

E into strongly convergent sequences in F. If the

linear mapping T in this definition is replaced by an

arbitrary weakly compact polynomial, we obtain the

definition of the Polynomial Dunford-Pettis Property

(PDP). We show that the properties DP and PDP are

equivalent, thus answering a question of Pelczynski [221.

- viii

We also consider the holomorphic version of this property,

and show that it, too, is equivalent to DP.

In Chapter V we study two problems concerning

spaces of continuous homogeneous polynomials on Banach

spaces. Firstly, we take E to be a Banach space with

a Schauder basis, and we ask when the monomials on E of

degree n with respect to this basis form a Schauder basis

for p(nE) We show that if the given basis for E is

shrinking, and if E has the Dunford-Pettis property, then

for each n the monomials of degree n, with a natural

ordering, form a Schauder basis for the Banach space

p(nE ) , where the norm is given by uniform convergence

on the unit ball of E. The Banach space co' with the

standard basis, is an example of a space to which this

result applies. We then consider the same problem for

the topology TO on p(nE). We show that the monomials of

degree n, with the above-mentioned natural ordering,

form a Schauder basis for (p(nE) ,T ) , where E is anyo

Banach space with a Schauder basis. Secondly, we take

E to be a reflexive Banach space, and we find for each n

conditions on E which are necessary and sufficient for

the Banach space p(nE) to be reflexive. These are

generalizations of classical linear theorems. For

example, p(nE) is reflexive if and only if the closed

unit ball of E is compact in the weak topology defined

by the family of all continuous n-homogeneous scalar

valued polynomials on E, or equivalently, if every such

polynomial on E attains its norm on the closed unit ball

of E.

The symbol c==J marks the conclusion of a proof, or the end of the statement of a Lemma or Proposition,

where no proof is given.

- ix

1

C HAP T E R I

Linearization of Polynomial and Gateaux-Holomorphic Mappings

on Vector Spaces

Throughout this chapter E, F will denote vector spaces

over a field 1K, which may be either the Real or Complex

numbers, and E*, F* their algebraic dual spaces. L(nEiF ) is

the vector space of n - linear mappings from En to F. When

n is zero this space consists of all the constant mappings

from E to F. When F is the field of scalars this space is

ndenoted by L( E).

We recall that the Tensor Product

(n times) n = 1,2, ..

is a vector space spanned by elements of the form xt:J .x ' n

where xl .. ,x are elements of E, subject to the relations: n

(i) x I (AX.) x = A(x x.

2

mapping may be regarded as being a universal n - linear

mapping on En, in the following sense (see [12J, Chapter I) :

Proposition 1.1:

Let T be an n - linear mapping from En into F. Then there

exists a unique linear mapping T : @nE ~ F such that

T 0 P = T. Conversely, if S is a linear mapping from nE n into F, then the equation T(xl, .... ,xn ) = S(Xl .@xn ) -defines an n - linear mapping T .. En ----?> F such that T = S. The correspondence T ~(--~) T establishes an isomorphism

between the vector spaces L(nEiF ) and L(gPEiF). In parti

cular, the vector spaces L(nE) and (nE)* are canonically

isomorphic.

T En --------------~) F

T

An n - linear mapping T En~ F is Symmetric if

T(Xa(l)''xa(n = T(Xl,,xn ) for every xl' .... ,x E E and every a E Sn0 n

Ls(nE : F ) will denote the subspace of L(nEiF ) consisting of

all the symmetric n - linear mappings from En into F. A

projection of L(nE: F ) onto Ls(nEiF ) is given by associating

3

with each n - linear mapping T its Symmetrization TS' where

1 Ts{Xl,,xn ) =nT L T(Xa{l)''xa{n)) (1)

asn

In terms of the associated linear mappings TS and T, (1)

reads:

TS(XlXn ) = ~(!I LS xa(l)@xa(n)) (2) a n

Equation (2) motivates the definition of the Symmetric Tensor

Product of elements xl' .. ,x of E by:n

x 0 .... 0 xn1 = !! a~s xa(l)xa{n) n

The mapping x l xn~ x l () .Ox defines a projectionn

on@nE The range of this projection is denoted by8nE and

is called the n - foZd Symmetric Tensor Product.

The mapping

a : En~ 8 nE n

defined by an (xl' . ,x ) = xlG .... 0 x ' is a symmetricn n n - linear mapping on En which' has a universal property similar

to that of the mapping p (see [12J, Chapter I):n

Proposition 1.2:

There is a canonical isomorphism of LS(riE;F) with L(cfE;F),

given by associating with a symmetric n - linear mapping T the

linear mapping T, with T = Toa In particular, Ls{nE) and n (fE)* are canonically isomorphic.

D Propositions 1.1 and 1.2 also hold for the case n = 0

if we define oE and 0 0 E to be the field of scalars, lK, and

4

define p and 0 to be the constant mappings with constant o 0

value 1. Note, too, that lE = OlE = E.

Linearization of Polynomials

We refer to [20J and [19J for basic facts concerning

pOlynomials. We recall that a mapping P:E ~ F is called an

n - Homogeneous PoZynomiaZ (n = 1,2, ) if there exists an

n - linear mapping T:En~ F such that

P(x) = T(x, .... ,x) for every x E E. (3)

p(nEiF ) will denote the vector space of all n - homogeneous

polynomials from E into F (n = 1,2, ), and P(oEiF) is

defined to be the vector space of all constant mappings from

E into F. When F = IK these spaces are denoted by p(nE) .

Equation (3) suggests the following definition:

Definition 1.1:

hnE is the subspace ofnE generated by the elements of the

form

(n)x = xx .... ~x.

Thus hnE consists of all elements u of nE which can be

written in the form

u = ~ A .x~n) , where k E

j=l J J

5

hOE is defined to be ]K. hnE will be called the n - tho

Homogeneous Product of E.

A mapping:

X :E ----;> hn

E n

(n)is defined by Xn(x) = x Xn is an n - homogeneous poly

nomial from E into hnE, since

X (x) = p (x, .... ,x) for every x E E. n n

Xn is then a universal n - homogeneous polynomial on E:

Pr~$ition 1.3:

Let P:E~ F be an n - homogeneous polynomial. Then

there exists a unique linear mapping p:hnE~ F such that

poX = P. Conversely, if S:hnE--+ F is a linear mappingn

then the equation P(x) = S(x(n defines an n - homogeneous

polynomial P from E into F such that P = S. The corres

pondence P ~F

~/

6

Proof:

Let P:E~F be an n - homogeneous polynomial. Choose

T L(nEiF ) such that P(x) = T(x, .... ,x) for all x E. - n

Then P(x} == T(X(n) for all x E, where T: E ----F is

the linear mapping associated with T by Proposition 1.1.

Let P be the restriction of T to hnE. Then P is linear,

and POX :::: P. n

P is uniquely defined by P, since if S E L(hnEiF) also

has the property that SoX == P, then for every element n

u = f A.X~n) of hnE, we have j=l ] ]

kS(u) = ~ A.S(X~n = 1: A.P(X.) =

j=l ] J j==l J J

~ k ()= PO; A. x . n } == P (u). Therefore S == P

j=l J J

Conversely, if S is a linear mapping from hnE into F,

let S be any extension of S to a linear mapping from nE

into F. Then the n - linear mapping T = Sop from En into n

F generates an n - homogeneous polynomial P with

- (n)P(x} == T(x, .. ,x) = S(x .. x} == S(X )

for every x E E. Therefore P = S.

Finally, it is easy to see that the bijection P< >P

is linear.

D For a given n - homogeneous polynomial, P, there may

be many different n - linear mappings, T, which generate P,

in the sense that

p(x) = T(x, .. ,x) for every x E E.

If, however, we require that T be symmetric, then, as is

well known [19J, T is uniquely determined by P. We establish

7

this by showing that the subspaces hnE and OnE of nE

are identical. This is proved by means of the classical

Polarization Formula:

Proposition 7.4:

( n)L: l' .. ( 1 X 1 + + x) ,

=+1 n n ni

and x 0 .... 0 x = x (n) Therefore hnE = OnE.

Proof:

That x0 .... 0x = x(n) is clear from the definition of

the symmetric tensor product. Accordingly, if u hnE,

we can write u as:

k ku=L: A.X.(n) = L: A.X .0....0x.

j=l J J j=l J J J

and therefore hnE c: OnE.

On the other hand, let x1' .... ,x E. Then n

1 (n)L: ( X + + X )

n I 2n i = 1 1 n 1 1 n n

1 = L: _+ 1 1 [L: I:: I:: nl2n .-- n 1 . . 1 ~ . ~ .

8

= A + B.

A may be written:

1 A = nl2n ~i=ll"'n ~Sn a(d"'a(n) xa(d

C9Xa (n)

= r S Qr +1"'' ( ) .. ( )JX ( )a ,=- 1 n a 1 a n a 1 n ~ Xa (n)

= Xl 0 ..... 0 xn

If we denote by D the set of all mappings of

n

{l, .... ,n} into itself which are not onto, then

1B =

1 = nl 2n z:: r,z:: =+ 1 1 . ( )] X (1) @ x ( )TDnLi - n TnT T n

If T is not onto, we may ~hoose j, 1 ~ j ~ n, which is not in

the image of T. Then

z:: =+lEl ... E E (l) E ( ) = E, - n T T n ~

9

Therefore B = 0, and the Polarisation Formula:

is proved. From this it follows that 0 nE C hnE. Therefore

onE and hnE are identical. D

Corollary 1.1:

n S nFor every P P( EiF) there is a unique TpL ( EiF) such

that P(x) = T (x, .... ,x) for every x E. T is given bypP

+E X )n n

The correspondence P< >Tp establishes an isomorphism

n S nbetween P( EiF) and L ( E;F).

D

Though hnE and OnE are identical, we shall see that

for the purpose of studying polynomials, it is more con

venient to work with homogeneous products then with sym

metric tensor products. This is exactly similar to the

situation in holomorphy, where one prefers to deal with a

Taylor series consisting of homogeneous polynomials rather

than symmetric multilinear mappings.

10

Polynomials of Finite Type

We consider first the case of scalar-valued mappings.

Let ~l""~k be linear forms on E, and let Al .. ,Ak be k n Enscalars. An n - linear form, E A.~., is defined on byj=l J J

k (x 1 ' , x ) ~ EA. ~ . (x 1) ~ . (x ).n j=l J J J n

This n - linear mapping generates the polynomial:

k

~=lAj [~j (x)] n ,

k n nwhich is also denoted by L A.~. The elements of p( E)

j=l J J

which arise in this way are called PoZynomiaZs of Finite

Type. The polynomials in p(nE) of finite type form a sub

space which we denote by Pf(nE)

Proposition 7.5:

Proof:

The mapping ~ ~ q,n from E* into P f (nE) is an n - homo

geneous polynomial. Therefore there exists a linear mapping

j : hn(E*) ~ P f (nE)

such that j(~(n = ~n for every ~ sE*. j is onto, since k n

the finite type polynomial L A.~. j=l J J

is the image under j of

11

The mapping P~ Tp is an isomorphism of Pf(nE) with

the vector space L~(nE) of symmetric n - linear forms on En

which are of finite type. With this identification, j can

be considered as the restriction to hn(E*) of the canonical

mapping of (ii9n(E*) into L(nE). Since this mapping is injec

tive ([12J, Chapter I, 7), so is j. Therefore j is an iso

morphism. D

We now consider vector valued functions. An n - homo

geneous polynomial from E into F which is of the form

x ~ k l: A [ . (x)] n y. ,

j=l J J J

where A1, ... ,A k E ~, l' .. 'k E E* and Yl""'Yk E F, is

denoted by

k nl: A. . y.

j=l J J J

The elements of p{nE;F) expressible in this form are called

n - homogeneous polynomials of Finite Type from E into F.

These polynomials from a subspace of p{nE;F) which is denoted

by P f {nE;F).

Proposition 7.6:

12

Proof:

The mapping

given by (u,y) ---+ u.y is bilinear, where u.y is the poly

nomial: x ~ u(x)y. Therefore there exists a linear

mapping

nsuch that j (ug, y) = u.y for every u e: h (E*), and every

k n n y e: F. j is onto, since the elementj~lAj~ .y of Pf ( EiF)

is the image under j of ~ A.~~n)~ y .. j J J ~ J

To see that j is injective, suppose that j(w) = 0 for

w e: hn(E*) F. If w ~ 0 then ([12J ,I,2) there exist

scalars A1, .. ,A k , none of which are zero, linearly inde

pendent elements Pl' ... 'Pk of hn(E*), and linearly inde

pendent elements yl' . 'yk of F, such that

k w =.L1A.P.~y.

J= J J J

But if j(w) = 0, then k L A.P. (x)y. = 0 for every x 'e:E.

j=l J J J

If we now choose x e: E such that P.(x) is not zero for J

some j, we obtain a contradiction.

D

We now begin to study continuous polynomials. If the

vector spaces E and Fare topologized, with topologies Ll and

13

'2 respectively, p(nE iF ), or p(n(E"l) i (F;'2)) will 'I T1

denote the set of all continuous n - homogeneous polynomials

from E into F, continuity being with respect to the topo

logies T1 and '2' If the topologies in question are clear

from the context, the notation p(nEiF) is used. p(nE) will

denote this space when F is the field of scalars.

As in [2~, a DuaZity, denoted by , is a pair of

vector spaces over the same field, lK, together with a bilinear

form

E x G ----7> Jl{, (x,y) ----.. ,

such that

= 0 for every y G implies x = 0

and = 0 for every x E implies y = 0

The weak topology, a(E,G), is then a locally convex topology

on E. It is the weakest locally convex topology for which

the dual space of E is G.

Proposition 7.?:

Let be a Duality_

n n'Then p( (E, cr (E,G)) is isomorphic to h G.

Proof:

Note first that G can be identified with a subspace of the

algebraic dual, E*, of Ei if ~ is an element of G,

~(x) =

14

defines a linear form on E, and the defining properties of

a duality show that this defines an embedding of G into E*.There

fore every element of h nG can be identified with a polynomial on

E of finite type, since hnG can then be identified with a subspace

of hn(E*). This identification provides the required isomorphism.

Suppose ~ A.~~n) is an element of hnG. Then

j=l J J

n n ~l""'~k G. Therefore the polynomials ~l,'cf>k are

k n all o(E,G) - continuous, and so the polynomial ~ A.~. is

j=l J J

o(E,G) - continuous.

Conversely, suppose P p(nE) is o(E,G) - continuous. k n

Then ([~, Section 2) p can be written in the form ~ A.~.,j=l J J

where ~l""'~k are elements of G. Hence P may be identified

with the element ~ A.*~)of hnG. j=l J J

In particular, with G = E*, we find that

Linearization of Gateaux-Holomorphic Mappings

For the rest of this chapter, all vector spaces will

be over the Complex field.

We recall that if E is a Complex vector space, and F

a Complex locally convex space, a mapping F:E~F is said

to be Gateaux - Hotomorphia if for every continuous linear

functional ~ on F and for every affine Complex line L in E,

15

(wof) IL is a holomorphic function of one Complex variable.

Kf(E) will denote the set of all finite - dimensional

convex balanced compact subsets of E.

The following result is well-known (see, for example,

[2lJ,1.2):

Lemma 1.1:

Let E be a Complex vector space and F a weakly sequentially

complete Complex locally convex space, with continuous dual

F'.

A mapping f:E---+F is Gateaux - Holomorphic if and only if

there exists a sequence {Pn}~=ol where P n E p(nEiF ) I such

that

f(x) = (Xj

E Pn(x) for every x E E. n=o

It follows that

(Xj

Conversely, if {Pn}n=o is a sequence of n - homogeneous poly

nomials from E into F which satisfies (1), then

rex) = (Xj

E P (x) n=o n

defines a Gateaux - Holomorphic mapping from E into F.

D

HG(EiF) will denote the vector space of all Gateaux

holomorphic mappings from E into F.

16

Our aim is to construct on E a universal Gateaux

holomorphic mapping which will "linearize" all other Gateaux

holomorphic mappings on E in the same way as the universal

n - homogeneous polynomials X :E~hnE linearize all other n

n - homogeneous polynomials on E. Since the definition of the

space HG(E;F) must involve a discussion of convergence of infi

nite series, it is necessary to introduce some topological ideas.

For each finite - dimensional convex, balanced compact sub

set K of E, E(K) denotes the subspace of E spanned by K, equipped

with the norm whose closed unit ball is K. p(nE(K is a Banach

space with the norm

II p II = sup{ Ip (x) I : X K}E

p(nE(K, being finite - dimensional, is reflexive, and so we

can define on hnE(K) the dual norm, which we denote by K II 11 n ; thus,

K II u II n = sup{ Ip (u) I

We now define

= { r ( h nE (K) , K 1\ II )}n=O n Co

= {(u ) E TI hnE (K) : limK.!I u II = O} , n n=O n-;..oo n n

the norm of an element u = (un) of hKE being

= sUPK II u 'II n n n Then hKE is a Banach space.

Now the set Kf(E) is directed under inclusion:

K ~ K' if and only if KeK'

If K ~ K' then E(K) is a vector subspace of E(K'), and so

17

hnE(K) is a vector subspace of hnE(K') for every n. Since then

for every P p(nE(K')) we have

it follows that the inclusion mapping of hnE(K) into hnE(K')

has norm at most equal to one. Therefore there is an inclusion:

j K I K : (hKE, K II . II )-----7> (hK, E, K I II . II )

with /I j K I K II ;:;: 1. Clearly, if K S K' ~ K", then

Therefore the family of Banach spaces hKE with the linking

mappings jK'K' indexed by the directed set Kf(E), is an

inductive system. We define the locally convex space hE to

be the locally convex inductive limit of this system:

For each K E Kf(E) there is a continuous linear inclusion

mapping

and these mappings commute with the linking mappings:

We now define a mapping

X : E~ hE

To do this, let x be an element of E. Choose any convex

balanced finite - dimensional compact subset K of E such

that x lies in the algebraic interior, int K, of K, where

18

If Y int K, then Kl ly III < I, and so

K1Iy(n)!l = [KllylllJn~ 0 as n~ co.n

Therefore a mapping

can be defined by

x (y) = (y(n K

It is clear that if K ~ K' then

and so a mapping X : E ~ hE is defined unambiguously by

setting

having chosen K Kf(E) so that x E int K.

Proposition 7.8:

(i) The mapping X : E~ hE is Gateaux - holomorphic.

(ii) Let F be a weakly sequentially complete Complex

locally convex space. For every Gateaux - holomorphic

mapping f : E~ F there exists a unique continuous

linear mapping

f : hE ~ (F,o{F,F'

such that f = foX.

Conversely, if T is a continuous linear mapping of hE

into (F,O{F,F' then the equation f(x) = T(X{x

19

defines a Gateaux - holomorphic mapping f from E into F

such that f = T.

The correspondence f ~

20

n - homogeneous polynomials {Pn,K}KEKf(E) satisfies:

P / P if K ~ K In,K ' E(K) = n,K

Hence for each n there is a unique n - homogeneous

polynomial P on E such that n

P / = P for every K E Kf(E)n E (K) n, K

From (3) it follows that

and from (2) we have

Therefore ~ 0 X is a Gateaux holormorphic func

tion on E for every continuous linear functional

~ on hE. Hence X is a Gateaux - holormorphic mapping.

(ii) It follows from part (i) of this proof that if

T : hE~ (F,a(F,F ' is a continuous linear mapping,

then T 0 X is a Gateaux - holormorphic mapping from

E into F. The mapping T --'" T 0 X is easily seen to

be a linear mapping from L(hEi (F,a(F,F ' ) into

To show that this mapping is injective, we must prove

that T 0 X = 0 implies T = O. Since T = 0 if and only

if ~ 0 T = 0 for every ~ E F', injectivity of the

mapping T ~ T 0 X will follow from:

21

Claim:

If ~ is a continuous linear functional on hE such that

~ 0 X = 0, then ~ = o.

To prove this, we follow the notation of the proof of (i) i

if the Gateaux - holomorphic function ~ 0 X vanishes iden

tically on E, then all of its derivatives at the origin,

Pn ' must vanish. Therefore Pn/E(K) = Pn,K vanishes for

every n and every K, and so for every K, each of the linear

mappings Pn,K vanishes. Therefore, from (3), ~K = 0 for every K, and so ~ = O. Thus our claim is proved.

Finally, we show that the mapping T~ T 0 X is sur

jective. Let f be a Gateaux - holormorphic mapping from E

into F. Then there exist n - homogeneous polynomials P n

from E into F such that

00

f(x) = L P (x) for every x E, andn=O n

00n!o II oPnil K < for every

22

sequentially complete, this series converges to an element

of F, which we denote by TK(U). Then TK is a continuous

linear mapping from hKE into (F,a(F,F')), and we have, by (4),

TK = TK, 0 jK'K if K ~ K'

Therefore there exists a continuous linear mapping T from

hE into (F,a(F,F')) such that

and it is clear from the definition of the mappings TK that

T 0 X = f. D

23

C HAP T E R II

Linearization of Continuous Polynomials

and Holomorphic Mappings on Locally

Convex Spaces

In this chapter, E, F will denote locally convex spaces.

E' is the continuous dual of E, and (E',C(E' ,E or E' will c

denote E' with the topology C(E',E) of uniform convergence on

convex balanced compact subsets of E. f(B) will denote the

convex balanced hull of a subset B of E, and feB) its closure.

nE denotes the completion of E. L( EiF) denotes the vector

space of all continuous n - linear mappings from En into F.

First, we recall some basic facts concerning topological

tensor products. We follow the notation and terminology of

Grothendick [13J as far as possible.

If p is a seminorn on a vector space E, a seminorm @np

is defined on the n - fold tensor product nE by

np(u) = inf{ . k E 'A, Ip (x 'I ). ~ . P (x, )

J=l J J In

k u = E LX'I x. }j=l J J In .

where the infimum is taken over all possible representations

nof u e:: E. np can be characterised as the largest of all

the seminorms, q, on nE satisfying

p(xl ) .. p(x ) for every xl' .. ,x e::E. n. n

24

nEquivalently, @P is the Minkowski functional of the set

r(nUp)' where

U = {x E: E .. p(x) ~ I} ,p and nu denotes the image of (U) n under the mappingp p

En -...;. nE p : n

If {p} A is a fundamental system of continuous semia aE:

norms for a locally convex topology on E, the seminorms

n@ Pa form a fundamental system of continuous seminorms for nlocally topology on E. The resulting locallya convex

convex space is denoted bynE , and is called the n - fold 'IT

Projective Tensor Product. A fundamental system of convex

balanced neighbourhoods of the origin in ~E is given by the 'IT

nsets r ( U ), where a

u = {x E: E p (x) ~ I}a a

is the closed unit ball of the seminorm p. The canonical a

n - linear mapping

is continuous, and has the following universal property:

For every locally convex space F, the canonical iso

n n'morphism of L ( E; F) with L ( Ei F) induces an isomorphism of

L(nEiF ) with L(@nEiF)i furthermore, this isomorphism estab'IT

lishes a one to one correspondence between the equicontinuous

subsets of L(nEiF ) and L(@nEiF).'IT

nIf we place on the symmetric tensor product 0 E the ninduced topology from E and denote the resulting locally'IT

n convex space bY0 E, then it is easy to see that the canonical

'IT

. I . . En 0 n . t .symme t r1C n - 1near mapp1ng on: ~. 'lTE 1S con 1nuous,

25

and has a universal property for continuous symmetric n

linear mappings on E similar to that of the mapping

p : En~ nE . n 7T

since0nE and hnE are identical subspaces of ~nE, we

have, therefore, obtained a locally convex topology on

hnE. The resulting locally convex space is denoted by

hnE. We shall call this space the Projective nth. Homo7T

geneous Product of E. Since a polynomial, P, is continuous

if and only if the symmetric mapping Tp which generates it

is continuous, we have:

Proposition 2.7:

Let E be a locally convex space. For each n, the mapping

Xn of Definition 1.1 is a continuous n - homogeneous poly

nomial from E into hnE. 7T

If F is a locally convex space, and P a continuous n

homogeneous polynomial from E into F, the linear mapping

p :_ h~E ~ F is continuous, and P = P 0 X Conversely, if n T is a continuous linear mapping from h~E into F then P=T 0 Xn

is a continuous n - homogeneous polynomial from E into F with

P = T.

The correspondence P ~

26

In the next proposition we describe a fundamental system

nof continuous seminorms on the locally convex spaces h E 'IT

which are more suitable then the seminorms np for dealing

with polynomials.

If U is a subset of E, we shall use the notation u(n)

for the subset X (U) = {x(n):x s U} of hnE. n

Proposition 2.2:

Let E be a locally convex space.

(i) If P is a seminorm on E then

(n) k n p (u) = inf{ . E IA . I [p (x. )] u = ~ A.x. (n) }

J=l J J j=l J J

where the infimum is taken over all representations of

u s hnE of the form ~ A.x. (n), is a seminorm on hnE. j=l J J

pen) is the Minkowski fUnctional of the set r(u(n,

where

U = {x s E : p(x) ~ l} .

Also, pen) (x(n = [p(x)]n for every x s E.

pen) is a norm if and only if p is a norm.

(ii) The seminorms~p and p(n) satisfy:

np(u) ~ pen) (u) ~ ~~np(U) for every u s hnE.

If {Pa}asA is a fundamental system of continuous semi

norms on E and {U} A is the corresponding fundamental a as

system of convex balanced neighbourhoods of the origin,

then {p (n)} is a fundamental system of continuous semia

norms on hnE and the corresponding fundamental system'IT '

27

of convex balanced neighbourhoods of the origin is

{fU(n)} a

(iii) Let F be a locally convex space, and let U be a convex

balanced neighbourhood of the origin in E. Let q be a

continuous seminorm on F.

Under the canonical isomorphism of p(nEiF) with

nL(h EiF), the equicontinuous subset If

{PEP(nE;F) : IIqo Pllu ~ I}

of p(nEiF) is identified with the equicontinuous subset

{TEL(h~E;F) : IlqoTllf(u(n ~ I} of L(hnE;F). In particular, the equicontinuous subset

If

{PEP (nE) : II P II u ~ I}

of p(nE) is identified with the polar set (fU{nO

in (hnE) I

If

Proof:

We use the methods of Grothendieck ([13J, prop.l, P. 28) for

(i) and of Gupta ([15J I 2, Prop. 6) for (ii)

(i) It is clear that pen) is a seminorm on hnE.

Now let u = ~ A.X~n) be any representation of u E hnE. j=l J J

We may assume, without loss of generality, that none of the

scalars A1 .... ,A are zero. We may also assume thatk

~ IA.' [P(x.)l n is not zero, for otherwise we have pen) (u) = 0, j=l J J and there is nothing to prove. We now define numbers A'~l"""~k'

28

and elements yl' .... 'yk of E, by:

k A = E IA I [p (x. ) ] n ,

j=l J J

A. rp{x.)]nJ - J if p{x.) :;t 0 ,

J

\J. = J

A. J k r IA. I [p (x. ) ] n

i=l 1 1

if p{x.) :;t 0 ,J

y. = J

x. if P (x .) = 0 ] J

We then have

u = A ~ \J.y. en) . J JJ

k with E I\J. I ~ 1, and y. U for every j.

j=l J J Therefore

pen) (u) = inf{A>O : u

and so pen) is the Minkowski functional of ru(n).

It is clear that pen) (x{n)~ [p{x)]n for every x E.

On the other hand, if we use part (ii) of this proposition,

29

Therefore p(n) (x(n = [p(x)]n for every x E.

It follows immediately from this that if p(n) is a

norm, then so is p. And if p is a norm, so also is np

([)3], Chapter I, Prop.l, p. 28), and again using part (ii),

since np ~ p(n) I it follows that p(n) is a norm.

(ii) Since the infimum which defines np(u) is taken over

all possible representations of u in nE , while the infimum

defining p(n) (u) is taken over representations of the form

f A.X~n) , we have j=l J J

np(u) ~ p(n) (u) for every u hnE.

k Now let u = L: A.X . ~ .. x . be any representation of u.

j=l J 1 J nJ Let be a positive Real number. If we define

if p (x .) " 0rJ

Yrj'= x .

..fl if p(x :) = 0

rJ

and

j.l. =A.p(X1}p(x.)J J J nJ I

where the term p(x .) in the definition of j.l. is replacedrJ J

by whenever p(x .) = 0, thenrJ

k u = L: j.l.yl. ... y .

j=l J J nJ

k L: j.l.yl.0 .... 0y .

j=l J J nJ

30

(n)E E E (E1Y 1 .+ +E Y .) E =+1 1 n J n nJ

].

Therefore, since p(Yrj) ~ 1 for every rand j,

p (n) (u) :!S 1 ~ IlJ. 1 E (p (Y .) +. . . +p (y . n 2nn I j =1 J E: =l 1 J nJ

].

nn k = -I E 1 A1 p {x .) E p (x .)

n.j=l J 1J nJ

Since this holds for every E: > 0, we may let E: tend to zero.

Then

~ nn -I n

k 1: IAI p(x1)p{x .).

j=l J J - nJ

Hence we have

p (n) (u)

This shows that the seminorms p{n) and np on hnE are equi

valent, and the remaining assertions of (ii) follow i~~ediately.

(iii) follows from (i)

D

Suppose now that E is a normed space, with norm II II , and (open or closed) unit ball B. Proposition 2.2 shows that hnE

1T

is a normed space for each n, with norm II II (n) and closed unit ball fB(n). Let Tb denote the topology of uniform convergence

on bounded sets. Then (p(nE) ,T ) is a Banach space with theb

31

norm:

Ilpll = sup{lp{x)I IIxll ~ l}.

From Proposition 2.2 we get:

Corollary 2.1:

Let E be a normed space. Then for each n, hnE is also a1f n {nnormed space, and the Banach spaces (P{ E) ,i b ) and h1fE)S

1

are isometrically isomorphic.

D

Next, we give some properties of locally convex spaces which

are preserved under the formation of projective homogeneous

An n "'nproducts. h E denotes the completion of h E and ~ E1f 1f' 1f

denotes the completion of nE 1f

Proposition 2.3

Let E be a locally convex space.

(i) hnE O;nE) is Quasi-Normable if E .(respectively, :8)1f 1f

is Quasi-Normable.

(ii) hnE is a Schwartz space if E is a Schwartz space.1f

(iii) hnE is a Nuclear space if E is a Nuclear space.1f

Proof:

(i), (ii) and (iii) folloW immediately from the corresponding

32

......n .....nresults for 1(9 E and E ([l3], Chap I, Prop. 7, p. 48 and

1T 1T

Chap. II, Theoreme 9, p. 47), and the fact that hnE is a 1T

ncomplemented subspace of E.

1T

D

The ~anonical isomorphism of suggests a

general method fOr tackling problems involving topoiogies an ~~e

given a locally convex topology 't . on

suppose that we can find a locally convex topology " on

which is a polar topology with respect to the duality

, such that, and " coincide under the canonical 1T 1T

isomorphism: If ,I turns out to be a familiar topology, such as the strong topology or the Mackey topology,

we can then apply what we know about,' to ,.

For example, we have seen (Corollary 2.1) that if E is a

normed space the topology 'b on p(nE) of uniform convergence

on bounded subsets of E, coincides with the strong dual topo

n nlogy S((h E) ',h E). A natural question to ask is whether this 1T 1T

is true in general.

Another much used topology on p(nE) is the topology '0 of

uniform convergence on compact subsets of E. If K is a compact

subset of E, then K(n) is a compact subset of hnE and the semi1T '

norm

P~ II P II K = sup{ IP (x) I : X E: K}

on p(nE ) coincides with the seminorm

33

(n) K (n) ,x : J

n on (h~E) '. Thus it is natural to ask whether TO corres

ponds to the polar topology c(hnE)', hnE) of uniform conver~ ~

gence on convex balanced compact subsets of hn

E. ~

Such questions appear to be very difficult in general.

Using the results of Grothendieck [l~, we will see that if

E is a DF space then Tb is equal to the strong topology An An

Sh E) ',h E), and if E is a Frechet space, that TO is equal ~ ~

An . An to the topology c(h E) ',h E).

~ ~

The seminorms pen) extend uniquely to seminorms on the

An ncompletion, hE, of h Ei we use the same notation for the

~ ~

extensions of these seminorms. Proposition 2.1 and Proposition

2.2 (ii) and (iii) remain true if we replace hnE by hnE. ~ ~

Proposition 2.4:

Let E be a DF space.

(i) h~E) and h~E are DF spaces.

(ii) Under the canonical isomorphism of p(nE) with (hnE) , ~

the topology Tb on p(nE) of uniform convergence on the

bounded subsets of E coincides with the strong dual

topology

S {(h~E) , , hnE) . ~

If {Bk}k is a fundamental sequence of bounded subsets of E then

(fB (n) } is a fundamental sequence of bounded subsets of hnE k k 'IT

(iii) If E is Bornological, or Barreled, or Infrabarreled, then

hnE has the same property. If E is a Montel space, so is ~

hnE. ~

34

Proof:

(i) (ii) and (iii) follow from the fact that hnE(hnE)1T 1T

n "'nis a complemented subspace of @ E (respectively, @ E), and

1T 1T

the corresponding results for n E and n E ([13] I Chap. I,1T 1T

Prop. 5, p.43 and Cor. 2, p.45).

D

As an example of an application of this Proposition, we

see that if E is a DFM space, then (p{nE) ,T ) is a Frechetb

Montel space, a result proved in a different way in [6J.

We now look at the projective homogeneous products of a

FrEkhet space.

Proposition 2.5:

Let E be a Frechet space.

An (i) Every element u of h E can be written as the sum of an

1T

absolutely convergent series of the form

u = E A.X~n) (1) j=l J J

where E IA.I < 1, and {x.} is a sequence which conj=l J J

verges to the origin in E.

An (ii) If J is a compact subset of h E, then there exists a

1T

fixed sequence {x.} converging to the origin in E such J

that every element of J can be written in the form (I),

35

with the sequence {A } ranging over a compact subsetj

of the unit ball of Zl' Hence every compact subset

of hnE is contained in a compact set of the form 1T

fK(n), where K is some compact subset of E.

Proof:

We prove (ii) from which (i) follows, An

(ii) If J is a compact subset of h E then J is also compact1T

"'nin E. Grothendieck ([13J, Chap I, Theoreme 1, p.51)1T

has shown that for such a set, there exist sequences

{Y1,k}k" ... ,{Yn,k}k' all converging to the origin in E,

such that every element u of J

form

00

u = L: llkY 1 k Y kk=l' n,

with the sequence {Ilk} ranging

subset of the unit ball of Zl'

hnE =0nE, we have 1T 1T

can be written in the

over a fixed compact

Since u is an element of

e lYl k+'" .+e Y k (n)(' n n, ) Ilk (n! )l/n

Since the sequences {Yr,k}k' 1 < r < n, are bounded,

this double series converges absolutely. Hence we may

36

order the terms as follows:

Choose an ordering

2n { (e 1 r' en r)} r=1, ,

of the set of all sequences (el' . ,en ) for which

~2' .

and

, I

e y + .. +e 2nYn1,2n 1,1 n, n / 2

(nl ) lin ,

lYl 2+' .+e lY 2e 1 I ,. n, n, , ... . . (nl)l/n

(n)Then u = 00L A.X. ,

j=l J J

and the sequences {A.}, {x.} have the desired properties.J J

D

Thus if E is a Frechet space, as K ranges over the compact

subsets of E, the sets rK(n) form a fundamental system of convex

37

nbalanced compact subsets of hnE. And for every P s P( E),

1T

IlplIK= IlpllrK(n)'

Therefore, we have

Corollary 2.2:

Let E be a Frechet space.

Then (p(nE),T ) is isomorphic to (hnE) ,o 1T c

D

Linearization of Holomorphic Mappings

In the rest of this chapter, E will be a Complex locally

convex space. Our aim is to construct a locally convex space

h E and a holomorphic mapping X from E into h E which will be 1T 1T

universal for holomorphic mappings on E in the same way as

the Gateaux - holomorphic mapping X:E~hE, constructed in

chapter I is universal for Gateaux - h'olomorphic mappings on E.

We shall assume that E is quasi-complete. This implies

that the closed convex balanced hull of any compact subset of

E is itself compact. Let K (E) d.enote the collection of all convex

balanced compact subsets of E. Then K(E) is a directed set under

inclusion:

K ~ K' if KCK', for K,K'sK(E).

For each KsK(E), let E(K) be the subspace of E spanned by K.

On E(K) we take as norm the Minkowski functional of K.

38

Since K is compact, E (K) is then a Banach space. \'Ve shall

denote by KII II the norm of E(K). We define the Banach

space h KE by:'IT ,

= {u={U } IT hnE(K) n n=Q IT

We denote the norm of hlT,KE by KII II also. Thus if

is an element of h KE,IT,

If K,K' are elements of K(E), and K ~ K', then the

inclusion of E(K) in E(K') induces, for each n, a continuous

linear mapping of hnE(K) into hnE(K') which has norm at most

IT IT

one, and so we may define a mapping continuous linear mapping

~n

with II jK'K 11 ~ 1, by defining jK'K on hlTE(K) to be the

natural mapping into h ~n

E(K') given above. If K,K',K" are IT

elements of K(E) and K~K'~K", then it is easy to see that

Therefore the family of Banach spaces h KE, indexed IT,

by the directed set K(E), together with the linking mappings

jK'K' forms an inductive system. We define the space h E IT

to be locally convex inductive limit of this system:

39

Then, for each K s K{E) there is a continuous linear mapping

j K : h KE -----;.hE,iT, iT

and these mappings commute with the linking mappings jK'K.

A Gateaux - holomorphic mapping

X : E '--7h iT E

is defined in exactly the same way as the mapping X : E--+hE

defined in chapter I. Thus if x is an element of E, we choose

a convex balanced compact subset K of E such that x lies in

the algebraic interior of K, and we define

X(x) = j K 0 X (x) ,K

where X : int K '--7h KE is the mapping y~{y{n)} .K iT, n

This definition is independent of K, and the mapping X so

defined is Gateaux - holomorphic.

The mapping X E ~h E is bounded on the compact subiT

sets of E. To see this, let K be a compact subset of E.

Choose a convex balanced compact subset J of E such that

KCAJ for some A such that O

40

space by Hk(E).

Suppose also that F is sequentially complete, and let

f E Hk(E;F). Let f = 00

E P be the Taylor expansion of f

n

n=O n at the origin, where P

n E p(nE;F) for every n. Then, using

the Cauchy inequalities, we find that

~ 'II qn=O

0 P 11K < n

00 for every K E K(E), and every

continuous seminorm q on F. Conversely, if {P } is a

sequence of n - homogeneous polynomials from E into F which

satisfies this condition, then it is easy to see that

f(x) = 00 E P (x) defines a k - holomorphic mapping f from E n=O n

into F.

Proposition 2.6:

Let E be a quasi - complete Complex locally convex space.

(i) The mapping x: E ----;.h E is k - holomorphic.'IT

(ii) Let F be a sequentially complete Complex locally convex

space. For every k - holomorphic mapping f:E ~F there

exists a unique continuous linear mapping f:h E~F such that'IT

f = foX. Conversely, if T is a continuous linear mapping

from h E into F, then the equation f(~) = T 0 x(x) defines a'IT

k - holomorphic mapping f from E into F such that f = T. The

correspondence f~~--~)f establishes an isomorphism between the

vector spaces Hk(E;F} and L(h'ITE;F).

In particular, the vector spaces Hk(E) and (h'ITE) I are canoni

cally isomorphic.

41

Proof:

(i) We have already seen that the mapping X is k - holo

morphic.

(ii) Let T be a continuous linear mapping from h~E into F.

Then T 0 X is a Gateaux - holomorphic mapping from E

into F, and since T maps bounded subsets of h E into ~

bounded subsets of F, T 0 X maps compact subsets of

E into bounded subsets of F. Therefore T 0 X is k

holomorphic. The mapping T~T 0 X is easily seen

to be a linear mapping from L(h~EiF) into Hk(EiF).

That this mapping is injective is proved in exactly the

same way as in the proof of Proposition 1.8.

Finally, we show that the mapping T~T 0 X is sur

jective. Let f be a k - holomorphic mapping from E into

F. Then there exist n - homogeneous polynomials P from n

E into F such that

f(x) = 00

l: P (x) for every x E E, (1)n=O n

and o PnIIK< for every K E K (E) , 00n!oll q

and for every continuous seminorm q on F. (2)

For each K E K(E) we define n - homogeneous polynomials

P K from E(K) into F byn, ,

It follows from (2) that P K is bounded on the unitn,

ball of E(K) I and hence P K is a continuous n - homon,

42

geneous polynomial from E(K) into F for every n.

If P K is the continuous linear mapping from hnE(K)n, ~

into F associated with P K' it follows from (2) that n, co ~

n!oll q 0 Pn,KI~K(n) < ~ for every continuous

seminorm q on F. (3)

Therefore, for every u = {u } e h KE,n ~,

is a Cauchy series in F, and so since F is sequen

tially complete, this series converges to an element

of F which we denote by TK(U). From (3), TK is a

continuous linear mapping from h KE into F, and it ~,

is clear from the definition of TK that

T T . 1.' f K < K'.K = K' 0 JK'K

Hence there exists a continuous linear mapping T

from h E into F such that ~

T 0 jK = TK for every K e K(E) I

and it follows from the definition of the mappings TK

that T 0 X = f.

D

43

Suppose now that E is a Frechet space, and let f be

an element of Hk(E), with Taylor expansion 00

I P at the n=O n

origin. The condition

~ II P II < for every K e: K (E)n=O n K

00

implies, in particular, that P is bounded on the compactn

subsets of E for every n. Therefore P is continuous for n

every n. Since the expansion 00

E P converges uniformly on n=O n

the compact subsets of E, f is continuous on compact sets.

But E is a k - space, and so f is continuous on E. Therefore

the h010morphic and the k - holomorphic scalar - valued func

tions on E are the same.

Lemma 2.1:

Let E be a Frechet space. If the sequence {xk}k converges

to the origin in E, there exists a convex balanced compact

subset J of E such that {xk } c. int J, and X (x ) converges toJ k

the origin in h JE. 'IT,

Proof:

There exists an increasing sequence of, Real numbers, { A } , k

Ak > 1 for every n, and lim Ak = + 00, such that AkXk con-k-+oo verges to the origin ( [19J , Prop. 16.5 ) . Let J be the closed convex balanced hull of the sequence {AkX }. Then JK

is compact, and

-1 = Ak for every k.

44

Therefore J II X J (xk ) II = Ak 1 for every k, and

so lim XJ(xk ) = 0 k"')'oo

Proposition 2.7:

Let E be a Complex Frechet space

(i) The mapping X: E -----;.h E is holomorphic.1T

(ii) The vector spaces H(E) and (h E)' are canonically1T

isomorphic. This isomorphism establishes a one-to

one correspondence between the 10 - bounded subsets

of H(E) and the equicontinuous subsets of (h E) '.1T

Proof:

(i) X is Gateaux - holomorphic, and Lemma 2.1 shows that

X is continuous at the origin. Hence X is holomorphic.

(ii) We have seen that Hk(E) and H(E) are identical. It

only remains to prove the last assertion of (ii).

A fundamental system of neighbourhoods of the

origin in h1TE is given by the s~ts of the form

U = r rU CKjK (Bl (h1T ,KE)~LKeK (E) J

where CK > 0 for every K, and Bt(h1T,KE) is the unit

ball of h KE. Therefore a fundamental system of equi1Tt

continuous subsets of (h E)' is given by the polars:1T

45

uO = n C-l[K jK(B l (h~,KE JO Kt:K(E) = C-

1 (.t)-l [B (h E)]O (1)n K J K 1 ~,KKt:K{E) where j~ is the transpose of jK' Therefore ,

if f is an element of B{E) with Taylor expansion 00

E P at n=O n

the origin then f is an element of uO if and only if

1E II p--IIK < CK- for every Kt:K{E} (2)n=O n

The set of all f t: B(E) satisfying (2) is bounded for TO:

and it is easy to see that as the family {CK}Kt:K{E) ranges

over all possible positive values of C ' the correspondingK

TO - bounded sets defined by (2) form a fundamental system

of TO - bounded subsets of B{E).

D

46

C HAP T E RIll

Duality for Spaces of Polynomials and Holomorphic Mappings

on Frechet Spaces.

Let F be a locally convex space. For every convex

balanced neighbourhood, U, of the origin in F, the associated

normed space F(U) is defined as follows: let PU be the

Minkowski functional of U. F(U) is the quotient of F by the

subspace p~l (0), and a norm, p , is given byu u

PU(x + P~ (0 = inf{pu(Y) : x - y E: ~l (o)}.

For every convex balanced bounded subset B of F, the asso

ciated normed space F(B) is defined to be the subspace of F

spanned by B, with norm, which we denote by B II II, equal to the Minkowski functional of B. The use of the same notation

for these two spaces is permissible since, for a subset of F

which is both bounded and a neighbourhood of the origin, they

coincide. We denote the quotient mapping of F onto F(U) by

TIU' and the inclusion of F(B) into F by iB. Both TIU and iB

are continuous. The dual space of F(U) can be identified

with the Banach space FI (Uo), since (TIu)t defines an isometric

isomorphism of F(U) I with FI (Uo). If U and V are convex

balanced neighbourhoods of the origin in F and U is a subset

of V, there is a canonical mapping TIVU:F(U)~F(V), given

by TIVU(x + puleo~ = x + pv1(0), and the norm of TIVU is at

47

most one. Similarly, if Band C are convex balanced bounded

subsets of F and if B is a subset of C, the inclusion mapping

iCB:F(B)~F(C) has norm at most one.

Let G be a normed space. We recall that an n - homo

geneous polynomial P:G~ C is said to be Nuclear if there is

a sequence {.} of continuous linear functionals on G, and a J

sequence {A.} of scalars , with +im II . II = 0,. ~ IA . I < co, J J+co J J=l J

and

P (x) = coE A.. (x) n for every x E G. j=l J J

The Nuclear Norm of P is defined to be the infimum of the

sums ~ IAI II J. lln , ranging over all representations of P j=l J

of the above form. This norm is denoted by II liN' and the resulting Banach space of all nuclear n - homogeneous poly

nomials on E is denoted by PN(nG} [15]. Thus PN(nG} is the

completion of the subspace Pf(nG} of p{nG} in the nuclear

norm. We have seen in Chapter I that there is a canonical

isomorphism

v : hnG' ~Pf(G}

co (n) k ngiven by v( E A. } = r A. . With the nuclear norm

j=l J n j=l J J on Pf(nE), and the projective norm on'hnGS' v is an isometry. Since the nuclear norm on Pf(nG) is bigger than the norm of

uniform convergence on the unit ball of G, v is a continuous

linear mapping of h~Ge into (p(nG},T ). Therefore v extendsb to a continuous linear mapping, which we also denote by v, of An n "nhnGS into (P{ G}"b)' The image of hnGS under v is the subspace PN(nG) of p(nG). Therefore v will be an isometric isomorphism

of h~Ge with (PN(nG), II.I~} if we know that it is injective.

48

We recall that if the Banach space GS has the Approximation Property, then the canonical mapping of ~GB into

(L(nG),T ) is injective ([131, Chap. I, Prop. 36, p. 167).b

It follows that the mapping v : h~GS ~(p(nG) ,T b ) is injec

tive. We have proved:

Proposition 3.1

Let G be a normed space, and suppose that the Banach space

G has the approximation property. Then the canonicals mapping

is an isometric isomorphism.

D

Now suppose that G is a locally convex space. The

nvector space P { G) is defined as follows: An n - homo-N

geneous polynomial, P, on G is Nuclear if there is a convex

balanced neighbourhood of the origin 9 such that P factors

through the canonical mapping TIU : G ~G(U) to give a

nuclear n - homogeneous polynomial on the normed space G{U).

PN(nG) is the vector space of all such polynomials on G.

Thus, for every convex balanced neighbourhood of the origin

U, we may identify P enG (Un with a subspace of P (nG), andN N

49

PN(nG) is the union of all these subspaces. We can suppose,

without loss of generality, in the above definition, that

the neighbourhoods U belong to a fixed fundamental system

of convex balanced neighbourhoods of the origin. Accordingly,

where U is any fundamental system of convex balanced neigh

bourhoods of the origin in G.

The topology JI on PN(nG) is obtained by taking the w

locally convex inductive limit 'topology of the normed spaces

PN{nG(U with the nuclear norm, as U ranges over a fundamen

tal system U of convex balanced neighbourhoods of the origin

[aJ . It is easy to see that that the definition of JI is w independent of the choice of U.

Recalling now that (G{US may be identified with

G' (UO) and using Proposition 3.1, we have

Proposition 3.2

Let G be a locally convex space, and suppose there exists

a fundamental system, U, of convex balanced neighbourhoods

of the origin in G such that the Banach space G' (UO) has

the approximation property for every U E U. Then

(PN(nG),JI ) is canonically isomorphic to the locally convex w

inductive limit lim ) h~G' (UO). U

D

50

The DuaZ Space of (P( n

E) ,TO)

Let E be a Frechet space. We know that the canonical

isomorphism of p{nE) with (hnE) , is an isomorphism of the'IT n An

locally convex spaces (P{ E),T ) and (h'ITE)~ ; where cO

denotes the topology of uniform convergence on all convex

balanced compact sets (Corollary 2.2). It follows from

the Mackey - Arens theorem (EI4], Th. 7, p. 68.) that the

dual space (p(nE) ,TO) I is algebraically isomorphic to

hnE. Explicitly, this isomorphism can be described as'IT

follows: if u is an element of h ~n

E, then by Proposition'IT

2.5 we can write u = 'f A.X~n) where < 00, and thejIIAjlj=l J J sequence {x j } converges to the origin in E. Then u corres

ponds to the linear functional on p(nE) given by

00

P ~ LA. P (x.) , j=l J J

and every TO - continuous linear functional is of this form.

We also know from Proposition 2.2 that the canonical iso

morphism of p(nE) with (hnE)' establishes a one-to-one corres'IT

pondence between the equicontinuous subsets of these spaces.

Now, since E is a Frechet space, the equicontinuous subsets

of p(nE) and the TO - bounded subsets are the same.

also a Frechet space, which implies that the equicontinuous An

subsets of (h E)I are the same as the weakly bounded subsets. 'IT

Therefore the strong dual space of (p(nE) ,TO) is isomorphic

lwith the strong dual space of (hnE)c which is hnE. ~'le haveI 'IT 'IT

proved:

51

Proposition 3.3:

n ,Let E be a Frechet space. Then (P( E)'~o)e is alge

braically and topologically isomorphic to h~E

D

Since hnE is the completion of hnE and this latter space1T 'IT '

can be identified with the space of all finite-type n

homogeneous polynomials on E, we would like to represent

the space hnE as a space of polynomials on El. We begin1T

with

Lemma 3.1:

Let E be a locally convex space. Then h ~n

E is isomorphic1T

to the locally convex inductive limit lim) hnE(K)

K (E) 'IT

Proof:

~n

Let iK denote the canonical mapping of h'ITE(K) into

Klim nnE(K), and let x denote the universal n - homo~ 'IT n K(E) An geneous polynomial from E(K) into h E(K).

1T

The inclusion mapping of E(K) into E induces a continuous

linear mapping of hnE(K) into hnE for each n; these map1T 1T

pings commute with the linking mappings:

and so there exists a continuous linear mapping, T, from

lim> hnE(K) into hnE such that T 0 i (x(n = x(n) when K(E) 1T 1T K

52

X E: E(K)

KThe family of n - homogeneous polynomials P = iK 0 Xn:K E(K) ~ lim hnE(K) define a continuous n - homogeneous

K (E 'IT An polynomial P from E into lim h E(K) such that P(x) = P (x)

K(Er 'IT K when x E: E(K). Hence there is a continuous linear mapping - An A P:h E---;. lim hnE such that, if x E: E(K) then p(x(n\=PK(X).

'If K(E) 'If The mappings P and T are inverse to one another, and

the Lemma follows. D We remark that Lemma 3.1 remains true if we replace

K(E) by any fundamental system of convex balanced compact

subsets of E.

nSo far, we have shown that the strong dual of (P( E) ,TO)

is isomorphic to lim hnE(K), where K is any fundamental K ) 'If

system of convex balanced compact subsets of E. The spaces

E(K) are the duals of the normed spaces E'(Ko ), and as K

o ranges over K, the sets K form a fundamental system of

neighbourhoods of the origin for the locally convex space E',c

Therefore Proposition 3.2 can be applied provided the Banach

spaces E(K) all possess the approximation property.

Definition 3.1:

The locally convex space E will be said to have the Stpiat

Apppoximation ppopepty if there is a fundamental system, K,

of convex balanced compact subsets of E such that the Banach

space E(K) has the approximation property for every K E: K.

Let E be a Frechet space, and let K be a fundamental

system of convex balanced compact subsets of E. Then E

53

coincides, both algebraically and topologically, with the

locally convex inductive limit lim>E(K). Furthermore, this KsK

inductive limit is Compactly-Regular, in the sense that

every compact subset of E is contained in E(K) for some K,

and is compact therein. It follows by a theorem of Bierstedt

and Meise ([3J,I, Satz 2) that if the Frechet space E has

the strict approximation property, then E has the approxi

mation property.

ExampZes:

We give some examples of spaces with the strict approxi

mation property.

Recall ([9],IV,13.9) that a subset K of Co is relatively

compact if and only if there exists a sequence {An} of

positive Real numbers converging to zero, such that

Ixnl ( An for every n, and for every x = {x } E K. n Let N be the collection of all sequences of positive Real

numbers which converge to zero. For each A = {A } E N, let n

Then the above condition shows that {KA}AEN is a fundamental

system of convex balanced compact subsets of co'

Let A E N. Then an element x of Co lies in CO(KA) if

and only if there exists u > 0 such that x E uK, which is

equivalent to the condition:

sup Ix I A-I < (1)00 n n n

54

Conversely, if x = {x } is any sequence of scalars which n satisfies (I), then it is clear that x e CO(KA). Therefore

< co}

The norm of Co (K A) is given by:

K II x II = inf{l.l >0 A

= inf{l.l>o Ix 1

55

{x={x }: suplx IA- 1 < oo} ct ct ct ct

and that the norm of (fT~) (KA) is given by cteA

II x II = sup Ix IA-I.KA ct ct ct

It follows that the mapping x ----;. {x A-1}ct ct

defines an isometric isomorphism of (f1~) (K~)cteA I\.

with loo(A). Since loo(A) has the approximation property,

the space f1~ has the strict approximation property. cteA

(3) EE1ct cteA

A subset K of the locally convex direct sum @~ is ct eA

compact if and only if there is a finite subset, B, of A

such that K is contained in the canonical image of Ef} in cteB

~~, and is compact therein. Therefore for every convex cteA balanced subset K of ~~, the Banach space ( ffict) (K) is

ctE:A cteA finite - dimensional, and so has the approximation property_

Hence @ has the strict approximation property.cteA

Proposition 3.4:

Let E be a Frechet space with the strict approximation

property. Then (p(nE) ,TO)S is isomorphic to (PN{nE~) ,~w)

Proof:

By Proposition 3.3 and Lemma 3.1, (p{nE) ,To'a is isomor

phic to lim hnE(K), where K is a fundamental system ofK ) 1T

convex balanced compact sets satisfying definition 3.1.

56

The Proposition now follows by applying Proposition 3.2 to

the space E'. c

From the remarks preceding Proposition 3.3, we see that

the above isomorphism can be described explicitly by means

of the Borel Transform: If T is a continuous linear functional

on (P( n E)" ), the corresponding nuclear polynomial on E' is o c

given by

It follows, in particular from Proposition 3.4 that the poly

nomials in p(nE) of finite type are dense in (p(nE)"o).

The DuaZ Space of (H(E)" )o

Let G be a locally convex space. We construct the space

of NucZear HoZomorphic Germs at the origin in G as follows:

Let V be a fundamental system of ~onvex balanced neigh

bourhoods of the origin in G. For each U V, HN,U is

defined to be the vector space of all functions f which are

holomorphic on some neighbourhood of the unit ball in G(U),

with ~I dnf(o) PN(nG(U) for every n, and satisfying:

lim II~, dnf(o)jjN = 0 n~ ex>

57

If v E U and V is a subset of U, composition with the

canonical mapping TIUv:G(V) ~ G(U} defines an injective

linear mapping

The vector spaces HN,V together with the linking mappings

rVU form an inductive system, and we define the vector space

~

HN(O(G)} = lim) (HN,u,rVU ) U

The elements of HN(O(G are called Nuclear Germs at the

origin in G. If we put on HN,U the norm:

o 1 AnII f II u = s ~p II Ii! d f (0) II N '

then the linking mappings rVU are continuous, and so we can

define a locally convex topology, TIw' on HN(O(G}} by forming

the locally convex inductive limit:

Proposition 3.5

Let E be a Complex Frechet space.

(i) (H (E) , TO) '8 is isomorphic to hTIE.

(ii) If E has the strict approximation property then

(H(E),T )~ is isomorphic to (HN(O(E' ,TI ).o ~ c w

Proof:

(i) We show first that h E is contained in (H(E),T )'.TI 0

Let u E hTIE; then u = jK(v) for some K E K(E}, and

some v = {v } E h KE. If f E H(E), let f be the n TI,

58

continuous linear functional on h E associated with'IT f

by Proposition 2.6.

Then the linear functional f~ f{u) is T o - continuous,

since

~ II vii

Next we show that (H{E),T ) I is contained in h E. o 'IT

Under the canonical isomorphism of H{E) with (h E) I, the'IT

topology T on H{E) corresponds to the topology on (h E) I o 'IT

of uniform convergence on the subsets of h E of the form'IT

XK, where K is a compact subset of E. We claim that the

closed convex balanced hull of XK in h E is compact if K'IT

is compact. It then follows (by Corollary 2, page 69 of

[14J) that (H{E),T ) I is contained in h E. . 0 'IT

To prove our claim, let K be a compact subset of E.

Since E is a Frechet space, K is contained in the closed

convex hull of a sequence which converges to the origin

in E. Therefore, by Lemma 3.1 there exists a convex

balanced compact subset, J, of E such that K is contained

in int J, and XJ{K) is compact inh'IT,JE. Since h JE is'IT,

a Banach space, rxJ{K) is compact in h'IT,JE jJ is a

continuous linear mapping, and therefore jJ(rxJ(K is a

convex balanced compact subset of h'ITE. But jJOXJ{K) = X{K),

and so

rxK = rjJoXJ{K) = jJ{rxJ{K)

jJ(rxJ(Kc:r(jJoxJ(K = rX(K)

59

It follows that jJ(fXJ(K is the closed convex

balanced hull of x(K), and therefore fX(K) is compact

in h E. 7T

Therefore (H(E),T )' is equal to h E. Now byo 7T

Proposition 2.7, the T - bounded subsets of H(E) coino

cide with the equicontinuous subsets of (h E) '. h E 7T 7T

is Barreled, and so the equicontinuous and the weakly

compact subsets of (h E)' are the same. Therefore the7T

strong dual topology on (H(E),T )' is equal to the o

topology of h 7T E.

(ii) If E has the strict approximation property let K be a

fundamental system of convex balanced compact subsets

of E such that E(K) has the approximation property for

every K E K. Then hnE(K) is isometrically isomorphic7T

to (PN(nE'(Ko,II.II N) for every n, and for every K E K.

From the definition of the Banach spaces HN,Ko , we have

oHN,KO = {n!o (PN(nE , (K , II liN) }c

o

Therefore

00 ~n

h E = lim) h KE = lim { E h E(K)}7T K 7T , ~ n=o 7T, Co

is isomorphic, both algebraically and topologically, to

= lim (EN KO ,11.11 ) = HN(O(E'),7T )-r' KO c W o The isomorphism of (H(E) ,TO) , with HN(O(E~ given by

this p~oposition can be described explicitly as follows:

http:PN(nE'(Ko,II.II

60

If T is a continuous linear functional on (H(E)/c ) I o

then

defines a nuclear holomorphic germ at the origin in

E' c

It is also possible to represent the dual space of

(H(E)/c ) as a vector space of entire functions on E~. O

If G is a Complex locally convex space, an entire func

tion f on G is said to be of NuoZear - ExponentiaZ Type

if there exists a convex balanced neighbourhood U in G

such that

An d f(o) is an element of PN(G(U for every n, and

An nlim sup II d f(o)II N

61

sequence of nuclear n - homogeneous polynomials on G(U)

for some U, such that lim II Pnil N = O. Again replacing n+oo 1

U by some scalar multiple AU, we have lim supll Pnll:n< 00' 1

Then f =

62

f E: H (E) ,

T*f(z) = T(T_Zf)

defines a holomorphic function on E, and the mapping

f~ T*f is a convolution operator on (H(E) ,TO)' The - ,

mapping v: (H(E),T ) ~ B(H(E),T ), where B(H{E),T )000

is the vector space of all convolution operators on

(H(E),T )' and VT = T*, is an isomorphism, with inverse O

the mapping v~B{H{E) ,TOJ ~(H(E) ,TO)' given by

(vA) (f) = Af{o).

B(H(E),T ) is an algebra under composition; by means ofO

the isomorphism v, a product (T ,T 2 ) ~ Tl*T2 is defined1

on (H{E),T )' so that v becomes an algebra isomorphism.O

With this product on (H{E),T )', the Borel transform isO

an algebra isomorphism of (H(E),TO

) with EXPNE~.

We define a Partial Differential Operator on (H(E) ,TO)

to be a convolution operator, A, such that there is a

positive integer, N, and a sequence {To, ..... ,TN} where n ,

Tn E: (P( E),TO) for each n, (0 , n , N), with

Af(x) = T(T_xf) for every f E: H(E) I

where T = To+ ..... +T Thus A ~s a partial differentialN

operator if and only if the corresponding function on Et

is an element of the space PN(EI) of all nuclear polynomials J. c

on E I c.

PN(E~) has the following division property: if p(l),

p(2) are elements of PN(E~) such that p(1)~(2)iS Gateaux

holomorphic on EI, then there exists an element P of PN(E~)

such that pel) = p.p(2). To see this we choose a convex

c

63

balanced neighbourhood of the origin U in E~ such that

p(l) and p(2) are both defined and nuclear on El (U)i we c

then apply the division theorem for nuclear polynomials

on normed spaces, ([7J, Lemma 8), to get a nuclear poly

nomial P on E~(U) with p(l) = p p(2).

Proposition 3.6:

Let E be a Frechet space with the strict approximation

property. If A is a partial differential operator on

(H(E),T )' then the kernel of A is the closed linear spanOof {pe~: P p(nE) for some n, ~ EI, and A(Pe~) = OJ.

Proof:

The proof is exactly as in [4J, using the division pro

perty of PN(E~) given above.

D

64

C HAP T E R I V

WEAKLY COMPACT POLYNOMIAL AND HOLOMORPHIC

MAPPINGS ON BANACH SPACES

Aron and Schottenloher [2J have made a study of compact

polynomials and compact holomorphic mappings between Banach

spaces. In this chapter we obtain some analogous results

for weakly compact mappings. We then answer a question of

Pelczynski [22J concerning weakly compact polynomials on

Banach spaces with the Dunford-Pettis property.

Throughout this chapter E, F will denote Banach spaces.

FI is the dual space of F with the Mackey topology T(FI,F).T

FI is the dual space of F with the topology of uniform con-c

vergence on convex balanced compact subsets of F. For each

n, (p(nE,F),T ) denotes the space p{nE;F) with the norm ofb

uniform convergence on the unit ball of E. We recall that

if E and F are Complex Banach spaces, a seminorm, p, on

H(EiF) is said to be ported by a compact subset K of E if

for every open subset U of E containing K, there is a posi

tive Real number, Cu' such that

p (f) .:;; C II f II u for every f e: H(EiF) .u T denotes the locally convex topology on H{E;F) defined by

w

65

all the seminorms which are ported by compact subsets of

E.

If G,H are locally convex spaces and t is a locally

G 1convex topology on for which every equicontinuous set is

bounded, Ls(GltiH) denotes the space of all continuous

linear mappings from Gt into H with the topology of uniform convergence on the equicontinuous subsets of G I

Let P be a continuous n - homogeneous polynomial from

E into F. P is said to be Weakly Compact (respectively,

Compact) if P maps the unit ball of E into a relatively weakly

compact subset of F (respectively, a relatively compact subset

of F) .

Lemma 4.1:

A continuous n - homogeneous polynomial P:E~F is weakly

compact (respectively, compact) if and only if the linear - An

mapping P:h E---?>F is weakly compact (respectively compact).'If

Proof:

This follows immediately from the fact that the closed unit

ball of the Banach space hnE is the closed convex balanced 'If

D

66

Aron and Schottenloher define the adjoint, pt, of a

continuous n - homogeneous polynomial P from E into F to be

the linear mapping from F' into p(nE) given by

n An nIf we identify (P( EiF) ,T b ) with (L(h~EiF) ,T b ) and (P( E) ,T b )

An twith (h~E)S then P becomes the adjoint, in the usual sense,

of the linear mapping P : hnE ~ F. ~

Proposition 4.1:

(cf. [2J. Proposition 3.2). Let E and F be Banach spaces,

and let P be a continuous n - homogeneous polynomial from

E into F.

The following are equivalent:

(i) p is weakly compact

(ii) pt F'~(p(nE),T )T 0 is weakly compact.

(iii)pt F ~ ~ (P (nE ) , T b) is continuous.

(iv) pt F6~ (p(nE) ,T b ) is weakly, compact.

Proof:

That (i), (ii), (iii) and (iv) are equivalent for linear

mappings is well-known ([lOJ, Theorem 9.3.1). The pro

position now follows from Lemma 4.1, using the fact that

67

- nP can be identified with the linear mapping P:h E~Ff

71"

and pt with the adjoint of P.

D

Aron and Schottenloher show that the Banach space

n(P K( EiF) "b) of all compact n - homogeneous polynomials

from E into F is isomorphic, by the mapping p~p t f to

the Banach space L (F'j (p(nE)"b where denotes thef c

topology of uniform convergence on equicontinuous sets.

PWK(nEiF) denotes the closed subspace of (p(nEiF) feb)

consisting of all weakly compact n - homogeneous poly

nomials from E into F. The following proposition is the

nanalogue for the space (PwK( EiF) feb) of the isomorphism

L (F I i (P (nE ) f 'b) . c

68

Proposition 4.2

(cf. [2J, Proposition 3.3)

Let E and F be Banach spaces.

The mapping p~pt establishes an isometric isomorphism

of (PwK(nEiF) ,T ) with LE(F~i (p(nE) ,T , where the normb b

on the latter space is the norm of uniform convergence

on the unit ball of FS'

Proof:

Proposition 4.1 shows that the mapping p~pt maps

pwK(nEiF) into L(F~i (p(nE),Tb . This mapping is clearly

linear and injective.

Let T be an element of L(F' (p(nE) T =L(F 1 (hnE) I).

T' , b T' 'If f3

t AnThen T is a weakly compact linear mapping of (h'lfE)S into

F, and restricting Tt to h~E defines a weakly compact

n - homogeneous polynomial p from E into F such that

pt T.

Now if p E PWK(nEiF), then

II ptll = sup{II1jJOP II: 1jJEB 1 (F 1 )} ~ II pll

The Mackey - Arens theorem shows that the mapping

t . n T~T is an isometric isomorphism of L (F'; (P( E) ,T )

E T b into L (P (nE ) , T ) I i F).

E b T

Therefore, II ptll = II pttll = sup{ II ptt(qi) :4iEB l

(p(nE) ,Tb ) I}

~ sup{ II ptt (x (n II: XEB (E) }1

= II pll

Thus, Ilptll = II pll for every P E pwK(nE;F)

D

69

We point out one further criterion for weak compactness,

which is contained implicitly in this Proposition.

Proposition 4.3:

nLet E and F be Banach spaces. P P( EiF) is weakly compact

if and only if ptt maps (p(nE ) ,T ) I into the canonical imageb

of F in FH.

Proof:

Apply the corresponding result for linear mappings

([10J, Corollary 9.3.2) to the linear mapping P. D We now discuss weakly compact ho1omorphic mappings.

Let E and F be Complex Banach spaces. A ho1omorphic

mapping f from E into F is Weakly Compaot if for every

element x of E, there is a neighbourhood, V, of x, such that

f(V) is a relatively weakly compact subset of F.

Following Aron and Schotten1oher, we define the Adjoint,

ft, of a ho1omorphic mapping f from E into F to be the linear

mapping from F' into H(E) given by

If f = E p is the Taylor expansion of f at the origin, pt n=o n n

nis a continuous linear mapping of Fa into (P( E) ,Tb ) for each

n. Since (p(nE) ,Tb ) is a subspace of (H(E) ,T ) it is immaw

terial whether we take the transpose of P as an n - homon

geneous polynomial, or as a ho1omorphic mapping.

70

Proposition 4.4:

(cf. [2J I 3 Remark (2

Let E and F be Complex Banach spaces. If f is a holomorphic

mapping from E into F, then ft is a continuous linear mapping

from Fa into (H(E)/L )' The mapping f~ft is a continuousW

linear mapping from (H(EiF)/L ) into L (FBI; (H(E) IL .w E w

Proof:

Let p be a seminorm on H(E) which is ported by the compact set

K. Let V be an open set containing K such that f is bounded on

V. There is a positive Real number C such that v

p(g) ~cvllgllv foreverygEH(E).

Therefore p (ft (ll! ~ C Ilft(1j!)11 V = C 1Ill! II f (V) , and so thev v t .

linear mapping f :F8~(H(E)/Lw) is continuous.

It is clear that f~ft is a linear mapping. To see that

it is continuous, let q be a continuous seminorm on L (FBI; (H(E),L . : w

We may assume that q is of the form

q(T) = sup{p(T(ll!:11 ll! II ~ I}

where p is a seminorm on H(E) which is ported by some compact set

K. If V is an open set containing K there exists C > 0 such v that

p(g) ~ cvll g Ilv for every g : H(E).

Therefore

sup{p(ftll!):IIll!II~ I} = sup{p(ll! 0 f) : 11ll!1I~ I}

~ sup {11ll! 0 f II V : II ll! II ~ I} = Cv II f II V Cv

71

l} is a T - continuousThus f ~ sup{p (f t 1jJ) : w

seminorm on H(E,F), and so f~ft is a continuous linear

mapping

D

Consider the mapping

IS : E~ (H(E),T )'W

which associates with x E E the evaluation functional IS (x) :

lS(x}f = f(x)

Lemma 4.2:

Let E be a Complex Banach space. Then IS is a holomorphic

mapping from E into (H(E),Tw)S

Proof:

We show that IS is Frechet differentiable at every point

of E. For x E E, let TI be the projection of (H(E),T )x W

onto the subspace ES = (P('E),T ) which associates withb

f E H(E) its first derivative at x, d1f(x). Let j denote

the canonical embedding of E into E".

We claim that the Fr~chet derivative of IS at x exists,

and is equal to the mapping from E into (H(E) ,TW)S given

by

This mapping is a continuous linear mapping since j and TIx t

72

are continuous and linear. We now show that

p(8(x+y)-8(x)-(1T t 0 j)(ylim X = 0IIYII~o II y II

for every continuous seminorm p on (H{E},LW)S

Accordingly, let A be a bounded subset of (H{E},L ), and w

let PA be the Minkowski functional of the set AO By [20J,

12, Prop I, there are positive real numbers C, c, such

that

nII ~I a.nf (x) II ~ Cc for every n, and every f E A. Then if 1\ y II < c- 1

tPA[8(x+y}-8{x)-{1T 0 j)(y)]x = II y II

Ily 11- 1 sUP[f{x+y)-f{x)-d 1 f(x) (y)] fEA

= II Y 11-1

sup I !2 ~I a.nf (x) (y) I fEA n- .

= Cc 2 II y II l-cllyll

which tends to zero with II y II .

D

We shall call a subset U of a Banach space E Circled

if for every x:U we have AX : U whenever II..I = 1. The following Lemma is a slight generalisation of a result

proved by Aron and Schottenloher [2] :

73

Lemma 4.3:

Let E and F be Complex Banach spaces and let f be a

holomorphic mapping from E into F. If the subset U of E

is circled and xEE then

for every n .

Proof:

Suppose this is false for some n. Then there is a continuous

linear functional,~, on F such thatl~(z) I~l for every 1 "'nZe: rf (x+U) , but I~ (b)l>l for some b = ~ f(x) (a)

ae:U. Consider the entire function of one Complex variable

g(A) = ~of(x+Aa)

We have ~Ig(n) (0) = ~(~! dnf(x) (a = ~(b) .Therefore

I~(b) I = 1~lg(n) (0) I ~ sup{ Igp) I: IAI = l}

~ {I ~ of (y) I: y e: x+U} ~ 1, which is a contradiction

In the next Proposition we show that a holomorphic

mapping is weakly compact if it is weakly compact near the

origin, and we show how weak compactness of a holomorphic

mapping is related to the weak compactness of its

derivatives

74

Proposition 4.5

(cf. [2 J, Proposition 3.4 ) Let E and F be Complex Banach spaces, and let f be a

holomorphic mapping from E into F The following are

equivalent

(i) f is weakly compact.

(ii) f maps some neighbourhood of the origin in E

into a relatively weakly compact subset of F.

(iii) d:nf{x) is a weakly compact polynomial for every

n, and for every x E.

(iv) dnf{O) is a weakly compact polynomial for every n.

Proof :

The implications (iii) ~ (iv) and (i) ~(ii) are trivial.

(i) :::=:::;'(iii) and (ii) )(iv) are proved exactly as in [21,

Proposition 3.4 . It only remains to show that (iv) implies

(i)

"nSuppose, then, that d f(O) is a weakly compact

polynomial for every n. Let X = {xEE : f(V ) is weaklyx relatively compact for some open neighbourhood Vx of x}

We shall show that X is a non-empty subset of E which is

both open and closed, from which it follows that f is weakly

compact

o E X .. choose r > 0 such that f is bounded on the ball B(o,r) Fix A , 0 < A < 1 , and let V = AB(o,r).

We claim that f{V) is a relatively weakly: compact subset

75

of F. To see this, let Kn = f(*!dnf(O} (B(O,r)

Since ~dnf(O) is a weakly compact polynomial, Kn is anl

weakly compact subset of F for every n. By Lemma 4. 3 K n

is a subset of the bounded set r(f(B(O,r) for every n.

Hence if Z E K for each n, the series! AnZ convergesn n 11=0 n

absolutely , and so

K = ~ AnK = { ~ AnZ . z nn=o n n=o n

is a well-defined bounded subset of F If x E V, then we

can write x = AY , where y E B(O,r) . Therefore

f(x) E K

Thus f(V) c: K. We show that K is relatively weakly compact

by applying the James criterion namely, that every

continuous linear functional on F achieves its norm on K

[24J . Let WE FI i since each of the sets Kn is weakly

there exist wn E Kn such that II~)II K = W (wn ) Then n

= W(n~o Anwn ) ~ I I WI I K

Therefore II WII K = W(nJ'o A nwn ) , and K is relatively weakly compact. Hence f(V) is relatively weakly compact.

X is open : if x EX, and Vx is an open neighbourhood

of x such that f(V ) is relatively weakly compact, thenx

clearly every element of Vx is an element of X.

X is closed : let {x } be a sequence of elements of Xn

76

converging to x E E . It follows from Proposition 4.1 that

n nPwK ( EiF) is a closed subspace of (P( EiF),Tb) and hence

since ~ldnf(Xk) ) ~ldnf{X) (k~oo) in (p(nEiF),T ) . . b

~!dnf(X) is a weakly compact polynomial for every n.

Consider the holomorphic mapping g:E~F defined by

1 An 1 "'ng ( z) = f (x+z ) Since for every n, ~ g(O) = ~ f{x),

n. .

it follows by the first part of this proof that g maps

some neighbourhood, U, of the origin into a relatively

weakly compact subset of F . Therefore f{x+U) = g(U) is

weakly relatively compact, and so x E X.

D

We now have the analogue for holomorphic mappings of

Propositions 4.1 and 4.3

Proposition 4 6 :

( cf. [2J, Proposition 3.6 )

Let E and F be Complex Banach spaces, and let f be a

holomorphic mapping from E into F The following are

equivalent:

(i) f is weakly compact.

(ii) ft:F'~(H(E)'T ) is continuous.T w

( iii)