Complex Analysis in Locally Convex Spaces

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COMPLEX ANALYSIS IN LOCALLY CONVEX SPACES

This Page Intentionally Left Blank

NORTH-HOLLAND MATHEMATICS STUDIES

57

Notas de Matematica (83) Editor: Leopoldo Nachbin Universidade Federal do Rio de Janeiro and University of Rochester

Complex Analysis in Locally Convex Spaces

SEAN DINEEN Department of Mathematics University College Dublin Belfield, Dublin 4, Ireland

NORTH-HOLLAND PUBLISHING COMPANY

- AMSTERDAM

NEW YORK

OXFORD

Q

North-Holland Publishing Company, 1981

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

ISBN: 0444863192

Publishers NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM NEW YORK OXFORD Sole distributors forthe U.S.A.and Canada

ELSEVIER NORTH-HOLLAND, INC. 5 2 VANDERBILT AVENUE, NEW YORK, N.Y. 10017

Library of Congress Cataloging in Publication Data

Dineen, Se&, 1944Complex a n a l y s i s i n l o c a l l y convex s p a c e s . (North-Holland mathematics s t u d i e s ; 57) B i b l i o g r a p h y : p. In cl u d es index. 1. Holamorphic f u n c t i o n s . 2 . L o c a l l y convex s p a c e s . I. T i t l e . 11. S e r i e s . QA33~D637 515.713 81-16885 ISBN 0-444-86319-2(U.S.) AACB

PRINTED IN THE NETHERLANDS

To Carol, Deirdre and Stephen

This Page Intentionally Left Blank

FOREWORD

The main purpose of this book, based on a course at Universidade Federal do Rio de Janeiro during the summer of 1978, was to provide an introduction to modern infinite dimensional complex analysis, or infinite dimensional holomorphy as it is commonly called, for the graduate student and research mathematician. Since we were more interested in communicating theqaturerather than the scope of infinite dimensional complex analysis and since it was clearly impossible to write a comprehensive account of the whole theory for such a short course we were obliged to limit our range and choose to develop a single theme which has made much progress in recent years and which exemplifies the intrinsic nature of the subject, namely the study of locally convex topologies on spaces of holomorphic functions in infinitely many variables. In retrospect, we feel we have provided a reasonably comprehensive view of the topoZogica2 nature of the theory, but have neglected to a large extent the geometric, aZgebraic and differential aspects. A l l of these aspects are equally important, interrelated and indeed a proper appreciation of the portion of the theory outlined in this book is not possible without an overall view of these other topics. To partially compensate for this deficiency we have written Appendix I in which we outline developments in other areas of infinite dimensional holomorphy. The main prerequisite for reading this book is a familiarity with the elements of functional analysis. An acquaintance with several complex variable theory is useful but not essential. In Appendix 11, we provide a resumk of results from these two areas for the non-specialist. On the other hand, much of the functional analysis that we use is not of the standard linear kind, but arises from the nature of infinite dimensional holomorphy and sothis text may also serve for functional analysts as a fresh view of vii

...

Foreword

Vlll

t h e i r own s u b j e c t . The p r i n c i p a l t o p i c d i s c u s s e d i n t h i s book i s t h e l o c a l l y convex s p a c e s t r u c t u r e s t h a t may b e p l a c e d on t h e set o f a l l holomorphic f u n c t i o n s d e f i n e d on a domain i n a l o c a l l y convex s p a c e .

To be s p e c i f i c , we are

p r i m a r i l y i n t e r e s t e d i n t h e p r o p e r t i e s o f , and r e l a t i o n s between, t h e T

and

T~

topologies.

T

0

i s t h e compact open t o p o l o g y ,

t o p o l o g y o f l o c a l convergence and c o u n t a b l e open c o v e r i n g s .

T&

T~

T

is a

0’

i s t h e t o p o l o g y dominated by t h e

Many o f t h e o t h e r t o p i c s d i s c u s s e d a r i s e from

avenues opened up by o u r i n v e s t i g a t i o n o f t h e s e t o p o l o g i e s . Our arrangement o f t h e m a t e r i a l i s a s f o l l o w s .

I n c h a p t e r 1, we

d i s c u s s polynomial mappings between l o c a l l y convex s p a c e s .

This, hopefully,

p r o v i d e s a g e n t l e i n t r o d u c t i o n t o i n f i n i t e dimensional complex a n a l y s i s s i n c e a holomorphic f u n c t i o n i s l o c a l l y a sequence o f homogeneous polynomi a l s which s a t i s f i e s c e r t a i n growth c o n d i t i o n s .

Furthermore, t h e t h e o r y o f

homogeneous p o l y n o m i a l s , which is e q u i v a l e n t t o t h e t h e o r y o f symmetric m u l t i l i n e a r forms, i s i n t e r m e d i a t e between t h e t h e o r i e s o f l i n e a r mappings and holomorphic mappings and t h e p r o p e r t i e s o f polynomials i n t e r v e n e a t various stages i n l a t e r chapters.

I n c h a p t e r 2 , w e d e f i n e and d i s c u s s t h e

d i f f e r e n t c o n c e p t s of holomorphic mapping between l o c a l l y convex s p a c e s . Our primary i n t e r e s t i s i n c o n t i n u o u s ( o r F r g c h e t ) holomorphic mappings, b u t w e f i n d t h a t our a n a l y s i s o f t h e c o n t i n u o u s c a s e r e q u i r e s t h e s e o t h e r concepts.

I n t h i s c h a p t e r , we a l s o d e f i n e t h e v a r i o u s t o p o l o g i e s on s p a c e s Chapter 3 i s devoted t o

of holomorphic f u n c t i o n s and g i v e some examples.

holomorphic f u n c t i o n s on b a l a n c e d s e t s .

In t h i s s i t u a t i o n , t h e Taylor

s e r i e s expansion l e a d s t o a t o p o l o g i c a l decomposition of t h e s p a c e o f holomorphic f u n c t i o n s and p r o p e r t i e s o f t h e u n d e r l y i n g s p a c e s o f homogeneous p o l y n o m i a l s are extended t o holomorphic f u n c t i o n s . holomorphic f u n c t i o n s on Banach s p a c e s .

I n chapter 4,we discuss

Here w e f i n d a n i n t e r p l a y between

t h e geometry o f t h e Banach domain, bounding s e t s and t h e t o p o l o g i e s on t h e

s e t o f holomorphic f u n c t i o n s . on n u c l e a r sequence s p a c e s .

C h a p t e r 5 d e a l s w i t h holomorphic f u n c t i o n s In t h i s chapter, w e construct a d u a l i t y theory

between t h e s e t o f holomorphic f u n c t i o n s on open p o l y d i s c s and holomorphic germs on compact p o l y d i s c s .

T h i s l e a d s t o a c l a r i f i c a t i o n o f a number o f

examples and counterexamples from p r e v i o u s c h a p t e r s and p r o v i d e s u s w i t h a holomorphic c l a s s i f i c a t i o n o f t h e s u b s p a c e s o f

s,

t h e r a p i d l y decreasing

s e q u e n c e s , w i t h i n t h e c a t e g o r y of F r g c h e t n u c l e a r s p a c e s w i t h a b a s i s .

In

ix

Foreword c h a p t e r 6 we d i s c u s s a number o f methods o f g e n e r a l i s i n g t h e p o s i t i v e r e s u l t s o f t h e p r e v i o u s c h a p t e r s t o more g e n e r a l classes o f s p a c e s , and we a l s o d e v o t e one s e c t i o n t o t h e t o p o l o g i c a l c l a s s i f i c a t i o n o f holomorphic f u n c t i o n s on power s e r i e s s p a c e s o f i n f i n i t e t y p e .

The r e s u l t s o f t h i s

c h a p t e r are g e n e r a l l y o f r e c e n t o r i g i n and, no d o u b t , w i l l b e improved i n t h e not too d i s t a n t f u t u r e . I n t h e f i f t h s e c t i o n o f each c h a p t e r , w e g i v e a s e t o f e x e r c i s e s . Some o f t h e s e might n o r m a l l y b e c o n s i d e r e d r e a s o n a b l e e x e r c i s e s , b u t o t h e r s are q u i t e d i f f i c u l t .

We i n c l u d e d t h e l a t t e r i n o r d e r t o i n t r o d u c e f u r t h e r

r e s u l t s w i t h o u t u n n e c e s s a r i l y c o m p l i c a t i n g t h e main body o f t h e t e x t , and i n Appendix I 1 1 we p r o v i d e r e f e r e n c e s and comments on t h e s e e x e r c i s e s .

A

serious attempt a t solving t h e exercises w i l l give t h e i n t e r e s t e d reader a much d e e p e r u n d e r s t a n d i n g o f t h e s u b j e c t and i n t r o d u c e him o r h e r t o many i n t u i t i o n s and s u b t l e t i e s which a r e o f t e n d i f f i c u l t t o communicate by t h e p r i n t e d word a l o n e .

Comments and r e f e r e n c e s on t h e t e x t a r e g i v e n i n t h e

f i n a l s e c t i o n o f each c h a p t e r . I t i s our o p i n i o n t h a t t o p o l o g i c a l c o n s i d e r a t i o n s w i l l e n t e r , t o a g r e a t e r o r l e s s e r e x t e n t , i n t o most problems i n i n f i n i t e d i m e n s i o n a l h o l o morphy.

On t h e o t h e r hand, we a l s o f e e l t h a t t h e t h e o r y o u t l i n e d i n t h i s

book w i l l b e more i m p o r t a n t a s a t o o l i n o t h e r b r a n c h e s o f i n f i n i t e dimens i o n a l holomorphy and a n a l y s i s r a t h e r t h a n as a n o b j e c t o f r e s e a r c h i n itself.

For t h i s r e a s o n , we f e e l it i m p o r t a n t t h a t t o p o l o g i c a l problems i n

i n f i n i t e dimensional a n a l y s i s b e motivated, i f a t a l l p o s s i b l e , e i t h e r d i r e c t l y o r i n d i r e c t l y , from o u t s i d e and t h a t t h e g e n e r a l d i r e c t i o n o f r e s e a r c h i n t o l o c a l l y convex s p a c e s s t r u c t u r e s on s p a c e s o f holomorphic f u n c t i o n s i n i n f i n i t e l y many v a r i a b l e s b e c o o r d i n a t e d and guided by d e v e l opments i n o t h e r a r e a s o f t h e s u b j e c t .

T h i s approach h a s , u n t i l now, l e d

t o t h e more i n t e r e s t i n g r e s u l t s . Most o f t h e r e s u l t s p r e s e n t e d i n t h i s t e x t have n o t p r e v i o u s l y a p p e a r ed i n book form.

The r e a d e r w i l l s e e t h a t t h e s u b j e c t i s s t i l l i n a s t a t e

o f t r a n s i t i o n and t h a t t h e r e a r e many open problems.

I t w i l l b e some time

b e f o r e t h e d e f i n i t i v e book on t h e s u b j e c t i s w r i t t e n , and t h e p r e s e n t work may b e r e g a r d e d a s a r e p o r t o f "work i n p r o g r e s s " .

The a r e a h a s , however,

been developed c o n s i d e r a b l y i n r e c e n t y e a r s , and it i s a p p r o p r i a t e t h a t t h e p r e s e n t s t a t e o f knowledge of t h e s u b j e c t be r e c o r d e d i n a r e a s o n a b l y

Foreword

X

organised fashion.

W e hope t h a t t h i s book w i l l s t i m u l a t e r e s e a r c h t o s o l v e

t h e open problems posed, and t h a t workers i n a l l i e d f i e l d s w i l l g a i n some i n s i g h t i n t o a t l e a s t one a s p e c t o f i n f i n i t e dimensional holomorphy. T h i s book would n e v e r have been w r i t t e n w i t h o u t t h e s u p p o r t , e n c o u r a g e ment and f r i e n d s h i p of J o r g e A l b e r t 0 Barroso, who a r r a n g e d my v i s i t t o Rio d e J a n e i r o i n 1978, and of Leopoldo Nachbin who f i r s t s u g g e s t e d t h e i d e a of w r i t i n g t h i s book.

To them I extend my s i n c e r e s t g r a t i t u d e .

To J . M .

Isidro

I a l s o extend my t h a n k s f o r g i v i n g me t h e o p p o r t u n i t y t o l e c t u r e on, more

o r less, t h e c o n t e n t s of c h a p t e r 5 i n S a n t i a g o d e Compostela d u r i n g J u n e of 1979.

Many o t h e r p e o p l e , by t h e i r a d v i c e , proof r e a d i n g , encouragement and

s u g g e s t i o n s enabled m e t o f i n i s h t h i s book.

I would e s p e c i a l l y l i k e t o

thank R. Aron, P . Boland, R . Ryan and R . Soraggi f o r t h e i r p a t i e n c e and i n t e r e s t and t o e n s u r e them t h a t a l l t h e e r r o r s are mine.

I thank R. Meise

and D . Vogt f o r t h e i r encouragement and f o r making a v a i l a b l e t o me t h e i r v e r y r e c e n t unpublished r e s e a r c h . J.F.

J.M. Ansemil, P . Barry, K-D.

Bierstedt,

Colombeau, C . Herves, J . Mujica, L . A . d e Moraes and Ph. Noverraz were

a l s o very helpful. The e x c e l l e n t l a y o u t and t y p i n g a r e due t o H i l a r y Hynes and Ann Lewis and I thank them v e r y s i n c e r e l y f o r a l l t h e h e l p t h e y have g i v e n m e .

/

Sean Dineen Dublin, October 3 , 1980.

CONTENTS

vii

FOREWORD CHAPTER 1.

POLYNOMIALS ON LOCALLY CONVEX TOPOLOGICAL VECTOR SPACES

1

1.1

Algebraic properties of polynomials

1

1.2

Continuous polynomials

9

1.3

Topologies on spaces of polynomials

22

1.4

Duality theory for spaces of polynomials

31

1.5

Exercises

42

1.6

Notes and Remarks

46

HOLOMOWHIC MAPPINGS BETWEEN LOCALLY CONVEX SPACES

53

2.1

Gzteaux holomorphic mappings

53

2.2

Holomorphic mappings between locally convex spaces

57

2.3

Locally convex topologies on spaces of holomorphic mappings

71

2.4

Germs of holomorphic functions

a4

2.5

Exercises

92

2.6

Notes and Remarks

99

CHAPTER 2 .

CHAPTER 3. HOLOMORPHIC FUNCTIONS ON BALANCED SETS

109

Associated topologies and generalized decompositions in locally convex spaces

110

3.2

Equi-Schauder decompositions o f (H(U;F),T)

119

3.3

Application of generalised decompositions t o the study of holomorphic functions on balanced open sets

124

3.1

xi

xii

Contents 3.4

S e m i - r e f l e x i v i t y and n u c l e a r i t y f o r s p a c e s of holomorphic f u n c t i o n s

141

3.5

Exercises

146

3.6

Notes and Remarks

153

HOLOMORPHIC FUNCTIONS ON BANACH SPACES

159

4.1

A n a l y t i c e q u a l i t i e s and i n e q u a l i t i e s

160

4.2

Bounding s u b s e t s o f a Banach s p a c e

172

4.3

Holomorphic f u n c t i o n s on Banach s p a c e s w i t h an unconditional bas i s

183

4.4

F u r t h e r r e s u l t s and examples concerning holomorphic f u n c t i o n s on Banach s p a c e s

196

4.5

Exercises

204

4.6

Notes and Remarks

210

HOLOMORPHIC FUNCTIONS ON NUCLEAR SPACES WITH A BASIS

217

5.1

Nuclear s p a c e s w i t h a b a s i s

218

5.2

Holomorphic f u n c t i o n s on f u l l y n u c l e a r s p a c e s w i t h a basis

236

5.3

Holomorphic f u n c t i o n s on DN s p a c e s w i t h a b a s i s

262

5.4

T o p o l o g i c a l p r o p e r t i e s i n h e r i t e d by s t r i c t i n d u c t i v e l i m i t s and subspaces

277

5.5

Exercises

288

5.6

Notes and Remarks

293

CHAPTER 4 .

CHAPTER 5 .

CHAPTER 6 .

GERMS, SURJECTIVE LIMITS, SPACES

1 -PRODUCTS

AND POWER SERIES

297

6.1

Holomorphic germs on compact s e t s

297

6.2

S u r j e c t i v e l i m i t s of l o c a l l y convex s p a c e s

316

6.3

€-Products

32 7

6.4

Power series s p a c e s of i n f i n i t e t y p e

336

6.5

Exercises

356

6.6

Notes and Remarks

360

Contents

...

Xlll

APPENDIX I

FURTHER DEVELOPMENTS I N I N F I N I T E DIMENSIONAL HOLOMORPHY

365

APPENDIX I1

D E F I N I T I O N S AND RESULTS FROM FUNCTIONAL ANALYSIS, SEVERAL COMPLEX VARIABLES AND TOPOLOGY

397

APPENDIX 111

NOTES ON SOME EXERCISES

41 1

Bib 1iography

433

Index

48 1

This Page Intentionally Left Blank

Chapter 1

POLYNOMIALS ON LOCALLY CONVEX TOPOLOGICAL VECTOR SPACES

T h e r e a r e two s t a n d a r d methods o f i n t r o d u c i n g p o l y n o m i a l s , by u s i n g t e n s o r p r o d u c t s o r by u s i n g

n

We have found it more

l i n e a r mappings.

c o n v e n i e n t t o a d o p t t h e l a t t e r approach.

T h e . r e a d e r familiar w i t h t h e

t e n s o r p r o d u c t approach w i l l have l i t t l e d i f f i c u l t y i n c o r r e l a t i n g h i s e x p e r i e n c e w i t h o u r approach b u t w e b e l i e v e t h a t t r a n s l a t i n g r e s u l t s i n t h e o t h e r d i r e c t i o n i s more d i f f i c u l t . between v e c t o r s p a c e s o v e r

C.

I n 5 1 . 1 we d i s c u s s polynomial mappings

In 11.2 w e discuss various kinds of contin-

uous polynomial mappings between l o c a l l y convex s p a c e s .

Our main i n t e r e s t

i s i n polynomials c o n t i n u o u s w i t h r e s p e c t t o t h e g i v e n t o p o l o g y on a

l o c a l l y convex s p a c e b u t we f i n d t h a t hypocontinuous and Mackey ( o r S i l v a ) c o n t i n u o u s polynomials a l s o p l a y a n i m p o r t a n t r o l e i n o u r s t u d y .

51.3 i s

d e v o t e d t o endowing s p a c e s o f p o l y n o m i a l s w i t h l o c a l l y convex t o p o l o g i e s For t h e n o n - s p e c i a l i s t we p r o v -

and i n 11.4 a d u a l i t y t h e o r y i s d e v e l o p e d .

i d e an i n t r o d u c t i o n t o f u n c t i o n a l a n a l y s i s i n Appendix 11. 51.1

ALGEBRAIC C,R,N

PROPERTIES

and

POLYNOMIALS

OF

d e n o t e r e s p e c t i v e l y t h e complex numbers, t h e r e a l num-

Z

b e r s , t h e n a t u r a l numbers and t h e i n t e g e r s . and

msN

and

m

then

copies of

In t h i s section,

and

B

E

If

A

and

and

xnyn F

w i l l d e n o t e t h e element

n

are sets,

B

w i l l denote t h e C a r t e s i a n product o f

AnBm

n

copies of

A

"FX' ' C Y I.

n times m times

w i l l d e n o t e v e c t o r s p a c e s o v e r t h e complex

numbers.

For each ings from

nsN

into

E

we l e t b a ( n E ; F ) F.

denote t h e space of n

The s u b s c r i p t

a

n o t assume any c o n t i n u i t y p r o p e r t i e s . d e f i n e d on

En

with values i n

o t h e r s remain f i x e d .

J!a(nE;F)

F

linear mapp-

refers t o algebraic s i n c e we do

Hence i f

L E ~ ~ ( ~ E ;t hFe n)

L

is

and i s l i n e a r i n each v a r i a b l e when t h e i s a v e c t o r s p a c e o v e r t h e f i e l d of complex 1

2

Chapter 1 1 l i n e a r mappings a r e j u s t l i n e a r mappings and i n t h i s c a s e w e

numbers.

use t h e n o t a t i o n da(E;F). 2 l i n e a r mappings are a l s o c a l l e d b i l i n e a r 2 i s sometimes used i n p l a c e o f "ea( E ; F ) . mappings and t h e n o t a t i o n @,(E;F) n When F=C w e w r i t e dea( E) i n p l a c e o f &a(nE;C) and E* i n p l a c e o f La(E;C).

E*

dLa('E;F)

as t h e s e t o f a l l c o n s t a n t mappings from

E.

i s called the algebraic dual o f

space can be i d e n t i f i e d with

When

n=O w e d e f i n e

F

into

E

and t h i s

i n a natural fashion.

F

The f o l l o w i n g a l g e b r a i c i d e n t i t y i s e a s i l y e s t a b l i s h e d and i s u s e f u l i n p r o v i n g r e s u l t s by i n d u c t i o n . P r o p o s i t i o n 1.1

then

E

If

and

F

are vector spaces over

and

C

nEN

z ~ ( ~ E ; F )I- L ~ ( E ; x ~ ( ~ - ' E ; F ) : ) B,(~-'E;~~(E;F)).

Proof

The mapping

T

T(xl'.-.Xn)

=

-f

T1

+

T2,

T

Xa(nE;F),

E

[Tl(xl)3(x2,...Jn)

=

g i v e n by

~ ~ , ~ ~ l ~ . . . , ~ n ~

e s t a b l i s h e s t h e r e q u i r e d correspondence. Definition 1 . 2

xl,

f o r any numbers.

L

An n-lineor mapping

symmetric i f

W l ' . . * ,xn)

..., xn E

E

=

.

L(xql)>

* *

and any permutation

We l e t ZS("E;F) a mappings from E i n t o

E

from

F

into

is said t o be

'Xo(n))

u

of t h e f i r s t

n

denote t h e vector space o f a l l symmetric F.

natural n

Zinear

By averaging over a l l permutations of t h e f i r s t

natural numbers we can associate, i n a canonical fashion, a symmetric n-linear mapping w i t h each n l i n e a r mapping and i n t h i s way we obtain a

n

st$(nE;F)

projection from f o l l o w i n g way.

where

L

Formally we a c h i e v e t h i s i n t h e

we s e t

E

s n denotes t h e s e t of permutations of the f i r s t We c a l l

s

IS

onto z:(nE;F).

s

t h e symmetrization o p e r a t o r .

if

L

E

%a(nE;F)

then

(b)

s(L) = L

i f and o n l y i f

(c)

s ( s ( ~ ) )=

S(L)

for all

s(L) L

L

Zz(nE;F)

E

E

bz(nE;F)

in

natural numbers.

The f o l l o w i n g p r o p e r t i e s of

are easily verified: (a)

n

x~(~E;F)

3

Polynomials on locally convex topological vector spaces (d) If

s

is a linear operator. i s a v e c t o r space we l e t

E

ing from

into

E

En

into

x

An)

denote t h e diagonal mapp-

onto t h e p o i n t

x".

A mapping from a ZocaZZy convex space

Definition 1 . 3

convex space

which maps

En

(or

A

E

t o a locally

which is the composition of the diagonal mapping from E

F

n

and an

E

Zinear mapping from

F

into

n

is called an

hom-

ogeneous poZynomia2. denote t h e vector space of a l l

We Zet QaCnE;F) nomials from

E

homogeneous poly-

F.

into

Thus we s e e t h a t i f t h e r e e x i s t s an

n

i s an

P:E+F

n

n

l i n e a r mapping

homogeneous polynomial i f and only from

L

F

into

E

such t h a t t h e

following diagram commutes E

~ ( x )= x

where

n

f o r every

A polynomial from

iaZs from

E

E

into

E

L

If

En

x

in

into

F

We l e t

F.

into

polynomials from Example 1 . 4

A ___f

E.

is a f i n i t e s m of homogeneous poZynom-

@,(E;F)

is a

l i n e a r (C-valued mapping on

2

it i s well known t h a t t h e r e e x i s t s an zAwt

for a l l If

z = (zl

A = (a. .) ij

,..., zn)

16idn 16 j6n

denote t h e v e c t o r space of a l l

F.

Cn

6

then

nxn

L(z,w)

=

P(Z) =

L(Z,Z) =

1_s i s_ n 1s j s n

1

1s i s n

L B,

B = - A + tA 2

zAzt = zBzt

f o r every

tA

z

L(z,w) =

C".

aijziwj. on

C"

'J

has t h e f a m i l i a r

J

s(L),

we o b t a i n a 2 - l i n e a r form

i s t h e symmetric matrix a s s o c i a t e d with where

E

then

a..z.z..

by i t s symmetrization,

I f we r e p l a c e whose matrix,

Since

1

such t h a t

w = (wl ,..., wn)

1s j s n Hence any C-valued 2-homogeneous polynomial, P , form

A

matrix

and a l l

ncN,

(c",

A,

i.e.

i s t h e transpose o f t h e matrix

in

Cn

d e f i n e t h e same 2-homogeneous polynomial.

it follows t h a t

L

and

A.

s(L)

Chapter I

4

More generally if L in E

dQa(nE;F)

E

then L(xn)

=

s(L)(xn)

for every x

and hence we do not, in general, have a one to one correspondence

between n-homogeneous polynomials and n-linear mappings. However, if we restrict ourselves to symmetric n-linear mappings we do obtain a unique correspondence. By the definition o f n-homogeneous polynomials and the symmetrization operator, the following diagram commutes

where the non-horizontal mappings are just restrictions to the diagonal. T+T

We denote the vertical mapping

by

is easily seen to be lin-

ear and we have already noted that it is surjective. As a consequence of the polarization formula we show that is injective and thus w e have a canonical bijective linear mapping between the space of symmetric n-linear A

mappings from E

E

into F

and the space of n-homogeneous polynomials from

into F. ( P o l a r i z a t i o n Formula)

Theorem 1.5 over

Proof

(c,

L

If E

and xl, ..., xn

Ex~(~E;F)

E

E

and F are vector spaces then

By linearity and symmetry

Zm.=n -

Hence 1 n 2 .n!

1

c=il

l,
El

. . . En

A

L(

n

1 EjXj)

j=l

=

Polynomials on locally convex topological vector spaces

- - 2"

1

1

L(Xl)

O,<m.sn m 1 ! . .. mn !

m

1

...

(x,)

m

n

.I

of

L(xl,. If

c l2

... e2n

and h e n c e t h e c o e f f i c i e n t

2"

=

i n t h e above e x p a n s i o n i s

f o r some

then

i

m.=O

m +1

ml+l

c

1 E . =+1

then

l,
. . ,xn)

m.>l

n

61

e. =?I 1

i sj s n

... = m n = l

ml=m2

m +1

... enn

1

Zm.=n If

ml+l El 6.=+1

5

E.

J1

l,
1.

f o r some

1

j

and hence mj+l+l E. J+1

1-1

=+1

..,'.

mn+l

Isisn,i#j

0.

=

Hence a l l t h e o t h e r terms v a n i s h and we have proved t h e p o l a r i z a t i o n formula. Corollary 1.6 Proof

The mapping

-:%:("E;F)

By t h e p o l a r i z a t i o n formula

i

and o n l y i f

i s i d e n t i c a l l y zero.

+

L

E

Qa("E;F)

i s a linear bijection. i s i d e n t i c a l l y zero i f

%-:(nE;F)

i s a l i n e a r mapping w i t h

Hence

z e r o k e r n e l and t h u s it i s i n j e c t i v e . T h i s completes t h e p r o o f . The p o l a r i z a t i o n f o r m u l a h a s many o t h e r a p p l i c a t i o n s , some o f . w h i c h we now d e s c r i b e .

These f o l l o w from t h e s p e c i f i c n a t u r e o f t h e formula which

a l l o w s u s t o compare t h e moduli o f symmetric n - l i n e a r forms and p o l y n o m i a l s S i n c e i t i s u s u a l l y e a s i e r t o compute t h e modulus o f an n

on c e r t a i n sets.

l i n e a r form, t h e n t h e modulus o f a p o l y n o m i a l , t h i s p r o v e s a e f u l . If

and

B

f

i s a f u n c t i o n d e f i n e d on a s u b s e t A o f

i s a semi-norm on

we let

F

llfll

u n d e r s t o o d from t h e c o n t e x t , we j u s t w r i t e Theorem 1 . 7

x&.A

and

F

are vector spaces over

baZanced subset of E

and

B

i s a semi-norm on

f o r any

Remark

L

in X;("E;F)

with values i n

= supB(f(x)).

E

I f

E

F

11

If

B

is f u l l y

/]A' CC,

A

is a convex

then

.

T h i s e q u a l i t y may t a k e t h e form

c a s e we o b t a i n i n f o r m a t i o n .

+ws+ms+-

F

b u t even i n t h i s

3

6

Chapter I

Proof

The l e f t - h a n d s i d e o f t h i s i n e q u a l i t y i s t r i v i a l s i n c e By t h e p o l a r i z a t i o n formula

i ( A ) C L(An).

lii,
x.

i = l , .. . , n

A,

E

and

C. = 2 1

then

,I &ixi

1 "

E

1=1

convex and b a l a n c e d , and hence

A,

since

A

is

Hence

T h i s completes t h e p r o o f . nn - i s t h e b e s t p o s s i b l e i n g e n e r a l (example 1 . 8 ) . In n! c e r t a i n c a s e s , however, i t i s p o s s i b l e t o improve t h i s e s t i m a t e - t h i s w i l l The c o n s t a n t

depend on

n

and on t h e geometry o f t h e s e t

A.

We g i v e a few examples t o

i l l u s t r a t e t h i s p o i n t (some w i t h o u t p r o o f ) and n o t e i n p a s s i n g t h a t t h e r e are s t i l l i n t e r e s t i n g open problems i n t h i s a r e a . Example 1 . 8 w i t h norm

where

L

Let

( t h e space o f a b s o l u t e l y convergent s e r i e s

E = R1

11 ( x ~ ) ~ I= / c n l x n l ) .

i

m

(Xm)m=l

xi =

Let

for

b e d e f i n e d by

L:En+C

.

i = l, . . , n .

i s n - l i n e a r and symmetric and one e a s i l y shows t h a t

l l L l l A :=

n

where

and t h e c o n s t a n t

A

is t h e u n i t b a l l of nn n!

R1.

llLlkn =

and

Hence

.

nn i n theorem 1 . 7 i s t h e b e s t p o s s i b l e . n!

Our n e x t example c o n c e r n s r e a l polynomials o v e r an i n n e r p r o d u c t s p a c e . The analogous problem f o r complex p o l y n o m i a l s i s s t i l l open a l t h o u g h some p a r t i a l r e s u l t s have been o b t a i n e d .

I

Polynomials on locally convex topological vector spaces Example 1 . 9 let

F

d e n o t e a real inner product space with u n i t b a l l d e n o t e a r e a l Banach s p a c e and suppose L E Y:(nH;F). Then

\]LIBn = llLlb

Let

H

where

B

i s t h e u n i t b a l l of

forms and r e a l p o l y n o m i a l s , w e always have <

l l ~ l b<

Case 1 Ow'

Dim(H)

and

< m

n=2.

Dim(H) < (xl,

...,x,)

aaB

n

m,

E

Bn

it f o l l -

arbitrary. such t h a t

# 0

such t h a t

J

(yl*, . . . , y n * ) E A

W e claim

y . * = 'yy

for all

such t h a t

y * + y:

# 0 and

= YD*

+

Yq*

IIYp*+ Y,*ll

=

j = l , ..., n .

n {(yl, ...,yn) E B ; b a ,

=

and Now choose

for

1

such t h a t t h e s e t A

and

11L(x,y) = ~ ( x + y -$(x-y) )

Since

L

Now choose

= z

and w e may suppose

IlLlbn

11

Case 2

z

6

1 IIL(x>Y) b T\lillB( IIx+Y112+(1x-YI(2)

that

Choose

IILIL

m.

and hence

a>O

H.

S i n c e t h e p o l a r i z a t i o n formula h o l d s f o r r e a l symmetric n - l i n e a r

Proof

o

B,

.

Hence t h e r e e x i s t s

j = l , ...,n.

( I L ( Y 1 , e . . 9 ~ n ) I I= I I L I I p

i s non-empty.

such t h a t

i

and

y* P

-

y;

j.

I f n o t t h e r e ex s t s

# 0.

Let

The f i r s t c a s e i m p l i e s t h a t

M + ( + < a , y

L

zi

= y r

if

(z l , . . . , z n )

- 1)

p

and

q

i # p , q and E

A.

Since

8

Clzapter 1

we have a r r i v e d a t a c o n t r a d i c t i o n and completed t h e p r o o f i n t h i s c a s e . Case 3

H

and

llLl/

arbitrary.

t h e r e e x i s t s a f i n i t e d i m e n s i o n a l subspace

Given €>O

that

n

' IILIIgn- t'

(Br\E)n

lfliB

c i s a r b i t r a r y we have shown t h a t

Since

= llLll

Bn

...

such

H

and completed t h e p r o o f .

The f o l l o w i n g r e s u l t r e d u c e s t o theorem 1 . 7 on t a k i n g nl=n2

of

E

By c a s e 2 it f o l l o w s t h a t

p=l,k=m,

n =l. k

P r o p o s i t i o n 1.10

Let

denote a Banach space and l e t

E

nl,

. . . ,nk

be

n +n +...+ n = m . Suppose x l , . . . ,xk are u n i t 1 2 k k \I z i x i / \ 6 ( \ z i I p ) l / P for a l l ( z , , . . . , z k ) E Ck i=l i=l

positive integers with vectors i n

E

and

l
where

"1

/Wl

"2

'X2

k

1

if L n

,... ' X k 1 I k

1

E

C(mE) nl!

6

...

nl/P

n1

A

and

nk! mm/P nk/P

.....nk

The n e x t p r o p o s i t i o n g i v e s , on t a k i n g

P r o p o s i t i o n 1.11

E

Let 2.

denote a power of

m!

p=2,

t o complex v a l u e d polynomials o f d e g r e e

m

i s t h e u n i t ball of E

Ilill,

a n e x t e n s i o n o f example 1.9

2".

denote a compZex

AP

space,

ldp<-,

and Let

Then

m! f o r any

L

where t h e norms are taken over t h e u n i t baZ1 o f

i n if,:(mE)

E.

We conclude t h i s s e c t i o n w i t h two r a t h e r s i m p l e b u t u s e f u l r e s u l t s . The f i r s t f o l l o w s r e a d i l y from t h e b i n o m i a l theorem and t h e second f o l l o w s from a n a p p l i c a t i o n o f t h e maximum modulus theorem f o r f u n c t i o n s o f one complex v a r i a b l e . Lemma 1 . 1 2

in

E

and

If L AEC

E

L;("E;F)

we have

and

P =

i

E

Oa("E;F)

then f o r any

x,y

9

Polynomials on locally convex topological vector spaces

P(x+y)

=

P(X)

If

E

and

i s a semi-norm on

F,

A

Lemma 1 . 1 3

Moreover, if

+

A#O,AXEA

P(Y)

c

n- 1 +

F

r =1

(;)L(x)n-r(Y)r

are vector spaces over

i s a balanced subset of

and

=

=

A

E

P

(E,

and

E

@a(nE;F), 6

XEE

then

is convex then

sup B(P(x+eiey)) YEA,BER sup B(P(eiex+y))

(since

i s balanced)

A

(by homogeneity)

YEA

0 ER

If

§1.2

AXEA

then

2

sup B(P(y)) YSA

=

1$\lb,A*

1 x+A C - A + A A

CONTINUOUS

=

( 1

+

(by t h e maximum modulus p r i n c i p l e )

x1 )A

and hence

POLYNOMIALS

Our main i n t e r e s t i n t h i s book i s i n s t u d y i n g c o n t i n u o u s polynomial and holomorphic mappings between l o c a l l y convex s p a c e s .

However, w e f i n d

i t u s e f u l (and n e c e s s a r y ) t o d e f i n e more g e n e r a l c l a s s e s o f mappings -

t h i s i s analogous t o t h e s i t u a t i o n i n f u n c t i o n a l a n a l y s i s where t h e weak t o p o l o g y i s used t o o b t a i n p r o p e r t i e s o f t h e norm topology.

10

Chapter 1 Let

and

E

b e t o p o l o g i c a l s p a c e s which are a l s o v e c t o r s p a c e s o v e r

F

U s u a l l y t h e r e w i l l b e some r e l a t i o n s h i p between t h e t o p o l o g i c a l and t h e

(E.

v e c t o r s p a c e s t r u c t u r e s b u t f o r t h e moment w e w i l l n o t make any such assumption.

We l e t

/ ; ( n E ; F ) , X(nE;F)

J ? ( ~ E ; F ) denote r e s p e c t i v e l y t h e

and

E

spaces of continuous n-homogeneous polynomials from E

tinuous n-linear mappings from E

n-linear mappings from

F

to

and the continuous symmetric

t h e above s p a c e s a r e v e c t o r s p a c e s and t h a t - ( d S ( n E ; F ) ) = S’(nE;F). spaces. a

cs(E)

E,

i s a semi-norm on

Note t h a t

Ea.

Ea

is

-

Ra

Ea

a(x) = 0

{ x e E ; a ( x ) < r ) and

B,(r)

i s the c l o s u r e of f

If

Ba(r)

Ba(r)

a r e l o c a l l y convex

F

in

II (x) = 0

E.

If

{xcE;a(x)
=

Since

E.

F

E

onto

and t h e open u n i t b a l l

i s a semi-norm on

a

from a t o p o l o g i c a l s p a c e i n t o

i s continuous for every

of

and

E

and

w i l l d e n o t e t h e normed l i n e a r s p a c e

i f and o n l y if

R a { x ~ E ; a ( x )< 1 ) .

=

nB

W e now suppose t h a t

J,’(~E;F)

=

w i l l d e n o t e t h e c a n o n i c a l s u r j e c t i o n from

Ba(r)

mapping

s(Y.(~E;F))

w i l l d e n o t e t h e s e t o f a l l c o n t i n u o u s semi-norms on

( E , ~ i ) / ~ - l ( ~ )and

of

the con-

I n a l l cases we c o n s i d e r we f i n d t h a t

F.

into

F,

into

=

we l e t

E

If

accs(E)

lim

ES (F)

then

(FB,IIB), a

i s c o n t i n u o u s i f and o n l y i f

F

B E C S ( F ) . We u s e t h i s f a c t i n e x t e n d i n g

r e s u l t s c o n c e r n i n g normed l i n e a r s p a c e v a l u e d polynomials and holomorphic f u n c t i o n s t o f u n c t i o n s w i t h v a l u e s i n a n a r b i t r a r y l o c a l l y convex s p a c e . Proposition 1.14

Let

E

be a l o c a l l y convex space over

C,

F

il

P E P ~ ( ( ” E ; F ) . The following are equivalent

normed l i n e a r space and suppose

(a)

P

i s everywhere continuous;

(bi

P

is continuous a t t h e o r i g i n ;

(el

P

i s bounded on some neighbourhood of the origin;

(dl

P

i s a l o c a l l y bounded f u n c t i o n i i . e . bounded on a neighbourhood of each p o i n t ) .

Proof 1.13

The i m p l i c a t i o n s ( c ) <=> ( d ) .

suppose

A=P.

( a ) => (b) => ( c )

W e now show

( c ) => ( a ) .

Let

By t h e p o l a r i z a t i o n formula and

b a l a n c e d neighbourhood o f z e r o be a r b i t r a r y .

are t r i v i a l and by lemma

Choose

a >O

V

such t h a t

such t h a t

A

(c)

XS(nE;F)

= M < m . Let Vn By Lemma 1 . 1 2

IlAIl

axo E V .

E

and

t h e r e e x i s t s a convex x0€ E

11

Polynomials on locally convex topological vector spaces

Hence

i s continuous a t

P

xo

and

( c ) => ( a ) .

This completes t h e proof.

Let E and F be ZocnZZy convex spaces over E and Corollary 1 . 1 5 n if and only if P i s continuous a t Zet P E Pa( E;F). Then P E @("E;F) one point. I t s u f f i c e s t o use p r o p o s i t i o n 1 . 1 4 and t h e p r o j e c t i v e l i m i t represent a t i o n of

by normed l i n e a r spaces.

F

We now look a t a very u s e f u l f a c t o r i z a t i o n lemma. l o c a l l y convex spaces, Hence

?(("E,;F)

a

E

cs(E)

and

P

!?(nE,;F)

E

may be i d e n t i f i e d with a subspace of

If

E

then

P

F

are

When

F

and 0

nLYE $fnE;F)

6'("E;F).

i s a normed l i n e a r space t h e f a c t o r i z a t i o n lemma says t h a t t h e union of a l l This i s not s u r p r i s i n g i n view of lemma such subspaces covers p("E;F). 1.13.

Lemma 1.16 F

and

(Factorization Lemma).

i s a ZocaZZy convex space

i s a nomed Zinear space then

f o r every p o s i t i v e i n t e g e r Proof

Let

P

E

B(nE;F)

n. and suppose

symmetric n - l i n e a r mapping.

Since

exists a

llpll

a ( x ) < 1, n $[

E

If

E

cs(E)

such t h a t

a(y) = 0

0EFI.

n A ( x ) ~ ( X ~ ) ~= - 1 ~]

1

R=O

polynomial from follows t h a t

g

Ax'

where

to

E

M<m.

The function

Now suppose

x , y ~E ,

g(A) = $oP(x+Ay) =

@ ( A ( X ) ~ ( ~ ) A~n - -R ~ ] . i s a

):(

of degree

i s the associated

i s a normed l i n e a r space t h e r e =

R=O

Ks(nE;F)

n.

sup a(x+Ay) = a ( x ) < 1 it AEC i s a bounded polynomial and hence has degree 0 . By t h e

C

Hahn-Banach theorem t h e form

and

F

A

C

R

Since

ObRsn-1. Since any z E E has R it follows t h a t A(z) (y)n-R = 0 i f Z E E

A(x) (y)n-R = 0 a(x')
Q

for

12

Chapter 1

and Y

a(y) = 0 , R#n.

E,

b e d e f i n e d on

P

n,(z1)

=

'TI

a

(?

) = z

Thus

2" P

u n i t b a l l of

E,

P(i,).

Hence

P(z+y) = P(z)

by

$(z)

then

71

=

P(3

if

(z2-z1) = 0 I

if

a(y) = 0 . Y

~ E Es a t i s f i e s

' T I J Z )= z.

Let If

and hence

_

i s well d e f i n e d and

z

f o r any

TO'TI=

P.

~ ( 2 =~ P) ( t l + ( ? 2 - i 1 ) ) = S i n c e n,(B,(l)) is the

w e have

T h i s completes t h e p r o o f . The f o l l o w i n g example shows t h a t t h i s r e s u l t d o e s n o t e x t e n d t o arbitrary

F.

Example 1 . 1 7

Let

d e n o t e t h e s e t o f a l l sequences o f complexnumbers

CN

i s g i v e n t h e p r o d u c t t o p o l o g y o r t h e t o p o l o g y o f c o o r d i n a t e convergence

CN

and i s a F r & c h e t s p a c e ( i . e . a c o m p l e t e m e t r i z a b l e l o c a l l y convex s p a c e ) . If

a

N

cs(C )

6

then

hence i s isomorphic t o

The i d e n t i t y L

of

E

'J

Cn

f o r some

belongs t o

I

&((CN),;CN)

a E C S (CN) C N . Hence

i s a f i n i t e dimensional normed l i n e a r s p a c e and

E

I

&![C

then

u

na.

N ;C N )

L((GN),)

and

We c l a i m

I(C N )

= C

N.

If

i s a f i n i t e dimensional subspace

a E cs (CN)

Our n e x t r e s u l t i s a f a c t o r i z a t i o n lemma f o r a r b i t r a r y p o l y n o m i a l s . The p r o o f f o l l o w s d i r e c t l y from l e m m a 1.16.

Lemma 1.18

If

G-J("E;F) =

and

E

F

are Locally convex spaces over

C

then

LJ P ( ~ E , ; F ~ ) BECS(F) a ~ c s ( E )

Before d e f i n i n g hypocontinuous and S i l v a c o n t i n u o u s polynomials we f i r s t p r o v e a r e s u l t a b o u t p o l y n o m i a l s on a Banach s p a c e .

This r e s u l t is t r u e

f o r any Baire s p a c e and i s used i n p r o v i n g Zorn's theorem i n t h e n e x t chapter. Lemma 1.19

Let

E

l o c a l l y eonvex space.

be a eompZex Banach space and

F

an arbitrary

If {Pm}i=l i s a sequence i n @("E;F)

and

13

Polynomials on locally convex topological vector spaces

We may suppose without l o s s of generality that F

Proof

linear space. Let Am E&'(~E;F)

where

im =

is a normed

Pm for every integer m.

By the polarization formula Am converges pointwise on E

to a symmetric

n-linear form A and hence P = E 2a(nE;F). We complete the proof by induction. Let B denote the unit ball of E. If n=l then Pm €;t(E;F) and by the uniform boundedness principle Hence IIPIIBs M and P is continuous by proposition 1.14. result holds for n=k. I f

Pm

E

6(k+1E;F)

for all m

= M <-.

Now suppose the

then, by induction,

Zsisk+l

By the uniform boundedness principle the collection of linear mappings

1 is bounded on the unit ball of E, (A (Y2,.*'3Yk+l) Yi'B 2fiSk+l i.e. . sup " (Y, .. .,Yn)II< YiEB ltick+l Thus P and A are continuous. This completes the proof. 9

The hypocontinuous and the Mackey continuous polynomial mappings between two locally convex spaces E and F can be defined in two different ways, either by defining new topologies on E o r by using properties of the compact and bounded subsets of E. We describe both methods.

A n element

p

of

@a(nE;F)

i s said t o be hypocontinuous i f

14

Chapter I

it i s continuous on t h e compact subsets o f

E.

pHy(nE;F)

We l e t

denote

the vector space o f a l l hypocontinuous n-homogeneous poZynomia2s from E

to

F.

Da(E;F)

An element of

i s said t o be Mackey ( o r S i l v a ) continuous F.

if it maps bounded subsets of E onto bounded subsets o f

We l e t

@M(nE;F) denote t h e vector space of a22 Mackey continuous n-homogeneous E

polynomials from

F.

into

Hypocontinuous and Mackey c o n t i n u o u s n - l i n e a r forms a r e d e f i n e d i n a n S i n c e t h e v e c t o r sum of compact ( r e s p . bounded) s u b s e t s o f

a n a l o g o u s way.

a l o c a l l y convex s p a c e i s compact ( r e s p . bounded) it f o l l o w s from t h e s y m m e t r i z a t i o n formula and t h e p o l a r i z a t i o n formula t h a t f o r any l o c a l l y convex s p a c e s

and

E

s(&,fE;F))

If

x

we have

F

giy(nE;F)

=

XHy(nE;F) n f;("E;F)

=

is a topological space ue l e t

Xk

denote the'space

with the

X

f i n e s t topology which coincides with the original topology on the compact subsets o f

If

X.

x

xk

=

we c a l l

X

a k-space.

If

is a locally

E

convex s p a c e t h e n it i s n o t t r u e i n g e n e r a l t h a t Ek i s a l s o a l o c a l l y convex space.

Any m e t r i z a b l e s p a c e i s a k - s p a c e .

A l o c a l l y convex s p a c e i s a

semi-Monte1 s p a c e if i t s c l o s e d bounded s e t s a r e compact. b a r r e l l e d semi-blontel s p a c e i s c a l l e d a Montel s p a c e .

An i n f r a -

The s t r o n g d u a l o f

a n i n f i n i t e d i m e n s i o n a l F r g c h e t Montel s p a c e ( a &I>mspace) o f a k-space which i s n o t m e t r i z a b l e .

i s a n example

The f o l l o w i n g p r o p o s i t i o n f o l l o w s

e a s i l y from t h e d e f i n i t i o n above.

Proposition 1.20

Hence i f convex s p a c e

E = Ek

If

E

=

B C ~ ( E;~ F ))

and

then

F

are ZocaZZy convex spaces then

Q H Y ( " ~ ; ~ )=

Q c ~ E ; F ) f o r any l o c a l l y

F.

W e now look a t Mackey c o n t i n u o u s p o l y n o m i a l s .

convex s p a c e w i t h t o p o l o g y

Mackey convergent t o scalars,

x,

T.

A sequence M

we w r i t e

(A~)E=~ \ i, n \ -f

01

as

xn-+ x,

n+-

Let

E

be a locally

( x ~ )i n~ E

is said t o be if there e x i s t s a sequence o f

, such t h a t

1 n(xn-x)

-f

0

in

(E,T)

15

Polynomials on locally convex topological vector spaces

-.

as

n

xn

y x as

-+

n-,

E

of

A

A subset

i s said t o be

( x ~ E) A,~

M-closed i f

The M-closed s u b s e t s o f

XEA.

implies

satisfy the

E

c l o s u r e axioms f o r a t o p o l o g y which w e c a l l t h e topology of the M-closure T ~ .W e

and d e n o t e by If

a l s o write

then t h e space

T = T~

EM

i n place of

(E,rM).

i s c a l l e d a superinductive space.

(E,T)

F r g c h e t s p a c e s and t h e s t r o n g d u a l s o f Frcchet-Schwartz s p a c e s a r e superinductive spaces. convex.

i s t h e s t r o n g d u a l o f a Frgchet-Monte1

(E,T)

For i n s t a n c e i f

(d.42 s p a c e s )

i s not necessarily l o c a l l y

The t o p o l o g y

space then t h e following a r e e q u i v a l e n t ;

is a 2 3 - 3 s p a c e ,

(a)

(E,T)

(b)

T = T~

(c)

( E , T ~ ) i s a l o c a l l y convex s p a c e .

on

E,

S i n c e c l o s e d sets are s e q u e n t i a l l y c l o s e d and Mackey c o n v e r g e n t sequences a r e convergent T~

in

T~

3

T~

on any l o c a l l y convex s p a c e

>,T

i s t h e k-topology a s s o c i a t e d w i t h M (E,rM) i f and o n l y i f xn x as

Proposition 1.21

any integer

n

F i r s t suppose

Proof

B(P(

F

and

E.

in

xn

+

x

as

be locally convex spaces.

Then f o r

P

E

@(n(EM);F) b u t

P

i s n o t bounded on t h e

B(P(xm)) > m

m 1 B(P(xm)) mn)) 1/2 X

=

>

m.

for all

1 m2 f o r a l l

m.

E,

(x,),,

This implies t h a t This contradicts t h e

n

fact t h a t

Hence

P

i s bounded on bounded s e t s and

8 Now suppose

P

n+-

(E,T).

Then we c a n f i n d a bounded sequence i n

B E C S ( F ) such t h a t

m

+

c ~ ~ ( ~ E ; =F )Q(~(E,);F).

bounded s u b s e t s of and

E

Let

n

-

Note a l s o t h a t

T.

-f

where

(E,T),

E

c~(E,);F)

c

(pM(nE;F).

IP~(~E;F). To show t h a t

P

E

@(n(EM);F)

it s u f f i c e s

16

Chapter 1 x as m - t m i m p l i e s P(xm) + P ( x ) as m b e a sequence o f scalars such t h a t 1x1, -t +

t o prove t h a t

Let hm(xm-x)

+

s u b s e t of

m+

x

as

0 E.

If

m

+.

A

ki(nE;F)

E

m

.

and

B = { A , ( X ~ - X ) }u~{XI i s a bounded

The s e t

m.

m

and

x=P

then

A

i s bounded on

Bn.

Hence

m=1,2,..

m = 1,2,.

i s a bounded s u b s e t o f

F.

I t now f o l l o w s t h a t P(xm)

- P(x)

shown t h a t

R n-R A(xm-x) (x)

P(x+xm-x)-P(x)

=

P(xm) + P ( x )

Corollary 1 . 2 2

as

=

m-+m

If E and

n

-t

1 R=l

as m + m if k & n . Since n R n-R (R) A(xm-x) (x) we have

0

and t h i s completes t h e p r o o f .

F

are ZocaZZy convex spaces then

~ , , ( " E ; F ) 3 P M ( " ~ ; ~3 )

for every

.

Q,,("E;F)

3 P ( n ~ ; ~ )

N.

Pa(nE;F)

may a l s o b e r e g a r d e d as a s p a c e o f c o n t i n u o u s polynomials i f we

p l a c e on

E

space, i s

the f i n i t e Z y open topoZogy tf

open i f V n F

tf.

A subset

V

of E

,

a vector

i s open for every f i n i t e dirnensionaZ subspace

F

of E where each f i n i t e dirnensionaZ space i s given i t s unique norm top0 zogy . When do we have e q u a l i t y i n c o r o l l a r y 1.22?

T h i s i s an i n t e r e s t i n g

q u e s t i o n and t h e answer p l a y s an i m p o r t a n t r o l e i n Chapter 5. a l r e a d y n o t e d cases i n which we do have e q u a l i t y .

We have

These r e s u l t e d from

c o n s i d e r i n g l o c a l l y convex s p a c e s as o b j e c t s i n t h e c a t e g o r y of t o p o l o g i c a l s p a c e s and c o n t i n u o u s mappings.

T h i s c a t e g o r y i s much t o o l a r g e f o r o u r

p u r p o s e s and r a r e l y p r o v i d e s n e c e s s a r y and s u f f i c i e n t c r i t e r i a .

In t h e

p r e s e n t case i t p r o v i d e s s u f f i c i e n t b u t n o t n e c e s s a r y c o n d i t i o n s f o r c e r tain equalities.

On t h e o t h e r hand, w e can a l s o look a t l o c a l l y convex

s p a c e s as o b j e c t s i n t h e c a t e g o r y of t o p o l o g i c a l v e c t o r s p a c e s .

T h i s cat-

egory i s t o o small and g e n e r a l l y o n l y g i v e s c o n d i t i o n s of t h e o p p o s i t e k i n d

-

i n t h e p r e s e n t s i t u a t i o n w e g e t c o n d i t i o n s which are n e c e s s a r y .

For example, i n o u r t e r m i n o l o g y a l o c a l l y convex s p a c e s p a c e if and o n l y if

6fM(E;F)

=

d(E;F)

E

is a bornological

f o r e v e r y l o c a l l y convex s p a c e

17

Polynomials on locally convex topological vector spaces I t i s , however, u s e f u l t o keep t h e s e t y p e

F.

o f r e s u l t s i n mind a s t h e y

p r o v i d e u s e f u l estimates and g i v e u s o u r p r e l i m i n a r y examples. The c o r r e c t s e t t i n g f o r t h e p r e s e n t c h a p t e r i s t h e "category" o f l o c a l l y convex s p a c e s and co n t i n u o u s m u l t i l i n e a r mappings and w e now g i v e some e x a m p l e s w h i c h r e f l e c t t h e

E = F x F' where F i s a l o c a l l y convex spa c e B i s i t s s t r o n g d u al ( i . e . F ' i s t h e spa c e o f a l l c ontinuous

Example 1 . 2 3 and

F' B

linear

Let

mappings on

F

bounded s u b s e t s o f Let

A

E

n a t u r e of t h i s "category".

A : E x E

and

f;(*E)

F). -+

Q: be d e f i n e d by

i ( x , x t ) = XI(%).

a normed l i n e a r s p ace. w e always have

w i t h t h e t o p o l o g y o f uniform convergence on t h e

iE

i s c o n t i n u o u s i f and o n l y i f

By t h e d e f i n i t i o n o f t h e s t r o n g topology on

PM(2E) and f r e q u e n t l y

form.

CN (C")

F'

i s hypocontinuous.

Our most u s e f u l counterexample i n t h i s book, CN x C(N), and

is

E

h a s t h e above

i s t h e s p a c e o f a l l complex s eq u en ce s w i t h t h e produc t topology i s t h e s p ace of a l l f i n i t e complex se que nc e s with t h e d i r e c t

sum to p o l o g y .

I n l a t e r c h a p t e r s we s h a l l u s e t h e f o l l o w i n g p r o p e r t i e s o f

it i s a r e f l e x i v e n u c l e a r s p ace w i t h a n a b s o l u t e b a s i s , it i s

CN x C " ) ;

a s t r i c t i n d u c t i v e l i m i t o f FrLchet n u c l e a r s p a c e s and a n open compact s u r j e c t i v e l i m i t of

( s t r o n g d u a l of F r 6 c h e t n u c l e a r ) s p a c e s , each

compact s e t i s c o n t a i n e d i n a s e t o f t h e form subset of

CN

and

KxL

where

K

i s a compact

i s a c l o s e d bounded f i n i t e dim e nsiona l s u b s e t o f

L

e v e r y neighbourhood o f t h e o r i g i n c o n t a i n s a neighbourhood o f t h e o r i g i n which h a s t h e . form U x CN-' x V where LEN, U i s a neighbourhood

(C(N),

i s a neighbourhood o f z e r o i n

CL and V

o f zero i n

_-Example 1.24

Let

E

(C").

b e a c o u n t a b l e i n d u c t i v e l i m i t o f normed l i n e a r

s p a c e s i n t h e c a t e g o r y o f l o c a l l y convex s p a c e s and c o n t i n u o u s l i n e a r

l i m En and E i s a b o r n o l o g i c a l spa c e which con-+ n m t a i n s a fundamental sequence o f bounded s e t s , (Bn)n,l. W e may suppose t h a t

mappings. each

Bn

Thus

E

=

i s convex and b al an ced .

S i n c e a l o c a l l y convex s p a c e i s b o r n o l -

o g i c a l if and o n l y i f ev er y convex b a l a n c e d s e t which a b s o r b s evEry bounded set i s a neighbourhood o f z e r o , w e f i n d t h a t s e t s o f t h e form

~n,lXnBn

Chapter 1

18

form a b a s i s of neighbourhoods of zero i n a l l sequences o f p o s i t i v e r e a l numbers

(ln=lAnBn m

m

ranges over m = Iln=lhnbn; bnEEn, m

i s convex and balanced and

IZ=lAnBn

a r b i t r a r y } ) . This follows s i n c e

as

E

absorbs every bounded s e t and hence i s a neighbourhood of zero.

Conversely

i s a convex balanced neighbourhood of zero then f o r every ntN m an t h e r e e x i s t s a n > 0 such t h a t anBn C V and hence V 3 Bn. Zn if

V

In=1

The countable d i r e c t sum of normed l i n e a r spaces and t h e strong dual I t can a l s o be

of a Frgchet Monte1 space a r e examples of such spaces.

e a s i l y shown t h a t t h i s c l a s s of spaces coincides with t h e c l a s s of bornological

DF

spaces.

E

We now show t h a t f o r such convex space

F.

$JM(nE;F) = @("E;F)

f o r every l o c a l l y

We may assume without l o s s o f g e n e r a l i t y t h a t

normed l i n e a r space.

Let

P

E

pM(nE;F)

and suppose

is a

F

A E%("E;F)

where

A = P.

B1 i s bounded I I P / I < M < B1 have been chosen so t h a t

Since

A2,

..., Am

If

x

m

E

I

AiBi, i=l

p(X+Xy)

1

i=l

=

and

YEB,,,+~

P(x)

XiBi

n

A >O

-.

Let

A1

= 1

and suppose

then

1

( " , A ( x ) " - ~ ( ~ ) X~R . and hence

IIAlL

i s f i n i t e and we can choose

+

R=l

+

where

ll A l l m Since

A

E

-

A;(~E;F),

s u f f i c i e n t l y small s o t h a t

X

=

A m+ 1

19

Polynomials on locally convex topological vector spaces m+l

s i=l

I

1 1

By induction that

?4(1 -i=l 2 i - 1

m

we can choose a sequence of p o s i t i v e numbers

llpll

such

(Xm)m=l

s 2M. m =1

Hence

P

i s bounded on a neighbourhood of zero and i s continuous by

p r o p o s i t i o n 1.14.

We a l s o n o t e t h a t t h e above proof shows t h a t i f

i s a subset of :J(nE;F), F i s a normed l i n e a r space, and ('cr)aE~ f o r every m then the collection is a locally s;p IIPall B, < (Pa)aEA bounded family of functions. Let

Example 1 . 2 5 convex space.

and only i f each I f each

E

=

Ci=lEm

PHy(nE;F')

where each

=

?("E;F)

Em

i s a normed l i n e a r space then

Em

i s a metrizable l o c a l l y

f o r every i n t e g e r

nz 2

if

i s a normed l i n e a r space.

Em

by example 1.24. Since

E

Then

Q("E;F)

=

pHy(nE;F)

i s not a normed l i n e a r space.

El

Conversely, suppose

i s a countable i n d u c t i v e l i m i t , it has a fundamental neighbourhood

system a t t h e o r i g i n c o n s i s t i n g of s e t s of t h e form a neighbourhood of zero i n

Em

i s contained and compact i n

f o r each

lm=iE

m

($m)m=2 denote a sequence i n (Elf and l e t Using t h e n a t u r a l embedding of each Em i n P

i t follows t h a t

P

Now suppose

=

E

P

lz=2$m$:-1

where

Pa("E).

Vm

and each compact s u b s e t of k.

is E

Let

Qm # 0 E EA f o r every m32. E and t h e p o l a r i z a t i o n formula

Since

@HY(nE) E

p("E).

neighbourhood of zero i n IIPII

Vm

Then t h e r e e x i s t s a sequence

Em, such t h a t

6 1.

Cm=1

E

Lo

f o r some p o s i t i v e i n t e g e r

m

i t follows t h a t

lm=lVm

m

(Vm)m=l,

Vm

a

20

Chapter I

For each

q,m(ym) # 0 .

choose

ym Vm such t h a t n-1 P(x+ym) = @,(XI ( ( ~ ~ ( y ~ ) ) and hence m22

Hence t h e r e e x i s t s a neighbourhood o f z e r o

II@ml)

v1 <

w

f o r every

ma2.

in

V1

a sequence i n

Hence we c a n c o n s t r u c t

Ei

11 @mil!,,m

such t h a t such t h a t

P

i n any l o c a l l y convex s p a c e

P

(Wm)i,2,

0,

and

m.

f o r every i n t e g e r

=

Since

CC

c a n b e embedded

as a c l o s e d complemented s u b s p a c e t h i s

F

q(nE;F) #

example can b e m o d i f i e d t o show t h a t

i s m e t r i z a b l e and n o t normable and

pH,(nE;F)

whenever

El

1122.

We now i n t r o d u c e a f u r t h e r s p a c e o f polynomials nomials.

then

V1

Q

such t h a t

El

@(nE).

g!

x

i s m e t r i z a b l e and n o t norm-

El

c o n t a i n s a neighbourhood system a t

a b l e t h e n it (@m):=z

However i f

If

-

t h e nuclear p o l y -

In c o n t r a s t t o t h e o t h e r s p a c e s o f polynomials we have i n t r o d u c e d ,

t h e s e form a s u b s p a c e of t h e s p a c e o f c o n t i n u o u s p o l y n o m i a l s .

The c o n c e p t

as a d u a l s p a c e and t o

o f n u c l e a r polynomial a l l o w s u s t o r e g a r d 6'("E)

The c o n c e p t i s a l s o u s e f u l i n d e v e l o p i n g t h e

develop a d u a l i t y theory.

t h e o r y of holomorphic f u n c t i o n s on n u c l e a r s p a c e s . D e f i n i t i o n 1.26

(a)

E

Let

F

and

be l o c a l l y convex spaces.

n

L E $ ( ~ E ; F )i s called a nuclear F

into

U

bourhood (A k ) i = l

k

E

and

R

yk

E,

in

(Yk>i,l

and

E

a bounded subset

from E (b)

E

and sequences where + i , k E B

for a l l

k

B

of

F,

m

( @ i , k ) k = l , i = l , .* q > n f o r aZZ i and

8

such t h a t

L ( X 1 > . . . > X n ) = l i = l A k @ l , k ( X 1 )' . * (X1, ...,X,) E E n .

We Let j t ' , f ~ ; ~ )

l i n e a r mapping from

i f there e x i s t a convex balanced zero neigh-

@n,k(xn)Yk f o r

denote t h e space of alL nuclear n

l i n e a r mappings

into F. P

E

@(%;F) i s called a nuclear

n

homogeneous polynomial

i f there e x i s t a convex baZanced zero neighbourhood

a bounded subset W

( @ k ) k = l IUo

and

B

of

F,

(Xk)L=l GB

E

R1

such t h a t

U

and sequences

in

E,

21

Polynomials on locally convex topological vector spaces

1"k = l A k @"(x)yk k

~ ( x )=

x

f o r every

E.

in

we l e t P 1 , ( " ~ ; ~ ) denote the space of aZZ nucZear n-homogeneous polyE

nomiaZs from Taking

F.

into

1.26(a) we obtain the d e f i n i t i o n of a nucZear Zinear

n=l i n

A ZocaZZy convex space

mapping between ZocaZZy convex spaces. t o be nucZear i f f o r every

a

E

cs(E)

B

there e x i s t s

EB

t h a t the carwnicaZ mapping from

Ea

onto

E

cs(E),BZa,

i s nucZear.

e x i s t s a sequence of

E'

such t h a t E

convex space

(An)n=l

in

p(x)

C n = l ( a n l I @ n ( ~ ) f o r every

and

.tl

1

EA

i s said t o be duaZ nucZear i f

Theorem 1 . 2 7

If E dN("E)

p

x

in

;k("E)

and

on

E

there

PN("E)

E.

A ZocaZZy

is nucZear.

i s a nuclear ZocaZZy convex space and

=

such

(@n)n a n equicontinuous s u b s e t

m

,<

i s said

This i s equi-

v a l e n t t o t h e c o n d i t i o n t h a t f o r e v e r y c o n t i n u o u s semi-norm m

E

nEN

then

p("E).

=

Proof

The second e q u a l i t y f o l l o w s immediately from t h e f i r s t s i n c e

-[x,("E))

= PN(nE).

show t h a t i f Since IL(xl

L

L

,..., xn) I

E

By d e f i n i t i o n c??~(~E)c- J?-(nE)

X(nE)

L

then

E

$("E).

is continuous t h e r e e x i s t s Q a(x,)

...

a(xn)

f o r any

f a c t o r i z a t i o n lemma, we may look upon E

is nuclear t h e r e e x i s t s a

from

EB

onto

(Yk)k C E a ,

x

For any

xl,

Ea

i s nuclear.

( @ k ) F = lc E '

=

in

E

L

cs(E)

~ E C S ( E ) such t h a t xl

,..., xn

E

Hence, by t h e

E.

as a n element o f

%(nEa).

Hence t h e r e e x i s t

in

w e have

Ea

Since

such t h a t t h e c a n o n i c a l mapping (Ak)kE k , ,

such t h a t

lE=lAk$k(x)yk

..., xn

in

B

and so it s u f f i c e s t o

f o r every

x

in

E

where

22

Chapter I

Since

t h i s completes t h e p r o o f .

51.3

TOPOLOGIES ON

SPACES OF

POLYNOMIALS

We now look a t t o p o l o g i e s on t h e v a r i o u s s p a c e s of polynomials we have S i n c e @(lE) = E '

defined i n t h e previous section.

( a l g e b r a i c a l l y ) , we

h a v e v a r i o u s g u i d e s from t h e d u a l i t y t h e o r y o f l o c a l l y convex s p a c e s .

The

most u s e f u l t o p o l o g y from t h e l i n e a r v i e w p o i n t i s t h e s t r o n g t o p o l o g y

-

t h a t i s t h e t o p o l o g y o f uniform convergence on t h e bounded s u b s e t s o f

E.

T h i s t o p o l o g y w i l l b e d e n o t e d by B .

We s h a l l s e e t h a t i t i s not t h e most u s e f u l t o p o l o g y from t h e holomorHowever, i t does s e r v e a p u r p o s e i n o u r development

p h i c p o i n t o f view. and m o t i v a t i o n .

For t h e s a k e o f e f f i c i e n c y we s h a l l always t r y t o d e f i n e

o u r t o p o l o g y on as l a r g e a s p a c e as p o s s i b l e . De finit i on 1.28

topology or the

Let B

E

and

F

be l o c a l l y convex spaces.

topology on @ ("E;F)

M

i s defined t o be the topoZogy of

uniform convergence on t h e bounded subsets o f (PM(nE;F),B) b y t h e semi-norms

The strong

E.

i s a l o c a l l y convex s p a c e and i t s t o p o l o g y i s g e n e r a t e d

I\

(/a,3

t h e bounded s u b s e t s o f

where

c1

ranges over

cs(F)

and

B

ranges over

E.

The f o l l o w i n g a r e e a s i l y proved. P roposi t i on 1.29

space, then

I f

E

(?("E;F),B)

P roposi t i on 1.30

If

l o c a l l y convex space then

i s a normed linear space and

F

i s a Banach

i s a Banach space. E

i s a ZocalZy convex space and

F

i s a complete

23

Polynomials on locally convex topological vector spaces

(~~,,("E;FI,B)

i s a complete ZocaZZy convex space.

we naturally consider the compact open topology

On @Hy(nE;F)

Let

Definition 1.31

E and

compact subsets of

F be ZocaZZy convex spaces.

We denote t h i s topoZogy by

E.

The compact

i s the topoZogy of uniform convergence on t h e

open topoZogy on $IHy("E;F)

T ~ .

(pHy(nE;F),~o) is a locally convex space and its topology is generated by the semi-norms 11 11 where c1 ranges over cs(F) and K ranges a,K

over the compact subsets of E. Since every compact subset of a locally convex space is bounded it follows that B fj =

on FHy(%;F).

3 T~

By the Hahn-Banach theorem we have

~ if and only if the closed convex hull of each bounded subset of E

T

is compact

i.e. if and only if E

is a semi-Monte1 space.

Since the uniform limit of continuous functions on a compact set is continuous the following is true. -Proposition 1.32

E be a ZocaZZy convex space and

Let

ZocaZZy convex space.

Then

space.

(BHy(n~;~) s ~ o )

By restriction B

convex topologies on

F a complete i s a compZete ZocaZZy convex and

T~

define locally

6 ("E;F) . However, we shall need a further

topology on

!?fE;F). This topology, denoted by T ~ , is the strongest topology we It is motivated by the factorization formula, the define on p(nE;F). definition of the strong topology on @("E;F)

when E' is a normed linear

space and certain properties of analytic functionals in several complex variables theory. It is perhaps more useful than T~ or B since it has stronger topological properties but it is more difficult to characterize in a concrete fashion. We first consider polynomials with values in a normed linear space. Let

E be a ZocaZZy convex space and l e t F be a T~ topology on Q("E;F) i s defined a s t h e i n d u c t i v e l i m i t topoZogy i n t h e category of ZocaZly convex spaces and conDefinition 1.33

normed Zinear space.

The

tinuous l i n e a r mappings

(B("E;F)

, T ~ ) =

o ~ P (;F), ~ Ea E CS(E) , t h a t i s lim a Zcs(E)

(P(~E~;F), B ) .

24

Chapter 1 p

Hence a semi-norm V

every neighbourhood P(P) ,<

on

d:

("E;F)

o f zero in C(V) I/pIIV

E

is .ru-continuous if and only i f f o r there e x i s t s

f o r every

P

in

c(V) 2 0

such t h a t

F("E;F).

W e w i l l subsequently s e e t h a t t h i s amounts t o saying t h a t a semi-norm on

i s .ru-continuous i f and only i f it i s ported by t h e o r i g i n and

,>("E)

f o r t h i s reason we c a l l

( 3 (nEa;F) ,B)

Since

t h e ported topology.

Tu

i s always a normed l i n e a r space

i s a bornological space when Banach space

F

i s a normed l i n e a r space.

i s a Banach space and

?(nEa;F)

(

( ?(nE;F)

When

, T ~ )

t i v e l i m i t of Banach spaces, i . e . an u l t r a b o r n o l o g i c a l space. ular,

(

? (nE;F) ,

(nE;F) ,.rW) F

is a

i s an inducIn p a r t i c -

i s then a b a r r e l l e d l o c a l l y convex space, t h a t i s

T ~ )

every closed convex balanced absorbing subset i s a neighbourhood of zero. For a r b i t r a r y

we use t h e weak form of t h e f a c t o r i z a t i o n lemma

F

(lemma 1.18) and d e f i n i t i o n 1.33 t o d e f i n e Definition 1.34

E

Let

Q("E;F)

defined on

and

F

.ru

on

P("E;F).

be l o c a l l y convez spaces.

Then

T*

as

The following elementary r e s u l t shows t h e r e l a t i o n s h i p between t h e topologies we have defined. Proposition 1.35

any p o s i t i v e i n t e g e r

(a) .ru Ibi

B

3

B

.ro

2,

and

For arbitrary locaZ7.y n we have

.ro

on

convex spaces

.

and

?("E;F).

d e f i n e t h e same bounded subsets o f

and hence have t h e same associated bornological topology

E

6(nE;F)

F

and

is

25

Polynomials on locally convex topological vector spaces We now g i v e a number o f e l e m e n t a r y examples r e l a t i n g t h e above t o p o l -

-

ogies

Afterwards, we d e f i n e a

f u r t h e r examples appear i n l a t e r c h a p t e r s .

topology on t h e s p a c e o f n u c l e a r p o l y n o m i a l s . Example 1 . 3 6

i s a n i n f i n i t e dimensional Banach s p a c e and

E

If

a l o c a l l y convex s p a c e , t h e n T

=

T~

2

6

have t h e same bounded sets and hence

a s s o c i a t e d with

T

Example 1 . 3 7

~

T

g'(lE)

If

bounded s u b s e t s o f

T~

i f and o n l y i f

= E'

Example 1 . 3 8

T~

and

Let

Example 1.39

Moreover, i f

If

subsets of

compact s u b s e t o f n

all

u

n

v

n

norm on P(*E) P(P) a

C")

E

=

cs(E)

-

..., 1

E

=

( P ,..., 4.0

d e f i n e d by

do n o t

and l e t

t h e n t h e r e e x i s t compact

such t h a t

K2,

K c K1 x K 2 .

Now e v e r y

IIPnIIK

i s a bounded s u b s e t o f

B

T~

= 0

for

(8( 2 E ) , T o ) .

0 . E and l e t C ntil p o s i t i o n

(0,

then

and

T~

P n ( ( x n ) n , ( y n ) n ) = xnyn,

Thus

=

I",=,

<'("E;F).

i s f i n i t e dimensional and hence

sufficiently large.

Let

?(2E)

and

K1

C"),

There-

and hence are n o t e q u a l .

i s a compact s u b s e t o f

K

and

EN

on

We show t h a t

E =

b e d e f i n e d by

Pn:E + E

B = (Pn):=l.

Using t h e method o f example

i s semi-Monte1 and hence a

E

= B = T~

T~

d e f i n e t h e same bounded s u b s e t s o f

Let

is a

E

E'.

d e f i n e t h e same bounded s u b s e t s of

T

s p a c e , it f o l l o w s t h a t

g 3 1'1

T~

on

= B

.?w

(j' ("E) ,@) i s m e t r i z a b l e and hence b o r n o l o g i c a l .

on ?(("E;F).

= @

R

~

c o n t a i n s a c o u n t a b l e fundamental system o f bounded s e t s

E

it follows t h a t

T~

T

Consequently i f

on

is a

be a c o u n t a b l e i n d u c t i v e l i m i t o f normed l i n e a r

T~,@

Since

> >

i s distinguished.

E

F

a r e l o c a l l y bounded.

,?(nE;F)

b e a normed l i n e a r s p a c e .

F

1.24 w e s e e t h a t 'j, ("E;F).

E

Let

s p a c e s and l e t

and

T

i s t h e b o r n o l o g i c a l topology

T~

i s t h e bornological topology a s s o c i a t e d with

w

is

F

Moreover,

i s a m e t r i z a b l e l o c a l l y convex s p a c e and

E

non-distinguished Frgchet space then

fore

?("E;F).

.

Banach s p a c e t h e n t h e Hence

If

on

To

I P(nun,vn)

,...

)

E

)

E

Let

C").

p

d e n o t e t h e semi-

nth position

- P(o,~,)

a(un,O) = 0

for all

I

for all n

P

E

~ ( 2 ~ 1 .

sufficiently large.

Hence

26

Chapter 1

P(nun,vn) = P(O,vn) P

i n P (2E).

n

and hence

T o

it f o l l o w s t h a t

for all

s u f f i c i e n t l y l a r g e and

S i n c e t h e semi-norm which maps

is

P((O,vn)(

n

for all

p

is a T

P(2E)

E

(Q( ~ E ) , T)

c o n t i n u o u s and B

( P (2E)

is not a

,T~]

T ~ , @

and

E

convex subset of a ZocalZy convex space

and

L

and

P

B

If

.

II

we l e t

$ k , i ~El}

8 N ( n E ) we l e t

E

where t h e i n f i m a are taken over a l l possible representations of IIB(L) IIB(L) hood

and

and

( r e s p . P) V

These

i s a balanced

$,("E)

E

and each and if

is

P(nE).

p N n( E ) , no, li B r e s p e c t i v e l y o n 8 ("E) .

T~

!?(2E). T~

on Y ( ~ E ) .

T~

We now d e f i n e t h r e e t o p o l o g i e s on correspond t o

p(P n ) = n

bounded s u b s e t o f

T~

-

is barrelled Since

i s a b o r n o l o g i c a l s p a c e and

t h e b a r r e l l e d topology a s s o c i a t e d with This r e s u l t extends e a s i l y t o

f o r every

m

I P(nun,vn)

to

c o n t i n u o u s semi-norm on :?(2E).

w e h a v e shown t h a t

We s h a l l s e e l a t e r t h a t

P

p(P) <

IIB(P)

may b e i n f i n i t e .

ItB(P)

are always f i n i t e .

&:,(%)

(respectively

E

of z e r o such t h a t

ITv(L)

However, i f

L

and

P.

i s bounded t h e n

B

Moreover, by d e f i n i t i o n , i f

L

t h e n t h e r e e x i s t s a neighbour-

Q,(nE)) <

and

TIv(€')

<

m.

These a l l o w us t o

give t h e following d e f i n i t i o n . Definition 1.40

(a)

The

IIo

topology on

t h e l o c a l l y eonvex topology generated by subsets of

AN(nE)

flK

PN(nE)I is K ranges over a l l compact

as

iresp.

E.

( b ) the

IIB

topology on J,("E)

( r e s p e c t i v e l y @,(nE)

locaZ2y convex topo2ogy generated by

nB

as

3

)

i s the

ranges over

all bounded subsets of E. ( c ) A semi-norm be

p

on J'",lnE)

IIwcontinuous

(respectiveZy ? N ( n E ) )

if for every neighbourhood V

is said t o in

of zero

21

Polynomials on locally convex topological vector spaces E

c(v) > 0

there e x i s t s p(L) s c(v)nV(L)

nu

11 LII B n 6

Since

of

B

E

it f o l l o w s t h a t for all

d N ( n E ) and '$"(nE) Proposition 1.41 TIo = IIR = B =

K1

ced subset

E

of

Since

~

,

n.

for every

%

3 T

w

on

Then

c K >0

I\L/I

(L) i c:

and a convex balan-

(Kl)n

for every

L

in

n.

i s q u a s i - c o m p l e t e and d u a l n u c l e a r i t s c l o s e d bounded

E

B =

and

T

II 0

= ItR.

E,

+Ei

such t h a t t h e c a n o n i c a l mapping from E' m a, K1 (xk)k,l E E l , ( a k ) k = l a sequence i n

i v e ) and a bounded sequence

@

Now l e t A

nu

Moreover, for each convex

exist

~1 E

and

E ' i s n u c l e a r w e c a n , g i v e n K a convex b a l a n c e d compact subB choose a convex b a l a n c e d compact s u b s e t K1 o f E c o n t a i n i n g K

Since

f o r every

f o r e v e r y convex b a l a n c e d

II R > B

there e x i s t

such t h a t

s e t s a r e compact and hence

set i n

;J~(~E))

be a quasi-complete dual nuclear space.

and for every non-negative

Proof

IIB(P) T

n.

K of E

balanced compact subset $N(nE)

E

Let

on d,("E)

T~

Xo 3

E

continuous serni-norms.

11 Pllgi

and

IIB(L)

P

irespectively

is t h e topology generated by a l l Xu

subset

p ( P ) 6 c(V)IIv(P))

(respectively

i n A.,("E)

L

f o r every

such t h a t

i n E'

K1

i n L ~ ( ~ E )NOW . suppose K.

E

Hence t h e r e i s semi-reflex-

such t h a t

E' K1

K.

There e x i s t s , by t h e Hahn-Banach theorem,

such t h a t

v e r g e s u n i f o r m l y on

(@k)k=l i n

(since

where t h e s e r i e s converges u n i f o r m l y on

L €AN("E).

(XN(nE),no)'

m

i s nuclear. K~

a ( L ) = IIK(L) L

=

I",l

We t h e n have

and

$m,l..

Ia(A)I

.

Q IIK(A)

f o r every

J,m,n where t h e series con-

28

...,kn =1xk 1

=

*

* . 'kn

1

...

@k 1 L(a , . . . , a 1 n kl kn

T h i s completes t h e p r o o f . Combining theorem 1 . 2 7 and p r o p o s i t i o n 1 . 4 1 w e o b t a i n t h e f o l l o w i n g result. C o r o l l a r y 1.42

then

X("E)

=

If

gN(nE)

If

Theorem 1 . 4 3

E

E

is a quasicomplete nuclear and dual nuclear space and

TIo =

T~

for every

n.

is a quasi-complete dual nuclear space then

( L ( ~ E ) , T ~is ) a nuclear space.

K

Proof

If

sequence

(Xm)m=l

f o r every

m

@

in

i s a compact s u b s e t of in

E'.

E and

Hence i f

m

n

E

E

R1

t h e n t h e r e e x i s t s a compact such t h a t

i s any p o s i t i v e i n t e g e r and

then

lrisn- 1

(and by induction)

L

E

&(("E)

29

Polynomials on locally convex topological vector spaces

a compact sequence

t h i s completes t h e proof.

The form o f t h e above i n e q u a l i t y w i l l b e u s e d i n c h a p t e r 3 Proposition 1.44 T

E

Let

on &("E)

= i7 w w

be a quasi-complete nuclear space.

Proof

nu

II~(L)c cn

such t h a t

C>O

Since

= T

IIw

2 T

w

W

of zero, contained i n V

Vn f o r every

L

in

, and

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

it s u f f i c e s t o p r o v e t h e above i n e q u a l i t y t o show

w'

By t h e n u c l e a r i t y o f bourhood

V

i c a l mapping m

($k)k=l in

IILI/

of zero, there

V

Moreover, f o r any convex balanced neighbourhood e x i s t a convex balanced neighbourhood

Then

n.

f o r any p o s i t i v e i n t e g e r

W"

E

E we can choose, g i v e n a convex b a l a n c e d n e i g h -

o f z e r o , a n e i g h b o u r h o o d of z e r o EW + EV

and

m

is nuclear.

L ~ d ( % ) and

Now suppose If

Y1,.

Hence t h e r e e x i s t

. . ,Yn

V

E

L1,

E

co

( x ~ ) c~ V= ~such t h a t

where t h e convergence i s i n

E

s u c h t h a t t h e canon-

WCV

x = rk=lXkQk(x)xk

for all

EV'

IILII

= 1.

V"

then

and h e n c e

=

Now

Ikl,

lL(xk , 1

...,kn . .. , x

kn

'kl )

1

... x

kn

,< 1

.

L(Xk ,. .,Xkn) bkl 1

f o r any c h o i c e o f Hence

x

. .. $k .

,. . . , x

kl

n

kn

in

V

since

x

Chapter 1

30

.,knIhkl ."

kn

I

=

then

Thus nw(L) if 11 LII vn

6 =

cn IlLll vn and since this inequality is trivially satisfied we have completed the proof. m

The preceeding results can be transferred to n homogeneous polynomials by using the inequality II*(L)

for any L B

of

E.

6 IIB(L)

in XS("E)

nn

c n! nB(L)

n A~("E)

=

~ki(~E) and any convex balanced subset (X ZPE) ,no)

In particular, we find

(A i(nE> ,nu)

=

($?N(nE)

=

WNN(nE),no)

and

,nu).

We now summarize results obtained in this way. Proposition 1.45 Let E be a quasi-complete Locally convex space and l e t n be a non-negative i n t e g e r . ( a ) If E i s dual nuclear then

no

= B =

and

('J(~E),T~)i s a nucZear space. ( b ) IS E i s nuclear then 2bN(n~) = 'J'("E and

= T

w

on J"(nE)

.

Moreover, the estimates given in propositions 1.41 and 1.44 are still valid, with minor modifications, for spaces of homogeneous polynomials on the appropriate locally convex spaces.

31

Polynomials on locally convex topological vector spaces 51.4

DUALITY THEORY FOR SPACES OF POLYNOMIALS I n t h i s s e c t i o n w e c o n s i d e r l i n e a r f u n c t i o n a l s on t h e l o c a l l y convex This t o p i c is currently t h e subject

s p a c e s o f polynomials d e f i n e d i n 5 1 . 3 .

o f r e s e a r c h and s h o u l d p l a y an i m p o r t a n t r o l e i n t h e development o f t h e Our p r e s e n t a t i o n of r e s u l t s i s n o t f u l l y com-

subject i n t h e near future.

p r e h e n s i v e b u t h o p e f u l l y o u t l i n e s t h e main developments and p r o v i d e s a g l i m p s e o f f u t u r e developments. We show t h a t c o n t i n u o u s l i n e a r f u n c t i o n a l s on s p a c e s o f polynomials Our main t o o l i n o b t a i n i n g

can themselves b e r e p r e s e n t e d by polynomials. t h i s r e p r e s e n t a t i o n i s t h e Borel t r a n s f o r m . D e f i n i t i o n 1.46

Let

A be a vector space of

.

form on h

The Bore2 transform of

T,BT,

vaZued n-homogeneous

C

E

poZynomiaZs defined on a ZocaZZy convex space

and Zet

i s defined on

T

be a Zinear

{$IEE*I+"EA >

by the formuZa BT($I) If BT

to

then

=

i s a subspace o f

F

T(99

EX

and

ms(Fn)c&

i s an n-homogeneous polynomial.

F

'4

if BT E $ ~ ( ~ E * and )

=

?(nE)

then t h e r e s t r i c t i o n of

For example i f

o r 3N(nE)

then

A

= qa("E)

BT E ? a ( n E 7 ) .

A i s a l o c a l l y convex s p a c e and T i s c o n t i n The Borel t r a n s f o r m w i l l o n l y be u s e f u l i f it i s i n j e c t i v e . T h i s

In t h e c a s e s w e consider uous.

w i l l always b e t h e case i f

p r o p e r t y and

fi

= PN(nE)

or if

E

h a s t h e approximation

'J'("E).

=

P r o p o s i t i o n 1.47

The Bore2 transform i s a vector space isomorphism from

( i ) ( 2 , ( n ~ ),n6)

1

onto D ( ~ E ; )

and

(ii)( @ N ( n ~,no) )

Under t h i s isomorphism the equicontinuous subsets o f espond t o the locaZZy bounded subsets o f (PN(%) ,no)

subsets of P(n(E,.ro) Proof

p("E;)

onto

P(~(E:T~)

(PN(("E),ITB)

corr-

and the equicontinuous

correspond t o the Zocally bounded subsets of

'1. S i n c e b o t h c a s e s a r e proved i n a s i m i l a r f a s h i o n , we o n l y c o n s i d -

er t h e case There e x i s t

(PN(E),i16).

B

i s l i n e a r and i n j e c t i v e .

and

B

an a b s o l u t e l y convex bounded s u b s e t o f

c>O

Let

such t h a t IT(P) I f

c nB(p)

f o r every

P

i n P,("E).

T E (@N(nE),TIbgll E

32

Chapter 1 In p a r t i c u l a r i f

c 11$"1/, 6 c .

$

Hence

tinuous subset of Now suppose

1 then

h

T h i s a l s o shows t h a t t h e image by

PI

and

P("Egt)

E

of

B

E.

If

/I PillBo P

E

T(P)

li=lP1($i).

=

ear o p e r a t o r on 3'N(nE), t h e representation of

c

6

of z e r o i n

V

ll$ill

Moreover, i f

P.

and

?(nEgt).

E,

then w e

T

is a well defined l i n -

T(P)

i s independent o f

One e a s i l y shows t h a t i.e. the definition of

Eb

o f an equicon-

f o r some c l o s e d a b s o l u t e l y m n and P = l i = l $ i , $ i E~l

PN(nE)

f o r some neighbourhood m

B

i s a l o c a l l y bounded s u b s e t o f

(?,("E),TIB)'

convex bounded s u b s e t

let

/ B T ( $ ) / = IT($")[ h

i s bounded on a neighbourhood o f z e r o i n

BT

so it i s continuous.

I/$ //

and

E E I

= 0

then

Pt(Oi) = 0

and

hence

=

Hence

T

cnB(P).

( ? N(nE) ,")

E

and

BT($)

=

T($n) = P I ( $ ) .

i s s u r j e c t i v e and a v e c t o r s p a c e isomorphism.

{P

=

E

v("Ei) ;

)I P \ \ Bo 8

C}

T h i s shows t h a t

B

The above a l s o shows t h a t

f o r any

c>O

and t h i s completes

t h e proof. Proposition 1.48 (?N(nE)

E'

,nu)

The Borel transform i s a vector space isomorphism from

onto

,FE ("El)

f t h e space of n

homogeneous poZynomiaZs on

which are bounded on the equicontinuous subsets of

El).

Under t h i s

isomorphism t h e equicontinuous subsets of subsets of

? ("El)

subsets of

El.

Proof T

E

5

6

(?N(nE) ,II) I correspond t o which are uniformly bounded on :he equicontinuous

The p r o o f i s v e r y similar t o t h e proof of p r o p o s i t i o n 1 . 4 7 .

(?N(nE) ,nu)

and l e t

V

Let

d e n o t e a n a b s o l u t e l y convex neighbourhood of

33

Polynomials on locally convex topological vector spaces 0

in

and

E.

BT

E

c(V)> 0

There e x i s t s

? ("El). 5

uous subset of

such t h a t

This a l s o shows t h a t t h e image by

i s a subset of

(yN("E),na)'

5

B

("El)

of an equicontin-

c o n s i s t i n g of fun-

E'.

c t i o n s which a r e uniformly bounded on t h e equicontinuous s u b s e t s of Now suppose

PI

(nE').

E

We d e f i n e

i s a neighbourhood of zero i n

Moreover, s i n c e isomorphism.

E

and

BT($) = PI($)

as i n proposition 1.47.

T P

E

?N(nE)

If

V

then

t h i s shows t h a t

B

i s a v e c t o r space

The r e s u l t about equicontinuous s e t s a l s o follows from t h e

above. We g i v e

Quite a number of c o r o l l a r i e s can be deduced from t h e above. The f i r s t i s perhaps t h e most i n t e r e s t i n g .

j u s t a few examples.

Since

t h e c o l l e c t i o n of spaces which occur i n t h i s c o r o l l a r y i s r a t h e r i n t e r e s t -

s),

ing, (see chapters 3and Definition 1.49 EL;

we g i v e them a s p e c i a l name.

A locally convex space

E

E

i s fuzzy nuclear i f

and

are both complete infrabarrelled nuclear spaces. A f u l l y n u c l e a r space i s a r e f l e x i v e nuclear space and t h e s t r o n g dual

of f u l l y n u c l e a r space i s f u l l y n u c l e a r .

Every FrGchet n u c l e a r space i s

f u l l y nuclear. Corollary 1.50 @("E),

n

I f

E

is a f u l l y nuclear space then

T~

= T*

a p o s i t i v e integer, i f and only i f F'M(nEA) = ?(nE;j)

B -bounded subsets of p:d(nEL)

are locally bounded.

on

and the

Chapter 1

34 Proof

Since

i s an i n f r a b a r r e l l e d l o c a l l y convex s p a c e t h e equicon-

E

c o i n c i d e w i t h t h e bounded sets and hence

tinuous subsets of

Ek

PM("Eb) = Pt(("E').

I t now s u f f i c e s t o a p p l y theorem 1 . 2 7 , and p r o p o s i t i o n s

1 . 4 5 , 1 . 4 7 and 1 . 4 8 t o complete t h e p r o o f . In particular w e note t h a t

o = Also t h i s shows t h a t

nuclear space.

T

T

w

T~

on

#

?(nE) on

T~

if

i s a Frgchet

E

if

F(nC(N) x CN)

1122,

a r e s u l t which w e h a v e a l r e a d y proved d i r e c t l y (example 1 . 3 9 ) . Corollary 1.51

If E (8(nE)

If E

Corollary 1.53

If E

ng

Q,(nE;)

I

C o r o l l a r y 1.52

FreTchet space then

is a r e f l e x i v e nuclear space then

=

i s an infrabarrelled l o c a l l y convex space then

i s an infrabarreZled

nw

DF s p a c e o r a distinguished

on P N ( n E )

We now look a t some examples i n which t h e Bore1 t r a n s f o r m g i v e s a t o p o l o g i c a l isomorphism.

We f i r s t need some p r e l i m i n a r y r e s u l t s .

E, a Locally convex space, has property E

of

K

subset

E K

such t h a t

EB.

is contained and compact i n

(EB i s t h e v e c t o r s u b s p a c e o f t h e norm whose u n i t b a l l i s

E

If

B).

g e n e r a t e d by B

EB

i s a Banach

S t r i c t i n d u c t i v e l i m i t s o f FrGchet s p a c e s and s t r o n g d u a l s o f

space.

f u l l y nuclear space has property Lemma 1 . 5 4 f,,(nE;F)

9M ("E;F)

If in

B

K E

i s compact i n such t h a t

continuous.

Lemma 1 . 5 5

If E

(EJ)

K

Hence

E, If

K. E

F

(s)

then

nEN.

and any

t h e n t h e r e e x i s t s an a b s o l u t e l y convex

i s compact i n P

In p a r t i c u l a r , every

has property

for any l o c a l l y convex space

i n d u c e t h e same t o p o l o g y on T

(5).

(s).

If the locally convex space =

Proof

hence

and endowed w i t h

B

i s complete t h e n

i n f r a b a r r e l l e d Schwartz s p a c e s have p r o p e r t y

set

if for each cornpact B of

(5)

there e x i s t s an absolutely convex bounded subset

P

E

EB.

Hence

PM(("E;F) t h e n

QHy(nE;F)

and

T , T ~

PIK

and

/I I / B

is

T~

and

$h(nE;F) = PH,,("E;F).

is a f u l l y nuclear space then @Hy(nE) is equal t o

35

Polynomials on locally convex topological vector spaces

t h e cornpZetion of Proof

The completion o f

convex s p a c e

.

~ )

( L 1 ( n E ) , ~ o ) l i e s i n j?Hy(nE) (j'("E)

and q u a s i c o m p l e t e w e c a n c h o o s e

KIC-K,

m

m

(yn)n=l C K 1

(Xn)n=l

E

(Qn)zX1

R1,

such t h a t f o r e v e r y x

x

A(xl,.

. .,xn)

. .,xn

xl,.

1

n

IA(yi

1

. . . . .yi

n

choose a f i n i t e s e t o f i n d i c e s

in

Kl

EK

E;<

with

.

Since

-+

E

i s nuclear.

K1

llQnll

C

1

and

m

m

6 (li=llxil)n

)

I

m.

<

m

and

Hence, f o r any 6 ' 0 ,

w e can

such t h a t

F

1 A . ... X i

<

i s c o n t i n u o u s on

A

. . ,&=lXi+ii(xn)Yi)

A(1.1=1X .1 + . 1( x l ) y i , .

1

-

E

K

E

m

=

A. , . . . , X i

IIA

and l e t

m

=

Now

?UP l , . . . ,in

LiY("E)

E

K

in

where t h e series converges u n i f o r m l y i n

we have f o r any

E

A

E. Since E i s dual nuclear a n a b s o l u t e l y convex compact

K1, such t h a t t h e c a n o n i c a l i n j e c t i o n

E

Hence t h e r e e x i s t

K:

Let

, T ~ ) .

b e a n a b s o l u t e l y convex compact s u b s e t o f

s u b s e t of

f o r any l o c a l l y

and t h u s t o p r o v e t h i s r e s u l t i t s u f f i c e s t o show

E

l i e s i n t h e completion o f

S'HY(nE)

K

,T

(t ("E)

A(yi ,..., yi ) $ i . . . $i ( 1 < 6/2. S i n c e E l i s d e n s e n 1 k 1 k K w e c a n choose a sequence of c o n t i n u o u s l i n e a r forms on E , ( $ i ) i = l , l1

ElK

such t h a t

Ill

F

Q

il

... Q

in

-

1 JI F

il

... 1L i II

< 6/2

n K

Combining t h e s e two i n e q u a l i t i e s we o b t a i n t h e d e s i r e d r e s u l t . If

(E,T)

iated with

i s a l o c a l l y convex s p a c e t h e n t h e Mackey t o p o l o g y a s s o c ( n o t t o b e c o n f u s e d w i t h t h e t o p o l o g y o f t h e M-closure

i s t h e f i n e s t l o c a l l y convex t o p o l o g y on

d u a l as T

(E,T).

E

If t h e Mackey t o p o l o g y a s s o c i a t e d w i t h

t h e n we s a y t h a t

(E,r)

T ~ )

which h a s t h e same c o n t i n u o u s

i s a Mackey s p a c e .

T

coincides with

An i n f r a b a r r e l l e d l o c a l l y

Chapter I

36

convex space is a !.lackey space.

If E i s a f u l l y nuclear space then

Proposition 1.56

is the completion o f a nuclear space and hence it @Hy(nE),~o) is a complete nuclear space. Let s s be the strong topology on

Proof

(nEb)

'

CJ?,,("E)

(proposition 1.47).

, T ~ )

($("E;)

, T ~ )

is the strong

dual of a semi-reflexive space and hence is a barrelled Mackey space. (jJ(nE;),~u) is also a barrelled space and hence a Mackey space. To complete the proof we need only show

'

(?(nEb) Since

' .

, T ~ )

(P(~E;) ,TB)

is semi-reflexive,

, T ~ )

osition 1.48,

(j)("EA)

=

, T ~ )

'

(5'(nEb)

, T ~ )

2

$'M(nE)

FHy(nE),

=

1

=

S;iY(n~).

BY prop-

and this completes the

proof. i s a fully nuclear space then

If E

Proposition 1.57

(!j"nE) ,TJ i f and only i f the If

Proof ((P("E)

,T,);)

(PHy ("E;)

, @(%)

su bounded subsets o f

(P (nE), g

;

T

~

@("E)

;a )

(PHY("E;),

and hence

T ~ )

then by proposition 1.56

(Q("E),T~)

space and thus it is reflexive. Hence

are Locally bounded.

is a barrelled semireflexive

(gHy(nEb),~o)

is also reflexive

and the equicontinuous subsets of the dual coincide with the strongly bounded subsets. The strong topology on (BHy(nE ; ~ ) , T ~ ) ' is the T topology by proposition 1.56 and the equicontinuous subsets are the locally bounded sets by proposition 1.47. Conversely if the su bounded subsets of

p("E) are locally bounded then the bounded subsets of (QHy("E;) , T ~ ) are equcontinuous (propositions 1.47 and 1.56). Hence (PHy(nEk),~o) is infrabarrelled and thus reflexive. By proposition 1.56, ((3(nE) , T ~ ) = (!!Hy(nE;),~o)~ is also a reflexive space and (OYEI,TJ;

=

((pHy("~;),

T ~ I ; ) ~

=

(pHY(n~t),~o~.

This completes the proof. We now look at linear functionals on spaces of homogeneous polynomials

37

Polynomials on locally convex topological vector spaces d e f i n e d on F r e c h e t s p a c e s w i t h t h e a p p r o x i m a t i o n p r o p e r t y . nomials s t i l l a p p e a r and p l a y a n i m p o r t a n t r o l e . uous forms as t h e c o r r e s p o n d i n g t h e o r y f o r developed and indeed t h e g e n e r a l t h e o r y f o r an u n t i l l e d f i e l d .

T~ T~

Nuclear p o l y -

We s t u d y o n l y

T~

contin-

c o n t i n u o u s forms i s n o t y e t c o n t i n u o u s forms i s a l m o s t

One c a n e a s i l y show t h a t t h e v e c t o r s p a c e isomorphisms

o f p r o p o s i t i o n 1.1 y i e l d a t o p o l o g i c a l isomorphism when t h e a p p r o p r i a t e s p a c e s are endowed w i t h t h e compact open t o p o l o g y .

This implies t h e f o l l -

owing r e s u l t .

Lemma 1.58

If E

i s a Frgchet space and F

If

P r o p o s i t i o n 1.59

T

convex space and

E

'

IT(L)I

for all L i n .Y.(E;F) then there e x i s t

E

i s a FrLchet spaee,

(x (E;F) ,T

'

~ )

F

i s an arbitrary locally

satisfies

CIILIly,K ~ E C S ( F ) and

where

( a ) a null sequence ($n)n

( b ) a sequence

i s a locally convex space

n

then f o r any p o s i t i v e integer

K

i s a compact subset of E

( x ~ )i n~ E ,

in

F'

such that

f o r all y i n F and all n and m (c) (hn)n=l E k1 where l n l h n l < l

I$,(y)

I S cY(y)

such that

m

T(L)

=

Cn,lXn+n(L(Xn))

for every Proof

L

i n X(E;F)

The compact s u b s e t

K

. of

E

i s c o n t a i n e d i n t h e a b s o l u t e l y con-

vex h u l l o f a n u l l s e q u e n c e , i . e . t h e r e e x i s t s

as n*

Hence

such t h a t

m

(Xn)n=lCE

where

Xn+o

Chapter 1

38

Thus

I

IT&) Since

c SUP Y(LX,). n

L

E

( L ( x ~ ) ) i~s a n u l l sequence i n

X(E;F),

{ ( Y ~ ) : = ~ ; y,

E

Fy and

y(yn)

-to

as

n -1.

F.

Let

co(Fy)

=

i s a normed l i n e a r

co(Fy)

s p a c e w i t n norm

II (Y,),"=, II

SUP Y(Yn). n

=

By t h e Hahn-Banach theorem w e can e x t e n d co(Fy)

s c supy(fn)

I?'(ifn}n)l Since

m

($n)n=l

T

t o a l i n e a r functional

'? on

such t h a t

n

co(Fy)l

=

R1((F

such t h a t %rfnIn) for all

(fn)n

Hence € o r any

in L

t h e r e exist

) I )

Y where

(Fy)l C F '

E

=

f o r every

$,(y)l

s cy(y)

{fnIn

in

m

E

co(FY). R1,

y

for all

lz=llh,,\ in

F

,< 1,

and a l l

n

In= m

co(Fy).

in

Z(E;F)

we h a v e

and t h i s c o m p l e t e s the p r o o f . Proposition 1.60

space.

If

in 2("E;F) B

E

cs(F)

T

E

Let

(d("E;F)

where

K

E

F

be a Frechet space and

'

,T~)

satisfies

I

I

C

a ZocaZZy convex

IIL//B,Kn f o r every

i s an absolutely convex compact subset o f

then there e x i s t an absolutely convez compact s e t

4

K

E

and

and:

L

39

Polynomials on locally convex topological vector spaces

(a)

for every

K

T(L)

such that

Proof

Nn

a sequence i n

L

, {x

lk

,. . .,xn

k k=l

,

1"

=

A~$~(L(X ,..., x ) ) k=l lk nk

%("E;F).

in

We proceed by induction on n .

Proposition 1.59 covers the case

n=l. Assume the proposition is true f o r the positive integer n . T

E

it a ,

x(n+lE;F).

4

Define T

of lemma 1.58.

? E

on

%("E;

f(E;F))

Let

by the isomorphism, call

( % ( n E ; ( ~ ( E ; F ) , ~ O ) ) , ~ O ) and '

Hence, by induction, there exist

-n

a sequence in K

n

By proposition 1.59, there exists a null sequence in K, (yj)y=l each k there exists a sequence of scalars

and f o r

Chapter I

40

Hence

in Q ~ + ' E ; F ) .

for every L

we may reorder the above to obtain a sequence with the required properties This completes the proof. Proposition 1.61

If

E

i s a FrLehet space w i t h t h e approximation prop-

e r t y , then t h e Bore1 transform,

Proof

T

E

( p ("E)

B , i s a l i n e a r isomorphism from

Since E has the approximation property B is injective. Let

'

, T ~ )

IT(P)

and suppose c/lpIIK

for all

P

E

P ("E)

where K is an absolutely convex compact subset of E . By the polarization formula and proposition 1.60 there exist a relatively compact sequence

such that

41

Polynomials on locally convex topological vector spaces A further application of proposition 1.60 shows that

that the equicontinuous subsets of the form {P

B

is surjective and

(~("E),T~)' correspond to sets of Since (E',T~)'= E and d cK}.

@N(n(Ef,~o));IIPI/KO 8 (nE) = BE(nE) it follows, by proposition 1.48, that the closed convex hull of sets of the form U(@~(~(E~,T~)), @(("E)) E

K compact are a fundamental neighbourhood system at 0 in ( @N(n(Eq,~o];flu) c ranges over all possible sets of positive numbers.

as

K

A fundamental neighbourhood system at

0 in

( 6'(nE),~o)h

by the polars of bounded sets. Since

K compact =

U((~("E),T~)',@

U { P E P PE); KCE

("E))

closed convex hull of 0

I I P Ic IcK} ~

K compact

=

U(((P(~E),T~)',

K compact This completes the proof.

@("E))

closed convex hull of

is given

42

Chapter I

51.5

EXERCISES

The f o l l o w i n g e x e r c i s e s d e v e l o p t o p i c s which w e s h a l l e n c o u n t e r i n

l a t e r c h a p t e r s and a l s o c e r t a i n material which we d i d n o t f i n d c o n v e n i e n t t o include i n t h e t e x t . difficult.

Consequently, some o f t h e s e e x e r c i s e s a r e r a t h e r

A s e r i o u s a t t e m p t a t s o l v i n g them, w i l l , however, p r o v i d e a

good d e a l o f i n s i g h t i n t o t h e t h e o r y f y i n g n o n t r i v i a l problems.

-

even i f o n l y as a means o f i d e n t i -

For t h e r e s e a r c h worker t h e y c o u l d e a s i l y l e a d

t o new t e c h n i q u e s and worthwhile r e s e a r c h p r o j e c t s .

Starred exercises are

commented o n i n Appendix 111. 1.62

Show t h a t

Dim(E) 6 1 o r 1.63* -

If

F =

If

E,F

for all and

show t h a t

1.65

E

If P

i f and o n l y i f e i t h e r

and

i f and o n l y i f

n

E

2

are v e c t o r s p a c e s o v e r

G

Q €!fa(F;G)

that

= X:(mE;F)

m=l,m=O,

i s a n i n f i n i t e d i m e n s i o n a l l o c a l l y convex s p a c e , show t h a t

E

Fa(nE) = @("E) 1.64 -

fa(mE;F)

to}.

a"). P

E,

E

pa(E;F)

and

Q o P E(~).(E;G).

a r e l o c a l l y convex s p a c e s and

F

i s c o n t i n u o u s a t one p o i n t i f and o n l y i f

P

P

E

Pa(E;F)

show

i s everywhere

continuous. 1.66

Replace c o n t i n u o u s by hypocontinuous ( r e s p . Mackey c o n t i n u o u s ) i n

e x e r c i s e 1 . 6 5. 1.67 I f E i s a m e t r i z a b l e l o c a l l y convex s p a c e , F i s a l o c a l l y convex s p a c e and P E p a ( E ; F ) show t h a t P E. @(E;F) i f and o n l y i f I$o P

E

1.68*

Let

E

and

mapping from

E

into

and d e f i n e

Ayl by2

nomial o f d e g r e e Y1'.

. . ,yn+l

+

f o r every

@(E)

and

F

F'.

b e r e a l Banach s p a c e s and l e t F.

...

Let

Aynf(x)

Ayf(x) = f ( x + y ) - f ( x ) inductively.

i f and o n l y i f

in x

in

in

E.

Ayl

...

f

be a continuous

for all

Show t h a t

f

x,y

in

i s a poly-

A ~ ~ + ~ f =( x0 ) f o r a l l

Show t h a t t h i s r e s u l t d o e s n o t e x t e n d t o

Banach s p a c e s o v e r t h e complex f i e l d .

E

43

Polynomials on locally convex topological vector spaces Let

1.69*

be a Frgchet space and suppose

E

P

E

Pa(nE).

Show t h a t

i s continuous i f i t s r e s t r i c t i o n t o a 2nd category subset of

P

is

E

continuous.

3

endow

0=

E =

Let

1.70*

m

,&

Space of

be t h e Dirac d e l t a function a t t h e point E

(

6('a)

=*

I:=,

. €in

(ansa)

Show t h a t t h e bounded s u b s e t s of

bounded and hence deduce t h a t

on @(nJl)

Show t h a t

for a l l

n

and

@Hy(nJl I ) =

#

T~

("a

I ) .

on @(rial)

T~

Let

spaces.

m Show t h a t t h e following a r e equivalent:

+

m

(b)

PPnEl;)

(c)

!?("El;) = PHy(nEl;)

E

E;

In=, $n$n

Em

E

E

a neighbourhood that

= !?Hy(nEb)

=

lm=o Em m

and l e t

m

$1 .

Conclude t h a t for all

T~ = T~

n22.

Ei

is a basis for

admits a continuous norm,

each

Let

('

be a s t r i c t inductive l i m i t of FrGchet Monte1

(a)

($n)n=l P =

E = lim E

?.!

(!?(n$jt),~O a r)e l o c a l l y

1.73* -

m

(ansa) .6n = f m

HY ( 2 B )\

E

1.72" I f E i s a Frgchet nuclear space and ($n)n show t h a t (9n@m)n>m=l i s a b a s i s f o r ( @('E),T,).

__ 1.74

We

st

E

m

Show t h a t

T~).

a.

Let

(fm)m=2 i s a Cauchy sequence i n

and t h a t

!?(*a) ,

R.

-functions of compact support i n

with i t s usual s t r i c t i n d u c t i v e l i m i t topology.

0

8H,(2E)

f o r some

n?2,

for a l l

nEN.

where each

Em

# J,n

E

n.

for a l l

EA

and t h a t

i s a l o c a l l y convex space.

P

'6 ( 2 E )

E

V

of zero such t h a t

(6 (2E) , T ~ )

i s not complete i f

I\$nIIV<

Let

Show t h a t

i f and only i f t h e r e e x i s t s f o r each

n.

Hence show

i s a non-normed metrizable

Eo

l o c a l l y convex space.

1.75

Let

E

=

lm=lEm m

where each

Em

i s a Banach space.

P E 8("E) and f o r each p o s i t i v e i n t e g e r m l e t m m E E j and y E Ej. Show t h a t Pm E

X

lj=l

as

m

+ m

lj=m+l

Let

Pm(x+y) = P(x) ("E)

uniformly on a neighbourhood of each point of

and t h a t E.

where Pm

+

P

Chapter I

44

1.76*

If

i s a m e t r i z a b l e l o c a l l y convex s p a c e and

E

i n t e g e r , show t h a t t h e compact open t o p o l o g y on l o c a l l y convex t o p o l o g y on

If

is a positive

is t h e finest

?! ("E)

which c o i n c i d e s w i t h t h e t o p o l o g y o f

("E)

p o i n t w i s e convergence on e v e r y e q u i c o n t i n u o u s s u b s e t o f 1.77 -

n

!?

(nE).

i s a l o c a l l y convex s p a c e i n which e v e r y n u l l sequence i s a

E

i s t h e s p a c e o f F-valued sequen-

Mackey n u l l sequence, show t h a t P M ( E ; F ) t i a l l y c o n t i n u o u s polynomials from

into

E

F.

denote an uncountable s e t .

1.78 __

Let

A

k-space,

but t h a t P ( ~ ( c ~ )=

p H yn(cA

=

CA

Show t h a t

8 ("cA)

for all

M

i s not a

n.

1.79 Show t h a t t h e f o l l o w i n g two c o n d i t i o n s on a l o c a l l y convex s p a c e E a r e equivalent: (a)

e v e r y compact s u b s e t o f

(b)

(i)

E

i s s t r i c t l y compact;

e v e r y n u l l sequence i n

i s a Mackey n u l l

E

sequence ; (ii)

e v e r y compact s u b s e t of

E

is contained i n t h e

a b s o l u t e l y convex h u l l o f a n u l l sequence. If

1.80 -

i s a l o c a l l y convex s p a c e , show t h a t

E

k-space a s s o c i a t e d w i t h 1.81 If B C !? ("E;F) x

in

E.

P

E

If

E

P,("(ExF))

1.83*

F

i s a Banach s p a c e , show t h a t

s u p I P(x) I < f o r every PEB C o n s t r u c t a counterexample which shows t h a t t h i s r e s u l t i s n o t is

Let X

bounded i f and o n l y i f

T~

true for arbitrary 1.82*

(E',u(E',E)).

i s a F r e c h e t s p a c e and

E

and

E. F

are b o t h F r 6 c h e t s p a c e s o r b o t h

i s s e p a r a t e l y c o n t i n u o u s , show t h a t

P

ayF/L

Show t h a t f o r each

n

s p a c e s and

i s continuous.

b e a c o m p l e t e l y r e g u l a r H a u s d o r f f s p a c e and l e t

t h e s p a c e o f E-valued c o n t i n u o u s f u n c t i o n s on topology.

( E ' , T ~ ) is the

h(X)

be

X with t h e compact open

45

Polynomials on locally convex topological vector spaces (a)

fM ("A (XI)

(b)

PHY(*,&(X))

=

p ( n J , (X))

E = Co(T),

r

uncountable,

1.84"

If

P E !?("E;F) -,

P

from

Co(r)

P F = II ( r l ) ,

r2

Co(r2).)

r

in

rl

u n c o u n t a b l e and

and

is t h e n a t u r a l p r o j e c t i o n

(Lr2

= Lr2.

=

i s Lindelbf.

X

.r/

such t h a t

onto

{ $ J n l $E JC O ( T ) ' } s p a n s a d e n s e

Show t h a t

( C? (nCo(r) 1 , 6).

subspace of

1.85

if

i s paracompact;

show t h a t t h e r e i s a c o u n t a b l e

P(nCo(r2 ) ; F )

E

x

B ( n a ( ~ ) ) if

=

K

Let

2 L ([O,lln+l)

E

i s symmetric w i t h r e s p e c t

K

and suppose

t o its coordinates. [P(x)l(t)

Let

j' ... jol

K tl,.

=

..,tn,t)x(tl).

.. x ( t n ) d t l . .

.dtn

0

f o r every

x P

1.86"

L2([0,1]).

E

If

6

8 (nL2[o,l];L2[o,ll) i s a s e p a r a b l e H i l b e r t s p a c e and

E

t h e r e e x i s t s an such t h a t

1.87" __

x

hP(x) If

Show t h a t

in

a

P

E

@ (nE;E)

C,

a r e Banach s p a c e s , we s a y t h a t

F

weakly compact i f i t maps t h e u n i t b a l l o f compact s u b s e t o f

in

show t h a t

1x1

= 1,

T

6

llpllx.

=

and

E

I / x ( I = 1 and

E,

F.

If

T

E

6 (E;F)

E

E

(E;F)

is

o n t o a r e l a t i v e l y weakly

show t h a t t h e f o l l o w i n g a r e

equivalent (i)

T

i s weakly compact,

(ii)

T"

E

(iii) 1.88" __

8

(F';E')

(the adjoint of

A Banach s p a c e

into

that

E

i s weakly compact,

T**(E") C F . E

i s s a i d t o have t h e polynomial D u n f o r d - P e t t i s

p r o p e r t y i f f o r e v e r y Banach s p a c e E

T)

F

F

t h e weakly compact polynomials from

map weak Cauchy s e q u e n c e s o n t o s t r o n g Cauchy s e q u e n c e s .

Show

h a s t h e polynomial D u n f o r d - P e t t i s p r o p e r t y i f and o n l y i f e v e r y

Banach v a l u e d weakly compact l i n e a r mapping maps weak Cauchy s e q u e n c e s o n t o s t r o n g Cauchy sequences.

Chapter 1

46

1.89* P

E

If

P(F)

i s a n u c l e a r subspace o f a l o c a l l y convex s p a c e

F

show t h a t t h e r e e x i s t s If

P c PN(E)

(v

such t h a t

i s a Banach s p a c e , show t h a t

E

=

E

and

P.

is also a

(@,("E),IIw)

Banach s p a c e .

Let

1.91*

d e n o t e t h e t o p o l o g y on

T~

8 a (nE)

o f uniform convergence

on t h e f i n i t e dimensional compact s u b s e t s of t h e v e c t o r s p a c e E:

phism from 1.92* let

( @ a ( n E ) , ~ f ) ' onto

m

6(nE*o)

( x ~ ) ~b e= an ~ orthonormal s u b s e t o f a H i l b e r t space

be a p o s i t i v e i n t e g e r .

only if

In=1I A n I <

__ 1.93*

If

03

E;

1.94

If

= F;iEAwhere

m

and

n

1.95*

(

8f

~ ) -c0) ,

(p (",&

1.96*

(X))

X

,B)

Show t h a t

xm

E

E

and

@N(mE) i f and

.

Ei = (E',T~) i f and

which c o n t a i n s

a s a subspace

E

{@EF';$I~=O}. msn,

and

E

is a locally

( @ ( m E ) , ~ o ) i s isomorphic t o a complemented sub-

A compact Hausdorff s p a c e

closed subset of that

EL =

F

are p o s i t i v e i n t e g e r s ,

convex s p a c e , show t h a t space of

In=,A

=

and a l s o t h a t

m

o n l y i f f o r each l o c a l l y convex s p a c e

w e have

P

Show t h a t

i s a l o c a l l y convex s p a c e , show t h a t

E

and l e t

.

m

Let

E

Show t h a t t h e Bore1 t r a n s f o r m i s an a l g e b r a i c isomor-

(E*,a(E*,E)).

=

-

i s s a i d t o be d i s p e r s e d i f e v e r y

X

contains an isolated point.

If

X

i s d i s p e r s e d , show

h a s t h e approximation p r o p e r t y . ( i f , ( " ~ ~ ) ,B) =. El

/-

QE

El

A

. . . 8,E l .

n times

51.6

NOTES AND REMARKS Mathematicians began e x p l o r i n g t h e c o n c e p t s o f polynomial and holomor-

p h i c mapping in i n f i n i t e dimensions a t a t i m e when i d e a s and t h e o r i e s such a s t h e t o t a l d e r i v a t i v e , p o i n t s e t topology and normed l i n e a r s p a c e , e t c .

Polynomials on locally convex topological vector spaces

47

were either still in their infancy o r not yet discovered. Moreover, it appears that the search for fundamental concepts in infinite dimensional differential calculus stimulated much of the work which resulted in the satisfactory linear theory that we now know as functional analysis. These pioneers were motivated by many different considerations, and at times were not aware of one another's work.

We provide here a brief outline of the

early development of polynomials, a similar treatment of holomorphic functions is given in 52.6, and refer to the historical survey of A . E . Taylor [680] for further details.

It is generally recognised that the definitive step in the creation of infinite dimensional analysis was taken by V. Volterra in 1887. I n a series of notes [705,706,707,708,709], which later evolved into the book [710], he developed a theory of scalar valued differentiable functions on ,&[a,b] and obtained the following Taylor expansion [705,p.105] for the real-valued analytic function y on &[a,b]

where

@,$

E

&[a,b].

The nth term in the above expansion is an n on ,& [a,b]. mappings.

homogeneous polynomial

Volterra did not, however, specifically discuss polynomial

The next step was taken by I). Hilbert who outlined a theory of holomorphic functions in infinitely many variables at the international congress in Rome in 1908 and published his results the following year, [332]. To Hilbert, each variable was a coordinate evaluation and he used a monomial expansion with absolute convergence on polydiscs as we do in chapter 5. Each holomorphic function had, in his notation, the following Taylor series expansion

48

Chapter 1

=

cc

n l . . .nk

n 1 x1

...

x

"k k

1 I IE

\x31 i / c 3 \ ,

IC 21

I

a b s o l u t e l y on x1 $ 1 , I x2 5 , I t i s c l e a r from t h e above t h a t H i l b e r t had a d e f i n i t e

t h e s e r i e s converging

. .. .

c o n c e p t o f polynomial i n i n f i n i t e l y many v a r i a b l e s . During t h e same y e a r , 1909, M. F r g c h e t p u b l i s h e d h i s f i r s t c o n t r i b u t i o n [240] t o t h e a b s t r a c t t h e o r y o f polynomials i n i n f i n i t e l y many variables.

Motivated by Cauchy's o b s e r v a t i o n t h a t any c o n t i n u o u s r e a l

valued function

o f a r e a l v a r i a b l e which s a t i s f i e d t h e ' e q u a t i o n

f

f(x+y) - f(x) had t o have t h e form

-

f(y) = 0

for a l l

x,y

in

R

f ( x ) = Ax, h e gave a n a b s t r a c t " d i f f e r e n c e " c h a r a c t -

e r i z a t i o n o f r e a l polynomials o f one o r s e v e r a l r e a l v a r i a b l e s ( s e e e x e r c i s e 1.68).

H e t h e n used t h i s c h a r a c t e r i z a t i o n t o d e f i n e r e a l polynomials

depending on a c o u n t a b l y i n f i n i t e number o f v a r i a b l e s . was

RN

H i s domain s p a c e

and on i t h e d e f i n e d , f o r c o n t i n u i t y p u r p o s e s , a m e t r i c which

g i v e s t h e u s u a l c o o r d i n a t e w i s e convergence t o p o l o g y .

The.following year

h e used t h e same method i n [241] t o d e f i n e r e a l polynomials on , & [ a , b ] and showed t h a t a r e a l n-homogeneous polynomial

U

on t h i s s p a c e c o u l d be

r e p r e s e n t e d as (xl,

"f

where

11;~)

. . ., x n ) f (x,) . . . f (xn)dxl. . .dxn, f c &,

[a,b] ,

..

i s a s e q u e n c e o f n-homogeneous polynomials i n

independent o f

f

xl,. 'n and t h e l i m i t i s uniform o v e r t h e compact s u b s e t s of

& [ a , b ] . He a l s o showed t h a t any polynomial c o u l d be r e p r e s e n t e d a s a f i n i t e sum o f homogeneous p o l y n o m i a l s . I n a subsequent p a p e r , [243], h e o b t a i n e d a R i e s z r e p r e s e n t a t i o n theorem f o r b i l i n e a r forms on The n e x t s t e p i s due t o R . GGteaux.

,&

[a,b]

.

H e made v e r y fundamental c o n t r i b -

u t i o n s t o t h e t h e o r y o f i n f i n i t e d i m e n s i o n a l c a l c u l u s ( s e e § 2 . 6 ) , and h i s s i m p l e e l e g a n t s t y l e makes f o r v e r y p l e a s a n t r e a d i n g .

G s t e a u x ' s work

c o n s i s t s e s s e n t i a l l y o f two p a p e r s [252,253], which he w r o t e d u r i n g t h e p e r i o d 1912-1914.

H e d i e d i n 1914, and h i s r e s u l t s were e d i t e d by P . L'evy

and p u b l i s h e d i n 1919 and 1922. KN

( K = IR

or

C),

R2

Gzteaux worked o n l y on t h e s p a c e s

and g [ a , b ] .

H e noted t h a t F r e c h e t ' s d e f i n i t i o n

o f polynomial was i n a d e q u a t e f o r f u n c t i o n s d e f i n e d on v e c t o r s p a c e s o v e r

49

Polynomials on locally convex topological vector spaces t h e f i e l d o f complex numbers and proposed i n s t e a d t h a t a c o n t i n u o u s function p

P

such t h a t

f o r any v e c t o r s

degree

n.

z

P(Xz+ut) and

t

i s a polynomial o f d e g r e e

n

in

and

X

i n t h e domain b e c a l l e d a polynomial o f

H e showed t h a t h i s d e f i n i t i o n c o i n c i d e s w i t h F r & h e t ' s f o r

real

v a l u e d f u n c t i o n s of r e a l v a r i a b l e s and went on t o prove v a r i o u s r e s u l t s such as t h e r e l a t i o n s h i p between t h e homogeneous p a r t s and t h e "G$teauxq' d e r i v a t i v e s o f a polynomial - w i t h h i s d e f i n i t i o n .

The development o f t h e

concept of normed l i n e a r s p a c e and a s s o c i a t e d i d e a s between 1910 and 1925 allowed F r g c h e t t o extend h i s d e f i n i t i o n of r e a l polynomial t o a r a t h e r g e n e r a l s e t t i n g i n [244] and [246]. I n 1931-1932, A . D . Michal, a s t u d e n t o f F r g c h e t , gave a s e r i e s o f l e c t u r e s a t t h e C a l i f o r n i a I n s t i t u t e o f Technology i n which h e o u t l i n e d t h e r e l a t i o n s h i p between symmetric n - l i n e a r forms and homogeneous polynomials. T h i s r e l a t i o n s h i p had been n o t i c e d e a r l i e r f o r b i l i n e a r forms and 2-homogeneous polynomials by M . F r g c h e t [243] and R . Ggteaux [252]. F u r t h e r work on t h e d e f i n i t i o n o f polynomial between Banach s p a c e s was c a r r i e d o u t by A . D . Michal and h i s s t u d e n t s A.H.

C l i f f o r d , R.S. M a r t i n , I . G . Highberg

and A . E . T a y l o r [331,492,493,449,677].

R.S. M a r t i n , i n h i s t h e s i s [449]

proved t h e p o l a r i z a t i o n formula and I . G . Highberg [331] c l a r i f i e d t h e r e l a t i o n s h i p between t h e d i f f e r e n t d e f i n i t i o n s and showed t h a t F r 6 c h e t ' s d i f f e r e n c e method could b e extended t o t h e complex case i f one added t h e h y p o t h e s i s o f G3teaux d i f f e r e n t i a b i l i t y . W . O r l i c z [481,482],

I n d e p e n d e n t l y , S. Mazur and

e s t a b l i s h e d t h e c o n n e c t i o n between t h e n - l i n e a r

approach and t h e now c l a s s i c a l approach of F r g c h e t and G2teaux f o r real Banach s p a c e s , and proved t h e p o l a r i z a t i o n formula.

T h i s ends o u r b r i e f

s k e t c h o f t h e development o f t h e concept o f a b s t r a c t polynomial.

For t h o s e

i n t e r e s t e d , we s t r o n g l y recommend t h e o r i g i n a l s o u r c e s as i n t e r e s t i n g reading

.

We r e t u r n now t o commenting on t h e t e x t and w i l l t r y t o a t t r i b u t e

r e s u l t s t o t h e i r o r i g i n a l sources.

There a r e b a s i c a l l y t h r e e approaches

t o s t u d y i n g polynomials, by c o n s i d e r i n g r e s t r i c t i o n s t o f i n i t e dimensional spaces,by means of t e n s o r p r o d u c t s

and by u s i n g m u l t i l i n e a r mappings.

of t h e s e methods a r e u s e f u l and none should be n e g l e c t e d .

All

The r e s t r i c t i o n

method, a s a l r e a d y n o t e d , was one o f t h e o r i g i n a l methods used and r e a p p e a r s i n o u r work e v e r y so o f t e n .

The t e n s o r p r o d u c t approach i s due

t o R . S c h a t t e n [626] and A . Grothendieck [287].

I n [620], R.A.

Ryan shows

50

Polynomials on locally convex topological vector spaces

t h a t most of t h e r e s u l t s we p r e s e n t can b e o b t a i n e d by t h i s method and t h e

same approach i s a l s o t o be found i n C . P . Gupta [295,296], [214,215,217,218,220,223],

T . A . Dwyer

A.Colojoar?i [138,139] and P . Krze [402].

We

f o l l o w t h e m u l t i l i n e a r approach i n t h i s book. Theorem 1 . 7 i s due t o R.S. M a r t i n [449] and an a l t e r n a t i v e proof i s g i v e n i n L.A. Harris [310].

Example 1 . 8 i s due t o L . Nachbin [SO91 and

example 1 . 9 was d i s c o v e r e d i n d e p e n d e n t l y by O . D .

Kellogg [379], J . G . van

d e r Corput and G . Schaake [169] and S . Banach [ 4 5 ] .

The proof g i v e n h e r e

i s due t o S . k o j a s i e w i c z and can be found i n [ 7 3 ] .

Prop.ositions 1.10 and

1.11 a r e due t o L.A.

found i n t h a t a r t i c l e

Harris [310] and f u r t h e r s i m i l a r r e s u l t s may a l s o b e

and i n [316].

The f a c t o r i z a t i o n lemma (and t h e c o r r e s p o n d i n g r e s u l t f o r holomorphic f u n c t i o n s ) h a s been i m p l i c i t i n t h e works o f many a u t h o r s , e . g . A . Hirschowitz [335], C . E . R i c k a r t [605] and L . Nachbin [ 5 1 4 ] .

A system-

a t i c s t u d y o f t h i s i d e a and i t s consequences i s u n d e r t a k e n i n S. Dineen [190] and E . Ligocka [443].

A l l r e s u l t s using s u r j e c t i v e l i m i t s (see

c h a p t e r 6) depend i n some way on a f a c t o r i z a t i o n p r o p e r t y . due t o S. Mazur and W. O r l i c z [482].

Lemma 1 . 1 9 i s

I t i s a l s o a consequence o f t h e

uniform boundedness p r i n c i p l e f o r polynomial mappings on a Banach s p a c e and more g e n e r a l r e s u l t s on t h e same t o p i c can be found i n J . Bochnak and J . S i c i a k [73], P . Lelong [431] and P . Turpin [687].

J . Bochnak and

J . S i c i a k [73,74] make e x t e n s i v e u s e of t h e "Polynomial lemma o f Leja"

[424] i n p r o v i n g t h e i r r e s u l t s . P r o p o s i t i o n 1 . 2 1 i s w e l l known, and p r o b a b l y due t o J . S a b a s t i a g e Silva.

The r e s u l t of example 1 . 2 4 i s proved f o r a c o u n t a b l e d i r e c t sum of

Banach s p a c e s i n S . Dineen [185] and f o r

33 N s p a c e s

i n S . Dineen [ 1 9 4 ] .

The p r o o f g i v e n h e r e i s modeled on t h o s e g i v e n i n [185] and [194] and r e l a t e d r e s u l t s are t o b e found i n [SO].

Example 1 . 2 5 i s due t o P . J .

Boland and S. Dineen [92] and t h e p r o o f i s s i m i l a r t o t h a t o f t h e p a r t i c N u l a r case C XE") which a p p e a r s i n S. Dineen [185]. The method o f p r o o f ha5 been f u r t h e r developed by L . A . de Moraes [498] i n h e r s t u d y of holomorphic f u n c t i o n s on s t r i c t i n d u c t i v e l i m i t s . Nuclear polynomials on Banach s p a c e s were i n t r o d u c e d by C . P . Gupta

Chapter 1

51

[295,296,297] i n o r d e r t o prove e x i s t e n c e and approximation p r o p e r t i e s of c o n v o l u t i o n o p e r a t o r s on Banach s p a c e s and m o t i v a t e d L . Nachbin [508,509] t o i n t r o d u c e t h e concept o f holomorphy t y p e .

T h i s a l l o w s one t o d i s c u s s

compact, i n t e g r a l , n u c l e a r , H i l b e r t Schmidt, e t c . polynomial and holomorp h i c mappings between l o c a l l y convex s p a c e s - c o n c e p t s which have proved u s e f u l i n l i n e a r f u n c t i o n a l a n a l y s i s and i n p r o b a b i l i t y t h e o r y on l o c a l l y convex s p a c e s ( s e e appendix I ) . P.J.

Boland [79,82,83,84]

i n i t i a t e d and developed t h e t h e o r y o f

holomorphic f u n c t i o n s on n u c l e a r s p a c e s and f o r such s p a c e s n u c l e a r p o l y nomials p l a y a fundamental r o l e ( s e e c h a p t e r s 3 and 5 ) . R.A.

Recent work by

Ryan [620] h a s shown t h e i r importance f o r holomorphic f u n c t i o n s on

F r g c h e t s p a c e s w i t h t h e approximation p r o p e r t y . The s t r o n g t o p o l o g y and t h e compact open t o p o l o g y are d e r i v e d from f u n c t i o n a l a n a l y s i s and p o i n t s e t topology r e s p e c t i v e l y .

The

T,,,

topol-

ogy i s more o r less s p e c i a l t o t h e t h e o r y of holomorphic f u n c t i o n s on l o c a l l y convex s p a c e s , a l t h o u g h some l i n e a r p r o p e r t i e s o f t h i s t o p o l o g y a r e d i s c u s s e d i n R . E . Edwards [229, p.511-5131, [397, p.4001 and J . A . B e r e z a n s k i i [ 5 7 ] .

K . F l o r e t [237], G . Kb'the

This t o p o l o g y was i n t r o d u c e d by

L . Nachbin [509] and was m o t i v a t e d by r e s u l t s of A. Martineau [450,453] on

a n a l y t i c f u n c t i o n a l s ( o f s e v e r a l complex v a r i a b l e s ) s u p p o r t e d by e v e r y neighbourhood o f a compact s e t b u t n o t by t h e compact s e t i t s e l f .

I t may

a l s o b e d e s c r i b e d as t h e t o p o l o g y o f l o c a l convergence. Example 1 . 3 8 i s g i v e n i n S. Dineen [185] f o r a c o u n t a b l e d i r e c t sum o f Banach s p a c e s . s e t s of a

$ 3 2 space i s 83% s p a c e s

[50] and f o r S . Dineen,

The same r e s u l t f o r holomorphic f u n c t i o n s on open subproved by J . A .

Barroso, M . C . Matos and L . Nachbin

by S. Dineen [ 1 9 4 ] .

Example 1.39 a p p e a r s i n

[185].

I n d e a l i n g w i t h n u c l e a r polynomials and t h e Bore1 t r a n s f o r m w e have a t t e m p t e d t o c o r r e l a t e v a r i o u s r e s u l t s on n u c l e a r and d u a l n u c l e a r s p a c e s by P . J .

Boland, [79,82,83,85],

r e s u l t s on f u l l y n u c l e a r s p a c e s by P . J .

Boland and S. Dineen [go] and r e s u l t s on F r g c h e t s p a c e s w i t h t h e approxi m a t i o n p r o p e r t y due t o R . A .

Ryan [620].

T h i s l e a d s t o a more compact

t r e a t m e n t b u t e s s e n t i a l l y t h e o n l y new r e s u l t s i n t h e f i n a l two s e c t i o n s

a r e p r o p o s i t i o n s 1.44 and 1 . 4 8 .

D e f i n i t i o n 1 . 4 0 ( a ) i s due t o P . J .

Boland

52

Polynomials on locally convex topological vector spaces

[ 7 9 ] . The fundamental i n e q u a l i t y needed i n d e f i n i t i o n 1 . 4 0 ( b ) i s a l s o due t o Boland [87]. The d e f i n i t i o n i s , however, new a l t h o u g h s p e c i a l c a s e s have p r e v i o u s l y been c o n s i d e r e d by C . P . Gupta [295] and R . A .

Ryan [620].

P r o p o s i t i o n 1 . 4 1 , c o r o l l a r y 1 . 4 2 and theorem 1 . 4 3 are due t o P . J . Boland [79,83].

P r o p o s i t i o n 1.44 i s proved f o r f u l l y n u c l e a r s p a c e s i n [ 8 7 ] .

The Bore1 ( o r F o u r i e r - B o r e l ) t r a n s f o r m was f i r s t used i n i n f i n i t e dimensional holomorphy by C . P . Gupta [ 2 9 5 ] .

Subsequently, i t h a s been

a p p l i e d by v a r i o u s a u t h o r s , see c h a p t e r s 3 , 5 , 6 and appendix I , i n t h e s t u d y o f c o n v o l u t i o n o p e r a t o r s and d u a l i t y t h e o r y .

Proposition 1.47 i s

due t o C . P . Gupta [295] f o r Banach s p a c e s and t o P . J . r e f l e x i v e n u c l e a r and d u a l n u c l e a r s p a c e s .

Boland [79] f o r semi-

Other p a r t i c u l a r cases o f

p r o p o s i t i o n 1 . 4 7 and 1.48 can b e found i n R . A . Ryan [ 6 2 0 ] .

Fully nuclear

s p a c e s ( d e f i n i t i o n 1 . 4 9 ) were d e f i n e d by P . J . Boland and S. Dineen [go] and f i g u r e p r o m i n e n t l y i n c h a p t e r s 5 and 6 .

C o r o l l a r y 1 . 5 0 and lemma 1.55

a r e proved f o r f u l l y n u c l e a r s p a c e s w i t h a b a s i s i n P . J . Boland and S. Dineen [go] and t h e l a t t e r r e s u l t h a s r e c e n t l y been extended t o q u a s i complete d u a l n u c l e a r s p a c e s by J . F . Colombeau, R . Meise and B. P e r r o t

Lemma 1.54 i s due t o S. Dineen [190].

[153].

P r o p o s i t i o n s 1 . 5 6 and 1.57 a r e due t o P . J . Boland and S . Dineen [go] when

E

h a s a b a s i s , and t o P . J .

Boland [87] i n t h e g e n e r a l c a s e .

p r o o f g i v e n h e r e o f p r o p o s i t i o n 1.57 i s new. 1 . 5 9 , 1.60 and 1 . 6 1 are a l l due t o R.A.

Ryan.

The

Lemma 1.58 and p r o p o s i t i o n s They appeared i n a p r e l i m -

i n a r y d r a f t o f h i s t h e s i s [620], b u t were r e p l a c e d by more e l e g a n t and p e r h a p s s l i g h t l y less g e n e r a l r e s u l t s i n t h e f i n a l v e r s i o n .

Chapter 2 HOLOMORPHIC MAPPINGS BETWEEN LOCALLY CONVEX SPACES

I n t h i s c h a p t e r , we g i v e t h e v a r i o u s d e f i n i t i o n s o f holomorphic mappi n g s between l o c a l l y convex s p a c e s , which we s h a l l u s e as well as t h e d i f f e r e n t t o p o l o g i e s on t h e s e s p a c e s o f mappings.

I n many c a s e s , i t i s o n l y

n e c e s s a r y t o c o n s i d e r Banach s p a c e v al u ed mappings and t h e r e s u l t s may be extended t o a r b i t r a r y v e c t o r v al u ed mappings q u i t e e a s i l y . The compact open t o p o l o g y and t h e

T

w

t opology e a s i l y e xte nd from

polynomials t o holomorphic f u n c t i o n s on l o c a l l y convex s p a c e s , b u t t h e s trong topology does n o t g e n e r a l i s e i n a s u i t a b l e fashion.

While t h e

T~

t o p o l o g y p l a y s a n i m p o r t an t r o l e i n o u r s t u d y , it doe s n o t , i n g e n e r a l , have good t o p o l o g i c a l p r o p e r t i e s .

We i n t r o d u c e t h e

-r6

topology which may b e

d e s c r i b e d as t h e t o po l o g y s u p p o r t ed by t h e c o u n t a b l e open c o v e r s .

We prove

e le m e n t a r y p r o p e r t i e s o f t h e s e t o p o l o g i e s and g i v e some sim ple examples.. The s i g n i f i c a n c e o f t h e s e examples and counterexamples i s c l a r i f i e d i n l a t e r chapters.

The r emai n i n g c h a p t e r s a r e d evote d t o a d e e p e r s t u d y o f

t h e s e t o p o l o g i e s on c e r t a i n classes of l o c a l l y convex s p a c e s and on s p e c i a l domains.

W e a l s o d e f i n e germs o f holomorphic f u n c t i o n s .

These are o f

i n t r i n s i c i n t e r e s t and a l s o p l a y a r o l e i n d u a l i t y t h e o r y . 52.1

G ~ T E A U X HOLOMORPHIC MAPPINGS

De finit i on 2.1 A subset U of a vector space E i s said t o be f i n i t e l y open i f UnF i s an open subset of t h e Euclidean space F f o r each f i n i t e dimensional subspace F of E. The f i n i t e l y open s u b s e t s o f ogy

tf.

The b a l a n c ed

neighbourhoods o f z e r o .

tf

E

define a translation invariant topol-

neighbourhoods o f z e r o form a b a s i s f o r t h e

On a l o c a l l y convex s p a c e

(E,T),

t h a n any o f t h e t o p o l o g i e s we have p r e v i o u s l y c o n s i d e r e d on Tf

3 TM 3 Tk 3 T.

53

T~

E,

is f i n e r i.e.

tf

Chapter 2

54 Pefinition 2.2

A function E

a vector space Gateaux or

f

defined on a f i n i t e l y open subset F

with values i n a l o c a l l y convex space

-

b EE

aE U,

G-holomorphic i f for each

of

i s said t o be

Q, E F '

and

U

t h e complex

valued f u n c t i o n of one complex variable A

$of(a+Xb)

i s holornorphic i n some neighbourhood of zero. s e t of a l l

G-holomorphic mappings from

U

HG(U;F)

We l e t

into

denote t h e

F.

Hartog's theorem i n f i n i t e dimensions says t h a t s e p a r a t e l y holomorphic f u n c t i o n s on

UxV

(UCK", V C C m )

a r e holomorphic.

Hence

f:UCEjF

i s G-holomorphic i f and only i f @ o f ) U n G i s a holomorphic function of

Q, i n

s e v e r a l complex v a r i a b l e s f o r each subspace

G

of

F'

and each f i n i t e dimensional

Consequently, one can u s e any of t h e equivalent

E.

f i n i t e dimensional conditions (e.g. Taylor s e r i e s expansions, Cauchy Riemann equations, e x i s t e n c e of t h e t o t a l d e r i v a t i v e ) i n t h e d e f i n i t i o n of G-holomorphicity. A t t h i s s t a g e , one may wonder why we demanded a l o c a l l y convex range

space i n d e f i n i t i o n 2 . 2 .

a:

v e c t o r space over

We could a l s o g i v e an analagous d e f i n i t i o n with a However, w e would then run i n t o

a s t h e range space.

d i f f i c u l t i e s i n showing t h a t a G2teaux holomorphic function has a Taylor s e r i e s expansion about each p o i n t s i n c e t h i s r e q u i r e s a convergence s t r u c t u r e on t h e range space and t h e f i n i t e open topology on

i s so f i n e t h a t

Even with a l o c a l l y

very few f u n c t i o n s would have t h e d e s i r e d expansion. convex range space w e s t i l l have t o be c a r e f u l .

F

We f i r s t show t h a t G-holo-

morphic functions a r e continuous f o r t h e f i n i t e open topology. Lemma 2 . 3

I f

E

i s a v e c t o r space,

i s a l o c a l l y convex space and

i s given t h e f i n i t e open topology. I t i s e a s i l y seen t h a t

i s a f i n i t e l y open subset of

f &HG(U;F)

F U

Proof

U

tf

then

f

i s continuous when

i s t h e inductive l i m i t topology, i n

t h e category of t o p o l o g i c a l spaces, given by t h e i n c l u s i o n mappings where

G

function

ranges over a l l f i n i t e dimensional subspaces of f

defined on a

tf

open subset

U

of

E

E.

G-E

Hence a

i s continuous i f and

only i f i t s r e s t r i c t i o n t o t h e f i n i t e dimensional s e c t i o n s of uous.

E,

U

a r e contin-

Since an a n a l y t i c f u n c t i o n of s e v e r a l complex v a r i a b l e s i s continuous

t h i s completes t h e proof. A

We now look a t Taylor s e r i e s expansions of Gateaux holomorphicfunctions.

Holomorphic mappings between locally convex spaces

55

If U i s a f i n i t e l y open subset of a vector space E, F i s a ZocaZly convex space and fEHG(U;F) then f o r each 5 E U there Proposition 2.4

e x i s t s a unique sequence of homogeneous polynomiak from

E into

A

F,

such t h a t

f o r all y i n some

Proof

where

Let

p,

P (x) m,S,@

neighbourhood of zero.

tf

E E U be fixed.

For each positive integer m,

@

EF' and

is chosen so that is independent of

By lemma 2.3, f is continuous and we may use Riemann's definition of the integral to define Pm,E(X)

=

dh .

The limit (of the Riemann sums) may not exist in F

but will always exist

A

in F , the completion of F. It will lie in F if F is sequentially complete o r if the closed convex hull of each compact subset of F is compact. F o r this reason, we sometimes place a completeness condition on the range space. Since f is continuous, Pm,S,@ (XI all 6 and x.

=

@(Pm,S(x))

for

Since the restriction of @of to any finite dimensional section of U is a holomorphic function of several complex variables, it follows that the function x -+ P (x) lies in @a(m~> f o r every positive integer m m,S,@ and each @ in F'. Let L be the associated symmetric m-linear m,S,@ mapping on E.

56 If

Chapter 2 xl,

...,xm

L

: Em

Let

m, 5

t h e n by t h e p o l a r i z a t i o n formula (theorem 1 . 5 )

E

E

-

h

F

b e d e f i n e d by 1

E

By t h e Hahn-Banach theorem Let

1

Lm, 5

i z (mE;F)

El...€

and

P n m,{

(1"i = lE.X.) 1 1

= A Lm,< E $,("E;?).

Pm,<

b e a t f - b a l a n c e d neighbourhood o f z e r o such t h a t

V

example, t a k e

V

{ x E E ; S + A X E U , I A ~ & ~ II)f.

=

a compact s u b s e t o f

S+{Ax;IA/&p} C U .

such t h a t

B ECS(F) let

If

Y E F ~ . By l e m m a 2 . 3

p>l

B = { $ E F ' ; l$(y)(sB(y) B i s c o n t i n u o u s and w e have

f

sup B(f(S+Ax))

sup

=

I$of(S+Ax)

I A I&P,OEBB

I A l W

I

(for

S + { X X ; I A ~ S ~i}s

then

XEV

and hence t h e r e e x i s t s

U

<+VCU

<

=

m

for all

.

Hence

f o r every p o s i t i v e integer

Hence f(S+x)

=

LmZo m

m.

T h i s shows t h a t

P

(x)

m, 5

f o r every

x

in

V.

Using t h e u n i q u e n e s s o f T a y l o r series expansions i n one complex v a r i a b l e we

57

Holomorphic mappings between locally convex spaces

see t h a t t h e sequence

m

i s u n i q u e l y d e t e r m i n e d by

('m, S)m=O

The f i n i t e d i m e n s i o n a l t h e o r y a l s o shows t h a t partial derivative of

f

at 5

i n the direction

f.

is the

Pm,s (x)

m

th

x, -and following

c l a s s i c a l terminology, P

The c o r r e s p o n d i n g

m, 5 m

m! l i n e a r form

Our expansion now becomes

L

i s d e n o t e d by

=

'mf(S) ___ (x) m!

m, 5

f(S+x)

proof.

.

dmf(S) ~

m!

T h i s completes t h e

I n p r o v i n g p r o p o s i t i o n 2 . 4 , w e h a v e a l s o shown t h e f o l l o w i n g : P r o p o s i t i o n 2 . 5 (Cauchy i n e q u a l i t i e s )

i s a balanced subset of

B

and

cs(F)

in

If f

such t h a t

and every non-negatiue integer

HOLOMORPHIC

12.2

E

Definition 2.6

MAPPINGS

Let

E

BETWEEN

and

a f i n i t e l y open subset of E.

F

HG(U;F),

E

c+pBCU

LEU;

p

LOCALLY

A function

5

in

~A { O }

m

CONVEX

SPACES

be locally convex spaces and l e t

if it i s G-hoZomorphic and f o r each

E

then for every

f:U U

+

F

U

be

i s calZed hoZornorphic

t h e function

converges and defines a continuous function on some r-neighbourhood of zero. We Zet U

into

H(U;F)

denote t h e vector space of a12 hoZomorphic functions from

F.

We u s u a l l y c o n s i d e r f u n c t i o n s d e f i n e d on open s u b s e t s o f

E

and i n

t h i s case, b e c a u s e o f t h e u n i q u e n e s s o f t h e T a y l o r series expansion and t h e

f a c t t h a t t h e f i n i t e open neighbourhoods of z e r o a r e a b s o r b i n g , a G-holomor-

Chapter 2

58

p h i c f u n c t i o n i s holomorphic i f and only i f it i s c o n t i n u o u s . The f o l l o w i n g o b s e r v a t i o n i s e a s i l y proved and f r e q u e n t l y a p p l i e d . Lemma 2 . 7

U

If

is an open subset of a ZocalZy convex space f

a Zocally convex space and naof

E

for every

H(U;Fa)

A continuous fu n ct i o n

HG(U;F)

E

f

then

E

i s bounded).

f

is

i f and only i f

w i t h v a l u e s i n a normed l i n e a r spa c e i s

f

l o c a l l y bounded ( i . e . each p o i n t i n t h e domain o f whose image under

H(U;F)

F

cs(F).

in

a

E,

f

h a s a neighbourhood

The co n v erse i s f a l s e i n g e n e r a l b u t it

i s t r u e f o r G-holomorphic f u n c t i o n s as o u r n e x t r e s u l t shows.

Lemma 2 . 8

U

I f

i s an open subset of a Zocally convex space

a normed Zinear space a d

f

E

HG(U;F)

f

then

E

H(U;F)

E,

F

i f and only if

is f

i s ZocaZZy bounded. Let

Proof

5

E

U

be a r b i t r a r y .

hood of z e r o such t h a t

5+VCU

By p r o p o s i t i o n 2 . 5 , f o r a l l

If

5,+5

a > a

.

as

a + -

m

and and

Choose f(5+V) O < S <1,

t h e n we can ch o o s e

Hence we h av e

T h i s completes t h e p r o o f .

V

ct 0

a convex ba la nc e d neighbouri s a bounded s u b s e t o f

F.

w e have

such t h a t

5-5,

E

6V

for all

59

Holomoiphic mappings between locally convex spaces S i n c e e v e r y l o c a l l y bounded polynomial i s c o n t i n u o u s , we a l s o have shown t h e f o l l o w i n g : C o r o l l a r y 2.9

E

Let

and E

U

be an open subset o f

U

and every p o s i t i v e i n t e g e r

F

be arbitrary ZocaZZy convex spaces. f

and suppose

is ( m ~ F; ) .

Zmf(E)

m,

E

8(mE;e)

E

A

Corollary 2.10

Let

E

be open i n

f

and

E

F

and

E

H(U;F).

Let 5

Then f o r every dmf(E)

and

E

be arbitrary ZocaZly convex spaces,

HG(U;F).

f

Then i f

in

U

i s ZocaZly bounded i t Zies i n

H(U;F). Apply lemmata 2.7 and 2.8

Proof

H

We Zet

LB

(U;F)

denote the vectop space o f a l l G-hoZomorphic ZocalZy

bounded mappings defined on t h e open subset U E

with vaZues i n t h e ZocaZZy convex space

MLB(U;F) C H ( U ; F )

f o r any

and

U,E

o f t h e ZocaZZy conuex space We have j u s t s e e n t h a t

F.

We now look a t t h e r e v e r s e

F.

inclusion.

Lemma 2 . 1 1

is an open subset of a ZocaZZy convex space E and f o r every ZocaZZy convex space F then E i s a normed U

If

H(U;F) = H LB (U;F)

l i n e a r space.

I t suffices t o take

Proof

i d e n t i t y mapping (from

F=E

and to n o t e t h a t t h e r e s t r i c t i o n o f t h e

into itself) t o

E

l o c a l l y bounded i f and o n l y i f

U

i s always c o n t i n u o u s b u t i s

i s a normed l i n e a r s p a c e .

E

There are, however, s e v e r a l n o n - t r i v i a l examples o f p a i r s o f s p a c e s and

F

i n g s from

into

E

F

coincide.

One can o b t a i n some i n f o r m a t i o n a b o u t t h i s

problem by e x t e n d i n g H a r t o g s ' theorem t o l o c a l l y convex s p a c e s .

f;U+G

E

for w h i c h , t h e holomorphic and t h e l o c a l l y bounded holomorphic mapp-

where

U

i s an open s u b s e t o f

ExF,

E,F

convex s p a c e s , i s separateZy hoZomorphic i f f o r each y + f(x,y)

i s holomorphic and f o r each

i s holomorphic.

y

in

F

and

G

x

in

A function

being l o c a l l y E

the function

t h e function x

+

g(x,y)

H a r t o g s ' theorem i n s e v e r a l v a r i a b l e s i m p l i e s t h a t s e p a r -

a t e l y holomorphic f u n c t i o n s a r e G-holomorphic. P r o p o s i t i o n 2.12

Let

E

and

F

be ZocaZZy convex spaces and suppose

60

Chapter 2

every separately holomorphic function defined on an open subset of i s holomorphic.

Proof f(x,$)

Let =

Then HLB(U;F) = H(U;F)

f

E

@(f(x)).

H(U;F).

We define

f o r every open subset

f; UxFb

+

B:

ExF'

U of

B E.

by the formula

f is obviously separately holomorphic and by our hypoth-

esis, it is holomorphic. Since Q: is a locally compact space f is a locally bounded function. Hence, if 5

E

U, we can find a neighbourhood of

5, VE, and a neighbourhood of zero in Fb, Bo,

where

B

is a bounded

Hence sup \@(f (x)) I < m xev for every $ in F' and by Mackey's theorem f(Vg) isc a bounded subset of F. This complets the proof. subset of F,

Example 2.13

Bo such that \ \ f \ \ v E

=

M <

-.

If E is a Frgchet space and

F

is a DF

space (the

strong dual of F is a Frgchet space), then HLB(U;F) = H(U;F) open subset U

for any

of E. This follows from proposition 2.12 since it is

known that separately holomorphic functions defined on open subsets of the product of Frgchet spaces are holomorphic. Example 2.14 If E is a $ 3 2 space and F is an 3 d space then HLB (U;F) = H(U;F) for any open subset U of E. This also follows from proposition 2.12 and the fact that separately holomorphic functions defined on open subsets of a product of $ 3 2 spaces are holomorphic. We do not know if the same result holds for

a3@ spaces although we

do know that separately continuous polynomials defined on a product of PTkyL spaces are continuous. We now look at functions which are holomorphic analogues of the hypocontinuous and the M-continuous polynomials defined in section 1.2 Definition 2.15

A function

ZocalZyconvex space

f defined on an open subset

U of a

E

with values i n a l o c a l l y convex space F i s said it i s G-holomorpkic and continuous on the compact subsets of E. We l e t HHY(U;F) denote the vector space of a l l hypoanalyt i c mappings from U i n t o F. t o be hypoanalytic i f

Some authors give a slightly more general definition of hypoanalytic functions - they consider functions which are G-holomorphic and bounded on the compact subsets of U. The following example shows that this can lead to a strictly larger class of functions.

61

Holomotphic mappings between locally convex spaces Example 2.16 (en)n

E

Let

be an i n f i n i t e dimensional H i l b e r t space and l e t

be a sequence of mutually orthogonal u n i t v e c t o r s i n

I : (E,u(E,E'))

+

/I 11)

(E,

E.

Then

maps compact s e t s onto bounded s e t s b u t i s not

hypoanalytic s i n c e t h e sequence

(en)n

i s weakly b u t not s t r o n g l y

conver-

convergent. If a

E

i s a k-space ( i n p a r t i c u l a r , i f

,33fM,space)

then

HHY(U;F)

f

A function

l o c a l l y convex space

i s metrizable o r i f

E

f o r any open subset

U

of

E

is

E

and

F.

any l o c a l l y convex space Definition 2.17

H(U;F)

c

defined on a

T~

open subset

of a

U

i s said t o be Mackey or S i l v a holomorphic i f

(EJ)

i t i s G-holomorphic and M-continuous. We l e t space of a l l Mackey holomorphic mappings from

%(U;F)

denote t h e vector

U

F.

into

The following r e s u l t gives an a l t e r n a t i v e d e f i n i t i o n of Mackey holomorphic f u n c t i o n s .

We omit t h e p r o o f .

Proposition 2.18

U

Let

be a

open subset of the l o c a l l y convgx

T~

space E and l e t F be a 1ocaZZ.y convex space. following are equivalent:

b)

for each 5 exists E>O

U

f(t+EB)

such t h a t

5

f o r each

E

U

g

and

E

If H(U;F)

(E,T) f o r any

H(V;E),

rdmf(5)

where

In p a r t i c u l a r , space o r a holds f o r

838

C

i s a superinductive space then , T

open subset

%(U;F)

= H(U;F)

space and

F

U

if

of

U

H (U;F) G

then the

B

V

and g(O)= 5, the f u n c t i o n on some neighbourhood of zero i n CC.

of zero i n

E

of E there i s a bounded subset of F,

and each bounded subset

5 E U and each m i n N ,

c ) for each d)

E

If f

E

E

@,("E;?).

i s a neighbourhood fog

T ~ = T

i s hoZaorphic

and hence

%(U;F)

and any l o c a l l y convex space

F.

is an open subset of a Frzchet

is arbitrary.

We do not know i f t h i s r e s u l t

3% spaces.

There are s e v e r a l o t h e r t y p e s of holomorphic f u n c t i o n s t o b e found i n the literature.

=

We s h a l l introduce them i f t h e need a r i s e s .

Our main

Chapter 2

62

i n t e r e s t l i e s i n t h e s t u d y o f holomorphic f u n c t i o n s and a l l o t h e r f u n c t i o n s p a c e s are i n t r o d u c e d s o l e l y t o h e l p o u r s t u d y i n t h i s d i r e c t i o n .

The d i f f -

e r e n t k i n d s of holomorphic f u n c t i o n s we have d e f i n e d s a t i s f y t h e f o l l o w i n g inclusions. E

Let E.

open subset of

and

F

U

be l o c a l l y convex spaces and

an

The following inclusions hold

An i m p o r t a n t q u e s t i o n which w i l l a r i s e i n t h i s book and which i s s t i l l undergoing a c t i v e r e s e a r c h i s t h e f o l l o w i n g :

f o r what

t h i s q u e s t i o n f o r polynomial mappings.

U,E

and

F

are

We have a l r e a d y looked a t

some (or a l l ) o f t h e above i n c l u s i o n s p r o p e r ?

For t h e moment, w e c o n s i d e r o n l y a

few s i m p l e examples. Example 2.19 %(U;F)

= H

If

(U;F)

has property

E

(s)

f o r any open s u b s e t

HY p o l y n o m i a l s i n c h a p t e r 1 can b e e x t e n d e d .

Example 2 . 2 0 of

A G-holomorphic f u n c t i o n

with val ues i n

E

and

F

of

E.

U

f

i s a r b i t r a r y , then The p r o o f g i v e n f o r

d e f i n e d on a n open s u b s e t

U

i s hypoanalytic i f e i t h e r of t h e following condit-

F

ions hold: a)

f i s bounded on compact sets and "m d f ( c ) E 8Hy(mE;F) f o r e v e r y 5 i n and e v e r y p o s i t i v e i n t e g e r

b)

s e p a r a b l e and criterion (i.e.

Since

E

(E,o(E,E'))

T

and

T~

d e f i n e t h e same conver-

E).

i s l o c a l l y convex and h y p o a n a l y t i c i t y i s a l o c a l prop-

e r t y , w e may suppose w i t h o u t l o s s o f g e n e r a l i t y t h a t anced and t h a t a)

Let

F

K C U

Hence there e x i s t s

is

s a t i s f i e s t h e Mackey convergence

E

g e n t sequences i n Proof

m.

i s bounded on compact s e t s ,

f

U

i s convex and b a l -

U

i s a normed l i n e a r s p a c e .

b e compact. X>1

such t h a t

We may suppose t h a t

XKCU.

K

i s balanced.

The Cauchy i n e q u a l i t i e s imply

63

Holornotphic mappings between locally convex spaces

lim sup( 1 mn! 3 f(0) By hypothesis ___ E PH,,(("E;F) and hence f is the uniform limit on K n! of a sequence of continuous polynomials. Thus f is continuous on K and

that

f

E

HHY(U;F). b)

By (a) we may suppose that

Since E

f is an m-homogeneous polynomial.

is weakly separable, the compact subsets of E are metrizable and

hence it suffices to show that f is sequentially continuous. Let be a null sequence in E . such that

(An\

There exists a sequence of scalars,

and

+ +m

m

(x~)~=~

Anxn

+

as n-.

0

Since f i s bounded on com-

pact sets

I An(

is a bounded subset of F. Now n

-t m

+ a

and this implies f(xn)

+

0 as

and completes the proof.

We now look at holomorphic versions of the Factorization Lemma proved for polynomials in the first chapter. The situation is much more complicat-

ed due to the fact the polynomials are always defined on the entire space and continuity at a single point implies continuity at all points. These properties are not necessarily true of arbitrary G-holomorphic functions. Here the topological and geometric properties of the set U and the global continuity properties of the function f have to be taken into consideration. E

Let

Theorem 2.21

be a l o c a l l y convex space and l e t

F be a normed

U i s a connected open subset of E and f E H(U;F) then there e x i s t s an a E cs(E) such t h a t f o r any XEU, YEE for which Zinear space.

If

a(y) = 0 and Ix

ue have

+

f(x+y) Proof

OCX,
xy; =

(*I

f(x). is convex and balanced. Condition (*)

We first suppose that U

is then satisfied if x and x+y

E

U.

Since F

is a normed linear space

there exists an a in cs(E) such that Ba = IxEE;~(x)< 11 C U and = M <=. By the Cauchy inequalities it follows that Ba

64

Chapter 2

"m

Ild)l/

m! Ba ials we have

E

U

for all m

in E

for every x x,x+y

M

6

a(y) = 0

and

"m

m

f(x+y)

=

and all

Cm=o

y

in E

for which a(y)

=

0. Hence if

then

f(o)

m!

and by the factorization lemma for polynom-

(x+y) = 1" m=O

This completes the proof when U

!L@l (x) m!

=

f(x3.

is convex and balanced. For arbitrary U

we choose S E U and V a convex balanced open set such that S + V C U . By the above there exists an a in cs(E) such that if x,y E E , x E V , x+y E V and a(y) = 0 then f(S+x) = f(€,+x+y). Moreover, if x,y E E , XEV, S+x+y E U, a(y) = 0 and (*) function of one complex variable

-

A

f(S+x+Ay)

is satisfied, then we may consider the

- f(S + X I

This function is constant, by the above, on some neighbourhood of zero, and hence it is constant on the connected interval

[0,1]. Hence

f(F+x+y)

=

f(S+x). Let Uo = {xsUlif YEE, a(y) = 0 and (*) is satisfied then f(x) = f(x+y)l. We have just shown that Uo is a nonempty open subset of U.

If xg

E

Uo

+

x

E

U

as

8

+

-, YEE and

{x+AylObh$lj C U , then)

since E i s a topological vector space, x8+Ay E U for all 9 sufficiently large and all A E [0,1]. Hence f(x+y) = lim f(xe+y) = lim f(x8) = f(x)

e-

Thus Uo is a non-empty open and closed subset of U . ected, this implies U=Uo and completes the proof.

e-

Since U

is connru

Our next step motivated by the polynomial case would be to define f on n a ( U ) . There are, however, several difficulties which cannot be surmounted without certain modifications. Without condition

(*)

N

f may not

be well defined.

It is possible to surmount this problem, in the general situation, by considering domains spread over E and using a pullback operator or by restricting oneself to pseudo-convex open sets. To simplify our presentation, we confine ourselves to convex open sets. A second difficulty arises from the fact that ila(lJ)

is not necessarily an open subset of E

65

Holomorphic mappings between locally convex spaces However,

w i l l always b e a

17 (U)

we c a n a s k , assuming

tf

and c o n s e q u e n t l y Ea N i s w e l l d e f i n e d , whether o r n o t f i s a holomor-

N

f

open s u b s e t of

N

The s e t o f p o i n t s o f c o n t i n u i t y o f

phic function.

may n o t b e a l l o f

na(U).

f

w i l l b e nonempty b u t

Example 2 . 2 2 i l l u s t r a t e s t h i s d i f f i c u l t y .

d i f f i c u l t y i s overcome by p l a c i n g e x t r a c o n d i t i o n s e i t h e r on

na

mappings

Example 2.22

We d e n o t e by

H(C)

t h e s p a c e o f e n t i r e f u n c t i o n s of one

Let

complex v a r i a b l e endowed w i t h t h e compact open t o p o l o g y . b e d e f i n e d by f u n c t i o n on

in

a

subset of

H(C)cl

f o r any

cs(H(C))

a

E

such t h a t

suplf(z)l

=

cs(H(E)). B

fn(z) fn(0) = 0,

where

I z1sR

n 2

=

Since If

O t h e r w i s e , t h e r e would

is a closed

z+z

=

R > 2.

let

+...

+ z

(2R) + . . . +

2 fn(2R) = 4R - 2R

n

(2R)

.

n

2

.

(Z

for all n

2

-2R)

n +

and

-.

F ( f + f n ) = ( f + f n ) ( f ( 0 ) + f n ( O ) ) = f (2R)+fn(2R) = 2 F ( f ) = f(2R) = 2R-4R # 0 w e h a v e shown t h a t no such a e x i s t s .

f(z)

U

2R-z

Then

i s convex and b a l a n c e d (or even pseudo-convex)

o f Theorem 2 . 2 1 i s s a t i s f i e d f o r a l l

(*)

C

i s an e n t i r e

{fEH(C);F(f) = 0 )

=

as

0.

+.

Without l o s s o f g e n e r a l i t y we may assume

For e a c h p o s i t i v e i n t e g e r

Let

F

F!H(C)

does n o t f a c t o r as a holomorphic

F

We claim t h a t

H(E),. a(f)

Then

I t is e a s i l y seen t h a t

F(f) = f ( f ( 0 ) ) . H(C).

func ti on through e x i s t an

Ea

and w e g i v e v a r i o u s s u f f i c i e n t c o n d i t i o n s .

This

o r on t h e

x

and

then condition

and we o b t a i n t h e

y

following f a c t o r i z a t i o n r e s u l t . Proposition 2.23 f

E

Proof

be a l o c a l l y convex space and l e t

F

be a

is a convex balanced open subset of E and r) then there e x i s t s an a i n c s ( E ) and f E HG(na(U);F) such

H(U;F)

that

E

Let

normed l i n e a r space.

If

U

J. .

f = fona.

We d e f i n e

ru

f

on

na(U)

by

.V

f(x) = f(2)

if

2E

U

and

66

Chapter 2 rrl

n,(x)

Theorem 2 . 2 .

= x.

f = f e n . Since f .-./ HG(na(U);F). N

shows t h a t

f"

i s w e l l d e f i n e d and by c o n s t r u c t i o n

i s a G-holomorphic f u n c t i o n , it f o l l o w s t h a t

W e now g i v e a s u f f i c i e n t c o n d i t i o n f o r t h e c o n t i n u i t y o f P r o p o s i t i o n 2.24

a directed s e t

If D

f

cs(E)

i s a ZocaZZy convex space and

E

4

of semi-norms which d e f i n e the topology of a in D then

contains

E

nO1

and

i s an open mapping f o r every

li.e.

if

such t h a t

f

E

H(U;F)

then there e x i s t s an

f = faofla)

to lie in

D.

Hence

na(U)

there exists a and

since

Ra

llfll

B

i s a n open s u b s e t o f

HG(na(U);F)

6

l o c a l l y bounded s i n c e

C U

fa

E

U

of

H(TIu(U);F) E

and

We first n o t e t h a t i n t h e p r o o f o f theorem 2 . 2 1 w e may choose

p r o p o s i t i o n 2.23, f a 0

and

F.

any normed l i n e a r space Proof

D

in

01

for any convQx balanced open subset

f

D m.

and

such t h a t

p>O

Hence

w

Ilfa//

BB,s,P

i s open, i t f o l l o w s t h a t

fa

and t h e r e e x i s t s , by

Ea

Now

= f O 1 olla.

i s a normed l i n e a r s p a c e .

F

in <

such t h a t

f

is

Hence f o r each

BB,S,P

d B B , S,P

a

) =

=

5 in IxEE; B(5-X)

IlfiiB

10

and,

B,S,P

i s l o c a l l y bounded and so b e l o n g s

t o H(nO1(U);F). T h i s c o m p l e t e s t h e p r o o f s i n c e t h e o p p o s i t e i n c l u s i o n i s obvious. The above p r o p o s i t i o n c o v e r s t h e case where each s p a c e and y i e l d s t h e f o l l o w i n g examples.

Let

Example 2 . 2 5

nnZl m

Eu

i s a Banach

En where each En i s a Banach s p a c e . n EO1 = II. E . f o r a l l n . I f f E H(E;F) and F i s a normed 1=1 I l i n e a r s p a c e , then t h e r e e x i s t s a p o s i t i v e i n t e g e r n and f n E H(E ; F ) an such t h a t f = f n o Rn where IIn i s t h e c a n o n i c a l p r o j e c t i o n o f E o n t o E

=

Let

Ea

n

.

As a p a r t i c u l a r example, we see t h a t

u ,

H((CN)

=

I

neN H((Cn).

67

Holomotphic mappings between locally convex spaces

Let X be a completely regular Hausdorff topological space

Example 2.26 and let E

=

& (X).

Then we can choose our directed set D

such that

Ea Y &,(K), K compact in X, for each a in D . Hence each E, is a Banach space and we obtain a factorization result for normed linear space valued holomorphic functions defined on convex balanced open subsets of

.

&(XI

Ea, is given in the following proposition. This condition also arises in various other A further sufficient condition, this time on the spaces

parts of infinite dimensional holomorphy, for instance analytic continuation and we show (theorem 2.28) that it is satisfied by any Banach space.

In

fact, we prove a more general result which we shall use later. This, then, I n more general

gives an alternative proof for examples 2.25 and 2.26.

factorization theorems it is applied to prove results unobtainable by using proposition 2.24. Let

E be a l o c a l l y convex space and l e t F be a normed linear space. I f there e x i s t s a directed family D of semi-norms which define t h e topology of E and each a i n D s a t i s f i e s the following condition: Proposition 2.27

f

i f

E

where

H(U,;F)

i s an open subset of

U,

then t h e s e t of points o f c o n t i n u i t y o f

Ea

f i s open and

ctosed. Then

H(U;F)

=

u

H(n,(U);F)

C~ED

f o r any convex balanced open subset

Proof

u

~ E C (SE l

U of

H(lla(U);F)

E.

This follows immediately from theorem 2.21.

Theorem 2.28 let

=

Let

U be a connected open subset o f t h e Banach space

F be a normed l i n e a r space and l e t f E HG(U;F). SEU and every p o s i t i v e integer m then f

f o r some

Proof

pf(E) H(U;F).

If E

Without l o s s of generality, we may assume that U

balanced set and that 5 = 0.

Let

E

E,

6(mE;/\F)

is a convex

Chapter 2

68

Am

Since each

i s continuous,

d f(0)

6

Vn = U s i n c e n=l say V some Vn,

i s a closed subset of

Vn

f

i s G-holomorphic.

,

has nonempty i n t e r i o r .

"0

V C V

no

.

n

1-m=O zm

L

f

qeU "0

and

i s a convex

V

then lemma 1.13 implies

2 f (0)

SUP

L

-V 2

Thus

If

n+VcV

Hence

Ilf Ill

and

By t h e Baire category theorem

balanced neighbourhood of zero such t h a t that

U

2no.

=

i s l o c a l l y bounded and continuous a t t h e o r i g i n .

By using t h e

Taylor s e r i e s expansion of G-holomorphic functions, we s e e t h a t

f o r any

0

in

f o r any

n

and any

that

i s continuous a t

f

have shown t h a t

and any

U

f

0

E

in

x

in

U.

Since

8.

H(U;F).

By lemma 1.19,

E.

Pf(0)

E

(P(nE;$)

By t h e f i r s t p a r t of o u r p r o o f , i t follows (3

was a r b i t r a r i l y chosen i n

U

we

This completes t h e proof.

6332

Since Frzchet spaces and

spaces a r e superinductive l i m i t s of

Banach spaces, one can e a s i l y prove a r e s u l t s i m i l a r t o theorem 2.28 f o r such spaces. So f a r we have been d e s c r i b i n g f a c t o r i z a t i o n r e s u l t s which use contin-

uous semi-norms on t h e domain space. solving t h e Levi problem on s o r t of f a c t o r i z a t i o n .

C e r t a i n s i t u a t i o n s ( f o r example, i n

&3mspaces

with a b a s i s ) r e q u i r e a d i f f e r e n t

We g i v e some r e s u l t s i n t h i s d i r e c t i o n .

A topological space

X

i s a L i n d e Z Z f space i f every open cover o f

contains a countable subcover.

X

i s s a i d t o be h e r e d i t a r y LindeZb'f i f

X

69

Holomotphic mappings between locally convex spaces every open subset of

i s Lindelbf.

X

533%

Separable FrGchet spaces and

spaces a r e examples of h e r e d i t a r y Lindelb'f l o c a l l y convex spaces. Proposition 2.29

F f

be a hereditary Lindelb'f l o c a l l y convex space,

H(U;F)

IJ

Proof

a metrizable l o c a l l y convex space

then there e x i s t f

U),

and

H(nf(U);F)

E

IIf(U)

in

5

t h e r e e x i s t s an

U

where

<

f

The semi-norms l o c a l l y convex topology on TIf

E.

Let

a5

U

cs(E)

If

(which E

onto

and

and

f

f = fonf.

such t h a t

5

5

it contains a countable sub-

g e n e r a t e a semi-metrizable

denote t h e a s s o c i a t e d m e t r i z a b l e

denote t h e q u o t i e n t mapping from

c o n s t r u c t i o n nf(U)

Ef

E. Ef

Ba (1)= { x E E ; ~(x) < 1 ) .

(a5 , )mn = l Ef

E

E

from

i s an open subset o f

I1 115+B a g ( l ) + Ba5(1) cu and Since { 5 + B a _ (1)IEEu i s an open cover of

c.

space and l e t

nf

a continuous s u r j e c t i o n

such t h a t

For every

a convex balanced open subset of

U

a normed l i n e a r space and E

depends on f

E

Let

i s an open s u b s e t of

Ef.

E

onto

Ef.

By

on

We now d e f i n e

i n t h e usual manner and s i n c e it i s l o c a l l y bounded, it l i e s i n

nf(U)

H(nf(U);F).

This completes t h e proof. Corollary 2.30 E

F

and

If U

space

i s a Banach space, then H(U;F)

Proof

is a convex balanced open subset of a

A bjq

=

u

~ E C (E) S

H(na(U);F).

space i s a h e r e d i t a r y Lindelb'f space and a l s o a DF-space.

The r e s u l t now follows e a s i l y by using t h e c o n s t r u c t i o n of p r o p o s i t i o n 2.29 and t h e following property of

DF

spaces:

if

m

i s any sequence o f

continuous semi-norms on a DF-space, then t h e r e e x i s t s a continuous semi-

norm

a

an 6 c na

on

E

and a sequence of p o s i t i v e r e a l numbers

for all

m

( c ~ ) ~such = ~t h a t

n.

Corollary 2.30 may be strengthened i n t h e c a s e of e n t i r e functions (see e x e r c i s e 2.105). Our f i n a l example f i r s t a r o s e i n f i n d i n g a counterexample t o t h e Levi problem.

The proof i s q u i t e d i f f e r e n t from t h o s e j u s t given and v a r i a t i o n s

of t h e technique used w i l l appear i n chapter 5 . Example 2.31

Let

r

denote an uncountable d i s c r e t e s e t .

If

Chapter 2

70 x = (x )

CL CiEr

c0(r)

E

( x E c o ( r ) ; s ( x ) cI',) /If11

and If

J

m

where

...,a n1

{al,

=

rl

f o r any s u b s e t

M <

=

s(x) = {acr;xa # O } .

we let

of

co(rl)

Let

Now suppose

I'.

i s t h e open u n i t b a l l of

B

r

i s any f i n i t e s u b s e t o f

=

f

E

(co(r),

H(2B;C)

11 1 1 ) .

t h e n , by u s i n g a monomial

e x p a n s i o n i n s e v e r a l v a r i a b l e s , w e see t h a t f(z) If

e,

has i t s

zero,lthen

where

IkENjw(k)zk

=

M2

z

for a l l

in

ith c o o r d i n a t e e q u a l t o 2 sup{)f(z)I2; z =

1

i=l

1

2Bn c o ( J )

and a l l o t h e r c o o r d i n a t e s

Ziei,\Zij=

1

il

all

I 1

i s n o r m a l i s e d Haar measure on { z i e i , z . =1} f o r i = l , .. . , n . iBm zm = e , m = l , ..., n, it f o l l o w s t h a t

dzi

By u s i n g the change of v a r i a b l e

Since

Jcr

where

N(')

Hence

{k;w(k)#O}

was a r b i t r a r y , i t f o l l o w s t h a t

= { $ : r + N, $ ( a ) = O

r l ={azr\ 3 s u b s e t o f r. I t

(x,A),

x

E

2B

and

EN'^),

k

E

2B,

co(r)

c o n t i n u o u s , we have shown t h a t E

B.

and

w(k)#O

is e a s i l y seen t h a t

x+Ae

i s a dense subspace o f

(xa)aer

a

in

r}.

is countable.

Let

able

f o r a l l e x c e p t a f i n i t e number o f

if

a

E

laer 1

i s a count-

ufor

f(x+Xea) = f(x)

r\rl.

Since

( i n t h e norm t o p o l o g y ) ( ("a

rl

k(a)#O}.

JCT , J f i n i t e C o ( J ) and f i s

= f((xaIaErl)

for a l l

By u s i n g t h e p r i n c i p l e o f a n a l y t i c c o n t i n u a t i o n ( i n s e v e r a l

71

Holomotphic mappings between locally convex spaces complex v a r i a b l e s ) one c a n e a s i l y e x t e n d t h e above p r o o f t o show t h e f o l l owing : if

then t h e r e e x i s t s a countable subset &

such t h a t onto

co (r I+ and

i s a convex b a l a n c e d open s u b s e t o f

U

f = f on

co(rl).

rl

where

'r

rl

of

r

and

f

E

f

E

H(U;K)

H(U n Co(i-l);E)

i s t h e c a n o n i c a l s u r j e c t i o n of c o ( r )

Many o f t h e above f a c t o r i z a t i o n theorems can be extended t o pseudoconvex domains (and t h i s e s s e n t i a l l y means t o a l l open s e t s ) by v i r t u e of t h e following r e s u l t : if and

U

U

i s a pseudo-convex open s u b s e t o f t h e l o c a l l y convex s p a c e

contains an a - b a l l , a

f i n i t e l y open s u b s e t o f

,E

E

and

cs(E),

then

n

(U)

E

i s a pseudo-convex

U = Uil(II,(U)T.

T h i s r e s u l t i s u s e d i n studying pseudo-convex domains, h o l o m o r p h i c a l l y convex domains and domains of holomorphy i n l o c a l l y convex s p a c e s . F a c t o r i z a t i o n r e s u l t s f o r Mackey holomorphic f u n c t i o n s a r e r e q u i r e d i n Chapter 6 .

The c o n c e p t s and methods needed t o p r o v e t h e s e r e s u l t s w i l l b e

given later. 52.3

LOCALLY CONVEX TOPOLOGIES ON SPACES OF HOLOMORPHIC FUNCTIONS Two t o p o l o g i e s a r e u s u a l l y c o n s i d e r e d on t h e s p a c e o f Mackey holomor-

phic functions;

t h e t o p o l o g y o f u n i f o r m convergence on t h e f i n i t e dimen-

s i o n a l compact s u b s e t s of t h e domain and t h e t o p o l o g y o f uniform convergence on t h e s t r i c t l y compact s u b s e t s o f t h e domain.

Since w e s h a l l not use

a n y r e s u l t s d e r i v e d by u s i n g t h e s e t o p o l o g i e s , we c o n f i n e our i n t e r e s t i n t h e s e t o p o l o g i e s t o some e x e r c i s e s a t t h e end of t h e c h a p t e r .

On t h e

s p a c e o f h y p o a n a l y t i c f u n c t i o n s , t h e n a t u r a l t o p o l o g y i s t h e compact open topology. Definition 2.32

Let

U

be an open subset of the l o c a l l y convex space

E and l e t F be a l o c a l l y convex space. The compact open topology ( o r t h e topology of uniform convergence on the compact subsets of U l on HHY(U;F) i s t h e l o c a l l y convex topology generated by t h e semi-norms

12

Chapter 2

ranges over the compact subsets of

K

where

F.

continuous semi-norms on Naturally

U

and

ranges over the

$

We denote t h i s topoZogy by

T

induces a l o c a l l y convex topology on

T~

again c a l l t h e compact open topology and denote by

T

~

~

.

which we

H(U;F) .

This i s t h e most

n a t u r a l topology t o consider on spaces o f holomorphic f u n c t i o n s .

We f i n d ,

however, it does n o t always possess very u s e f u l p r o p e r t i e s and f o r t h i s reason w e introduce t h e

topology.

T~

This topology has good topological

p r o p e r t i e s b u t can be d i f f i c u l t t o d e s c r i b e i n a concrete fashion and i t s r e l a t i o n s h i p with t h e

topology may not always be c l e a r .

T~

duce a f u r t h e r topology,

T

w’

We a l s o i n t r e

whose d e f i n i t i o n was motivated by c e r t a i n

p r o p e r t i e s o f a n a l y t i c f u n c t i o n a l s i n several complex v a r i a b l e s . topology i s intermediate between t h e

T

0

and t h e

f u l l y it w i l l s h a r e t h e good p r o p e r t i e s o f both

topologies.

T~

and

T~

This

r o l e appears t o be a s a t o o l i n proving r e s u l t s about

T~

T

6 and

Hope-

but i t s main (see

T~

f o r i n s t a n c e chapter 5 ) . D e f i n i t i o n 2.33 F

and l e t

Let

U

be an open subset of a ZocaZZy convex space

be a normed Linear space.

to be ported by t h e compact subset V,KCVcU,

The

T~

C(V)>O

there e x i s t s

topoZogy on

H(U;F)

K

A semi-norm

of

U

p

on H ( U ; F )

E

i s said

i f , f o r every open s e t

such t h a t

i s t h e LocaZZy convex topology generated by

a22 seminorms ported by compact subsets of U . D e f i n i t i o n 2.34

Zet

F

D e f i n i t i o n 2 .3 5

and l e t

Let

u

be an open subset of a ZocaZZy convex space and

be a ZocaZZy convex space.

F

Let

U

We d e f i n e

T~

on

H(U;F)

by

be an open subset of a ZocaZly convex space

be a normed l i n e a r space.

A semi-norm

p

on

H(U;F)

E

i s said

13

Holomorphic mappings between locally convex spaces

t o be (Vn);=l’

T&

P(f)

The

6

C

I1 f l l V

n

topology on H ( U ; F )

T&

the

U,

continuous i f for each increasing countable open cover o f there e x i s t s a p o s i t i v e i n t e g e r no and c , O such t h a t f o r every

f

in

H(U;F).

0

i s t h e l o c a l l y convex topology generated by

continuous seminoms.

Definition 2.36 F

and l e t

be an open subset of a l o c a l l y convex space

U

Let

be a l o c a l l y convex space.

(H(U;F)

We d e f i n e

T&

on H ( U ; F )

E

by

=

,T&)

The g e n e r a l r e l a t i o n s h i p between t h e t h r e e t o p o l o g i e s j u s t d e f i n e d i s g i v e n i n t h e f o l l o w i n g lemma.

Lemma 2 . 3 7

E.

F

and

be l o c a l l y convex spaces and l e t

On H ( U ; F )

we have

T

~

~

T

~

~

We may suppose, w i t h o u t l o s s o f g e n e r a l i t y , t h a t

Proof

l i n e a r space.

Since

follows t h a t on

E

Let

open subset o f

H(U;F)

T~ 2 T

0

.

IlfllK

<

llfllv

Now suppose

p

is a

exists

no

such t h a t

such t h a t

c>O

shows t h a t

T

p

is

T&

containing of

K

Since

U.

i s a neighbourhood o f

Vn

p(f)

cll€lIvno

F

be an

~

i s a normed K

it

c o n t i n u o u s semi-norm

w

which i s p o r t e d by t h e compact s u b s e t

d e n o t e an i n c r e a s i n g c o u n t a b l e open cover o f can choose

v

f o r every

U

T

f o r every

f

U. K

Let

(Vn)i=l

i s compact w e

K.

Hence t h e r e

in

H(U;F).

This

c o n t i n u o u s and completes t h e p r o o f .

Our n e x t r e s u l t shows t h a t

T&

has good t o p o l o g i c a l p r o p e r t i e s when

t h e r a n g e s p a c e i s a Banach s p a c e ( a s l i g h t l y less g e n e r a l r e s u l t h o l d s when t h e r a n g e s p a c e i s a normed l i n e a r s p a c e ) . a l t e r n a t i v e description of t h e Proposition 2 . 3 8

Let

U

T&

T h i s r e s u l t a l s o g i v e s an

topology.

be an open subset o f t h e l o c a l l y convex space

E and l e t F be a Banach space. Then ( H ( U ; F ) , T & ) is an inductive l i m i t o f Frgchet spaces and hence i t is barrelled, bornological and ultrabornol-

ogical .

Chapter 2

74 Proof

For each i n c r e a s i n g c o u n t a b l e open c o v e r o f

l e t HU(U;F) = I f EH(U;F); llfll < m Vn t o p o l o g y g e n e r a t e d by t h e semi-norms

m e t r i z a b l e l o c a l l y convex s p a c e .

n).

all

L') = (Vn);=l,

U,

We endow

.

p n ( f ) = Ilflb

HV(U;F) w i t h t h e

H,(U;F)

is then a

I t i s i n f a c t a 8 r & h e t space s i n c e

F

i s a Banach s p a c e and l o c a l l y bounded G-holomorphic f u n c t i o n s a r e holomor-

phic.

We c l a i m

H(U;F) = U$Ho(U;F)

c o u n t a b l e open c o v e r s o f i s open and 0

Wn

Since

f

H(U;F)

x m Hb(U;F).

( H ( U ; F ) , T ~ )=

follows t h a t

H(U;F) w e l e t

E

Wn

=

CXEU; [lf(x)(\ < n3

We now l e t

w e h a v e proved our claim.

i n d u c t i v e l i m i t t o p o l o g y on i.e.

f

ranges over a l l increasing

i s an i n c r e a s i n g c o u n t a b l e open c o v e r o f

= (Wn)mnZ1

Hw(U;F)

E

If

U.

where

T~

d e f i n e d by a l l t h e s p a c e s

Since

Hv(U;F)

U.

denote the H,(U;F),

i s a F r g c h e t s p a c e it

( H ( U ; F ) , T ~ ) i s a n u l t r a b o r n o l o g i c a l s p a c e and hence it i s

b a r r e l l e d and b o r n o l o g i c a l . We complete t h e p r o o f by showing t h a t

mapping from

Hw(U;F)

into

a b l e open c o v e r CJ o f Let not

p

of

denote a

(Vn):=l

U,

/lfnlIVn$ Let

Wn

=

~

.

Since t h e i d e n t i t y

i s c o n t i n u o u s f o r any i n c r e a s i n g count-ii

and a sequence

p(fn) > n

for all

(1

n

CXEU; ( ( f m ( x )

denotes t h e i n t e r i o r o f

W,"

T

>

7

6' H(U;F).

+,

(U;F) (fn)iz1

i s continuous.

(fn)EZ1

for all Since

Wn.

ml.

in

Let

]]fmIIVQ

H(U;F)

is

for a l l

n.

such t h a t

= (W")"

where n n=l 1 f o r a l l min, it

s i p llfmlb ,<

BY c o n s t r u c t i o n

i s a bounded s u b s e t o f

p(fn) > n

fact that

p

n.

f o l l o w s t h a t D i s an i n c r e a s i n g c o u n t a b l e opt% c o v e r o f 'IH Hence

Suppose

Then t h e r e e x i s t s an i n c r e a s i n g c o u n t a b l e open c o v e r

,

=

1 and

=

c o n t i n u o u s semi-norm on

T i

continuous.

T&

H(U;F)

we have

U

T.

Ho(U;F).

Hence

7.

=

7&

U.

n

Hence f o r every n .

This Fontradicts t h e and we have completed

t h e proof. Since 'o,b with

H(U;F)

and T~

T~

and

w,b and T

T

w

a r e n o t i n g e n e r a l b o r n o l o g i c a l t o p o l o g i e s , we l e t

d e n o t e t h e b o r n o l o g i c a l t o p o l o g i e s on

T

w

respectively.

H(U;F)

associated

(Note t h a t t h e t o p o l o g y induced on

by t h e b o r n o l o g i c a l t o p o l o g y a s s o c i a t e d w i t h t h e compact open

topology o f

HHY(U;F) need n o t b e

o g i e s t h a t can b e p l a c e d on

H(U;F)

T ~ , ~ T ) h. e r e a r e a l s o f u r t h e r t o p o l -

-

such a s t h e t o p o l o g y o f uniform

convergence of t h e f u n c t i o n and i t s f i r s t

n

d e r i v a t e s on t h e compact

75

Holomotphic mappings between locally convex spaces

...,

U, n = 1 , 2 ,

s u b s e t s of

but s i n c e we s h a l l not u s e t h e s e topologies

we w i l l not go i n t o any f u r t h e r d e t a i l s . A g r e a t p o r t i o n of t h i s book i s concerned with f i n d i n g conditions on

U,E

and

F

which imply e i t h e r

=

T~

T

w

, T~

= T~

or

= T

T~

(together

6

The remainder of t h i s s e c t i o n

with t h e i m p l i c a t i o n s of t h e s e c o n d i t i o n s ) .

is devoted t o a number of b a s i c f a c t s , concerning t h e s e t o p o l o g i e s , which we s h a l l f r e q u e n t l y u s e and t o a few examples and counterexamples which w i l l prove u s e f u l i n l a t e r c h a p t e r s .

lVe f i r s t n o t e t h a t t h e compact open topology i s a sheaf topology, i . e .

it i s l o c a l l y defined. T&

We do not know i f t h i s i s t r u e f o r t h e

T

and t h e

w

topologies and any r e s u l t s i n t h i s d i r e c t i o n would c e r t a i n l y h e l p t h e

general development of t h e t h e o r y .

The l o c a l c h a r a c t e r o f t h e compact open

topology i s contained i n t h e following lemma. Lemma 2 . 3 9 (Ui)iEI

Let

U

U

an open cover o f (H(U;F),T~) i n t o

from

and

F

a locally convex space.

IIisI(H(Ui;F),~o)

i s t h e r e s t r i c t i o n of f (where f ' U i onto a subspace of IIiEIH(Ui;F)

H(U;F) Proof

I t suffices t o note t h a t

K

only i f t h e r e e x i s t s a f i n i t e subset of a compact subset of

E,

be an open subset of a l o c a l l y convex space

f o r each

UL.

1

j,

which maps

to

.

Uil

f

to

The mapping (f(Ui)isI

i s an isomorphism o f

i s a compact subset of I,

Ll,

..., Ln

such t h a t

K

and

U

i f and

(Kj)Y=l,

=U3,1Kj.

K.

1

I t i s obvious t h a t a s i m i l a r r e s u l t holds f o r hypoanalytic f u n c t i o n s . We now show t h a t (H(U;F),ro)

( p ( m E ; F ) , ~ o ) i s a closed complemented subspace of f o r any open subset U of t h e l o c a l l y convex space E , any

complete loca?.ly convex space (H(U;F),ro) spaces then

F

m.

Hence i f

( 6 ( m E ; F ) , ~ o ) must a l s o have t h e same property. U

i s an open subset of a ZocalZy convex space E i s a complete locaZZy convex space then ( 6 (mE;F) ,TO) i s a cZosed If

compZemented subspace o f Proof

and any p o s i t i v e i n t e g e r

has any property i n h e r i t e d e i t h e r by subspaces o r by q u o t i e n t

Proposition 2 . 4 0

and

F,

Since

(H(U;F),T~) for any p o s i t i v e integer m.

(H(U;F),T~) = (H(U-S,F),T~) f o r any

5

E

E

we may suppose

76

Chapter 2 As u n i f o r m convergence on t h e compact s u b s e t s o f

O E U.

is equivalent t o

E

uniform convergence on t h e compact s u b s e t s o f some neighbourhood o f z e r o f o r

elements o f

(mE;F)

it f o l l o w s t h a t

t h e compact open t o p o l o g y . A

dm -

H(U;F)

:

m!

f

( H ( U ; F ) , T ~ ) i n d u c e s on @(mE;F)

Now c o n s i d e r t h e mapping

-

H(U;F) ;;mf ( 0 )

_ _ _ _ _ f -

m!

A

T h i s i s a l i n e a r mapping and s i n c e it i s a p r o j e c t i o n from

p(mE;F)

dm - (P)(O) = P f o r every P i n m! H(U;F) o n t o @ (mE;F): To complete t h e

p r o o f w e must show t h a t it i s a c o n t i n u o u s p r o j e c t i o n . convex b a l a n c e d open s u b s e t o f balanced s u b s e t o f

f o r every

in

E

such t h a t

V C U .

Let If

denote a

V

i s a compact

K

t h e n , by t h e Cauchy i n e q u a l i t i e s

V

and hence t h e p r o j e c t i o n i s c o n t i n u o u s .

cs(F)

This

completes t h e p r o o f . For a r b i t r a r y

w e do n o t have any u s e f u l r e p r e s e n t a t i o n o f t h e

U

@

t o p o l o g i c a l complement o f

(%;F)

in

H(U;F)

b u t we s h a l l see, i n t h e

n e x t c h a p t e r , t h a t t h e T a y l o r s e r i e s r e p r e s e n t a t i o n o f holomorphic f u n c t i o n s g i v e s u s a means o f i d e n t i f y i n g a u s e f u l t o p o l o g i c a l complement when

We now p r o v e t h e a n a l o g u e o f p r o p o s i t i o n 2.40 f o r t h e

balanced.

xS

topologies.

T

is

U

w

and

Our p r o o f i s f o r Banach s p a c e v a l u e d mappings b u t t h e same

r e s u l t f o r a n a r b i t r a r y complete l o c a l l y convex r a n g e s p a c e c a n be proved i n a similar f a s h i o n . Proposition 2.41

and

F

If

U

((3 ( m E ; F ) , ~ w ) is a cLosed complemented

i s a Banach space then

subspace of (H(U;F),-cw ) and of ( H ( U ; F ) , T ~ ) . I n particular induce the same topoZogy on 8 (mE;F). Proof

We f i r s t show t h a t

denote a

T~

T~

i n c r e a s i n g c o u n t a b l e open c o v e r o f N

and

c o n t i n u o u s semi-norm on

b a l a n c e d neighbourhood o f z e r o i n integer

and

C>O

E

is an open s u b s e t of a l o c a l l y convex space

such t h a t

E. U

T~

c o i n c i d e on

H(U;F)

and l e t

The sequence

l$'(mE;F). V

T

w

and

T6

Let

p

d e n o t e a convex m

(UnnV)n=l

i s an

and h e n c e t h e r e e x i s t a p o s i t i v e

71

Holomorphic mappings between locally convex spaces

for a l l

f

E

H(U;F).

'i;

Hence

Ptopology I PPE;F)on

is a

continuous semi-norm on

T , ~

4

=

8(

i)/p(mE;F) m ~ ; ~f o) r a l l

it follows t h a t

T~

H(U;F) and

and s i n c e induce t h e same

T~

m.

The above a l s o shows t h a t t h e mapping given i n p r o p o s i t i o n 2.40 i s a continuous p r o j e c t i o n f o r both

T~

and

T ~ .

We now look a t t h e l o c a l l y bounded o r equicontinuous s u b s e t s of H(U;F). D e f i n i t i o n 2.42 E

F

and l e t

be an open subset of the locally convex space

U

Let

be a ZocaZly convex space.

locally bounded i f f o r every c,Vg,

Lemma 2.43

A subset

2

of H(U;F)

is

there e x i s t s a neighbourhood of

E

where

Proof

F.

A locally bounded subset of

(H(U;F),T6)

and

F

U

an open subset of

are l o c a l l y convex spaces, i s a bounded subset of

We may assume, without l o s s of g e n e r a l i t y , t h a t Let r) be a l o c a l l y bounded subset of

continuous semi-norm.

wn and l e t

H(U;F),

*

l i n e a r space. T~

in U

such t h a t

i s a bounded subset of

E

5

=

IXEU;

For each p o s i t i v e i n t e g e r

(IfCx)((

Vn = I n t e r i o r (Wn).

c n f o r every f Since

3

i s a normed

F

H(U;F), n

and

p

a

let

i n 31

i s l o c a l l y bounded

(Vn)n

i s an

78

Chapter 2

i n c r e a s i n g c o u n t a b l e open c o v e r of

Hence t h e r e e x i s t s

U.

and

C>O

N,

a p o s i t i v e i n t e g e r such t h a t

cllfll

p (f) s

f E

f

in

H(U;F)

and t h i s completes t h e p r o o f .

sup p ( f ) s C.N

Hence

f o r every

vN

3

If U i s an open subset of a ZocaZZy convex space E, F is a l o c a l l y convex space and every bounded subset of (H(U;F),ro) is l o c a l l y bounded then T ~ , T and ~ T & have the same bounded subsets in C o r o l l a r y 2.44

H(U;F). In particular

etc. i n place o f

C o r o l l a r y 2.45

then

and

(H(U),-ro)

H(U;F) = H(U;Fu)

H(U;C), YIY(U;E), e t c . F

be ZocaZly convex spaces.

Let

Fa = ( F , o ( F , F ' ) ) .

where

.3

= (@of)@EB

lies i n

H(U).

Thus

:y

Consequently

in

Vg

f(Vg)

containing

U

5

f @

i s a l o c a l l y bounded f u n c t i o n .

H(U;Fu).

E

in

(H(U),ro)

Hence f o r each such t h a t

sup

then

U

F'.

The

Hence

5 E U there V5 < a.

exists

and once more

i s a bounded s u b s e t of

f(Vg) Hence

and s o by

I($of((

i s a weakly bounded s u b s e t o f @ E B F

by Mackey's theorem it f o l l o w s t h a t f

f o r every

C

i s a bounded s u b s e t o f

o u r h y p o t h e s i s , it i s l o c a l l y bounded. a n open s e t

and l e t

and by Mackey's theorem i t i s

F

i s a weakly bounded s u b s e t o f

s t r o n g l y bounded.

i s a normed

F

i s a compact s u b s e t o f

K

If

@cf(K) = @ ( f ( K ) ) i s a bounded s u b s e t o f f(K)

F' B

be t h e u n i t b a l l o f

B

I f the

are ZocaZZy bounded f o r every open subset U

We may assume w i t h o u t l o s s o f g e n e r a l i t y t h a t

l i n e a r space. set

E

Let

bounded subsets of

Proof

.

T

I f t h e r a n g e s p a c e i s t h e f i e l d o f complex numbers w e w r i t e

Notation H(U), HHy(U)

of E

is t h e bornoZogicaZ topoZogy associated with

T~

F, i . e .

i s a holomorphic f u n c t i o n and

f

t h i s completes t h e p r o o f s i n c e t h e c o m p o s i t i o n o f holomorphic f u n c t i o n s i s holomorphic and so we always h a v e Example 2.46

Let

s p a c e and l e t subset,

3,

F of

U

H(U;F)

C

b e a n open s u b s e t o f a m e t r i z a b l e l o c a l l y convex

We claim t h a t any

b e a normed l i n e a r s p a c e . H(U;F)

H(U;F,).

i s l o c a l l y bounded.

SEU such t h a t f o r e v e r y open s e t

V,

,€ E

T~

bounded

I f not, then t h e r e e x i s t s

VCU,

w e have

sup I l f l ) fE

3

=

a.

79

Holomotphic mappings between locally convex spaces Hence we can choose IIfn(Sn)II > n

S,,

n.

for a l l

5,

U,

E

+

Since

5

n

as

and

+ m,

(fn)n

c3such

that

i s a compact subset of

{5n}nUIC)

U

t h i s i s impossible and we have proved our claim. The remaining examples given i n t h i s s e c t i o n d e a l with holomorphic and on l o c a l l y 8 3 % spaces, Banach spaces, aN x C")

f u n c t i o n s on

convex spaces which do not admit a continuous norm.

These examples a r e

elementary i n s o f a r a s t h e proofs a r e r a t h e r d i r e c t .

However, they a r e of

i n t e r e s t s i n c e they show t h e divergence between l i n e a r and holomorphic f u n c t i o n a l a n a l y s i s and a l s o because many of t h e examples and methods encountered h e r e have e x p l i c i t l y and i m p l i c i t l y motivated t h e development These examples a l s o provide a good

of t h e theory a s o u t l i n e d i n t h i s book.

i n t u i t i v e guide t o t h e t y p e of behaviour we may look f o r i n d e l i c a t e situations. Example 2.47 F

Let

U

be an open subset of a

b e a normed l i n e a r space.

We show t h a t t h e

space

and l e t

E

bounded s u b s e t s of

a r e l o c a l l y bounded (we have already proved t h i s r e s u l t f o r homo-

H(U;F)

geneous polynomials i n chapter 1 ) . ity, that

We may assume, without l o s s of genera

i s a convex balanced open subset of

U

bounded subset

T~

83 T~

3

of

H(U;F)

E

and we show t h a t t h e

i s l o c a l l y bounded a t t h e o r i g i n .

Let

be a fundamental system of convex balanced compact s u b s e t s of

(BJn

L-

E.

As we have previously noted, i t s u f f i c e s t o f i n d a sequence of p o s i t i v e

r e a l numbers,

(in),., such t h a t

B1

f E 3

B = kl 1

If

11 IIA

1.1=1A.B. 1 1

=

6>0

choose and

we l e t

f

M < m .

L(6) = L + 6Bk+l.

such t h a t

in

inequalities.

Now suppose

H(U;F) Hence

L(S1)CU

B1 B2L(61)

where

c(B1)

11

<

m.

'i;=l~n%

il> 0 such

...,

that

Hence

AIBICU.

i2, ik have been chosen

is. a compact s u b s e t of

such t h a t

61>0

B 2 '1,

for a l l

f~3-

i s compact, we can choose

Since

L

sup Ilf

U

Let

E

and >O

sup fE3

((fl(

S M +

be a r b i t r a r y .

and next choose

B1

so t h a t

I,,=,

--

-MI.

2"

We f i r s t

and

i s a l s o a compact subset of

B2, U.

i s derived by using t h e Cauchy

B1>

1

Hence

Chapter 2

80

continuous semi-norm on

___ dnf(o)

Let

n! If

f

E

H(U;F)

and s i n c e

H(U;F)

we can f i n d a p o s i t i v e i n t e g e r

and 6 > 0

s2

T

-bounded and

61,

B2>1

fin d f(O)/n!.

then

n. <

0,

is

be t h e symmetric n - l i n e a r form a s s o c i a t e d with

f o r any non-negative i n t e g e r

62

3

such t h a t

N

m

f o r each

n

and

6

we can choose

so t h a t

Hence, by induction, we can choose a sequence of p o s i t i v e real numbers,

3 and

i s a l o c a l l y bounded family o f f u n c t i o n s . T

6

d e f i n e t h e same bounded s u b s e t s o f

This implies t h a t H(U;F).

T ~ ,T~

Since B 8 3 b s p a c e s

a r e h e r e d i t a r y Lindelzf spaces and contain a fundamental sequence of compact s e t s it follows t h a t every open subset of a 3 3 M s p a c e contains

81

Holornorphic mappings between locally convex spaces a fundamental system of compact s e t s . Since

T6

that

T~

643%

spaces a r e

=

This, i n t u r n , implies t h a t

i s a metrizable, and hence a bornological l o c a l l y convex space. i s a l s o a bornological topology on H(U;F) we have i n f a c t shown

(H(U;F),ro)

T

-

w

-

on

6'

H(U;F).

F i n a l l y we remark t h a t open s u b s e t s of

k-spaces and s o

i s a Frgchet space i f

(H(U;F),ro)

F

is

a Banach space. Example 2.48

Let

Banach space m

E

is a

T

0

c0

be an open subset of an i n f i n i t e dimensional

U

Let

E.

5

continuous semi-norm on

be t h e u n i t b a l l of

If

E.

H(U).

H(U) Hence

which i s not continuous f o r t h e (H(U) ,

T ~ )#

(H(U) ,.cw)

f o r any open

o f any i n f i n i t e dimensional Banach space.

U

Example 2.48

Let

CN

E = CN

((? (2E) , T ~ ) .

( @ (2E) , T ~ )#

s e t of

B

( t h e space of n u l l sequences of complex numbers) then

compact open topology on subset

and l e t

U

E

x

C(N).

x

a").

Hence

We have a l r e a d y seen t h a t (H(U),ro) # (H(U),ru)

Example 1.39 shows, a l s o , t h a t

T

f o r any open sub~

#

,T

~~

on ,

~H(U).

For our next example we need a concept which f r e q u e n t l y a r i s e s i n i n f i n i t e dimensional holomorphy

-

t h e concept of very s t r o n g s e q u e n t i a l '

convergence - b u t which does not a r i s e i n l i n e a r f u n c t i o n a l a n a l y s i s .

Since

t h e dual concept - very weak s e q u e n t i a l convergence - w i l l a l s o be needed l a t e r , we t a k e t h e opportunity of giving i t s d e f i n i t i o n h e r e .

Further

information on t h e s e concepts i s o u t l i n e d i n t h e e x e r c i s e s .

D e f i n i t i o n 2.50

A sequence

( x ~ )i n~ a locally convex space

said t o be very strongZy convergent if sequence o f scalars x

Xnxn+ 0

in

E

as

n-

E

is

f o r every

The sequence is said t o be nontrivial i f

n # 0 f o r each n. A sequence i s obviously very s t r o n g l y convergent i f and only i f f o r

each

p

p(xn) = 0

in

cs(E)

for a l l

there exists a positive integer,

n 2 n(p).

n(p),

such t h a t

A metrizable l o c a l l y convex space

a n o n t r i v i a l very s t r o n g l y convergent sequence i f and only i f

E

E

admits

does n o t

-

82

Chapter 2

admit a c o n t i n u o u s norm.

un

(0, ...,1 , O ...)

=

Definition 2.51

CN

For example, i n

t h e sequence

i s a n o n t r i v i a l v e r y s t r o n g l y convergent sequence.

nth position

For example, i n

E

( x ~ )i n~ a l o c a l l y convex space

A sequence

t o be very weakly convergent if A n xn + 0 i n of non-zero scalars ( A ~ ) ~ .

E

as

i s said

f o r some sequence

M

u = (0 ,..., 1,0 ,...) i s n o t a n L- n t h p o s i t i o n In f a c t , i f E i s any Fre‘chet s p a c e which

t h e sequence

v e r y weakly convergent sequence. i s n o t a Banach s p a c e t h e n

c o n t a i n s a sequence which does n o t converge

E’

v e r y weakly.

Let

Example 2 . 5 2

b e a l o c a l l y c o n v e x s p a c e which c o n t a i n s a non-

E

( x ~ ) ; = ~ . Let

t r i v i a l v e r y s t r o n g l y convergent sequence

E.

For each

If

f

f

f(x+w) = f ( x )

Since

00

such t h a t

n > n

for all

a

f + If(ny+x )

uous semi-norm on

n H(E)

a b a r r e l l e d t o p o l o g y on semi-norm on

f o r every

x

and

w

and

-

for all

a(xn) = 0 p(f)

E

C,

all

defines a

f(ny)(

?

H(E)

n,.

it f o l l o w s t h a t

Hence f

in

and hence a

-co

B

be a

H(E), T~

i.e.

n

2

no

p

is a

= 0.

p

f(ny+xn) = H(E).

-c6

nu and s i n c e is a

-i6

The

continT~

is

continuous

and a l l

f

fn(xny + xn) # f n ( x n y )

i s bounded on t h e

T~

c o n t i n u o u s semi-norm on o,b H(E). We b e g i n by showing t h a t

n

such t h a t

in

B.

for all

p

‘I

bounded s u b s e t o f

f(iy+xn) = f(hy)

for all

I f t h i s were n o t t r u e , t h e n by

u s i n g subsequences i f n e c e s s a r y , w e can choose that

n

f o r every p o s i t i v e i n t e g e r

there exists a positive integer h

cs(E)

.(w)

H(E).

bounded s u b s e t s of Let

CL E

such t h a t

E

i s f i n i t e f o r every

W e now improve t h i s r e s u l t by showing t h a t

H(E).

in

i s a v e r y s t r o n g l y convergent sequence t h e r e e x i s t s a p o s i t -

(Xn)n=l i v e i n t e g e r n, function

w e c o n s i d e r t h e sum

H(E)

such t h a t

f(ny)

belong t o

t h e n , by t h e F a c t o r i z a t i o n Lemma, t h e r e e x i s t s an

H(E)

E

in

y#O

n.

An E C and f E B such n For each p o s i t i v e i n t e g e r n l e t

83

Holomorphic mappings between locally convex spaces

By t h e i d e n t i t y theorem f o r f u n c t i o n s o f one complex v a r i a b l e we may

s e l e c t a sequence o f complex numbers, gn(Xh) # 0.

Now

hn

For each i n t e g e r

H(E)

E

f o r each

n,

n.

Hence

and each

and s i n c e m

sequence o f complex numbers, for all

n

IXAl

(wn)n=l,

Ifn(X;y+wnxn)I

>

n

let

C

w E

6

% , n

hn(0) # h n ( l ) , for all

we c a n choose a

I hn(wn) I

such t h a t n.

such t h a t

Since

I

I

n+ fn(X,!,y)

>

( x ~ )i s~ a

v e r y s t r o n g l y c o n v e r g e n t sequence, (w x ) i s a n u l l sequence. Since n n-n IhAl s - f o r a l l n it f o l l o w s t h a t XAy+wnxn -t 0 as v and hence n2 03 K = I01 u IA;Y+wnxnln=l i s a compact s u b s e t o f E . As IlfnllK > n f o r

all

n

t h i s contradicts the fact that

B

is a

H(E).

Hence t h e r e e x i s t s a p o s i t i v e i n t e g e r

f(Xy)

for all

n

We now show t h a t

n

5

p

all 1

0'

C

is not a

otherwise, i . e . t h a t

p

T~

n

and a l l

f

bounded s u b s e t o f

such t h a t in

T h i s shows t h a t

B.

c o n t i n u o u s semi-norm on

T

i s p o r t e d by t h e compact s u b s e t

H(E).

$,(y)

# 0,

+ o ( ~ n )= 0

for all

n,

$,(y)

of

K

u s i n g subsequences i f n e c e s s a r y we c a n choose a s e q u e n c e i n such t h a t

f(Xy+xn) =

= 0

Suppose By

E.

E', for all

n>O

$ n ( ~) # 0 i f and o n l y i f n=m. Let m $,,, = $o$m n f o r any p o s i t i v e i n t e g e r s n and m . $n,m E P("+'E) and n n P($n,m) = Im $ o ( y ) $ m ( x m ) ~ f o r a l l n and lil. I f V i s any neighbourhood and f o r

of

H(E).

ll~olIvm < m'

O

m($,(Y)

I

follows t h a t

# 0.

$,(y)

Let

V

=

ll$mllvm <

II$,1IVm.

llgoIIvm.

6

c(V) > 0

such that.

Choose a n a r b i t r a r y neighbourhood and

II$ I l n

c(V,,,)

positive integers

then there e x i s t s

K

in

f

m,n

p

Hence

Taking Since

ml+o(y)I 6 Hence

-.

nth

Vm

II$oIlx.

is not

{f € H ( E ) ; p ( f ) ,< l}.

p ( f ) 6 c(V) llfllv Vm

of

K

w

V

such t h a t

mnl~o(Y)lnl$m(xm)l 6

r o o t s and l e t t i n g

we get

n-

was a n a r b i t r a r y neighbourhood o f T h i s cannot hold f o r a l l

T

f o r every

m

K

it

since

continuous.

i s a convex b a l a n c e d

T

(and h e n c e

T

w

.)

Chapter 2

84

bounded subset

of

Since

E.

neighbourhood of zero (because that neither spaces.

(H(E) ,

p

nor

T ~ )

i s not a

V

i s not

T T

w

(and hence not a

T ~ )

continuous) we have shown

(H(E) , T ~ ) a r e i n f r a b a r r e l l e d l o c a l l y convex

The above can e a s i l y be modified t o show t h a t t h e same r e s u l t

holds f o r

H(U:F\,

an a r b i t r a r y open subset of

U

E,

and

F

any l o c a l l y

convex space.

12.4

GERMS OF

HOLOMORPHIC

FUNCTIONS

We now introduce t h e space of holomorphic germs on a compact subset of Apart from i t s c l o s e r e l a t i o n s h i p with spaces of

a l o c a l l y convex space.

holomorphic f u n c t i o n s defined on open sets, t h e space

of germs i s a l s o an

important t o o l i n developing a s a t i s f a c t o r y d u a l i t y theory.

The problems

t h a t a r i s e i n studying t h e topological vector space s t r u c t u r e of t h e space of germs a r e of a d i f f e r e n t kind from those which a r i s e i n function space theory and t h i s d i f f e r e n c e a r i s e s p r i m a r i l y from t h e d i f f e r e n c e between p r o j e c t i v e and inductive l i m i t s . Let

K

be a l o c a l l y convex space.

relation which

f

4

u

be a compact subset o f a l o c a l l y convex space

f

where g

and

N

g

On

H(V;F) V 3K V open

We denote by

H(K;F)

and l e t

F

we define the equivalence

i f there e x i s t s a neighbourhood

are both defined and

E

W

K

of

on

flw = g l w .

the r e s u l t i n g vector space o f equivalence classes

If f is and the elements of H(K;F) are called holomorphic germs on K. an F-valued holomorphic f u n c t i o n defined on an open subset of E which contains

K

then we a l s o denote by

determined by

f.

f

t h e equivalence c l a s s i n

The natural topology on

H(K;F)

H(K;F)

i s given by t h e

l i m (H(V;F),rcw) ( t h e inductive l i m i t being taken i n t h e --f V 3 K Vopen category of l o c a l l y convex s p a c e s ) .

inductive l i m i t

for

I f F i s a normed linear space we l e t Hm(V;F) = {f E H(V;F);I) f l I V < m j V open in E and on t h i s space we define a topology by means

of the norm F

11 11 v.

i s a Banach space.

see that

H"(V;F)

i s a normed l i n e a r space which i s complete i f

Using t h e same equivalence r e l a t i o n s h i p we e a s i l y

85

Holomorphic mappings between locally convex spaces

=

H(K;F)

V3 K V open

If K i s a compact subset o f a l o c a l l y convex space

Lemma 2 . 5 3

E and

F i s a normed l i n e a r space then

V3K V open

V3K V open

Proof If V is an open subset of E which contains K then the natural injection from Hm(V;F) into (H(V;F) ,T,,,) - is continuous and hence the identity mapping from lirn (Hm(V;F), 11 ) into lirn (H(V;F) , T ~ )

IIv

-+

VDK V open is also continuous. Conversely, if p lim

(Hm(V;F),

IIv)

11

then for each V

3

V> K V open is a continuous semi-norm on open, V 2 K

there exists

c(V)>

0

-----f

V 2K V open such that p(f)

c(V) IIfllV for every f in Hm(V;F). If then llfllv = m and the same inequality holds. Hm(V;F) ,<

f E H(V;F)\ Hence the restriction of p to H(V;F)

is a

ported by the compact subset K

Thus p

norm on

lim

+

(H(V;F),T~)

of V.

T

continuous semi-norm is also a continuous semi-

and this shows that the two topologies coincide

V> K V open

on H(K;F)

and completes the proof.

It follows that H(K;F) is a bornological space if F is a normed linear space and an ultrabornological (and hence a barrelled) space if F is a Banach space.

If E

is a metrizable space and F

is a Banach space

then H(K;F)

will be a countable inductive limit of Banach spaces and hence a bornological DF-space. Thus we see that the space of germs will always have some good topological properties since it is an inductive limit and indeed the main topological problems connected with H(K;F) are those generally associated with inductive limits (as opposed to those connected with projective limits) such as completion, description of the continuous semi-norms (sometimes we only need a description of sufficiently many

86

Chapter 2

continuous semi-norms) and a c h a r a c t e r i z a t i o n of t h e bounded s e t s . encounter a l l of t h e s e problems i n l a t e r c h a p t e r s . s e l v e s t o c h a r a c t e r i z i n g bounded s e t s when E = limE

Let

E

We s h a l l

Here we confine our-

i s metrizable.

be an i n d u c t i v e l i m i t of l o c a l l y convex spaces.

The

4

a

inductive l i m i t i s said t o be regular i f each bounded subset of E contained and bounded i n some

is

Ea.

Lemma 2 . 5 4 W

let B

=

Let K be a compact subset o f a l o c a l l y convex space be a convex balanced open subset of E . Then

{f EH(K+W); ( ( f ( \ K + ,W < 1) i s a cZosed subset o f

Proof

Let

{falaEA be a convergent n e t i n

H(K)

which l i e s i n

show t h a t

{fa)aEA i s a Cauchy n e t i n K+W

L CK+pW.

By using t h e Cauchy i n e q u a l i t i e s we s e e t h a t

(H(K+W) , T ~ ) .

'm, K , L

p,

If

L

O
H(K)

and a l s o on

We

i s a compact

such t h a t

(H(K+W),

YEL

f o r every nonnegative i n t e g e r

m.

{ f a l a E A C B t h e Cauchy i n e q u a l i t i e s imply t h a t

Hence, f o r some

B.

XEK YEL

d e f i n e s a continuous semi-norm on

Since

and

H(K).

subset of

then t h e r e e x i s t s a r e a l number

E

00

and a l l

a,B E A ,

T ~ ) .

Hence

87

Holomorphic mappings between locally convex spaces this

t e n d s t o z e r o as in

a+=

H(K).

ci,B

since

+ m

By t h e above, t h e r e e x i s t s a

u n i f o r m l y on t h e compact s u b s e t s of every non-negative i n t e g e r m h f o l l o w s t h a t s m f ( F ) = d g(C) g

Proof

Since

in

Since

K+W.

m

for all

+

such t h a t

B

as

f fa

+

g

Pm,K,L(f) = Pm,K,L(g)

and a l l H(K)

fa

5

of

L

K.

in

K+W

it

Hence

f

and

and t h i s completes t h e p r o o f .

i s a compact subset of a rnetrizable locally i s a regular inductive Z i m i t .

H(K)

i s m e t r i z a b l e it c o n t a i n s a c o u n t a b l e fundamental

E

for

K

If

convex space then

g

and e v e r y compact s u b s e t

d e f i n e t h e same e q u i v a l e n c e class i n

Proposition 2.55

Now suppose

O
neighbourhood system a t z e r o ,

(Vn)n,

and hence

\In)

H(K) = l i m (Hm(K+iTn),ll --f

n

i s a bornological

DF

space.

Hence each bounded s u b s e t o f

c o n t a i n e d i n t h e c l o s u r e o f a bounded s u b s e t o f some

is

Hm(K+Vn). S i n c e t h e

Hm(K+Vn) i s a l s o a c l o s e d s u b s e t o f

closed u n i t b a l l of

H(K)

H(K), by lemma

2 . 5 4 , t h i s completes t h e p r o o f A semi-norm c h a r a c t e r i z a t i o n o f t h e bounded s u b s e t s o f

H(K)

is given

i n t h e following p roposition Proposition 2.56

convex space

E.

K

Let A subset

be a compact subset of a rnetrizabze ZocalZy B

of

H(K)

i s bounded i f and onZy i f it

s a t i s f i e s each of the foZlowing conditions: ( a ) f o r each continuous semi-norm

(b) i f

03

( x ~ ) ~and = ~ (x;):=~ m

p

on

H(0)

are two convergent sequences i n

m

(ynInz1

and (yA)n=l are nu22 sequences i n E, (kn);=l a s t r i c t l y increasing sequence of p o s i t i v e integers and xn+yn

Proof

=

xA+y'

n

for a22

n

K, is

then

W e f i r s t show t h a t t h e semi-norms g i v e n i n

(*)

and

(**)

are

Chapter 2

88

continuous with r e s p e c t t o t h e i n d u c t i v e l i m i t topology

T

on

H(K).

Using t h e above n o t a t i o n w e l e t

f o r every

f

Since and

n

in (H(K)

i s a b a r r e l l e d l o c a l l y convex s p a c e and b o t h

T)

induce t h e

H(0)

T~

t o p o l o g y on

8

(nE)

H(K)

f o r each p o s i t i v e i n t e g e r

it s u f f i c e s t o show each o f t h e above semi-norms i s f i n i t e .

Let such t h a t

f

E

f

H(K

be a r b i t r a r y .

c a n be i d e n t i f i e d w i t h an element o f

M = sup lf(x+4y) XEK,YEV

Since

p

C(V) > 0

i s a c o n t i n u o u s semi-norm on

such t h a t

p(g)

n>N

o f zero

Hm(K+4V). L e t

(H(O),T)

t h e r e e x i s t s a constant

c C(V) / / g / / v f o r e v e r y g i n H(0).

Hence

is continuous.

(*)

Now choose a p o s i t i v e i n t e g e r For a l l

V

I .

and each semi-norm o f t h e form

n>N.

There e x i s t s a neighbourhood

N

such t h a t

yn

and

yn '

E

V

for a l l

89

Holomorphic mappings between locally convex spaces

and hence

Thus each semi-norm o f t h e form

set of

H(K)

i s c o n t i n u o u s and any bounded sub-

(**)

s a t i s f i e s c o n d i t i o n s ( a ) and ( b ) . i s a s u b s e t o f H(K) which s a t i s f i e s (fa)aar Using semi-norms o f t h e form (*) we see t h a t

Conversely, suppose ( a ) and ( b ) .

$"fa (x) (-

n!

1

a~y,x~Kn , arbitrary

i s a bounded s u b s e t o f

Since

H(0).

t h e r e e x i s t s a neighbourhood o f

f o r every If

a XEK,

in

r,

x

and

YEW

in aE.T

0

H(0)

in

and a l l

K

E,W,

i s r e g u l a r ( p r o p o s i t i o n 2.55) and

such t h a t

&O

n.

let

To complete t h e p r o o f , it s u f f i c e s t o show t h a t t h e r e e x i s t s a neighbourbood x,x'

V E

o f zero i n

K, y,yf E V

E,

with

VCW,

such t h a t

x+y = x ' + y '

f,(x)(y)

and e v e r y

a in

= fa(x')(y')

r.

for all

90

Chapter 2

If not, there exist sequences in E ,

xn+yn

such that

two sequences in m

(yn)n=l

and

(Y;);=l,

= x’+y’ f o r all

n

K,

n

n

m

03

( x ~ ) ~ =and ~

( x A ) ~ = ~ ,two null

and a sequence

(an):=1

in

r

and

Now choose inductivelz a strictly increasing sequence of positive integers m

(kn)n= 1

such that

2 nSn > n+2M

for all n.

Let

Since

(faIaEr

satisfies condition ( b ) ,

sup q(fa)

<

m.

c1

On the other hand,

This is a contradiction, and completes the proof Condition (a) says that bounded subsets of H(K)

satisfy Cauchy

K while condition (b) says that the Taylor series expansions

estimates on

satisfy certain coherence properties. If K

satisfies certain connected-

ness conditions, then condition (a) is sufficient. That some conditions on K are necessary for boundedness is shown by the following example. Example 2.57

Vn Let

=

Let

IzEE; z

=

E

= C.

x+iy, x <

F o r each positive integer

-1

E

H(V

n

u Wn)

let

and let tun = I z E C ; z=x+iy, x

n + -2

fn

n

be identically one on Vn

1 1 n+z

> __

and identically zero on Wn

91

Holomorphic mappings between locally convex spaces 1 K = { --} n ~ ul { O }

If

n.

for all

then

The sequence

i s a compact s u b s e t o f

K

{fn};=l

and

C

fn

E

H(K)

s a t i s f i e s c o n d i t i o n (a) o f p r o p o s i t i o n

2.56 b u t i s n o t bounded by p r o p o s i t i o n 2.55. We s h a l l u s e p r o p o s i t i o n 2 . 5 6 and i t s method o f p r o o f a g a i n i n a l a t e r

chapter.

The f o l l o w i n g r e s u l t f o l l o w s e a s i l y from t h e c o r r e s p o n d i n g r e s u l t

f o r holomorphic f u n c t i o n s on open s e t s .

a compact subset of a l o c a l l y convex space i s a cornpZete ZocaZly convex space, then ( 8 ( m E ; F ) , ~ , ) is a

P roposi t i on 2.58 E

and

F

K

If

is

closed complemented subspace o f H(K;F)

f o r every p o s i t i v e i n t e g e r

I n d i s c u s s i n g holomorphic f u n c t i o n s on a compact s u b s e t l o c a l l y convex s p a c e complete.

A

If not, l e t

E

b a l a n c e d open s u b s e t o f into

6

d e n o t e t h e completion o f

E.

If

of a

K

w e may always suppose t h a t t h e s p a c e

E

is

E

i s a convex

V

t h e n t h e n a t u r a l r e s t r i c t i o n from

m.

Hm(K+V)

H m ( K + V ~ E ) i s a b i j e c t i v e i s o m e t r y and hence t h e s p a c e s

H(KE)

H(KA) are isomorphic as l o c a l l y convex s p a c e s , ( K E ( r e s p e c t i v e l y E t h e s e t K c o n s i d e r e d as a s u b s e t o f E ( r e s p e c t i v e l y g ) ) .

Kg)

and is

We s h a l l a l s o need h y p o a n a l y t i c germs i n l a t e r c h a p t e r s and t h e d e f i n i t i o n s are e x a c t l y as one would e x p e c t .

If

K

u

i s a compact subset of a l o c a l l y convex space

ence r e l a t i o n s h i p f

N

and

v

by g

and

F

is a

HHY(V;F) 7')e define t h e equivalKCV, V open fNg i f there e x i s t s a neigkbourhood W of K

l o c a l l y convex space then on

on which both

E

are defined and coincide.

open

We l e t

/

and t h e elements of t h i s vector space are caZled F-valued hypoanaZytic germs

on K.

On

Yly(K;F)

we d e f i n e t h e l o c a l l y convex i n d u c t i v e l i m i t t o p o l o g y

W e have used an i n d u c t i v e l i m i t o f t o p o l o g y on

H(K;F).

T

topologies t o define t h e natural

I t i s n o t known i f t h e

as a p r o j e c t i v e l i m i t o f t h e s p a c e s

H(K;F).

T

topology can be recovered

T h i s problem i s r a t h e r impor-

t a n t and a p p e a r s t o b e i n t i m a t e l y r e l a t e d t o t h e l o c a l i z a t i o n problem

92

Chapter 2

is

mentioned p r e v i o u s l y :

T

a l o c a l topology?

w

i n q u i r y , w e c a n d e f i n e a "new'' t o p o l o g y on

a l o c a l l y convex s p a c e and T

.

D e f i n i t i o n 2.59 F

and

U

If

H(U;F),

a n open s u b s e t o f

U

a l o c a l l y convex s p a c e , which w e d e n o t e by

F

I t i s conjectured t h a t

Following t h i s l i n e o f

and

T~

always c o i n c i d e .

T

E

i s an open subset o f a locaZly convex space

i s a Zocally convex space then the

defined as the projective l i m i t o f

topoZogy on H(U;F)

T~

is

H(K;F)

where

on

H(U;F).

I n c h a p t e r s i x , we s h a l l

and

coincide.

K

ranges over the

compact subsets of U. One e a s i l y sees t h a t

T ~ '
~

TS

g i v e examples o f s i t u a t i o n s i n which

52.5

EXERCISES

2.60

Show t h a t

HG(U;F)

dimensional space E .z; (E"). 2.61*

T

w

f o r any open s u b s e t

and a n y l o c a l l y convex s p a c e

E

Show t h a t

H(U;F)

=

T

(E,tf)

U

o f an i n f i n i t e

i f and o n l y i f

F

i s a t o p o l o g i c a l v e c t o r s p a c e i f and o n l y i f

E

has a countable algebraic b a s i s . 2.62 E

onto

If

and

E

TI : ( E , t f )

2.63

If

f o r any 2.64* E.

f o r every 2.65*

@

i s a l i n e a r mapping from

fl

Let

E

i s a n open mapping.

a r e v e c t o r s p a c e s and

E

E

and

A

E

Show t h a t

f

E

it(mE;F)

show t h a t

n6m.

b e Banach s p a c e s and l e t

F

HG(U;Fi). in

(F,tf)

y1 ,..., yn

and

E E

---t

F

E,

in

Let f

and

E

x

Let

a r e v e c t o r s p a c e s and

F

show t h a t

F

H(U;F;)

U

b e a n open s u b s e t o f

i f and o n l y i f

@ o f EH(U)

F. and

F

b e Banach s p a c e s and

U

a connected open s u b s e t

93

Holomorphic mappings between locally convex spaces of

E.

2.66 -

E.

If

by

f(U)

f : U

Let

f(V) C F '

Let f

2.67* -

F*

-f

and suppose

@

0

f EH(U)

f o r some nonempty open s u b s e t

H(U;F)

E

F

and

E

f o r every

of

V

b e Banach s p a c e s and l e t

Let

show t h a t t h e c l o s e d v e c t o r s u b s p a c e o f

and

E

F

b e Banach s p a c e s .

such t h a t

f(Vx)

F

A mapping

generated

F

g e n e r a t e d by

is said

f : E-tF

i f t h e r e e x i s t s a neighbourhood

x

and

F

f EH(U,; Fb).

b e a n open s u b s e t o f

U

i s equal t o t h e c l o s e d v e c t o r subspace o f

t o b e compact a t t h e p o i n t

in

@

Show t h a t

U.

i s a r e l a t i v e l y compact s u b s e t o f

F.

If

of

Vx

f

x

H(E;F)

E

show t h a t t h e f o l l o w i n g are e q u i v a l e n t : f

i s compact a t e v e r y p o i n t o f

(b)

f

i s compact a t some p o i n t o f

(c)

t h e r e e x i s t s a compact s u b s e t

(a)

f(E)

E, E,

K

of

F

such t h a t

i s c o n t a i n e d i n t h e v e c t o r s p a c e spanned by

K,

(d)

t h e t r a n s p o s e mapping

2.68* Suppose

Let fn

f * ( @ ) = $of)

dnf(0)

i s compact f o r a l l

A

(e)

E

=

H(En)

E

f*; (F',To)

(where

m

where each

En

f o r each

n

I:=,

(H(E),ro)

n.

U

i s a l o c a l l y convex s p a c e .

En

and l e t

If t h e r e e x i s t s a neighbourhood

-f

i s continuous,

gn

E

o f zero i n

8 E

Ek)

such t h a t

f o r each

n.

Ilgnllu

+

for all

n

2.69

Show t h a t , t h e c o m p o s i t i o n o f holomorphic ( r e s p e c t i v e l y

show t h a t

F =

gnfnE

c

m

H(E).

h y p o a n a l y t i c ) f u n c t i o n s i s holomorphic ( r e s p e c t i v e l y

M-holomorphic,

M-holomorphic, h y p o a n a l y t i c ) .

2.70

If

i s a l o c a l l y bounded holomorphic f u n c t i o n ,

f

c o n t i n u o u s holomorphic f u n c t i o n and

gof

i s a Mackey

g

i s d e f i n e d , show t h a t

gof

is a

( c o n t i n u o u s ) holomorphic f u n c t i o n . 2.71 -

Let

U

be a n open s u b s e t o f a l o c a l l y convex s p a c e

b e a l o c a l l y convex s p a c e .

If

f

HG(U;F)

and e i t h e r

E

or

and l e t

E F

is

F

Chapter 2

94

separable, show t h a t

f

HHY(U;F)

E

if

gof

E

H(U)

f

HG(E) show t h a t

If

E

i s a Frgchet space and

and only i f

f

i s a Bore1 measurable f u n c t i o n .

2.72*

E

f o r every

g f

in

H(F).

H(E)

E

if

2.73* L e t U be an open subset o f a Banach space E and l e t F be a Banach space with a b a s i s , I f f i s a mapping from U i n t o F

then

f(x)

The mapping of

m

=

f

fn(x)en

where

fn : U

+

C.

i s s a i d t o be normal i f , f o r each compact subset

U

and each p o s i t i v e

fn

E

6,

there exists a positive integer

n(K)

K

such

that

If

H(U)

n

for a l l

show t h a t

f

H(U;F)

E

i f and only i f

f

is a

normal mapping. Let

__ 2.74*

in

be a l o c a l l y convex space and l e t

E

(a)

If

E

i s a Frgchet space o r a83fLvl space show t h a t

l z = l ( $ n y ~ H ( E ) i f and only i f f o r every (b)

If of

x

in

($n)n

show t h a t

bounded s e t s

8

such t h a t

i s a connected

being

as

E

H(E)

n-tm

i f and

contains a fundamental system of

E

( t h e vector span o f

EB B

in 6

HG(U;F)

in

f

f

E

%(U;F).

normed by t h e gauge E

is

TM

complete.

complete l o c a l l y convex show t h a t

f

E

%(U;F)

and every p o s i t i v e i n t e g e r

complete l o c a l l y convex spaces, and

U

is a

U

If

continuous show t h a t

where

T!,

f c HG(U;F)

T~

E

B

then we say t h a t

open subset of t h e

T

space E , F i s a Banach space and if 'dhf ( x ) E 6 M ( n E ; F ) f o r some x 2.76*

0

i s a n u l l sequence and t h e vector span ($n)n i s f i n i t e dimensional.

of B ) i s complete f o r every

F

+

1"n=l( @n )"

I f t h e l o c a l l y convex space

2.75* -

U

@,(x)

E.

E = (E,o(E,E'))

only i f

If

($n)n be a sequence

E'.

T~

open subset of f

ExF, E

n. and

i s s e p a r a t e l y Mackey

95

Holomoiphic mappings between locally convex spaces __ 2.77

If

i s an open s u b s e t of a l o c a l l y convex s p a c e

U

l o c a l l y convex s p a c e , show t h a t f o r each

f

in

HG(U;F)

E

and

is a

F

t h e following are

equivalent: (a)

f E H(U;F)

(b)

f o r each

(respectively in

5

HLB(U;F)),

U

is a n e q u i c o n t i n u o u s ( r e s p e c t i v e l y l o c a l l y bounded) f a m i l y of

mappings. 2.78 -

Let

b e a Banach s p a c e and l e t

E

sequences i n

2.79 -

If

(WE) ,

T~)

f

H( II Ea)

E

where each

If

J

fon

and &(Y), that

If

m

(0,)

n

.

of

A1

i s a l o c a l l y convex s p a c e , show A

and

II

%(afAEa) __ 2.80*

L(($n)n) =

f

i s t h e n a t u r a l p r o j e c t i o n from A1 i s m e t r i z a b l e f o r each a i n A show t h a t

where

A1

Ea

X

E

H( JI Ea) a d1

TI Ea a EA

such t h a t

onto

aZAIEa.

=

and

Y

a r e c o m p l e t e l y r e g u l a r Hausdorff s p a c e s

and

& , (X)

each w i t h t h e compact open t o p o l o g y , are i n f r a b a r r e l l e d , show HLB(,&(X); QCY);)

2.81 __

Ea

~ E A

t h a t there e x i s t s a f i n i t e subset f =

b e d e f i n e d by

i s w e l l d e f i n e d and holomorphic.

L

Show t h a t

-

be t h e s p a c e o f a l l n u l l

endowed w i t h t h e sup norm t o p o l o g y .

E'

L : co(E')

Let

co(E')

Show t h a t

and any open s u b s e t

=

H(U;F)

=

of

& ,

U

H(A(x);

,&(Y)~l-

H (U;F) f o r any l o c a l l y convex s p a c e F HY (X) where X i s a paracompact t o p o l o g i c a l

space. __ 2.82

Show t h a t

any open s u b s e t

U

H(U;F) = %(U;F) of

@,

(X)

where

f o r any l o c a l l y convex s p a c e X

i s a Lindel'df s p a c e .

F

and

96

Chapter 2

If

2.83*

r

is an uncountable discrete set and F

is a separable

Banach space, show that

where the topology on co(r)

is given by the sup norm and

r 'cr T ' countable 2.84* -

F

Let U be a

T~~

open subset of a locally convex space and let

be a locally convex space. Let

T~

denote the topology on $(U;F)

of uniform convergence on the finite dimensional compact subsets of U.

is metrizable if and only if E

(%(U;F),rF)

contains

a countable fundamental system of finite dimensional compact sets and F is metrizable. Show that $(U;F)

T~

induces on each

T~

bounded subset of

the topology of pointwise convergence.

Show that

(%(U;F),T~) is a semi-Monte1 space if and only if F is a semi-Monte1 space. Show that ($(U;F); T ~ ) is quasi-complete if and only if F is quasi-complete. is an arbitrary set, show that the bounded subsets of locally bounded. 2.86 -

If E

is a complete locally convex space and the

subsets of H(E 2.87*

the T~ U of E.

Let E and T

are locally bounded, show that E =

T~

bounded

is barrelled.

En where each En is a Banach space. Show that bounded subsets of H(U) coincide for every open subset

We say that a countable increasing cover 3 = (Fn)n=l of a topological space U is k-dominating if each F, is closed and each compact subset of U is contained in some 'n. Let U be an open subset of 2.88*

50

97

Holomorphic mappings between locally convex spaces a l o c a l l y convex s p a c e all

n}.

H

"

(U)

t h e semi-norms

E

and l e t

Hg(U)

I f cHG(U);

=

llfllF<

pn(f) =

Ildg .

Let

n

(H(U),T

6K

)

l i m H3(U)

=

2+

r a n g e s o v e r a l l c o u n t a b l e i n c r e a s i n g k-dominating c o v e r s o f T

6K

is t h e bornological topology a s s o c i a t e d w ith

(H(U),T

i s a n u l t r a - b o r n o l o g i c a l space i f

)

6K

-

for

i s endowed w i t h t h e l o c a l l y convex t o p o l o g y z e n e r a t e d by

U

T

0

on

U.

where

.3

Show t h a t

H(U).

Show t h a t

i s a k-space.

2.89 On H(U), U a n open s u b s e t o f a l o c a l l y convex s p a c e , l e t T P d e n o t e t h e t o p o l o g y o f p o i n t w i s e convergence. I f = (Fn):=l is a

3

countable i n c r e a s i n g cover o f

c o n s i s t i n g o f c l o s e d sets and

U

H3(U)

is

t h e a s s o c i a t e d m e t r i z a b l e s p a c e o f holomorphic f u n c t i o n s , show t h a t

is t h e bornological space associated with

l i m H 3(U)

(H(U) , T ~ ) . (

3

-3+ r a n g e s o v e r a l l p o s s i b l e c o v e r s o f t h e above t y p e ) .

If

2.90 and

i s a n open s u b s e t o f a m e t r i z a b l e l o c a l l y convex s p a c e

U

i s a Banach s p a c e , show t h a t

F

E

(H(U;F),T&) i s a r e g u l a r i n d u c t i v e

l i m i t of Frgchet spaces. If t h e l o c a l l y convex s p a c e

2.91*

c o n v e r g e n t sequence, show t h a t

Ei

E

contains a nontrivial very strongly

c o n t a i n s a sequence which i s n o t v e r y

weakly c o n v e r g e n t . __ 2.92

If

i s a s e p a r a b l e l o c a l l y convex s p a c e i n which e v e r y sequence

E

i s v e r y weakly c o n v e r g e n t , show t h a t 2.93

Let

E

be a F r g c h e t s p a c e .

EL

a d m i t s a c o n t i n u o u s norm.

Show t h a t t h e f o l l o w i n g are e q u i v a l -

ent : (a)

E

c o n t a i n s a n o n t r i v i a l v e r y s t r o n g l y c o n v e r g e n t sequence,

(b)

E

c o n t a i n s a s u b s p a c e isomorphic t o

(c) (d)

i s a quotient of

CN,

E'

a'

e v e r y c o n t i n u o u s semi-norm on

E

has an i n f i n i t e dimensional

kernel. Show t h a t ( a ) and (d) a r e n o t e q u i v a l e n t f o r a r b i t r a r y l o c a l l y convex spaces.

Chapter 2

98

Let

2.94*

~

Let

E,

E A

{xgE;

=

3 (x~):=~ E

.

c o n v e r g e n t sequence}. f

E

A

b e a m e t r i z a b l e l o c a l l y convex s p a c e w i t h completion

E

such t h a t

i s a v e c t o r subspace o f

E,

Show t h a t

E.

( x ~ - x ) i~s a v e r y s t r o n g l y

-

A

E.

If

show, by u s i n g t h e f a c t o r i z a t i o n lemma o r o t h e r w i s e , t h a t t h e r e

H(E),

d

e x i s t s a unique

f

in

H(E,)

such t h a t

flE = f .

Generalise t h i s r e s u l t

t o a r b i t r a r y l o c a l l y convex s p a c e s .

~

Let

2.95 (H(U),-ro)

b e a n open s u b s e t of a l o c a l l y convex s p a c e .

U

i s a l o c a l l y m-convex a l g e b r a .

m u l t i p l i c a t i v e semi-norm on

~

Let

2.96*

f

Let

E

H(U;F).

F-valued holomorphic germs a t rJ

f(x)

show t h a t

N

f

If

2.97"

T~

Show t h a t e v e r y

=

{germ o f

If in

0 f

?: E)

at

T~

continuous

continuous.

be a n open s u b s e t of a Banach s p a c e

U

a Banach s p a c e .

is

H(U)

Show t h a t

U

-f

E

H(0;F)

F

and l e t

be

( t h e s p a c e of

i s d e f i n e d by

x

translated t o the origin)

HG(U;H(O;F)). i s a compact s u b s e t o f a l o c a l l y convex s p a c e

K

i s a l o c a l l y convex s p a c e , show t h a t

H(K;F)

and

E

F

i s a Hausdorff l o c a l l y convex

space. Show t h a t

__ 2.98*

H(K)

i s n e v e r a s t r i c t i n d u c t i v e l i m i t when

is a

I<

compact s u b s e t of a l o c a l l y convex s p a c e . If

__ 2.99*

space

E

neighbourhood

f o r every 2.100*

i s a convex b a l a n c e d s u b s e t of a m e t r i z a b l e l o c a l l y convex

K

show t h a t

5 If

V

in U

B CH(K)

i s bounded i f and o n l y i f t h e r e e x i s t

of z e r o and two p o s i t i v e numbers

K

and e v e r y n o n - n e g a t i v e i n t e g e r

c

and

C

2.101"

F' B

'2

If

such t h a t

m.

i s a n open s u b s e t of a Banach s p a c e , show t h a t

a d u a l Banach s p a c e ( i . e . show t h a t t h e r e e x i s t s a Banach s p a c e that

a

Hm(U)

F

is

such

Hm(U)).

K

i s a compact s u b s e t o f a m e t r i z a b l e l o c a l l y convex s p a c e ,

99

Holomorphic mappings between locally convex spaces show t h a t

m -convex a l g e b r a .

is a l o c a l l y

H(K)

Show t h a t a convex b a l a n c e d s u b s e t of

2.102

o f a l o c a l l y convex s p a c e

is a

E,

T&

i f it a b s o r b s e v e r y e q u i c o n t i n u o u s s u b s e t o f 2.103*

Let

integer

n

rn

let

an open s u b s e t

U

H(U).

b e a l o c a l l y convex space.

E

H(U),

neighbourhood o f z e r o i f and o n l y

d e n o t e t h e t o p o l o g y on

For each n o n - n e g a t i v e g e n e r a t e d by t h e semi-

H(E)

norms

where

K

r a n g e s o v e r t h e compact s u b s e t s o f

bounded s u b s e t s o f

B

ranges over t h e

E.

rn =

Show t h a t

and

E

T

~

+f o r~

some (and hence f o r a l l )

n

i f and o n l y i f

each bounded s u b s e t o f

E

i s c o n t a i n e d i n t h e c l o s e d convex h u l l o f a

compact s e t . A subset

2.104"

K

of a l o c a l l y convex s p a c e

i s s a i d t o be s t r i c t l y

E

compact i f t h e r e e x i s t s a convex b a l a n c e d bounded s u b s e t that

K

i s c o n t a i n e d and compact i n

EB.

On

%(E)

B

let

of rS

E

t o p o l o g y o f uniform convergence on t h e s t r i c t l y compact s u b s e t s of Show t h a t

(%(E),rS)

2.105"

If

f

t h e r e e x i s t s a neighbourhood

52.6

E.

i s a semi-Monte1 s p a c e .

i s a n e n t i r e f u n c t i o n on t h e

every p o s i t i v e i n t e g e r

such

denote t h e

n.

V

a3w

o f z e r o such t h a t

space

/If

lLv

E,

show t h a t

is f i n i t e for

NOTES AND RtPIARKS

The h i s t o r i c a l n o t e s o f A . E . T a y l o r [680] a r e p r o b a b l y t h e most comprehensive g u i d e t o t h e development o f i n f i n i t e dimensional holomorphy up t h e mid-nineteen f o r t i e s a v a i l a b l e and w e have made e x t e n s i v e u s e o f t h e s e n o t e s i n 11.6 and s h a l l do s o a g a i n i n t h i s s e c t i o n .

In another

p a p e r , [681], A . E . T a y l o r documents t h e r o l e of a n a l y t i c i t y i n o p e r a t o r and s p e c t r a l t h e o r y and h i s h i s t o r y of t h e d i f f e r e n t i a l i n t h e n i n e t e e n t h

Chapter 2

100

and t w e n t i e t h c e n t u r i e s [682] c o n t a i n s much o f i n t e r e s t c o n c e r n i n g t h e growth o f i n f i n i t e d i m e n s i o n a l a n a l y s i s .

E . H i l l e and R.S. P h i l l i p s ( [ 3 3 4 ] ,

s e c t i o n 3.16 and c h a p t e r 26) a l s o p r o v i d e a summary o f t h e fundamentals known a t t h a t t i m e

-

1958

-

t o g e t h e r w i t h a s h o r t b i b l i o g r a p h y on t h e sub-

j e c t ( c h a p t e r 26 was w r i t t e n i n c o l l a b o r a t i o n w i t h M.A.

Zorn).

D. Pisanelli

[578] g i v e s a s u r v e y o f most o f t h e d i f f e r e n t c o n c e p t s o f holomorphic f u n c t i o n t h a t are c u r r e n t l y i n u s e and documents t h e i r o r i g i n . V.I. Averbukh and O . G . Smolyanov ( [ 4 0 ] , § 2 ) g i v e a comprehensive a c c o u n t of t h e development o f t h e concept of t h e d i f f e r e n t i a l i n t o p o l o g i c a l v e c t o r spaces.

They a r e m a i n l y concerned with the r e a l t h e o r y b u t n a t u r -

a l l y t h i s h a s many consequences f o r t h e complex c a s e .

An i n t e r e s t i n g p o i n t

t h a t emerges from [40] i s t h a t t h e s e a r c h f o r a d e f i n i t i o n o f d i f f e r e n t i a b l e f u n c t i o n between r e a l l i n e a r t o p o l o g i c a l v e c t o r s p a c e s h a s been i n p r o g r e s s more o r l e s s c o n t i n u o u s l y f o r t h e l a s t s e v e n t y y e a r s w i t h many d i f f e r e n t d e f i n i t i o n s b e i n g proposed - t h e a u t h o r s l i s t t w e n t y - f i v e i n [40] - and s i n c e a number o f d e f i n i t i o n s were n o t s e e n t o b e e q u i v a l e n t and i n d e e d , v a r i o u s a u t h o r s were n o t aware o f one a n o t h e r ' s work, t h i s l e d t o a c e r t a i n

a mount o f chaos - o r a t l e a s t a p p a r e n t chaos - which h a s o n l y been r e c t i f i e d i n [40].

F o r t u n a t e l y f o r d i f f e r e n t i a t i o n o v e r complex s p a c e s we

have power series e x p a n s i o n s and t h i s h a s l e d t o a f a i r l y unanimous a c c e p t a n c e o f Frgchet d i f f e r e n t i a b i l i t y as t h e s t a n d a r d d e f i n i t i o n w i t h minor r o l e s b e i n g p l a y e d by GGteaux holomorphic, Mackey o r S i l v a holomorp h i c and h y p o a n a l y t i c f u n c t i o n s . The names c h i e f l y a s s o c i a t e d w i t h t h e development o f r e a l i n f i n i t e d i m e n s i o n a l d i f f e r e n t i a l c a l c u l u s a r e ( s e e [40] f o r more d e t a i l e d i n f o r mation) V . V o l t e r r a , J . Hadamard, M . F r g c h e t , R . Ggteaux, P . L&y,

A.D.

Michal, L . M . Graves, E.W. Paxson, J . G i l d e Lamadrid, J . Sebasti'go e S i l v a , H.H.

Keller and G . Marinescu.

I n p a r t i c u l a r , we would l i k e t o mention

J . Hadamard, M. F r g c h e t and A . D . Michal.

Hadamard was one o f t h e f i r s t t o

r e c o g n i s e t h e importance o f V o l t e r r a ' s work, h e e x e r c i s e d a d e e p i n f l u e n c e

on h i s s t u d e n t s FrGchet and Gsteaux and was always v e r y i n s i s t a n t on l i n e a r i t y b e i n g i n c o r p o r a t e d i n t o any d e f i n i t i o n o f t h e d e r i v a t i v e .

M . F r g c h e t and A . D . Michal b o t h s p e n t a c o n s i d e r a b l e p o r t i o n of t h e i r l i v e s developing t h e t h e o r y

-

r o u g h l y f o r t y y e a r s by F r g c h e t - and opened many

avenues s t i l l being explored. b e found i n P . L&y

([442],

Readable a c c o u n t s o f t h e r e a l t h e o r y a r e t o

b o t h t h e 1922 and 1950 e d i t i o n s ) , A . D . Michal

101

Holomoiphic mappings between locally convex spaces [491], V.I. Averbukh and O . G .

Smolyanov [39], J . P . Penot [567], A.

Frb'hlicher and W . Bucher [247] and S. Yamamuro [718]. Surveys of a more s p e c i a l i s e d n a t u r e but which a r e more r e l e v a n t t o t h e t o p i c s discussed i n t h i s book a r e L . Nachbin [518], M . Schottenloher B i e r s t e d t and R . Meise [70], P . Krce [405] and S . Dineen [198].

[642], K.D.

We now p r e s e n t our b r i e f h i s t o r y . We have a l r e a d y noted i n 51.6 t h a t t h e d e f i n i t i v e s t e p s i n t h e c r e a t i o n of r e a l and complex i n f i n i t e dimensional d i f f e r e n t i a l c a l c u l u s were taken by V . V o l t e r r a (1887) and D . H i l b e r t (1909) r e s p e c t i v e l y . P r i o r t o t h e s e , t h e works of J . Bernoulli on t h e curve of quickest descent, L . Euler on t h e c a l c u l u s of v a r i a t i o n s (see J . Hadamard [298]) and H. Von

Koch [393], on i n f i n i t e systems of d i f f e r e n t i a l equations may a l s o be mentioned a s of p a r t i c u l a r importance f o r l a t e r developments.

D. Hilbert

[332] discussed a n a l y t i c continuation and t h e composition of holomorphic f u n c t i o n s but h i s work i s not easy t o read, and a t times r a t h e r vague. Next came R. G2teaux [251,252,253] who looked a t both r e a l and complex analysis

-

a s opposed t o M. Frgchet who only s t u d i e d t h e r e a l c a s e

s e t out t o c l a r i f y and extend t h e works of Frgchet and H i l b e r t . he succeeded b r i l l i a n t l y .

-

and

In t h i s

I n [252], he defined complex polynomials and

n o t i c e d t h e i r r e l a t i o n s h i p with b i l i n e a r and q u a d r a t i c forms, defined t h e "G$teaux" d e r i v a t i v e , extended t h e Cauchy i n t e g r a l formula and t h e Cauchy i n e q u a l i t i e s , proved an i d e n t i t y theorem and v a r i o u s convergence theorems f o r holomorphic functions, e s t a b l i s h e d t h e now standard correspondence between d e r i v a t i v e s of a holomorphic function and t h e homogeneous polynomials i n i t s Taylor series expansion and gave various r e s u l t s about a n a l y t i c continuation and power s e r i e s expansions - a l l i n an i n f i n i t e dimensional s e t t i n g .

H i s o t h e r paper [253] i s a l s o q u i t e i n t e r e s t i n g and

contains b r i e f o u t l i n e s of a number of t o p i c s which h i s untimely death prevented him from developing.

CN,

&[a,b]

and

G2teauxIs work was confined t o t h e spaces

.Q2.

In 1920, S . Banach gave, i n h i s t h e s i s , t h e axioms f o r a complete normed l i n e a r space (a Banach space) and i n 1923, N.Wiener

[715], who

i n c i d e n t a l l y gave independently t h e same axioms t h r e e months a f t e r Banach,

102

Chapter 2

n o t i c e d t h a t t h e Cauchy i n t e g r a l formula e x t e n d s t o Banach v a l u e d holomorp h i c f u n c t i o n s o f one complex v a r i a b l e and i n t h i s s e t t i n g , many o f t h e c l a s s i c a l r e s u l t s such as M o r e r a ' s theorem, A b e l ' s theorem and t h e r e s i d u e theorem are a l s o v a l i d .

L . FantappiGIs long a r t i c l e [234] appeared i n 1930

I n i t h e proposed t h a t a c o n t i n u o u s f u n c t i o n aBanach s p a c e be c a l l e d holomorphic i f

f

d e f i n e d on a domain

f o $ i s holomorphic ( a s a f u n c t i o n

o f one complex v a r i a b l e ) f o r a n y holomorphic f u n c t i o n @ t h e complex p l a n e i n t o

D.

in

D

from a domain i n

F a n t a p p i g ' s work s e r v e d as m o t i v a t i o n f o r A . E .

T a y l o r 116751 and was developed and p u t i n a modern c o n t e x t u s i n g t o p o l o g i c a l v e c t o r s p a c e s by J . S e b a s t i r o e S i l v a [649,650,651,652,653]

(see a l s o

D. P i s a n e l l i [578]).

I n t h e n i n e t e e n t h i r t i e s , an i n t e n s i v e s t u d y o f t h e whole f i e l d o f a n a l y s i s a n d geometry i n a b s t r a c t s p a c e s was c a r r i e d o u t by A . D . Michal and h i s s t u d e n t s R . S . M a r t i n , A . H . C l i f f o r d , I . E . Highberg and A . E . T a y l o r .

In

1932, R . S . M a r t i n [449] developed t h e t h e o r y o f holomorphic mappings f o r Banach s p a c e s u s i n g a power series approach ( s e e a l s o A . D . Michal and A.H. C l i f f o r d [492]).

The f i n a l s t e p towards t h e c u r r e n t d e f i n i t i o n of holomor-

p h i c mapping between Banach s p a c e s was t a k e n i n d e p e n d e n t l y by L.M. Graves i n 1935 [275] and A . E . b r i e f , p.649-653,

T a y l o r i n 1937 [ 6 7 5 ] .

Graves' t r e a t m e n t i s r a t h e r

and i n it h e n o t e s t h a t Ggteaux d i f f e r e n t i a b i l i t y p l u s

c o n t i n u i t y are e q u i v a l e n t t o F r g c h e t d i f f e r e n t i a b i l i t y ( d e f i n e d f o r normed l i n e a r s p a c e s by F r g c h e t i n 1925, "2451) when d e a l i n g w i t h f u n c t i o n s between complex normed l i n e a r s p a c e s and a l s o t h a t t h i s l e a d s t o a s a t i s f a c t o r y t h e o r y o f holomorphic f u n c t i o n s between normed l i n e a r s p a c e s . T a y l o r ' s work i s much more e x p l i c i t .

H e d e f i n e s a holomorphic f u n c t i o n as

a c o n t i n u o u s f u n c t i o n whose one d i m e n s i o n a l s e c t i o n s are holomorphic.

This

d e f i n i t i o n i s a n a t u r a l outgrowth o f t h e work o f G2teaux and b r i n g s t o g e t h e r t h e i d e a s o f many o f h i s p r e d e c e s s o r s .

H e proved t h e f o l l o w i n g

result: If

f

i s a mapping between normed l i n e a r s p a c e s , t h e n t h e f o l l o w i n g

are equivalent a)

f

i s c o n t i n u o u s and h a s a Ggteaux d e r i v a t i v e a t each

point; b)

f

h a s a F r g c h e t d e r i v a t i v e a t each p o i n t ;

c)

f

h a s a power s e r i e s expansion which converges

103

Holomorphir mappings between locally convex spaces u n i f o r m l y i n a neighbourhood o f each p o i n t . T a y l o r goes on t o g e n e r a l i s e Riemann's theorem on removable s i n g u l a r i t i e s , M i t t a g - L e f f l e r ' s theorem, L i o u v i l l e ' s theorem and t h e CauchyRiemann e q u a t i o n s .

He a l s o shows [675,p.292] an awareness o f t h e f a c t t h a t

t h e r a d i u s o f uniform convergence i s n o t e q u i v a l e n t t o t h e r a d i u s o f a n a l y t i c i t y and t h i s i s t h e f i r s t s t e p i n a n e x t e n s i v e s t u d y o f t h e r a d i u s o f boundedness which w e u n d e r t a k e i n c h a p t e r 4 .

I n a f u r t h e r p a p e r [678],

A . E . T a y l o r d i s c u s s e s a number o f i n t e r e s t i n g examples o f holomorphic

f u n c t i o n s on

s p a c e s and g i v e s a g e n e r a l i z a t i o n of H a r t o g s ' theorem on

kP

separate analyticity.

The P o l i s h s c h o o l o f f u n c t i o n a l a n a l y s i s (S. Banach,

S. Mazur, W . O r l i c z , e t c . ) a l s o c o n t r i b u t e d d u r i n g t h i s p e r i o d ( s e e c h a p t e r 1) and a l t h o u g h t h i s t r a d i t i o n a l

interest

i n non-linear functional

a n a l y s i s was m a i n t a i n e d u n t i l r e l a t i v e l y r e c e n t l y ( s e e A . Alexiewicz and W . O r l i c z [ 7 ] , W .

[563,564,565]),

f o r instance

Bogdanowicz [76] and A . P e l c z y n s k i

it was l a r g e l y overshadowed by t h e r a p i d development of

t h e l i n e a r theory.

T h i s predominance o f l i n e a r f u n c t i o n a l a n a l y s i s may b e

t r a c e d t o t h e i n f l u e n c e o f S. Banach's book [44], p u b l i s h e d i n 1932, a l t h o u g h Banach h i m s e l f hoped t o d e v e l o p t h e n o n - l i n e a r t h e o r y i f we a r e t o j u d g e from h i s comment [ 4 4 , p . 2 3 1 ] on complex v e c t o r s p a c e s

"Ces espaces c o n s t i t u e n t Ze point de de'part de l a the'orie des op&ations

Zingaires cornpZexes e t d ' m e cZasse, encore pZus

vaste, des op6rations anazytiques, qui prgsentent une g'enerali s a t i o n des fonctions anaZytiques ordinaires icf. p . ex. L . Fantappi;,

I . funz?:onaZi a n a l i t i c i , G i t t a d i Caste220 1 9 3 0 ) .

Nous m u s proposons d 'en exposer Za thebrie dans un autre vo ZTme

".

See a l s o t h e p r e f a c e t o [ 4 4 ] . M . A . Zorn [723,724,725] made a number o f i m p o r t a n t c o n t r i b u t i o n s i n

t h e m i d - f o r t i e s and w e refer t o h i s r e s u l t s l a t e r i n t h e t e x t .

The n i n e -

t e e n f i f t i e s saw t h e a p p e a r a n c e of A . G r o t h e n d i e c k ' s memoir [287] on t o p o l o g i c a l t e n s o r p r o d u c t s and n u c l e a r s p a c e s and t h i s h a s had and w i l l c o n t i n u e t o h a v e a c o n s i d e r a b l e i n f l u e n c e on t h e development o f i n f i n i t e d i m e n s i o n a l holomorphy.

During t h i s p e r i o d , J . S e b a s t i z e S i l v a a l s o

developed h i s t h e o r y o f i n f i n i t e d i m e n s i o n a l holomorphy and H . J .

Bremermann

[103,104,105] proved v a r i o u s r e s u l t s on pseudo-convex domains and t u b u l a r

104

Chapter 2

domains o f holomorphy i n Banach s p a c e s . T h i s b r i n g s u s t o t h e modern p e r i o d , t h e l a s t s i x t e e n y e a r s , d u r i n g which time most o f t h e work we d i s c u s s h a s been d i s c o v e r e d .

The con-

t r i b u t i o n s o f t h e various authors during t h i s period are d e t a i l e d i n t h e f i n a l s e c t i o n o f each c h a p t e r . The concept o f Mackey holomorphic f u n c t i o n i s u s u a l l y a t t r i b u t e d t o J . S e b a s t i a o e S i l v a [652], a l t h o u g h an e a r l i e r e q u i v a l e n t d e f i n i t i o n (con-

d i t i o n 2 . 1 8 ( d ) ) i s due t o L . F a n t a p p i g [234].

The t h e o r y o f Mackey holo-

morphic f u n c t i o n s f i n d s i t s most n a t u r a l e x p r e s s i o n w i t h i n t h e language of b o r n o l o g i e s - s e e f o r i n s t a n c e D . P i s a n e l l i [578], D . Lazet [423] and J . F . Colombeau [141]. E . H i l l e [333].

L o c a l l y bounded holomorphic f u n c t i o n s were i n t r o d u c e d by I t i s d i f f i c u l t t o locate t h e o r i g i n of hypoanaliticity

( d e f i n i t i o n 2.15) b u t t h e name u s u a l l y a s s o c i a t e d w i t h t h e r e a l analogueh y p o c o n t i n u i t y i s J . L . K e l l e y who a l s o i n t r o d u c e d t h e concept of k-space ( s e e , f o r i n s t a n c e , S. W i l l a r d [716] o r Z . Semadeni [654]).

Examples 2.16,

2.19 and 2 . 2 0 t o g e t h e r w i t h f u r t h e r examples o f a similar k i n d are g i v e n i n S. Dineen [190] ( s e e a l s o c h a p t e r 5 ) .

The c o n n e c t i o n between H a r t o g s '

theorem and l o c a l l y bounded holomorphic f u n c t i o n s ( p r o p o s i t i o n 2 . 1 2 , examples 2 . 1 3 and 2.14) a p p e a r s i n [190] and i n f i n i t e dimensional g e n e r a l i z a t i o n s o f H a r t o g s ' theorem a r e d i s c u s s e d i n t h e comments on e x e r c i s e 2.76. F a c t o r i z a t i o n r e s u l t s (most o f which depend on L i o u v i l l e ' s theorem c o n c e r n i n g bounded e n t i r e f u n c t i o n s ) are i m p l i c i t i n A. Hirschowitz [335],

C . E . R i c k a r t [605] and L . Nachbin [514]. S. Dineen [188,189,190,191]

and

E . Ligocka [443], developed i n d e p e n d e n t l y a g e n e r a l t h e o r y o f f a c t o r i z a t i o n

which we d i s c u s s i n more d e t a i l i n 16.2 and a p p l i e d i t t o such t o p i c s as t h e Levi problem, RungeS theorem, H a r t o g s ' theorem, holomorphic c o m p l e t i o n , Zorn's theorem and t o p o l o g i e s on s p a c e s o f holomorphic f u n c t i o n s .

Factor-

i z a t i o n methods h a v e a l s o been a p p l i e d t o t h e Levi problem by Ph. Noverraz [540,544,546],

S. Dineen [183,186],

B. J o s e f s o n [358], M . S c h o t t e n l o h e r

[640], J . F . Colombeau and J . Mujica [154], J . Mujica [506],to t h e c o n s t r u c t i o n o f t h e envelope o f holomorphy by M . S c h o t t e n l o h e r [640] and P. Berner [59,60], i n s t u d y i n g meromorphic f u n c t i o n s and t h e Cousin I problem by V . Aurich

[35] and S. Dineen [192], t o t h e t h e o r y of c o n v o l u t i o n o p e r a t o r s

by P. Berner [ 6 2 ] and M.C. Matos [463,467] and i n s t u d y i n g l o c a l l y convex t o p o l o g i e s on s p a c e s of holomorphic f u n c t i o n s by P . Berner [61], S. Dineen

105

Holomotphic mappings between locally convex spaces

[194], and Ph. Noverraz [552]. Theorem 2 . 2 1 and proposition 2.33 are given in [190;. Example2.22and proposition 2.24 aredue to L. Nachbin [514] and a generalization can be found in S. Dineen [190]. subset U

in (EN and

A. Hirschowitz [335] gives an example of an open

f &H(U)

which admits a local but not a global

factorization. This example shows that condition (*) of theorem 2 . 2 1 is not sufficient to obtain a global factorization result. The difficulties posed by Hirschowitz's example were overcome by P. Berner [59] and M. Schottenloher [640]. Examples 2.22, 2.25 and 2.26 are due to L. Nachbin [514], and the proof of example 2 . 2 2 appears in [193].

Proposition 2.27 is

due to S. Dineen [190]. Theorem 2.28 is due to M.A. Zorn [724].

Generalizations to other

locally convex spaces are due to Ph. Noverraz [536,537], D. Pisanelli [578], A. Hirschowitz [341], D. Lazet [423], J.F. Colombeau [141], E. Ligocka [443]

and S. Dineen [190].

A locally convex space E

for which the con-

clusion of proposition 2.29 is valid is called an w-space (S. Dineen

[184],

[190]). E . Grusell [292,293] and M. Schottenloher [645] give examples of locally convex spaces which are not w-spaces. Corollary 2.30 is due to J.F. Colombeau and J. Mujica [154], who use it together with the solution to the Levi problem for Hilbert spaces (L. Gruman [290]) to solve the Levi problem in strong duals of Frgchet nuclear spaces (&3?:spaces). Example 2.31, which arose in constructing a counterexample to the Levi problem, is due to B. Josefson [358] andthe proof given here is taken from

S. Dineen [193].

The inequality of example 2.31 is used in A. Renaud

([604, proposition 4) to obtain a vector-valued Schwarz lemma.

Other

factorization results may be proved by considering locally bounded holomorphic functions and in such cases one gets results for non-normed range spaces [190]. The topological vector space structure of H(U),

U an open subset of

Cn, has been studied by a number of authors including A. Grothendieck [285], G. Kothe [396] and A. Martineau [451]. The compact open topology was first investigated on spaces of holomorphic functions in infinitely many variables by H. Alexander [S] and L . Nachbin [509]. They found, however, that it did not possess very good topological properties and s o , motivated by properties of analytic functionals in several complex variables

Chapter 2

106

due t o A. Martineau [450], L . Nachbin I5091 i n t r o d u c e d t h e ( d e f i n i t i o n s 2 . 3 3 and 2 . 3 4 ) .

The

topology

T~

t o p o l o g y ( d e f i n i t i o n s 2.33 and 2.34)

T&

was f i r s t i n t r o d u c e d f o r holomorphic f u n c t i o n s on s e p a r a b l e Banach s p a c e s by G . Coeur;

[128,129] i n c o n n e c t i o n w i t h problems o f a n a l y t i c c o n t i n u a t i o n

and t h e g e n e r a l d e f i n i t i o n i s due t o L. Nachbin [510]. and 2.41 are g i v e n i n S. Dineen [185]. known and a p r o o f i s g i v e n i n K - D . c o r r e s p o n d i n g problems f o r t h e

T~

P r o p o s i t i o n s 2.38

Lemma 2.39 a p p e a r s t o b e w e l l

B i e r s t e d t and R . Meise [ 6 9 ] . and

T~

The

topologies, i . e . are t h e s e

l o c a l o r s h e a f t o p o l o g i e s ? are p e r h a p s t h e most i n t e r e s t i n g c u r r e n t open problems i n t h e g e n e r a l t h e o r y o f l o c a l l y convex s p a c e s of holomorphic f u n c t i o n s i n i n f i n i t e l y many v a r i a b l e s .

A r e l a t e d open problem i s t o

c h a r a c t e r i z e t h e open sets i n which

and

l6

7

continuous m u l t i p l i c a t i v e

l i n e a r f u n c t i o n a l s are p o i n t e v a l u a t i o n s (by e x e r c i s e 2 . 9 5 t h e the

T

continuous m u l t i p l i c a t i v e linear f unc t i o n a l s c o i n c i d e ).

T~

and This

problem arises i n s o l v i n g t h e Levi problem and i n c o n s t r u c t i n g t h e envelope R e s u l t s d e a l i n g w i t h t h i s problem are t o be found i n

of holomorphy.

H. Alexander [S], G . Coeur:

641,6431, J . M .

[129,131], M. S c h o t t e n l o h e r [632,633,634,635,

I s i d r o [352] and J . Mujica [502,503,504,505].

Lemma 2.43,

c o r o l l a r y 2.44 and example 2.46 are proved i n S. Dineen [185]. 2 . 4 5 and example 2.47 a r e t o be found i n S . Dineen [194].

For

Corollary

832

( s t r o n g d u a l s o f F r g c h e t Schwartz) s p a c e s , example 2.47 i s due t o J . A . Barroso, M . C .

Matos and L . Nachbin [ 5 0 ] .

Reference [194] i s devoted t o t h e

t h e o r y o f holomorphic f u n c t i o n s on &$hl ( s t r o n g d u a l s o f Fr‘echet Monte1 s p a c e s ) and i t i s i n t e r e s t i n g t o n o t e t h a t a l t h o u g h many r e s u l t s on 8 3 8 s p a c e s ( f o r i n s t a n c e , example 2.47) can be extended t o holomorphic f u n c t i o n s on83K;spaces, t h e methods o f proof a r e q u i t e d i f f e r e n t .

OnB&f

s p a c e s , one u s e s f r e q u e n t l y t h e p r o p e r t y t h a t such s p a c e s are i n d u c t i v e l i m i t s o f Banach s p a c e s w i t h compact l i n k i n g mappings i n t h e c a t e g o r y o f

t o p o l o g i c a l s p a c e s ( t h i s i s n o t t r u e o f &J-h’L

s p a c e s ) w h i l e on

8 3%

s p a c e s , one u s e s t h e f a c t t h a t such s p a c e s a r e k-spaces and h e r e d i t a r y Lindelb’f s p a c e s .

Example 2.48 i s due t o L . Nachbin [SO91 and example 2.49

can b e found i n S . Dineen [185]. Very s t r o n g l y convergent s e q u e n c e s ( d e f i n i t i o n 2.50) were i n t r o d u c e d by S . Dineen [184] i n s t u d y i n g holomorphic completions ( s e e 5 4 . 4 ) .

Subsequent-

l y t h e y were used by S . Dineen [185] i n counterexample 2.52 and i n [190] t h e y were a p p l i e d t o g e t h e r w i t h v e r y s t r o n g l y convergent n e t s and v e r y weakly c o n v e r g e n t sequences ( d e f i n i t i o n 2 . 5 1 ) i n o t h e r areas o f i n f i n i t e

107

Holomorphic mappings between locally convex spaces dimensional holomorphy.

F u r t h e r a p p l i c a t i o n s a r e t o b e found i n P . Berner

[58,59] and S. Dineen and Ph. Noverraz [ 2 0 5 ] . T

0

= T

0

A on H(C )

example 2 . 5 2 ,

i f and o n l y i f

A

J.A.

Barroso [46] shows t h a t

i s c o u n t a b l e and t h i s , t o g e t h e r w i t h

completes t h e p i c t u r e o f t h e r e l a t i o n s h i p between t h e d i f f e r -

e n t t o p o l o g i e s on

A

H(C*),

a r b i t r a r y ( s e e a l s o c h a p t e r 5, J . A .

Barroso

and L . Nachbin [53] and V . Aurich [ 3 3 ] ) . Holomorphic germs on compact s u b s e t s o f Banach s p a c e s were f i r s t i n v e s t i g a t e d by S.B. Chae [119,120] and A . Hirschowitz [339,343].

Subse-

q u e n t l y , J . Mujica [503] developed t h e t h e o r y on m e t r i z a b l e l o c a l l y convex spaces ( s e e a l s o R.R. 3. Mujica [41], K-D.

Baldino [ 4 3 ] , A . J . M .

Wanderley [714], P . A v i l e s and

B i e r s t e d t and R . Meise [69,70] and E . Nelimarkka [525])

Lemma 2.54 and p r o p o s i t i o n 2.55 are due t o J . Mujica [503].

Earlier,

S.B. Chae [120] and A . H i r s c h o w i t z [339] had g i v e n incomplete p r o o f s f o r t h e Banach s p a c e case.

In p r o v i d i n g a complete p r o o f , A. Hirschowitz

i n t r o d u c e d semi-norms s i m i l a r t o t h o s e which a p p e a r i n p r o p o s i t i o n 2.56 ( s e e a l s o S. Dineen [ZOO]).

The q u e s t i o n o f whether o r n o t c o n d i t i o n (b)

o f p r o p o s i t i o n 2.56 i s redundant i s answered n e g a t i v e l y by example 2.57 (due t o R.M. W.R.

Aron) and h a s l e d t o some i n t e r e s t i n g i n v e s t i g a t i o n s by

Z a m e [721], K. Rusak [617] and J . T . Rogers, J r . and W.R.

f i n i t e dimensions and by R . L .

Z a m e [609] i n

S o r a g g i [664,665,666] i n i n f i n i t e dimensions.

Rogers and Zame show t h a t c o n d i t i o n (b) i s n o t n e c e s s a r y f o r compact subs e t s of

which have o n l y a f i n i t e number o f connected components and

(c

t h i s r e s u l t does n o t extend t o h i g h e r dimensional s p a c e s .

For a r b i t r a r y

l o c a l l y convex s p a c e s , s u f f i c i e n t c o n d i t i o n s f o r t h e removal o f (b) a r e o b t a i n e d by p l a c i n g r a t h e r t e c h n i c a l l o c a l c o n n e c t e d n e s s c o n d i t i o n s on t h e compact s e t . The

T~

t o p o l o g y was i n t r o d u c e d by S.B. Chae [120] and it h a s a l s o

been s t u d i e d by J . Mujica [SO31 and K . D . chapter 6).

B i e r s t e d t and R . Meise [70], ( s e e

This Page Intentionally Left Blank

Chapter 3

HOLOMORPHIC FUNCTIONS ON BALANCED SETS

I n many b r a n c h e s o f f u n c t i o n t h e o r y t h e r e a r e c o l l e c t i o n s o f s e t s which a r i s e n a t u r a l l y and p o s s e s s p r o p e r t i e s which e n s u r e t h a t t h e y p l a y an i m p o r t a n t r o l e - f o r e x a m p l e c o n v e x s e t s i n f u n c t i o n a l a n a l y s i s and e f f i l g s e t s i n p o t e n t i a l theory.

I n t h e t h e o r y o f s e v e r a l c o m p l e x v a r i a b l e s many

d i f f e r e n t k i n d s o f s p e c i a l s e t s a r i s e - pseudoconvex s e t s , p o l a r s e t s , p o l y n o m i a l l y convex s e t s , p o l y d i s c s and S t e i n manifolds. B a l a n c e d open s e t s a r i s e a s t h e n a t u r a l domain o f convergence o f t h e Taylor s e r i e s expansion a t t h e o r i g i n f o r holomorphic f u n c t i o n s .

I n t h i s c h a p t e r we c o n s i d e r a l l t h e

h o l o m o r p h i c f u n c t i o n s on a b a l a n c e d o p e n s e t U a n d show t h a t t h e T a y l o r s e r i e s e x p a n s i o n s l e a d t o a t o p o l o g i c a l decomposition of H(U) chapter 2.

f o r t h e d i f f e r e n t t o p o l o g i e s we d i s c u s s e d i n

We t h e n u s e t h i s d e c o m p o s i t i o n t o d e d u c e t o p -

-0logica1 properties of H(U)

and t o e x t e n d r e s u l t s a b o u t

s p a c e s o f homogeneous p o l y n o m i a l s t o s p a c e s o f h o l o m o r p h i c functions.

Our main t o o l s a r e a s s o c i a t e d t o p o l o g i e s and

Schauder d e c o m p o s i t i o n s and s i n c e t h e s e t o p i c s a r e n o t v e r y w e l l known we d e v o t e t h e f i r s t s e c t i o n t o t h e i r e x p o s i t i o n . The t h e o r y o f h o l o m o r p h i c f u n c t i o n s on b a l a n c e d o p e n s e t s may b e r e g a r d e d a s a l o c a l t h e o r y a n d i f we c o u l d show t h a t T~

and T& were l o c a l

(or sheaf) topologies then the

r e s u l t s o f t h i s c h a p t e r would e x t e n d t o a r b i t r a r y open s e t s .

109

110

Chapter 3

1 3 . 1 . ASSOCIATED

TOPOLOGIES

DECOMPOSITIONS I N

AND

GENERALISED CONVEX

LOCALLY

SPACES

I n l i n e a r f u n c t i o n a l a n a l y s i s we e n c o u n t e r t w o m u t u a l l y exclusive but not exhaustive types of properties

-

those

p r e s e r v e d under l o c a l l y convex i n d u c t i v e l i m i t s and t h o s e preserved under p r o j e c t i v e limits.

For example t h e l o c a l l y

convex i n d u c t i v e l i m i t o f b a r r e l l e d l o c a l l y convex spaces i s b a r r e l l e d

(respectively bornological) (respectively bornological)

and t h e p r o j e c t i v e l i m i t o f complete l o c a l l y convex s p a c e s i s complete

(respectively nuclear)

(respectively n'uclear).

blany o t h e r p r o p e r t i e s a r i s e a s a c o m b i n a t i o n o f p r o p e r t -

i e s from t h e s e two t y p e s , as semi-reflexivity infrabarrelledness

f o r example r e f l e x i v i t y i s d e f i n e d

(preserved under p r o j e c t i v e l i m i t s ) plus (preserved under l o c a l l y convex inductive

l i m i t s ) and t h e combined p r o p e r t y i s n o t p r e s e r v e d under A g e n e r a l method used i n showing t h a t

e i t h e r kind of l i m i t . a l o c a l l y convex s p a c e , associate with

(EJ)

has a given property is t o

(E,T),

a n o t h e r l o c a l l y convex s p a c e ( F , T ' ) with

t h e g i v e n p r o p e r t y a n d t h e n t o show t h a t

(E,T)

= (F,T').

For

e x a m p l e we c a n a s s o c i a t e w i t h a g i v e n l o c a l l y c o n v e x s p a c e i t s completion and a l s o i t s a s s o c i a t e d b o r n o l o g i c a l s p a c e ,

A

s y s t e m a t i c t h e o r y h a s been developed f o r p r o p e r t i e s which are p r e s e r v e d u n d e r l o c a l l y c o n v e x i n d u c t i v e l i m i t s a n d we now d i s c u s s t h i s t h e o r y and a p p l y i t i n l a t e r s e c t i o n s t o t h e theory o f holomorphic functions. Definition 3.1

i s a f a m i l y ?of (a)

A Q f a m i l y of

ZocaZZy c o n v e x t o p o l o g i e s

l o c a l l y convex topologies such t h a t

3 i s s t a b l e under l o c a l l y convex i n d u c t i v e Z i m i t s ,

( b ) on any v e c t o r s p a c e t h e f i n e s t ZocaZly c o n v e x t o p o l o g y

belongs t o 3 . L e t ( E , T ) b e a Z o c a Z l y c o n v e x s p a c e and l e t Q f a m i l y of

ZocalZy c o n v e x t o p o l o g i e s .

3 be a

The ZocaZly c o n v e x

i n d u c t i v e Z i m i t o f a l l t o p o l o g i e s on E w h i c h l i e i n s a n d a r e f i n e r than

T

also l i e s i n

3.

We d e n o t e t h i s t o p o l o g y by

and c a l Z i t t h e 3 t o p o l o g y a s s o c i a t e d w i t h

T.

T~

111

Holomorphic functions o n balanced sets The t o p o l o g y

c a n a l s o b e c h a r a c t e r i z e d as t h e s o l u t i o n 3 t o a universal problem. S p e c i f i c a l l y i f (E,T) i s a l o c a l l y T

is

convex space and 3 i s a Q-family t h e n

t h e unique

;C

t o p o l o g y on E s u c h t h a t a n y c o n t i n u o u s l i n e a r m a p p i n g f r o m (F,T!) into E,

~ ~ € f a3c t o,r s

For c e r t a i n Q-families

through

(EJ.rg).

t h e r e a r e a l s o o t h e r known

c h a r a c t e r i z a t i o n s which can be u s e f u l . Let b denote t h e family o f a l l bornological

Example 3 . 2 topologies.

Then b i s a Q - f a m i l y and f o r a n y l o c a l l y convex

topology,

we l e t

T ,

topology.

If

T~

denote the associated bornological

( E J ~ )i s a l o c a l l y c o n v e x s p a c e a n d B i s a

c l o s e d c o n v e x b a l a n c e d b o u n d e d s u b s e t o f E t h e n we l e t E B d e n o t e t h e v e c t o r s u b s p a c e o f E g e n e r a t e d by B and normed s o that B i s its closed unit b a l l . ( E , T ~ )=

I t i s w e l l known t h a t

l i m

7E B

where B r a n g e s o v e r a l l c l o s e d convex b a l a n c e d bounded s u b s e t s of E. Example 3 . 3

Let ub d e n o t e t h e f a m i l y o f a l l u l t r a -

b o r n o l o g i c a l l o c a l l y convex t o p o l o g i e s .

Then ub i s a Q - f a m i l y .

F o r a n y l o c a l l y c o n v e x s p a c e ( E , T ) we l e t

T

a s s o c i a t e d u l t r a b o r n o l o g i c a l t o p o l o g y on E .

-

example 3 . 2 ,

ub

denote t h e

We h a v e , a s i n

( E , T ~ ~= ) l i m EB B where B r a n g e s o v e r a l l c o m p l e t e b a l a n c e d convex bounded subsets of E. Example 3 . 4

A l o c a l l y convex space,

(E,T) is c a l l e d a

K e l l e y s p a c e i f a n y f i n e r l o c a l l y convex t o p o l o g y on E h a s l e s s compact s e t s . topology.

An u l t r a - b o r n o l o g i c a l

topology is a Kelley

The c o l l e c t i o n o f a l l l o c a l l y convex K e l l e y

t o p o l o g i e s i s a Q - f a m i l y a n d we l e t

T~

denote the Kelley

topology a s s o c i a t e d w i t h t h e l o c a l l y convex topology

T .

Chapter 3

112

The b a r r e l l e d a n d i n f r a b a r r e l l e d t o p o l o g i e s a l s o f o r m Q - f a m i l i e s a n d f o r a g i v e n l o c a l l y c o n v e x s p a c e ( E , T ) we l e t and T d e n o t e t h e a s s o c i a t e d b a r r e l l e d ( t o n n e l e ) and t i infrabarrelled topologies respectively. An a l t e r n a t i v e

T

description of these topologies, induction,

by means o f t r a n s f i n i t e

i s given i n proposition 3.5.

The c o l l e c t i o n o f a l l

b a r r e l l e d and b o r n o l o g i c a l l o c a l l y convex t o p o l o g i e s ,

bt,

a l s o a Q - f a m i l y a n d we d e n o t e b y

and

T

b a r r e l l e d topology associated with

the bornological

bt T.

For any Hausdorff l o c a l l y convex s p a c e ( E , T )

-

is

we h a v e t h e

following i n c l u s i o n s between t h e various a s s o c i a t e d topologies, where a

b means t h a t a i s f i n e r t h a n b ,

We c o m p l e t e o u r d i s c u s s i o n o f a s s o c i a t e d t o p o l o g i e s b y

showing t h a t t h e b a r r e l l e d topology a s s o c i a t e d w i t h a complete topology i s a l s o complete. Proposition 3.5

I f

( E , T ) is a c o m p l e t e l o c a l l y c o n v e x s p a c e

t h e n (E,-rt) i s a l s o a complete l o c a l l y convex space. Proof

We f i r s t c o n s t r u c t a n o r d e r e d f a m i l y o f l o c a l l y c o n v e x

Let

t o p o l o g i e s on E . number and (E,T,)

=

l i m B Z

T

Now s u p p o s e a i s a n o r d i n a l

~ T= .

h a s b e e n d e f i n e d f o r a l l o r d i n a l s B
T~

(E,T ) B

d e f i n e d by a l l

-

i.e.

.

Let

Ta i s t h e p r o j e c t i v e l i m i t t o p o l o g y

T ~ , B < ~ . We d e f i n e - r a a s t h e t o p o l o g y w h i c h h a s

a neighbourhood b a s e a t z e r o c o n s i s t i n g o f a l l T a c l o s e d convex balanced absorbing subsets of E.

By t r a n s f i n i t e i n d u c t i o n

i s defined f o r every ordinal a.

Since t h e c a r d i n a l i t y of t h e

T

s e t of a l l convex balanced absorbing subsets o f E i s less than o r equal t o

*IE1

it f o l l o w s t h a t t h e r e e x i s t s an o r d i n a l

number 8 s u c h t h a t

T~

=

'eel

for all

el 2

0.

Since a locally

convex s p a c e i s b a r r e l l e d i f and o n l y i f each c l o s e d convex balanced absorbing s e t i s a neighbourhood o f zero it follows that

( E , T ~ ) i s a b a r r e l l e d l o c a l l y convex s p a c e .

S i n c e T t ->

T

113

Holomorphic functions on balanced sets

t h e above c o n s t r u c t i o n shows t h a t

T

> T ~ . t -

w e a k e s t b a r r e l l e d t o p o l o g y on E f i n e r t h a n

T

is the t it follows t h a t As T

We now s h o w b y t r a n s f i n i t e i n d u c t i o n t h a t

= Tee (E,T") i s a c o m p l e t e l o c a l l y c o n v e x s p a c e f o r e a c h o r d i n a l n u m b e r a. Tt

By h y p o t h e s i s ( E , T ~ ) = ( E , T ) i s c o m p l e t e . is complete for a l l

c1<

al.

Now s u p p o s e

(E,T )

Since the projective l i m i t of

complete l o c a l l y convex spaces i s complete t h e space (E, Ta ) =

lim "<"l

c -

1

(E,T")

is complete.

Let (x,) B E

.

B

b e a Cauchy

) Hence (xg) i s a l s o a Cauchy n e t i n t h e "1 B E B c o m p l e t e s p a c e (E,T ) and s o c o n v e r g e s t o an e l e m e n t x i n "1 Now l e t V b e a TC1 - n e i g h b o u r h o o d o f z e r o i n E . With(E,Tal). 1 closed o u t l o s s o f g e n e r a l i t y we may s u p p o s e t h a t V i s a T "1 convex balanced s u b s e t o f E . Since ( x ) is T - Cauchy B E B "1 t h e r e e x i s t s P o i n B s u c h t h a t x B 1 - x B 2 E V f o r a l l B ~ , B ~B ~ . Since x x i n ( E , T ) a s B2and V i s T closed 82 "1 "1 it follows t h a t xB x E V for all B 13,. H e n c e x -+ x i n B T h i s shows t h a t ( E , T " ~ ) i s c o m p l e t e . ( E , T a l ) a s -8 By

n e t i n (E.T

-

-.

-

-

t r a n s f i n i t e i n d u c t i o n (E,.ra) i s complete f o r each o r d i n a l number a and h e n c e ( E , T ~ )= (E,rt) i s a c o m p l e t e l o c a l l y c o n v e x space.

This completes t h e proof.

A m o d i f i c a t i o n o f t h e above proof

yields the following

r e s u l t and a l s o g i v e s an a l t e r n a t i v e d e s c r i p t i o n o f t h e associated infrabarrelled topology. Proposition 3.6

L e t ( E , T ) be a l o c a l l y c o n v e x s p a c e and

b e , r e s p e c t i v e l y , t h e b a r r e l z e d and i n f r a l e t T~ and T~ b a r r e l l e d t o p o l o g i e s a s s o c i a t e d w i t h T. I f ( E , T ) i s compZete ,

then (E,ri) ( E , T ~ )and

is a l s o c o m p l e t e . (E,.ri)

I f

( E , T ) i s quasi-complete

are both quasi-complete.

sequentiaZZy complete then (E,.rt)

I f

then

(E,T) i s

and ( E , T ~ )a r e b o t h s e q u e n t -

i a l l y complete. We now l o o k a t v a r i o u s k i n d s o f S c h a u d e r d e c o m p o s i t i o n s .

These are g e n e r a l i z a t i o n s o f t h e c o n c e p t o f a b a s i s and i n d e e d a l o c a l l y convex s p a c e h a s a Schauder b a s i s i f and o n l y i f it

Chapter 3

114

has a Schauder decomposition c o n s i s t i n g of one dimensional subspaces. Definition 3.7

conoex space

(E,

[Enjn

of a ZocaZZy i s a S c h a u d e r d e c o m p o s i t i o n of ( E , T ) i f

A s e q u e n c e of s u b s p a c e s T)

e a c h x i n E can be w r i t t e n i n a u n i q u e way as x =

-

C m x ) w h e r e xn n = l 11

l i m m--.,

urn : E

u m (:zl

E,

E

E,

xn) =

If t h e s e q u e n c e of m a p p i n g s ,

;zl

xn ( i . e .

a 2 2 n and if t h e p r o j e c t i o n s

zm

xn a r e aZZ c o n t i n u o u s .

n =1

(um),",,,

f o r m an e q u i c o n t i n uous f a m i Z y t h e n we s a y t h a t t h e d e c o m p o s i t i o n i s e q u i -

If E c o n t a i n s a f u n d a m e n t a 2 f a m i Z y of s e m i - n o r m s ,

Schauder.

'1,

such t h a t p(E" xn) = Yl p ( x n ) f o r e v e r y p i n n=l

and e v e r y

C" Xn i n E t h e n we s a y t h a t t h e d e c o m p o s i t i o n i s a b s o b u t e . The n =1 decomposition i s s a i d t o be s h r i n k i n g i f ( ( E ) ' ) " - 1 i s a nB n Schauder d e c o m p o s i t i o n f o r E h .

I t i s e a s i l y s e e n t h a t e v e r y a b s o l u t e S c h a u d e r decompos-

i t i o n i s an equi-Schauder decomposition.

s e t of aZZ s e q u e n c e s of c o m p t e x n u x b e r s

We l e t (an)

A

denote t h e

such t h a t n

l i m s i p lanll'n 5 1. If i s a Schauder d e c o m p o s i t i o n nfor ( E , T ) t h e n we s a y { E n I n i s an A-Schauder d e c o m p o s i t i o n i f W

Xn E E and imply anxn E . The d e c o m p o s i t i o n n=l n=l i s an A - a b s o Z u t e d e c o m p o s i t i o n i f f o r e v e r y p E c s ( E ) and (an)E=1

A

E

t h e semi-norm ;(c"

continuous. Proposition 3.8

n=1

xn) = xm n=l

a fundamentaZ s y s t e m of semi-norms = SU nP P,(Un(x))

f o r e v e r y a i n A and x

Proof

is

A S c h a u d e r d e c o m p o s i t i o n {En}n of a ZocaZZy

convex space (E,T) i s equi-Schauder P,(x)

bnI P ( X n )

i f and onZy i f t h e r e e x i s t s A for

(E,-r)such t h a t

(*I i n E.

We f i r s t s u p p o s e t h a t { E n ) ,

i s an equi-Schauder

-

d e c o m p o s i t i o n o f E a n d p i s a c o n t i n u o u s s e m i - n o r m on ( E , T ) .

-

Let q ( x ) as n

= sup p(un(x)) f o r every x i n E. Since un(x) x n and p i s c o n t i n u o u s i t f o l l o w s t h a t q ( x ) i s f i n i t e

115

Holomorphic functions on balanced sets f o r every x i n E. w h e r e m 5 n we h a v e q(x) = Let

SIIP

B = {x E E;

Then B = { x

E

E;

S i n c e u (u x)=u ( x ) f o r e v e r y x i n E m m n

q(un(x)) f o r every x i n E. q ( x ) 5 11 and

8

= {x; p(x) 5 1).

p(un(x)) 5 1 f o r a l l n}

=

n n Ix e E ;

=

Qun-'

p ( u n ( x ) ) 5 1)

($1.

-

-

i s an e q u i c o n t i n u o u s f a m i l y o f mappings q u n - l i s a neighbourhood o f zero i n E and hence q i s a continuous

S i n c e (un),

semi-norm on E .

Since un(x)

i n E t h i s shows t h a t q 2 p .

x as n

As p r a n g e s

m

(5)

f o r every x

over cs(E) q ranges

o v e r a fundamental f a m i l y of semi-norms which s a t i s f y ( * ) . This completes t h e proof i n one d i r e c t i o n . C o n v e r s e l y i f E c o n t a i n s a fundamental f a m i l y o f semi-

*,

n o r r n s , ( ~ ~ ) w ~ h~i c h s a t i s f y ( * ) t h e n f o r e v e r y a i n A . I x E E ; p a ( x ) 5 11 = a n d h e n c e (un),"=,

Ba

=G

un-'

(B,)

i s an equicontinuous f a m i l y o f mappings.

This completes t h e proof. Proposition 3.9

A S c h a u d e r d e c o m p o s i t i o n ~ f 'a b a r r e Z Z e d

ZocaZZy c o n v e x s p a c e is an e q u i - S c h a u d e r Proof

Let {En),

decovposition.

be a Schauder decomposition f o r t h e

b a r r e l l e d l o c a l l y c o n v e x s p a c e (E,T).

If p i s a continucus

semi-norm on E l e t q ( x ) = s x p p ( u ( x ) ) f o r e v e r y x i n E .

n

As

i n t h e p r e v i o u s p r o p o s i t i o n q i s a semi-norm on E ,

B = {x EE;

u

q ( x ) 5 11 = O { x E E ; p ( u n ( x ) ) 5 1 ) .

i s a c o n t i n u o u s l i n e a r mapping

Let

Since each

I x E E ; p ( u n ( x ) ) 5 11 i s a

n c l o s e d s u b s e t o f E and hence B i s a c l o s e d convex b a l a n c e d absorbing subset of E.

S i n c e E is b a r r e l l e d B i s a n e i g h b o u r -

hood o f z e r o and h e n c e q i s a c o n t i n u o u s semi-norm on E .

As

i n t h e p r o o f o f t h e p r e c e d i n g p r o p o s i t i o n i t now f o l l o w s t h a t q L P a n d q ( x ) = S,UP q ( u n ( x ) ) . An a p p l i c a t i o n o f p r o p o s i t i o n 3 . 8 now c o m p l e t e s t h e p r o o f . The a b o v e method o f p r o o f i s a l s o u s e d i n t h e f o l l o w i n g proposition.

Chapter 3

116 Proposition 3.10

i s a n /J1-SchazAer d e c o m p o s i t i o n for t h e b a r r e Z Z e d ZocaZly c o n v e x s p a c e ( E , r ) t h e n {En}, i s a n A-absoZute decomposition. Proof

Let p

E

If

cs(E) and l e t

E

4.

The s e q u e n c e

(n 2 a n r 2 a l s o l i e s i n a a n d s o i f 1" x E E then n a x n n n=l n n=l 2 also lies i n E. H e n c e En a x la' i s a bounded s u b s e t c f E . n I? n = l - 1 2 T h e r e f o r e s x p p ( n a x ) = M
ifl

%

%

shown t h a t p i s a s e m i - n o r m on E . S i n c e B = { X EE ; p ( x ) 5 1 ) n n m = l x m E E ; ;z1 l a m l p i x , ) 5 11 i s a c l o s e d c o n v e x

=n{c-

b a l a n c e d a b s o r b i n g s u b s e t o f E and E i s a b a r r e l l e d s p a c e i t 2,

f o l l o w s t h a t B i s a neighbourhocd o f z e r o and hence p i s a c o n t i n u o u s s e m i - n o r m on E .

This completes the proof.

We now l o o k a t a s s o c i a t e d t o p o l o g i e s f o r s p a c e s w i t h a n A-absolute decomposition. I f i s an j - c b s o Z u t e d eco mp o s itio n Proposition 3.11 for ( E , T ) t h e n { E n , ~ b } nis a Z s o an $ - a b s o l u t e d e c o m p o s i t i o n

for

(E>Tb).

Proof of

For each A i n

(E,T).

p is a

T

p

xn) =

'L

[4zl

A

Let B = (x ) A s r

d e n o t e an a r b i t r a r y bounded s u b s e t

r

l e t xA =

;Z1

x:

c o n t i n u o u s s e m i - n o r m o n E a n d (a , )

norm on E .

:zl

kzl

n21an/ p(xn)

T

E

(E,T).

4

If

then

n continuous semi-

is a a n d {n2 a x A } n n n,X I f q i s a rb c o n t i n u o u s semi-norm b o u n d e d s u b s e t o f (E,T) o n E t h e n t h e a b o v e s h o w s t h a t s u [ n 2 l a n l q ( x n1 ) ] = M 1 < n x a 1 and h e n c e s p xn) 5- M1 $ = 1 < m . Hence t h e

Hence syp

2,

n

2

is also a

in

,K

ifl

x

+

-

-

lanl

q(Xn)

o as A m.

w

-

-*

i s b o u n d e d on t h e

so i s rb continuous.

bounded s u b s e t s o f ( E , T ~ )a s A

M <

.

semi-norm q ( C m n = 1X n ) t h a t x:

=

l a n l p[x:)

in(E,Tb)

T

T h i s a l s o shows

i f xA

o in

T o c o m p l e t e t h e p r o o f we m u s t s h o w t h a t

PI

xm

If p is a

T

Zx i n ( E , T ~ )a s n -m. m=l m c o n t i n u o u s semi-norm on E and x =

&zl

x m €E then

117

Holomorphic functions on balanced sets

i s a bounded s u b s e t o f

(E,Tb). q(x

-

as n

i!yl

If q is a

-

n2 q ( n 2

Xm) = q ( m c"= n Xm) = 1

-.

(E,'I)

and o f

c o n t i n u o u s semi-norm on E t h e n

Tb

zrn

m=n xm) __+

O

This completes t h e proof.

The s i t u a t i o n f o r t h e a s s o c i a t e d b a r r e l l e d t o p o l o g y i s s l i g h t l y morc c o m p l i c a t e d .

T h i s i s d u e t o t h e f a c t t h a t we

d o n o t , i n g e n e r a l , know i f a S c h a u d e r d e c o m p o s i t i o n ,

i s a l s o a S c h a u d e r decompos-

(E,T),

f o r a l o c a l l y convex s p a c e ,

i t i o n f o r the associated barrelled space (F,Tt). a p o s i t i v e answer t o t h i s problem i f t o p o l o g i e s on E i . e . and only i f o(E',E).

(E,'I)'

(E,T)'

if

=

'I

and

'I

We o b t a i n

are compatible

t ( E , T ~ ) 'a n d t h i s o c c u r s i f

i s q u a s i - c o m p l e t e f o r t h e weak t o p o l o g y

On t h e o t h e r h a n d we d o h a v e t h e f o l l o w i n g u n i q u e i s a Schauder decovposition

n e s s r e s u l t ; i f {En), and i f each E

for (E,r)

i s a b a r r e l l e d s p a c e t h e n t h e r e e x i s t s a t most

n one b a r r e l l e d t o p o l o g y on E which c o i n c i d e s w i t h and h h i c h h a s {En}

n

T

on e a c h En

a s a Schauder decomposition.

A l t h o u g h we d o n o t know i f s u c h a t o p o l o g y e x i s t s t h e a b o v e s a y s t h a t when i t d o e s i t m u s t b e t h e a s s o c i a t e d barrelled topology.

he p r o v e a l e s s g e n e r a l r e s u l t which i s

adequate f o r our applications. Proposition 3.12

~ e {t E ~ be ) ~a r ~ R - . d e e o m p o s i t i o n for (E,T)

and suppose each En i s a b a r r e l l e d l o c a l l y convex space.

I f

t h e r e e x i s t s G b a r r e l l e d t o p o l o g g T~ o n E w h i c h c o i n c i d e s w i t h 'I on each. Erl and i.f { E n } n i s c S c h a u d e r d e c o m p o s i t i o n fcr ( E , T ~ )t h e n

Tl

=

Tt

a x d a f u n d c m e n t a l s g s t e n : of sen;<-norms

f o r ( E 3 r t ) i s g i v e n by G Z Z s e m i - n o r m s , following conditions; (a)

PIE,

is ~ - e o n t i n ~ o u s

( b ) p ( , ? l l x n ) = ;:l Cm x i n E. n=l n Proof

P , w h i c h satisfy t h e

p(xn) f o r every

By h y p o t h e s i s

'I

5

T~

5

' I ~ and

hence

Chapter 3

118

=

"En

T

=

.

T

S i n c e {En}n i s a S c h a u d e r decomposit follows t h a t it is also a

i t i o n f o r ( E , T ) and ( E , r l )

S i n c e I E n I n i s an

Schauder d e c o m p o s i t i o n f o r (E..rt). A-decomposition

f o r (E,T) it i s a l s o an 2-decomposition f o r

a n d ( E , T ~ )a n d h e n c e b y p r o p o s i t i o n 3 . 1 1 i t i s a n

(E,rt)

d - a b s o l u t e and i n p a r t i c u l a r an a b s o l u t e d e c o m p o s i t i o n f o r both of these topologies.

Hence b o t h o f t h e s e t o p o l o g i e s a r e

g e n e r a t e d by semi-norms which s a t i s f y ( a ) and ( b ) .

On t h e

o t h e r hand i f p i s a semi-norm on E which s a t i s f i e s ( a ) and p(x) 5 1 ) is a

(b) then V = {x E E ;

closed c o n v e x b a l a n c e d

a b s o r b i n g s u b s e t o f E a n d h e n c e p i s a -rt c o n t i n u o u s s e m i norm.

Hence

' I ~ =

r t and

T~

i s t h e l o c a l l y c o n v e x t o p o l o g y on

E g e n e r a t e d by a l l t h e semi-norms on E which s a t i s f y ( a ) and

(b). We now l o o k a t d e c o m p o s i t i o n s o f t h e s t r o n g d u a l . Proposition 3.13

E

a

a n d BI' = { n 2 C m a m=n 'm)n,X

n

= TIE

n

i s bounded

E

El

i s an

Since

decomposition it follows t h a t

I f T E E ' we l e t T

I f T8

are bounded s u b s e t s

f o r every positive integer n .

f o r e a c h 8 i n s o me d i r e c t e d s e t R a n d B

projection o f B i n E then A n 1 IiTenIIB = s u p I T e ( x n ) 5 -2 I l ~ ~ l l Bf t o r a l l n * n a an Hence, s i n c e En i s a complemented s u b s p a c e o f E , T i

-

i f Te

B

& ( w e may a n d s h a l l a s s u m e w i t h o u t l o s s o f

B ' = ( n 2 an ' n ) n , a of E. B"

T) I .

a

g e n e r a l i t y t h a t an 2 1 f o r a l l n ) . A-absolute

decomposition f o r

(x )aET.b e a bounded s u b s e t o f E and l e t

Let B

a = (a,),

{ E n I n is a n & a b s o l u t e

I n is an ~ - a b s o Z u t e d e c o m p o s i t i o n for ( E ,

{ (En);

(E,T) t h e n Proof

I f

o i n (E o in

I

) I

nB

as 0

-

m

n

Since

is the

f o r each n

E l .

B

T h i s shows t h a t i s a Schauder decomposition f o r To c o m p l e t e t h e p r o o f i t s u f f i c e s t o n o t i c e t h a t

E l .

B

Holornorphic functions on balanced sets

119

C"

n=1

< C"

n=

f o r every T i n E'.

A n d - a b s o Z u t e d e c o m p o s i t i o n of a ZocaZZy

Corollary 3.14

convex space i s a shrinking d e c o m p o s i t i o n . F u r t h e r p r o p e r t i e s o f a s s o c i a t e d t o p o l o g i e s and S c h a u d e r decompositions a r e given without proof a s t h e need a r i s e s . For f u r t h e r d e t a i l s we r e f e r t o t h e N o t e s a n d Remarks a t t h e e n d of t h i s chapter,

Apart from t h e immediate a p p l i c a t i o n ' s o f t h e above r e s u l t s w h i c h we g i v e i n t h i s c h a p t e r we w i l l a l s o f i n d t h a t t h e techniques developed t o o b t a i n t h e s e r e s u l t s a r e useful i n s t u d y i n g h o l o m o r p h i c f u n c t i o n s on c e r t a i n n u c l e a r s e q u e n c e spaces.

13.2

EQUI-SCHAUDER DECOMPOSITIONS O F ( H ( U ; F )

,T).

In t h i s s e c t i o n U w i l l d e n o t e a b a l a n c e d open s u b s e t o f a l o c a l l y c o n v e x s p a c e E and F w i l l d e n o t e a Banach s p a c e (Most o f t h e r e s u l t s ,

quasi-complete

however, can be e x t e n d e d t o a r b i t r a r y

l o c a l l y convex s p a c e s ) .

P r o p o s i t i o n 3.15

If U is a b a l a n c e d o p e n s u b s e t o f a Z o c a l Z y c o n v e x s p a c e E , F i s a Banach s p a c e , f E H ( U ; F ) and a =

(an)*

A then g = 1" a n=o n

E

Let K C

Proof

s

E

bT e Uc o m p a c t .

H(U;F) We may s u p p o s e w i t h o u t l o s s

o f g e n e r a l i t y t h a t K i s b a l a n c e d , t h e n t h e r e e x i s t A > 1 and V ,

a c o n v e x b a l a n c e d n e i g h b o u r h o o d o f z e r o , s u c h t h a t A(K+V) C U and l

l

IlfIIA(K+v) = ~

\

<

"'

By t h e C a u c h y i n e q u a l i t i e s

l (K+V) A < M for a l l n.

S i n c e (an),

E

3

t h e r e e x i s t s C > o such t h a t

120

<

Chapter 3 n

1+A :zo(x)

C.M

<m.

Thus g i s a l o c a l l y b o u n d e d f u n c t i o n on U a n d s i n c e i t i s G-holomorphic i t

in H(U;F).

is

Proposition 3.16

This completes t h e proof.

I f U i s a baZanced o p e n s u b s e t of a ZocaZZy

c o n v e x s p a c e E , F is a Banach s p a c e and f a H ( U ; F )

then the

TayZor s e r i e s e x p a n s i o n o f f a t t h e o r i g i n c o n v e r g e s t o f i n (H(U;F)

*

For each positive integer m l e t

Proof =

"m

,T6)

{x E U ;

n

2 d f o )

the interior of V i t i o n shows t h a t

m

.

(W

Since

5 m f o r a l l n l a n d l e t Wm d e n o t e 2 "

(n ) n = l

Ed t h e

preceding propos-

i s an i n c r e a s i n g c o u n t a b l e open c o v e r

)" m m=l

of U.

Now l e t p b e a

exist

a positive integer m

c o n t i n u o u s semi-norm on H ( U ; F ) .

T~

for every f i n H(U F).

\I

(x)

a n d C>o s u c h t h a t p ( f ) < C

Ilfll

There

wm 0

IIence f o r a l l n w e h a v e

^.

J = O

j!

d j f (0) j!

Ic

Fl'n+l

"Wmo

m0

j2

c

*

Therefore p(f

- j=o

djf(0) )j!

0

as n

-

and t h i s completes t h e p r o o f . Theorem 3 . 1 7

L e t U b e a baZanced o p e n s u b s e t of a ZocaZZy

c o n v e x s p a c e E and Z e t F be a Banach s p a c e . {&"E;

m

F ) ,~~l~~~

d e c o m p o s i t i o n for Proof

Then

i s a n , # - d e c o m p o s i t i o n and an A - a b s o Z u t e (H(U;F)

,T

By p r o p o s i t i o n 2 . 4 1

complemented s u b s p a c e o f

6) (P("E;F),T,)

(H(U;F)

, T )~

is a closed

and s o p r o p o s i t i o n

121

Holomorphic functions on balanced sets m

3.16 implies that {!?(nE;F),~w}n=O

for (H(U;F),T&).

is a Schauder decomposition

Proposition 3.15 implies that it is an

,$-decomposition

and since

T~

is a barrelled topology

proposition 3.10 implies that it is an A - a b s o l u t e decomposition.

This completes the proof.

We now obtain the same result for the

-c0

and

T~

topologies and

their associated bornological topologies. L e t U b e a baZanced o p e n s u b s e t of a

Proposition 3.18

l o c a l l y c o n v e x s p a c e E , l e t F be a Banach s p a c e , l e t K b e a Then t h e semi-norm compact s u b s e t of U - a n d l e t ( a n ) n E 8 . n f (0 p(f) = I I W b K

Efola,l

is

T

Proof

continuous on H ( U ) .

We may assume without loss o f .generality that K

is a balanced subset o f U . compact subset o f U and n lanl 5 ( I +T A)

for all n 1. n

For any f in H(U;F)

.

we have by the Cauchy inequalities

for every f in H ( U ; F ) Theorem 3.19

0

Choose X > l such that X K is a a positive integer such that

and this completes the proof.

L e t U b e a b a l a n c e d o p e n s u b s e t of a ZocaZZy

c o n v e x s p a c e E and l e t F b e a Banach s p a c e . Then {' 6 (nE ;F) ,T ~ } : = ~ i s a n 8 - d e c o m p o s i t i o n and an d - a b s o Z u t e

d e c o m p o s i t i o n f o r (H(U;F) , T ) . By proposition 3.16, since T 6 > T the Taylor Proof 0' series expansion at the origin o f a holomorphic function

converges t o the function in the compact open topology. Since (@("E;F) , T ~ )is a closed complemented subspace o f n (H(U;F) , T ~ ) (proposition 2.40) this s h o w s that I(?( E;F) ,

OD

T ~ } ~ = ~

is a Schauder decomposition for (H(U;F) , T ~ ) . Proposition 3.15 implies that it is a n A -decomposition and proposition 3 . 1 8 shows that it is a n d - a b s o l u t e decomposition. This

Chapter 3

122 completes t h e proof. Corollary

3.20

L e t U b e a b a l a n c e d o p e n s u b s e t o f a ZocaZZy

c o n v e x s p a c e E and l e t F be a Banach s p a c e . i s and-decomposition

and an 4 - a b s o l u t e

( H ( U ; F ) , T ~ , ~i ) f e a c h p ( " E ; F )

Then { P ( n E ; F ) l z = o

d e c o m p o s i t i o n for

i s given t h e bornoZogical

t o p o l o g y a s s o c i a t e d w i t h t h e compact o p e n t o p o l o g y . Proof

Apply p r o p o s i t i o n

3.11 and theorem 3 . 1 8 .

L e t U be a b a l a n c e d o p e n s u b s e t of a

Proposition 3.21

l o c a l l y c o n v e x s p a c e E , Z e t F b e a Banach s p a c e , be a

T

w

Zet p

c o n t i n u o u s semi-norm on H ( U ; F ) and l e t ( a )

n n

Then t h e semi-norm

E

A.

Proof

Suppose p i s p o r t e d by t h e compact b a l a n c e d s u b s e t K

o f U.

We s h o w t h a t p i s a l s o p o r t e d b y t h e same c o m p a c t s e t .

%

Let V b e a n e i g h b o u r h o o d o f K which l i e s i n U .

ChooseX> 1

and a b a l a n c e d neighbourhood o f z e r o W such t h a t K A(K+W)CV i+x n s u c h t h a t la 5 (-1 for all

I

Choose a p o s i t i v e i n t e g e r n n > no.

2

T h e r e e x i s t s a p o s i t i v e n u m b e r C(W) s u c h t h a t Hence, f o r e v e r y f o r e v e r y f i n H(U;F) p ( f ) 5 C(W) llf

IIK+W

.

f i n H ( U ; F ) , we h a v e

%

and p i s a

T~

c o n t i n u o u s semi-norm on H(U;F).

Theorem 3 . 2 2

L e t U b e a baZanced o p e n s u b s e t o f a ZocaZZy

c o n v e x s p a c e E and Z e t F be a Banach s p a c e . n

Cp(

E; F), T

m

~

}

i s =a n ~ A-decomposition ~

decomposition f o r (H(U;F)

Then

and a n / j - a b s o Z u t e

,T~).

Proof

By p r o p o s i t i o n 3 . 1 6 , s i n c e T 6 > T the Taylor w' series expansion a t t h e o r i g i n o f a holomorphic function

converges t o t h e function i n t h e m

T

w

t o p o l o g y . BY P r o P o s i t i o n

2.41 {@(nE;F),~w)n=O i s a Schauder decomposition f o r I t i s an,J'-decomposition

by p r o p o s i t i o n

3 . 2 1 shows t h a t i t i s a n d - a b s o l u t e

(H(U;F);:)

3.15 and p r o p o s i t i o n

decomposition.

This

123

Holomoiphic finctions on balanced sets completes t h e proof. m

( p ( n E ; F) , T ~ n} = o is m / j ' - d e c o r n p o s i t i o n and an

Corollary 3 - 2 3

2 - a b s o Z u t e d e c o m p o s i t i o n f o r (H(U;F) , T ~ , ~ ) . Proof

Since T

w

-<

T

w,b <

Tc6

and

TwI p ( n E ; F )

=

T

i n d u c e s t h e T~ t o p o l o g y on w,b The r e q u i r e d r e s u l t now f o l l o w s b y a p p l y -

f o r a l l n it follows t h a t T P("E;F)

f o r a l l n.

i n g p r o p o s i t i o n 3 . 1 1 and t h e o r e m 3 . 2 2 . The e x i s t e n c e o f a n d - a b s o l u t e d e c o m p o s i t i o n o f H(U;F)

by

s p a c e s o f homogeneous p o l y n o m i a l s i s a d e q u a t e f o r a l l o u r applications i n t h i s chapter.

However we s h a l l n e e d ,

in

c h a p t e r 4 , a s l i g h t l y s t r o n g e r r e s u l t which a p p e a r s t o h o l d o n l y f o r t h e compact open and t h e

T

w

topologies.

L e t U b e a b a l a n c e d o p e n s u b s e t of a

Proposition 3.24

l o c a l l y c o n v e x s p a c e E , l e t F b e a Banach s p a c e and l e t p b e a T c o n t i n u o u s semi-norm on H ( U ; F ) , T = T 0 o r T w . T h e n t h e r e e x i s t s A > 1 such t h a t

is a l s o a Proof

T

c o n t i n u o u s semi-norm

If p is a

T~

on H ( U ; F ) .

c o n t i n u o u s s e m i - n o r m on H(U;F)

p ( f ) 5 C llfIIK f o r e v e r y f i n H(U;F) such t h a t AK i s a subset of U.

2,

a n d h e n c e p i s T~

continuous.

and

t h e n we may c Q o o s e X > 1

For e v e r y f

=&zo

d " f 0 E

n!

This completes t h e proof

H ( U ;F)

for

t h e compact open t o p o l o g y . Now s u p p o s e p i s a

T

w

c o n t i n u o u s s e m i - n o r m on H(U;F)

i s p o r t e d by t h e compact b a l a n c e d s u b s e t K o f U . s u c h t h a t X K i s a compact s u b s e t o f U .

W a balanced neighbourhood o f K such t h a t

L e t C(W)>o b e s u c h t h a t p ( f ) 5 C(W) I f Em n=o

anfo n!

E

H(U;F)

Choose X > 1

Let V d e n o t e an open

b a l a n c e d s u b s e t o f U which c o n t a i n s A K .

H(U;F).

which

T h e r e e x i s t a > l and AK C a A W C V C U .

\ ( f \ l Wf o r e v e r y f i n

124

Chapter 3

2

c . Ilfll" .

Hence p i s p o r t e d b y X K a n d s o i t i s

continuous.

T~

This

completes t h e proof. We c o m p l e t e t h i s s e c t i o n b y c o n s i d e r i n g s p a c e s o f g e r m s . Theorem 3.25

If K is a baZanced compact s u b s e t o f a ZocaZZy

c o n v e x s p a c e E and F is a Banach s p a c e t h e n {6'("E;F)

,T

lm

w n=o

decomposition for

is a n & - d e c o m p o s i t i o n and a n B - a b s o l u t e H(K;F). Proof

I f f EH(K;F) t h e n f €H(U;F)

subset U of E.

f o r some b a l a n c e d o p e n (H(U;F),T ) as U r a n g e s

S i n c e H(K;F)=="

w

o v e r a l l b a l a n c e d open s u b s e t s o f E c o n t a i n i n g K and s i n c e t h e Taylor s e r i e s converges in(H(U;F),T

w

)

(theorem 3 . 2 2 )

it

follows t h a t t h e Taylor s e r i e s of E a t t h e o r i g i n converges t o f i n H(K;F).

By p r o p o s i t i o n 2 . 5 8

complemented s u b s p a c e o f H(K;F)

( p ( n E ; F ) , ~ w )i s a c l o s e d m

a n d h e n c e I ~ ' ( " E ; F ) , T ~ i~s~ = ~

a Schauder decomposition o f H(K;F).

P r o p o s i t i o n 3.15 shows

t h a t it i s a n $ - d e c o m p o s i t i o n and an a p p l i c a t i o n o f p r o p o s i t i o n 3 . 1 0 c o m p l e t e s t h e p r o o f s i n c e H(K;F)

is a

b a r r e l l e d l o c a l l y convex s p a c e . 53.3 A P P L I C A T I O N S O F G E N E R A L I S E D D E C O M P O S I T I O N S T O T H E STUDY OF H O L O M O R P H I C F U N C T I O N S O N BALANCED OPEN S E T S

The r e s u l t s o f t h e t w o p r e v i o u s s e c t i o n s a r e a p p l i e d t o H(U;F)

where U i s a b a l a n c e d open s u b s e t o f a l o c a l l y convex

s p a c e and F i s a Banach s p a c e .

The f i r s t p a r t o f t h i s s e c t i o n

i s devoted t o topologies associated with t h e

'cU

topology.

Our

first r e s u l t motivated t h e introduction of associated

t o p o l o g i e s i n t h e t h e o r y o f holomorphic f u n c t i o n s on l o c a l l y convex s p a c e s .

125

Holomorphic functions on balanced sets L e t U b e a b a l a n c e d o p e n s u b s e t of a Z o c a l Z y

Theorem 3 . 2 6

c o n v e x s p a c e E and l e t F b e a Banach s p a c e . = T

T6

w,t

=

T

w,bt =

'Iw , u b .

S i n c e ( H ( U ; F ) ,T ) i s a n u l t r a b o r n o l o g i c a l s p a c e i t 6 T i s t h e b a r r e l l e d topology associated

Proof

s u f f i c e s t o show t h a t with

On H ( U ; F )

each n .

6

By p r o p o s i t i o n

T ~ .

2.41

T~

and

T

6

agree on6)(nE;F)

for

An a p p l i c a t i o n o f p r o p o s i t i o n 3 . 1 2 now c o m p l e t e s t h e

p r o o f s i n c e {P("E;F) ( H ( U ; F ) , T ~ )a n d

i s an.3 -decomposition

,T,}:=~

f o r both

(H(U;F),-r ) b y t h e o r e m s 3 . 1 7 and 3 . 2 2 .

6

P r o p o s i t i o n 3 . 1 2 a l s o shows t h a t

T~

i s t h e f i n e s t topology

f o r w h i c h we h a v e a b s o l u t e c o n v e r g e n c e o f t h e T a y l o r s e r i e s expansion and which c o i n c i d e s w i t h T

w

on s p a c e s o f

Formally t h i s i s expressed as

homogeneous p o l y n o m i a l s . follows.

L e t U b e a baZanced o p e n s u b s e t of a ZocaZZy c o n v e x s p a c e E and Z e t F b e a Banach s p a c e . The Proposition 3.27

t o p o Z o g y o n H ( U ; F ) i s g e n e r a t e d b y aZZ s e m i - n o r m s ,

T~

p , which

s a t i s f y t h e foZZowing c o n d i t i o n s ;

is

( b ) 'I@("E;F)

continuous.

T

T h e f o l l o w i n g lemma i s a n i m m e d i a t e c o n s e q u e n c e o f t h e existence of an,8-absolute

An a n a l o g o u s

decomposition.

r e s u l t f o r t h e compact open t o p o l o g y i s a l s o t r u e . Lemma 3 . 2 8 L e t U b e a baZanced o p e n s u b s e t of a ZocaZZy c o n v e x s p a c e E and l e t F b e a Banach s p a c e . L e t be a T (respectively T , T & ) bounded n e t i n H ( U ; F ) . Then fa

w

-0

w ,b

- -

a s a-

m

i n (H(U;F),rw) (respectively

( H ( U ; F ) , T ~ , ~ ) (, H ( U ; F ) , T ~ ) ) if and o n l y i f An o a s c1 = i n (@("E;F) d f,(O) In!

,T

w

I f o r e v e r y non-

negative i n t e g e r n. This means,

in particular,

s a m e t o p o l o g y on t h e 3.26 implies,

T~

that

T ~ , T ~ , a ,n d

'c6

induce t h e

bounded s u b s e t s o f H(U;F).

among o t h e r t h i n g s , t h a t T~

and

T~

Theorem

define the

same c o n v e x b a l a n c e d c o m p l e t e b o u n d e d s u b s e t s o f H ( U ; F ) .

126

Chapter 3

U s i n g lemma 3 . 2 8 we show t h a t t h e same r e s u l t h o l d s f o r compact b a l a n c e d convex s e t s .

L e t U be a baZanced o p e n s u b s e t of a

Proposition 3.29

ZocatZy c o n v e x s p a c e E and l e t F be a Banach s p a c e . c o n v e x b a l a n c e d compact s u b s e t s o f ( H ( U ; F ) (H(U;F)

with

i s t h e KelZey t o p o l o g y a s s o c i a t e d

) c o i n c i d e and 6 on H ( U ; F ) .

T

Since

s u f f i c e s t o show t h a t a n y c o n v e x

,T

‘cw

Then t h e

, T ~ )and

Proof

T

> T it 6 w

6

b a l a n c e d compact s u b s e t K o f ( H ( U ; F ) , T ~ )i s

compact. By 6 bounded s u b s e t o f T

theorem 3 . 2 6 K i s a complete balanced T 6 I f ( f a l a c r i s a n e t i n K t h e n i t c o n t a i n s a T~ H(U;F). convergent subnet. By lemma 3 . 2 8 t h i s s u b n e t i s a l s o T 6 convergent. Hence K i s a T c o m p a c t s u b s e t o f H ( U ; F ) . Since -i6

6 i s an u l t r a b o r n o l o g i c a l topology i t i s a l s o a Kelley

t o p o l o g y and h e n c e

T

6 = Tw , K .

One c a n a l s o show t h a t topology associated with

T

w

T is the infrabarrelled w,b on H(U;F). Thus we s e e t h a t t h e r e

are,

i n g e n e r a l , t w o t y p e s o f t o p o l o g i e s t h a t we may a s s o c i a t e

with

T

w

.

On t h e o n e h a n d t h e r e a r e t h e a s s o c i a t e d b a r r e l l e d ,

u l t r a b o r n o l o g i c a l , b a r r e l l e d and b o r n o l o g i c a l , t o p o l o g i e s a l l o f which a r e e q u a l t o

T~

and K e l l e y

and t h e a s s o c i a t e d

i n f r a b a r r e l l e d and b o r n o l o g i c a l t o p o l o g i e s which a r e e q u a l t o I t i s an o p e n q u e s t i o n w h e t h e r o r n o t t h e s e t w o ‘w,b’ t o p o l o g i e s c o i n c i d e i . e . is T w , b = T6?. Theorem 3 . 2 6 a n d p r o p o s i t i o n 3 . 2 9 i n d i c a t e t h a t t h e y a r e v e r y c l o s e t o one another.

The f o l l o w i n g r e s u l t g i v e s n e c e s s a r y a n d s u f f i c i e n t

c o n d i t i o n s u n d e r w h i c h t h e s e t o p o l o g i e s c o i n c i d e a n d we s h a l l i n t h i s and l a t e r c h a p t e r s , e n c o u n t e r v a r i o u s s u f f i c i e n t conditions for t h e i r equality. Proposition 3.30

L e t U be a baZanced o p e n s u b s e t of a

l o c a l l y c o n v e x s p a c e and l e t F be a Banach s p a c e .

The

f o Z Z o w i n g a r e e q u i v a l e n t on H ( U ; F ) ; (a) (b)

T w , b = T6 ‘c6 and T d e f i n e t h e same bounded s e t s

(c) (d)

T~ T

w

and and

w

T~ T&

d e f i n e t h e same compact s e t s , i n d u c e t h e same t o p o Z o g y o n T

w

bounded s e t s

127

Holomorphic functions on balanced sets T T

w

i s a barrelled topology

,b

i s t h e f i n e s t l o c a l l y convex topology f o r which t h e

W,b

Taylor s e r i e s expansion a t t h e o r i g i n converges a b s o l u t e l y and w h i c h i n d u c e s t h e T @ t o p o l o g y o n P ( n E ; F ) f o r e v e r y positive integer n, i f T~ E (@(:E ;F) ,T 1 1 f o r every non-negative i n t e g e r n W dnf o and T n ( . e ) converges f o r every f = Zm n=O n!

:lo

( a ) , ( b ) , ( e ) and ( f ) a r e e q u i v a l e n t by t h e o r e m 3.26

Proof

and p r o p o s i t i o n 3 . 2 7 . and B i s a

T

(a)*(c)

w

bounded t h e n t h e r e e x i s t s a (f,),,

b y lemma 3 . 2 8 .

If

b o u n d e d s u b s e t o f H(U;F) w h i c h i s n o t

-

6

c o n t i n u o u s semi-norm p and

6

such t h a t p(fn)

a sequence i n B , -

T

(c) holds T

m

as n

I m.

U { o } i s T~ c o m p a c t b u t n o t T b o u n d e d . 'he s e t { f " / J p ( f n ) } n = l 6 T h i s c o n t r a d i c t i o n shows t h a t ( c ) - ( b ) . ( c ) and ( d ) a r e e q u i v a l e n t b y lemma 3 . 2 8 .

Now s u p p o s e ( a ) h o l d s a n d t h e

sequence CTn}nAsatisfies t h e conditions of ( g ) .

5=o m

I

ITn (-1 n! By p r o p o s i t i o n 3 . 2 7 3.15

:Io

<

m

2zo

f o r every f =

By p r o p o s i t i o n E

H(U;F)

n!

I

d e f i n e s a -c6 a n d h e n c e a

T

c o n t i n u o u s s e m i - n o r m on H ( U ; F ) .

i t f o l l o w s t h a t ,Zzo

( H ( U ; F ) , T ~ , ~a)n~d h e n c e ( a )

p(f1 =

IT,

1-(

Tn

E

w.b

(9)

Conversely i f (9) i s s a t i s f i e d then ( H ( U ; F ) ; T ~ , ~ ) ' = (H(U;F),r6)'

and s i n c e

T

i s a Mackey

,b topology ( i t i s i n f r a b a r r e l l e d ) t h i s implies t h a t and (g) 4 ( a ) .

W

T

~

=

,T &~

This completes t h e proof.

Some a n a l o g u e s o f t h e a b o v e r e s u l t s c a n a l s o b e p r o v e d f o r t h e compact open t o p o l o g y .

The r e s u l t s , h o w e v e r ,

are not as

c o m p l e t e i n t h i s c a s e s i n c e ( ~ ( , , E ; F ) , T ~i )s n o t i n g e n e r a l a b a r r e l l e d l o c a l l y convex s p a c e . Proposition 3.31

We g i v e o n e e x a m p l e .

L e t U b e a b a l a n c e d o p e n s u b s e t of a

l o c a l l y c o n v e x s p a c e E and l e t F be a Banach s p a c e . f o l l o w i n g a r e e q u i v a l e n t on H(U;F);

The

128 (a) (b)

Chapter 3 T

is, a ~ b a r r e Z Z e d t o p o l o g y ,

~

(i)

($'(nE;F),~o,b)

( i i ) i f Tn

E

Em n = o Tn (

(i) (ii)

,T

inf(0) ) <

0

then

;=,

,b

f o r e a c h n and

) I

for e a c h f = nC = o

n!

i n H(U;F) (c)

i s barreZZed f o r each i n t e g e r n ,

(f"(nE;F)

m

Tn

E

(H(U;F)

Pf(0) n!

, T ~ , I~ , )

( 6 ' ( n ~ ; F ) , ~ o , b is ) barreZZed f o r each i n t e g e r n ,

is t h e f i n e s t ZocaZZy c o n v e x t o p o Z o g y on for w h i c h t h e T a g l o r s e r i e s c o n v e r g e s and w h i c h i n d u c e s on Q ( n ~ ; ~t h)e T ~ t o,p o Z~o g y for e a c h n . T

,b H(U;F) 0

-

We now i n t r o d u c e a weak f o r m o f c o m p l e t e n e s s

-

s e r i e s completeness

Taylor

which a l l o w s us t o extend v a r i o u s

t o p o l o g i c a l p r o p e r t i e s o f s p a c e s o f homogeneous p o l y n o m i a l s t o h o l o m o r p h i c f u n c t i o n s on b a l a n c e d o p e n s e t s . Definition 3.32

L e t E and F be ZocaZZy c o n v e x s p a c e s and

Z e t U be a b a l a n c e d o p e n s u b s e t o f E . space ( H ( U ; F ) , - c )

is T . S .

T

The ZocaZZy c o n v e x

complete (T.S.

i f t h e f o Z l o w i n g c o n d i t i o n is s a t i s f i e d ;

s e q u e n c e o f homogeneous poZynomiaZs, nE m = o p ( P n ) < m for e a c h Em E H(U;F). n = o 'n

Pn

TayLor s e r i e s )

=

m

( P n ) n = o is a

i f E

$' ( " E ; F ) ,

and

c o n t i n u o u s semi-norm p , t h e n

T

We h a v e a l r e a d y s e e n e x a m p l e s o f T . S .

completeness.

is T.S.

F o r example theorem 2 . 2 8 s a y s t h a t H ( U ; F )

T

P

i f U i s a b a l a n c e d open s u b s e t o f a Banach s p a c e and t h e topology o f pointwise convergence. Let

T~

and

T~

and suppose

i s a l s o T.S.

complete 'I

P

is

d e n o t e two l o c a l l y c o n v e x t o p o l o g i e s on H(U;F) >

T T~

T

2.

I f H(U;F)

complete.

i s T.S.

T~

complete then it

The f o l l o w i n g r e s u l t d e s c r i b e s a

s i t u a t i o n i n which t h e converse h o l d s .

L e t U b e a b a l a n c e d o p e n s u b s e t of a ZocaZZy c o n v e x s p a c e and l e t F b e a Banach s p a c e . I f T i s a ZocaZZy m R ~ an ,8 - a b s o Z u t e c o n v e x t o p o z o g y o n H ( U ; F ) and (6(E ; F ) , T ) ~ = is Lemma 3 . 3 3

decomposition f o r H ( U ; F ) and o n l y i f H ( U ; F ) Proof

then H(U;F)

is T . S .

Suppose H ( U ; F )

T,,

i s T.S.

T

complete i f

compzete

i s T.S.

T~

complete.

Let

m

(Pn)n=o

129

Holomorphic functions on balanced sets b e a s e q u e n c e o f homogeneous p o l y n o m i a l s ,

Pn€ @(nE;F)

,

and

< m f o r e v e r y T c o n t i n u o u s s e m i - n o r m p on s u p p o s e nz m =o p(pn) i s a T and hence a T bounded H(U;F). The s e q u e n c e { P 1 b n n m n s u b s e t o f H(U;F). S i n c e I @ ( E ; F ) , T ~ } ~i s= ~a l s o a n d - a b s o l u t e

d e c o m p o s i t i o n f o r ( H ( U ; F ) , T ) ( p r o p o s i t i o n 3 . 1 1 ) we h a v e b zm p ( p n ) < m f o r e v e r y T b c o n t i n u o u s semi-norm on H ( u ; F ) . n=o This Hence Em PncH(U;F) and H(U;F) i s T . S . T c o m p l e t e . n=o completes t h e p r o o f . Corollary 3.34

L e t U be a b a l a n c e d o p e n s u b s e t o f a l o c a l l y

c o n v e x s p a c e and l e t F be a Banach s p a c e . T.S.

T~

(respectively

T.S.

T

w

(respectiuelyr

0 ,b Proposition 3.35

Then H ( U ; F ) i s

it i s

) c o m p l e t e i f and o n l y i f

Sb

) comptete.

L e t U be a b a l a n c e d o p e n s u b s e t of a

ZocaZZy c o n v e x s p a c e and l e t F be a Banach s p a c e . i s T.S.r complete t h e n r = ' c 6 on H ( U ; F ) . w

Proof

Let

W, b We s h o w t h a t r w a n d

(fa)aEr denote a T

T~

d e f i n e t h e same b o u n d e d s e t s .

bounded s u b s e t o f H(U;F).

homogeneous p o l y n o m i a l s form a n 4 - a b s o l u t e ( H ( u ; F ) , ~ ~ ) if to l l o w s t h a t E m s;p n=o c o n t i n u o u s semi-norm p on H ( U i F ) . complete it follows

n!

<

m

for every

<

i s T.S.

Sincz!H(U;F) E

Since the

decomposition f o r

p(anfa(o))

t h a t E m dnfC'-n(o) n=o n!

sequence (a ) in r. m n ; Hence q (dnfan(o))

If H ( U ; F )

H(U;F)

T

w

T~

f o r any

f o r e v e r y -r6 c o n t i n u o u s s e m i - n o r m q

and any sequence ( a n ) n i n r . I t i s now e a s y t o s e e -n SLIP q ( d f a ( o ) ) < m f o r a n y T & c o n t i n u o u s s e m i - n o r m t h a t Cm n=o n! q and t h i s completes t h e p r o o f .

on H(U;F)

Our n e x t r e s u l t s h o w s t h e c o n n e c t i o n b e t w e e n T . S . completeness and completeness. Proposition 3.36

Let U be a balanced

o p e n s u b s e t of a l o c a l l y

c o n v e x s p a c e E and l e t F b e a Banach s p a c e . c o n v e x t o p o l o g y o n H (U ;F) s u c h t h a t { p ( n E ; F ) d e c o m p o s i t i c n for H ( U ; F ) (H(U;F)

(e.g.

Let

, T I is

T

be a l o c a l l y an

4- a b s o l u t e

T = T ~ , T ~ , ~ , T ~ o , r T T~~ ,) ~.

Then

,T ) i s c o m p l e t e f r e s p e c t i v e l y q u a s i - c o m p l e t e , (p(nE;F)

,TI

is complete ( r e s p e c t i v e l y quasi-complete, s e q u e n t i a l l y complete) f o r e v e r y n and H ( U ; F ) i s T . S . T c o m p l e t e . s e q u e n t i a l l y c o m p l e t e ) i f and o n l y i f

130

Chapter 3

Proof

The c o n d i t i o n s a r e o b v i o u s l y n e c e s s a r y .

are sufficient.

cases a r e handled i n a s i m i l a r fashion.

f

Cauchy n e t i n ( H ( U ; F ) , T ) . Then

Let

--

-

).er

n!

H(U;F).

n! f o r each n .

We may s u p p o s e p ( f )

f o r a l l a,,

zk

I-ience

n=o

Let p be a

~ r a,

ao,

6

-

Pn)

p[d"fa(o) n!

T

Pn

E

@(nE;F) a s

c o n t i n u o u s s e m i - n o r m on p(

=

(fa)aEr be a i s a Cauchy n e t i n

;i"fa(o)

( @ ( " E ; F ) , ~ )f o r e a c h n a n d h e n c e a n f a ( o ) a

We p r o v e t h e y

We c o n s i d e r o n l y t h e c o m p l e t e c a s e , t h e o t h e r

anfo) n! for

every

ao.

5

for a l l a

E

a

and e v e r y

In particular

positive integer k. m

anf

n=o

(O)

n!

1

+

F

f o r a l l k and s o

zm p ( p n ) < f o r e v e r y T c o n t i n u o u s semi-norm p . Since n=o m E H(U;F). i s T.S..r c o m p l e t e t h i s i m p l i e s t h a t f P n H(U;F) -n The a b o v e a l s o shows t h a t p ( d f a ( o ) - Pn) 5 for all n! T h i s c o m p letes a n d h e n c e f a f a s a m . a 1. a the proof.

- -

=AFo

izo

O u r a i m now i s t o show t h a t

( H ( U ) , Tw ) i s c o m p l e t e when-

e v e r U i s a b a l a n c e d open s u b s e t o f a m e t r i z a b l e l o c a l l y convex s p a c e .

S i n c e ( H ( U ) , T ~ )i s c o m p l e t e f o r a n y o p e n s u b s e t

U o f a m e t r i z a b l e l o c a l l y convex s p a c e p r o p o s i t i o n

i m p l i e s t h a t H(U) i s T.S..co b a l a n c e d open s e t U . show t h a t

( @ ( n E ),

T ~ )i

3.36

a n d h e n c e T . S . T ~c o m p l e t e f o r a n y

H e n c e t o p r o v e t h i s r e s u l t we must s c o m p l e t e for any p o s i t i v e i n t e g e r n .

F i r s t we n e e d some p r e l i m i n a r y r e s u l t s w h i c h a r e a l s o o f independent i n t e r e s t . Proposition 3 . 3 7

Let

u

b e a n o p e n s u b s e t of a ZocaZZy

convecc k - s p a c e . T h e n ( H ( U ) , r o ) is a semi-Monte2 s p a c e ( i . e . t h e r,-bounded s u b s e t s of H(U) a r e r e Z a t i v e Z y c o m p a c t ) .

131

Holomorphic functions on balanced sets Proof

Let & ( U )

d e n o t e t h e c o n t i n u o u s complex v a l u e d

f u n c t i o n s on U endowed w i t h t h e c o m p a c t o p e n t o p o l o g y . B be a s u b s e t o f H(U).

Now ( H ( U )

, T ~ )i

Let

s a closed subspace o f

and h e n c e B i s a c l o s e d bounded ( r e s p e c t i v e l y compact)

,&(U)

subset of

i f and o n l y i f i t i s a c l o s e d bounded

(H(U),.ro)

( r e s p e c t i v e l y compact) s u b s e t o f , & ( U ) . e x p a n s i o n s we s e e t h a t a n y

T

0

By u s i n g T a y l o r s e r i e s

b o u n d e d s u b s e t o f H(U) i s e q u i -

c o n t i n u o u s on t h e c o m p a c t s u b s e t s o f U a n d h e n c e a n a p p l i c a t i o n o f Ascoli’s t h e o r e m c o m p l e t e s t h e p r o o f . Corollary 3.38 (H(U)

, T ~ )i

Proof

I f U i s an o p e n s u b s e t o f a 2 3 h Z s p a c e t h e n

s a Frgchet-Monte1 s p a c e .

A a j r s p a c e i s a k - s p a c e a n d h e n c e (H(U) Example 2 . 4 7 shows t h a t

Montel s p a c e .

(H(U)

,To)

, T ~ )

i s a semi-

i s a Frgchet

s p a c e and t h i s c o m p l e t e s t h e p r o o f . Corollary 3.39

If

u

i s an open s u b s e t o f a m e t r i z a b l e

Z o c a Z l y c o n v e x s p a c e t h e n ( H ( U ) , - r o ) i s a semi-Monte1 s p a c e . The a b o v e r e s u l t s a n d s i m i l a r M o n t e l t y p e t h e o r e m s c o u l d a l s o b e p r o v e d by u s i n g S c h a u d e r d e c o m p o s i t i o n s .

Some o f

t h e s e a r e t o be found i n t h e e x e r c i s e s a t t h e end o f t h i s chapter. We now n e e d a l i n e a r r e s u l t w h i c h w i l l a l s o b e u s e f u l i n chapter 6. P r o p o s i t i o n 3.40

Let

T

~

T, *

and

T~

be t h r e e Hausdorff

l o c a l l y convex t o p o l o g i e s on a v e c t o r space E such t h a t (a) (b)

L

2 T ~ ; i s a b o r n o l o g i c a l DF s p a c e f o r e q u i v a l e n t l y a c o u n t a b l e i n d u c t i v e l i m i t of normed l i n e a r s p a c e s ) w i t h a countable fundamental s y s t e m o f c l o s e d convex balanced bounded s e t s ( B n ) n ; (c)

‘cl

T~

(E,rl)

( E , r 2 ) i s a b a r r e l l e d locaZZy c o n v e x s p a c e ;

B n is ‘ c 3 compact f o r all n . (d) T h e n T~ = T 2 * Proof A fundamental system o f neighbourhoods o f zero i n ( E , r l ) i s g i v e n by s e t s o f t h e form Cw

n =1

XnBn

= {Em

n=l

X

x

*

n n’

x

n

E

B

n

and m a r b i t r a r y ) where h

n

Chapter 3

132 is positive for all n EN.

5

ill

Let V =

denote the algebraic closure of V i n E,

%

V =

{ X EE ;

X > o and l e t

AnBn,

i.e.

S i n c e Bn

Ax E V f o r o 5 X < 1 ) .

i s a compact s u b -

AnBn i s a l s o a c o m p a c t s u b s e t s e t o f (E,-r3) i t f o l l o w s t h a t Z k n =1 o f ( E , T ) and h e n c e a c l o s e d s u b s e t o f (E,.r2) f o r e v e r y

3 positive integer k. %

such t h a t x integer

and hence x

AV

f!

k.

#

Now l e t x

F o r e a c h k c h o o s e $,

I

k and +k(&l AnBn) 5 1. S i n c e of ( E , r 2 ) it fcllows t h a t {$k)k

1

'L

Then t h e r e e x i s t s X > 1 k XnBn f o r every

V.

,d

k

E(E,T~)' such t h a t $k(x) m

A

=

i s an a b s o r b i n g s u b s e t

XnBn

i s a p o i n t w i s e bounded and

hence a r e l a t i v e l y weakly ccmpact s u b s e t of

(E,r2)'.

+

If

i s a l i m i t o f a weakly convergent subnet of t h e sequence {+k}k = and so t h e n $ ( x ) = A and ] $ ( V ) l 5 1. Hence

vT2 C ( ~ + EV )f o r a b s o r b i n g and

vT2

every

neighbourhood.

Since

0 .

vT2 i s

convex balanced

c l o s e d it i s a neighbourhood o f zero i n

T~

and s o every

(E,r2)

>

E

5

T~

neighbourhood o f zero c o n t a i n s a

T h i s shows t h a t

~

-

and c o m p l e t e s t h e

=

T~

T

proof. Propositicn 3.41 L e t E be a r n e t r i z a b z e L o c a l l y c o n v e x s p a c e ( i . e . Tw clnd l e t n b e a p o s i t i v e i n t e g e r . On B(nE) , T =~ T o,t i s t h e b a r r e l l e d topoZogy a s s c c i a t e d w i t h T ~ ) . ( @ ( n E ) , ~ w i) s a b o r n o l o g i c a l D F s p a c e w i t h Proof fundamental system o f bounded sets

Bm = {P

8

E

(nE);

IIP

lIvm -<

1 ) w h e r e Vm r a n g e s o v e r a

fundamental neighbourhood system of zero i n E c o n s i s t i n g of c l o s e d convex balanced s e t s . compact s u b s e t o f topology

T

'

u - T O, t L i n proposition 3.40. T

01

=

T

By C o r o l l a r y 3 . 3 8 B m i s a

( P ( n E ) , ~ o ) . Since T

~

.

L e t T~

=

T~

T

~

is a barrelled T,

~

= , T~

~

a n d -c3 =

T

0

Since a l l the requirements are s a t i s f i e d

and t h i s conpletes t h e p r o o f .

O , t

Corollary 3.42

If E i s a m e t r i z a b l e l o c a l z y c o n v e x s p a c e t h e n ( @ l n E ) , Tw ) i s a compZete l o c a l l y c o n v e z s p a c e f o r e v e r y nor1 n e g a t i v g i n t s y e r n . Proof

S i n c e (@("E) , T o )

t h a t @(nE)

i s c o n : p ~ e t ep r o p o s i t i o n 3 . 5 i m p l i e s

endowed w i t h t h e a s s o c i a t e d b a r r e l l e d t o p o l o g y i s

a l s o complete.

The a s s o c i a t e d b a r r e l l e d t o p o l o g y i s

p r o p o s i t i o n 3 . 4 1 and t h i s completes t h e p r o o f .

T

by

133

Holomorphic functions on balanced sets As a f u r t h e r c o r o l l a r y we o b t a i n a g e n e r a l i z a t i o n t o

h o m o g e n e o u s p o l y n o m i a l s o f t h e w e l l known l i n e a r c h a r a c t e r ization of distinguished metrizable spaces. Corollary 3.43

If E i s a m o t r i z a b l e l o c a l l y c o n v e x s p a c e t h e n

t h e following a r e e q u i v a l e n t ; (a) ( p ( " E ) p 0 ) ( r a e s p e c t i v e l y (6'("E) , @ ) ) i s a b a r r e l l e d l o c a l l y convex space, (b)

(@("E) , T ~ ) ( r e s p e c t i v e l y ( 8 ( n E ) , @ ) ) is a b o r n o l o g i c a l ZocaZ 7y c o n v e x s p a c e , (@("E), T o )

(c)

( r e s p e c t i v e l y ( Q ( ~ E, )B ) ) is an u l t r a -

bornological l o c a l l y c o n v e x s p a c e . Proof

I C suffices to notice that

on @ ( n E ) a n d , s i n c e T

= Bb

= 6,

= 'ub

'c0

2 6 5

T

~

,

T = T o,b = 'o,t we ~ , a l s o have

'

=

o,ub

*

I f U is a b a i a n c e d o p e n s u b s e t o f a m e t r i z -

-C -orollary 3.44

a b l e Z o c a l l g c o n v e x s p a c e t h e z (ti(u),'cw) and ( H ( U ) , r ) a r e 6 b o t h c o m p i e t e ZocaZZy c o a v e x s p a c e s . Proof

I t s u f f i c e s t o a p p l y p r o p o s i t i o n 3 . 3 6 and c o r o l l a r y

3.42.

i s coniplete

I f U i s a b a l a n c e d open s e t and ( H ( U ) , ' c w )

[H(U),'c ) i s a l s c c o m p l e t e , we may p r o v e t h i s i n t h r e e 6 d i f f e r e n t ways;

then

since

T

i s t h e b a r r e l l e d topology associated with

6

'c

w

( t h e o r e m 1 . 2 6 2 we may a p p l y p r o p o s i t i o n 3 . 5 w h i c h s a y s t h a t t h e t a r r e l l e d topology associated with a complete l o c a l l y convex t o p o l o g y i s a l s o c o m p l e t e , if

(H(U),r

w

)

i s complete then H(U)

( p r o p o s i t i o n 3 . 3 6 ) and h e n c e s i n c e , T.S.

T~

complete.

Since

T~

and

T

w

i s T.S. 'c6

2

'cw

complete

T ~ , ,H ( U )

agree on@("E)

is

for all

n we may t h e n a p p l y p r o p o s i t i o n 3 . 3 6 t o c o m p l e t e t h e proof, if

(H(U),.rw)

i s complete then H ( U )

and h e n c e a l s o T.S.-c

i s T.S.rw c o m p l e t e

c o m p l e t e ( c o r o l l a r y 3 . 3 4 ) and an

w,b a p p l i c a t i o n o f proposition 3 . 3 6 completes t h e proof.

I f U i s an open s u b s e t o f a q u a s i - n o r m a b l e m e t r i z a b l e

Chapter 3

134

s p a c e t h e n i t i s k n o wn t h a t ( H ( U ) , - c U )

is complete.

However

t h e g e n e r a l problem f o r open sets i n m e t r i z a b l e spaces i s s t i l l open.

We r e t u r n t o t h i s q u e s t i o n i n c h a p t e r 6 .

R e s u l t s s i m i l a r t o t h e a b o v e may a l s o b e p r o v e d f o r h o l o m o r p h i c germs b y u s i n g t h e same t e c h n i q u e s .

In t h i s

manner w e o b t a i n t h e f o l l o w i n g r e s u l t s . Proposition 3.45

L e t K be a compact b a l a n c e d s u b s e t of a

l o c a l l y c o n v e x s p a c e E and l e t F b e a Banach s p a c e .

Then

H(K;F)

is a c o m p l e t e ( r e s p e c t i v e l y q u a s i - c o m p l e t e , s e q u e n t i a l l y n c o m p l e t e ) ZGCaZZy c o n v e x s p a c e i f and onlyv if ($( E ; F ) , r U ) is comp 7 e t e l r e s p e c t i v e 2 y q u a s i -camp t e t e , s e q u e n t i a l Z y comp 1 e t e 1 f o r a l l n and for any s e q u e n c ~o f homogeneous p o l y n o m i a t s m ( P ~ ) ~ p=n ~E , P ( ~ E ; F ) ,n z="o p ( p n ) < m f o r e a c h c o n t i n u o u s s e m i n 0 y . m p on H ( K ; F ) i m p Z i e s Zm n = o 'n

E

H(K;F).

Corollary 3.46 I f K i s a compact b a l a n c e d s u b s e t of a m e t r i z a b l e l o c a l l y convex space E t h e n H ( K ) i s a complete l o c a l l y convex space. B Y c o r o l l a r y 3 . 4 2 ( ~ ( " E ) , T ~ i)s c o m p l e t e f o r a l l n . Proof If

m

(Pn)n=o i s a s e q u e n c e o f homogeneous p o l y n o m i a l s , m

Pn E ? l ( n E ) , a n d =;, p(Pn) < m f o r each continuous semi-norr Since p o n H ( K ) t h e n CPn);=o i s a bounded sequence i n H ( K ) . H(K)

is a regular inductive l i m i t

( p r o p o s i t i o n 2.55)

there

e x i s t s a n e i g h b o u r h o o d V o f K and X > 1 s u c h t h a t m 1 S,UP l l P n l l A V = M m * hence IIPr!IIV 5. M p < co a n d s o Zm Pn E H ( K ) . An a p p l i c a t i o n o f p r o p o s i t i o n 3 . 3 6 now C O m F l e t e s n=o the proof.

iZC

hZC

I n t h e a b o v e c o r o l l a r y we u s e d t h e r e g u l a r i t y o f H ( K ) w h i c h was p r o v e d f o r a r b i t r a r y c o m p a c t s u b s e t s o f a m e t r i z a b l e space i n chapter 2.

T h i s r e s u l t can a l s o b e proved independ-

e n t l y f o r b a l a n c e d c o m p a c t s e t s by u s i n g S c h a u d e r decomposi t i o n s ; s p e c i f i c a l l y o n e u s e s t h e semi-norms

where

( x ~ r )a n~g e s o v e r a l l s e q u e n c e s w h i c h t e n d t o K ,

135

Holomotphic functions on balanced sets

I n c h a p t e r 5 , which d e a l s w i t h holomorphic f u n c t i o n s o n n u c l e a r s p a c e s , we s h a l l s e e t h a t r e g u l a r i t y a n d c o m p l e t e ness o f s p a c e s o f germs a r e e q u i v a l e n t i n a number o f nonYe g i v e h e r e a n e x a m p l e o f a s p a c e o f

trivial situaticns.

germs which i s n o t r e g u l a r .

L a t e r r e s u l t s w i l l show t h a t i t

i s a l s o n o t complete. Exaniple 3 . 4 7

cn

we i d e n t i f y

a").

Let E =

For each p o s i t i v e i n t e g e r n

with t h e subspace o f

spanned by t h e f i r s t

g")

Let 0 d e n o t e t h e o r i g i n i n

n coordinates.

b e a c o n t i n u o u s semi-norm on H ( 0 ) .

@(')

and l e t p

Ye c l a i m t h e r e e x i s t s a n

C n = o f o r some H(0) and f l u neighbourhood U o f z e r o i n Cn t h e n p ( f ) = 0 . We may s u p p o s e i n t e g e r n such t h a t i f f

E

:Io

by theorem 3.25 t h a t p ( f ) = Since p

I p(kE)

is

T~

and hence

p ( m ) f o r every f i n H(0).

n.

T~

continuous

(E i s a

3zw

he c a n f i n d f o r e v e r y i n t e g e r k a n o t h e r p o s i t i v e k i n t e g e r k ' s u c h t h a t i f P E 6(E ) a n d P I tk ' = 0 t h e n p ( P ) = H e n c e i f o u r c l a i m i s n o t s a t i s f i e d t h e n we c a n f i n d a

space)

sequence o f homogeneous p o l y n o m i a l s , P

=

0

and p(P.)

#

0.

Let Q

j l,j 1 i s a T o b o u n d e d s u b s e t o f H(

j

a:

m

( P j ) j = l , such t h a t

jP

2

=

p(pj)

for all j .

{Q.)" 3 j=1

s i n c e e v e r y compact s u b s e t

CC ( N ) i s c o n t a i n e d a n d c o m p a c t i n C n f o r s o m e p o s i t i v e integer n. H e n c e , s i n c e T~ = T w on H( C(N)) ( e x a m p l e 2 . 4 7 ) . ,

of

m

IQjljZl i s a boupded s u b s e t o f H(0).

But p ( Q . ) = j 3 claim.

for all j

and t h i s c o n t r a d i c t i o n proves o u r X

n For each i n t e g e r n l e t f , ( ( ~ ~ ) = ~ i) q q -

.

Since fn = o i n

f o r a l l n t h e a b o v e shows t h a t I f 1 i s a bounded Q: n n s u b s e t o f H ( @ ) . I f H(0) was a r e g u l a r i n d u c t i v e l i m i t t h e n t h e r e w o u l d e x i s t a n e i g h b o u r h o o d W o f z e r o i n CE ('1 on

H(O

w h i c h e a c h f n was d e f i n e d a n d b o u n d e d .

1

(--,

o

. . .

.,o,

)

1

we c o n c l u d e t h a t H(O =

v3 0 V open

W f o r all n. (i:

( H m ( V ) , 11

By

OUT

construction

Since t h i s i s impossible

(N))

[Iv)

is not a regular inductive l i m i t .

0 .

136

Chapter 3 We now c o n s i d e r

a r b i t r a r y E)

linear

and on H ( E )

f u n c t i o n a l s on HN(E) ( f o r

f o r E a m e t r i z a b l e l o c a l l y convex

We t h e n c o m b i n e t h e s e

space with t h e approximation property. r e s u l t s t o show t h a t

T

= T

o n H ( E ) when E i s a F r g c h e t

W

n u c l e a r s p a c e a n d ( H ( E ) , T ~ ) '2~ H(OEI@)a l s o u n d e r t h e s a m e conditions.

These r e s u l t s g e n e r a l i s e r e s u l t s a l r e a d y proved

f o r homogeneous p o l y n o m i a l s

i n chapter 1 (proposition 1.61).

The p r o o f s u s e S c h a u d e r d e c o m p o s i t i o n s and e s t i m a t e s p r e v i o u s l y obtained i n proving t h e corresponding r e s u l t polynomials.

f o r homogeneous

The r e s u l t s p r e s e n t e d h e r e a r e r e l a t i v e l y r e c e n t

as these topics are currently the object of research.

We

d i s c u s s h e r e two d i f f e r e n t s i t u a t i o n s a n d i t i s p r o b a b l e t h a t a g e n e r a l t h e o r y which c o v e r s b o t h s i t u a t i o n s s i m u l t a n e o u s l y w i l l appear i n the not too d i s t a n t f u t u r e .

A similar theory

f o r balanced open s e t s has n o t y e t been developed. Definition 3.48

L e t E b e a ZocaZZy c o n v e x s p a c e .

If V i s a

c o n v e x b a l a n c e d o p e n s u b s e t of E we d e f i n e m

HN(V) = c f

E

H(V); inf(o)CfN("E)

f o r e a c h n and

( H i ( V ) , a ) is a Banach s p a c e . V

HN(E), t h e nuclearly e n t i r e

f u n c t i o n s on E, is d e f i n e d a s { f E H ( E ) ; d n f ( o ) & N ( n E )

all n

and for e a c h compact s u b s e t K of E t h e r e e x i s t s an o p e n s u b s e t V of

E,

KCV,

such t h a t a V f f ) <

m),

The r o t o p o Z o g y on H N ( E ) is g e n e r a t e d by a l l semi-norms

d"f

m

T K ( f )

=

n=o

T

T

~

(

( 0 )

n!

)

a s K r a n g e s o v e r t h e compact s u b s e t s of E . A semi-norm p on H ( E ) is s a i d t o be n u c o n t i n u o u s if t h e r e N

e x i s t s a compact s u b s e t K o f E s u c h t h a t for e v e r y o p e n s u b s e t V of E c o n t a i n i n g

p(f)

K there e x i s t s c(V)>o such t h a t

2 c ( V ) n v ( f ) for e v e r y

We l e t H ~ ( o )=

5

O E V . oDen

f i n HN(E).

( H ~ ( v ) , T ~and ) c

a t h~ i s ~t h e s p a c e of

conirejt balanced

n u c l e a r h o l o m o r p h i c germs a t 0 . I t i s immediate t h a t

IT

0

<

-

T

w

on HN(E) a n d H N ( 0 )

i s an

137

Holomorphic functions on balanced sets u l t r a b o r n o l o g i c a l l o c a l l y convex space. Proposition 3.49 {

L e t E be a ZocaZZy c o n v e x s p a c e

m

(? N ( n E ) , 1 ~ ~ i ls ~a n = . 8 -~d e c o m p o s i t i o n and an A - a b s o 2 u t e

decomposition f o r (HN(E)

, T ~ )

a n J - d e c o m p o s i t i o n and a n

and H N ( 0 ) .

4 -absoZute

{

6 N(

n

E)

,

T

m ~

i~ s =

}

~

decomposition f o r

(HN(E) ,no)

We l e a v e t h e p r o o f o f t h i s p r o p o s i t i o n t o t h e r e a d e r . I t p r o c e e d s i n a s i m i l a r manner t o t h a t u s e d f o r t h e t o p o l o g i e s on H ( U ; F )

T

and

0

(theorem 3 . 1 9 and p r o p o s i t i o n 3.18)

also uses the following equality;

'cW

and

i f B is a subset of E ,

a

i s a p o s i t i v e r e a l number and n i s a p o s i t i v e i n t e g e r t h e n n 71 71 (PI for a l l P E P N ( n ~ ) . aB ( P I = a B P r o p o s i t i o n 3.50

L e t E b e a ZocaZZy c o n v e x s p a c e .

transform, B , i s a Zinear isomorphism from (H ( E ) N

(HN(E)

, T ~ )

onto

) and u n d e r t h i s i s o m o r p h i s m e q u i c o n t i n u o u s s u b s e t s

H ( o ( 1 ~, T o )

of

The Bore2

,T

~

)

correspond t o subsets o f

H(O(E

which are

,To))

d e f i n e d and bounded on some n e i g h b o u r h o o d o f z e r o i n ( E ! , T ~ ) . Proof

f o r (HN(E)

, T ~ )p

r o p o s i t i o n 3 . 1 3 i m p l i e s t h a t {(@,(

is and-absolute

E

decomposition f o r (HN(E) ,v0);i.

( 8 N ( n E ) , ~ o ) 1f o r a l l n . m

BT = ;=o

BT,.

m IT^)^)^=^

Hence a n y .

We d e f i n e BT b y t h e f o r m u 2 a

IT(f)l 5 crK(f) f o r every f i n

( E ) t h e n t h e p r o o f o f p r o p o ~ i t i o n 1 ~ 4s7h o w s t h a t

N f o r a l l n. Since Em (BT,) 5

io i s a n e i g h b o u r h o o d ~ i p== 2~c < this m

of zero i n i m p l i e s BT

n = o '+KO and t h e image by B o f an e q u i c o n t i n u o u s s u b s e t o f i s a bounded s u b s e t of Hm(V) Also,

for

Morever i f K i s a compact s u b s e t

integer n.

o f E and c >o a r e such t h a t

( E ' , T ~ ) .

E)

B Y p r o p o s i t i o n 1 . 4 7 B T ~E P ( " ( E * , T ~ ) )

every non-negative H

n

c a n b e r e p r e s e n t e d a s nE m = o Tn w h e r e

T E (HN(E),so)l Tn

m

n o ) }n=o i s a n A - a b s o l u t e d e c o m p o s i t i o n

Since { (@N(nE),

TKO(BTn) 5 c (El, T E H(0)

~ )a n d

(H (E) ,no) ' N f o r some n e i g h b o u r h o o d V o f 0 i n

s i n c e t h e Bore1 t r a n s f o r m i s an isomorphism

o n e a c h s p a c e o f n - h o m o g e n e o u s n u c l e a r p o l y n o m i a l s , we h a v e shown t h a t B i s a w e l l d e f i n e d i n j e c t i v e l i n e a r m a p p i n g . C o n v e r s e l y l e t A = CgE H

m

(KO);

izo -,[lI

-< 11

where K i s

138

Chapter 3

a compact s u b s e t o f E .

F o r e a c h g i n A t h e r e e x i s t s , by

proposition 1 . 4 7 , a unique sequence o f l i n e a r f u n c t i o n a l s , Tn E ( 8 N ( n E ) , R ~ )I , s u c h t h a t BTn = "n d g(O),,! for a l l (Tnl;=OI

n and lTn(P)

1

2

CT

K

(P ) f o r e v e r y P i n

@N(nE).

If f

E

HN(E)

m

Tn) = g a n d s o t h e B o r e 1 t r a n s f o r m i s a b i j e c t i v e onto H(O ). Since the l i n e a r mapping from ( H ( E ) , n o ) '

Now B(;=,

N

( E 1 , T O )

a b o v e a l s o shows t h a t A i s t h e i m a g e u n d e r B o f a n e q u i continuous subset of (H

N

(E),ao)l

we h a v e c o m p l e t e d t h e p r o o f

L e t E b e a ZocaZZy c o n v e x s p a c e and Z e t V b e a f i n i t e Z y open s u b s e t o f E l c o n t a i n i n g t h e o r i g i n .

We Z e t

I f T i s a ZocaZZy c o n v e x equicontinuous subset A of V l . t o p o l o g y o n E l we l e t H ( 0 ) d e n o t e t h e s p a c e o f germs 5 (E',T) a r i s i n g from t h e usual equivalence r e l a t i o n s h i p i n U H ( V ) , V r a n g i n g o v e r aZZ c o n v e x baZanced n e i g h b o u r h o o d s

v

5

o f zero i n ( E ~ , T ) . Proposition 3.51

L e t E b e a ZocaZZy c o n v e x s p a c e .

The

Bore1 t r a n s f o r m i s a t i n e a r i s o m o r p h i s m f r o m ( H N ( E ) , v u ) ('(E

,TO)

onto

) and u n d e r t h i s i s o m o r p h i s m e q u i c o n t i n u o u s s u b s e t s

of ( H N ( E ) , ~ u ) l c o r r e s p o n d t o s u b s e t s of H 5 ( 0 ( E ' 7 o )) w h i c h a r e d e f i n e d and bounded o n t h e e q u i c o n t i n u o u s s u b s e t s of some neighbourhood o f z e r o i n Proof

( E l

,to).

Use t h e same m e t h o d a s i n t h e p r e c e d i n g p r o p o s i t i o n

and t h e e s t i m a t e s g i v e n i n p r o p o s i t i o n 1 . 4 8 .

Note t h a t a s V

r a n g e s o v e r t h e convex b a l a n c e d open neighbourhoods o f t h e compact s u b s e t K o f E , V o

ranges over t h e closed equicontinuous

s u b s e t s of t h e i n t e r i o r of

KO.

The a b o v e p r o p o s i t i o n s y i e l d a number o f i n t e r e s t i n g c o r o l l a r i e s s i n c e two t o p o l o g i e s o n a l o c a l l y c o n v e x s p a c e a r e e q u a l i f a n d o n l y i f t h e y h a v e t h e same d u a l and d e f i n e t h e same e q u i c o n t i n u o u s s u b s e t s i n t h i s d u a l . Corollary 3.52

I f E i s a Fre'chet MonteZ s p a c e t h e n

139

Holomorphic functions on balanced sets IT

=

IT

on H N ( E ) .

w

If E i s a fully n u c l e a r s p a c e t h e n T = IT = n = T o n H ( E ) if H ( V ) = H(V) and T~ bounded 0 w w 5 s u b s e t s of H(V) a r e l o c a l l y bounded f o r e a c h o p e n s u b s e t V Corollary 3.53

of E A . I f o u r hypotheses are s a t i s f i e d then p r o p o s i t i o n s 3.50

Proof

__.

and 3 . 5 1 imply t h a t

IT

0

=

IT

w

on HN(E).

o f p r o p o s i t i o n 1 . 4 1 we s e e t h a t T~

=

By u s i n g t h e e s t i m a t e on HN(E).

IT

Since

T < T < IT on H ( E ) f o r any l o c a l l y c o n v e x s p a c e E i t f o l l o w s 0 w w N By p r o p o s i t i o n t h a t a l l o f t h e s e t o p o l o g i e s a g r e e on H N ( E ) .

1 . 4 1 HN(E) i s a d e n s e s u b s p a c e o f H(E) a n d h e n c e a l l o f t h e s e t o p o l o g i e s a g r e e on H ( E ) .

This completes t h e proof.

c o r o l l a r y 3 . 5 3 w e r e c o v e r a number o f

As a consequence o f

r e s u l t s previously proved by o t h e r methods; e . g . H(E) w h e r e E i s a a 3 t s p a c e ( e x a m p l e 2 . 4 7 )

and

T

=

0

#

T~

T

w

on

on

T~

H ( E x E ' ) when E i s a n i n f i n i t e d i m e n s i o n a l F r G c h e t n u c l e a r s p a c e B ( e x a m p l e 2 . 4 9 ) a n d a l s o o b t a i n t h e f o l l o w i n g new r e s u l t . Corollary 3.54

on H ( E )

I f E is a F r g c h e t n u c l e a r s p a c e t h e n

-

O

Proof

Since

n

c (6( E) ,

B

;i,i=oi s

T ~ )

H ~ ( o ( , ~T o')

w

1.

a Schauder decomposition f o r

( H ( E ) , T ~ ) ~( c; o r o l l a r y 3 . 1 4 a n d t h e o r e m 3 . 1 9 )

any T

c a n b e r e p r e s e n t e d as nZ = o Tn w h e r e Tn

,To)

m

i t i o n 1 . 6 1 BTn n.

T

The BoreZ t r a n s f o r m is a l i n e a r t o p o l o g i c a l

isomorphism from (H(E),T ) ' onto _c

=

L e t E b e a FrGchet s p a c e w i t h t h e a p p r o x i m a t i o n

Theorem 3 . 5 5

property.

T

E

gN(n(E',To))

E

(@("E)

E

I .

f o r every non-negative

(H(E),T~)' By p r o p o s integer

I f L i s a compact s u b s e t o f E and c > o are s u c h t h a t

I T ( f ) l 5 c llflL f o r e v e r y f i n H ( E ) t h e n , b y p r o p o s i t i o n 1 . 6 0 , t h e r e e x i s t s K compact s u c h t h a t aKO(BTn) 5 c f o r a l l n . S i n c e KO i s a n e i g h b o u r h o o d o f z e r o i n ( E ' , r o ) a n d - 1 Cm nSKO(BTn) 5 c yn = 2 c < t h i s i m p l i e s t h a t BTE H N ( 0 ) n=o and t h e image by B o f a n e q u i c o n t i n u o u s s u b s e t o f ( H ( E ) , T ~ ) ' m

i s a b o u n d e d s u b s e t o f H (V) f o r s o m e n e i g h b o u r h o o d V o f 0 i n N (E' , T o ) . m

C o n v e r s e l y l e t A = {(P,).=.E

m

H N ( ~ ) ; $ " o r K 0 ( P n ) 5. 1 ) w h e r e K

140

Chapter 3

i s a convex b a l a n c e d compact s u b s e t o f E .

If

W

(Pn)n=o F A

then f o r each non-negative i n t e g e r n t h e r e e x i s t s a unique T~

E

( P ( n ~, )T

~ )

s u c h t h a t BTn = Pn a n d

lTn(P)

every P i n Q("E).

Since

it follows t h a t T =

i=o Tn E ( H ( E ) , T ~ ) 'a n d W

I

5 lIPIIK f o r

BTE A.

This a l s o

shows t h a t A i s t h e image o f a n e q u i c o n t i n u o u s s u b s e t o f ( H ( E ), T ~ )

Since

I (@("E) l

~ o )

W

i s an

d -decomposition

for

( H ( U ) , T ~ )p r o p o s i t i o n s 1 . 6 1 a n d 3 . 1 3 i m p l y t h a t m

i s a n d - d e c o m p o s i t i o n f o r ( H ( E ) ,T ) (E T ~ 1) n u ) l n Z o 0 B ' S i n c e (H(E),-co) i s a semi-Monte1 s p a c e i t s s t r o n g dual

(8, ("

Hence,

(H(E),.ro)f; i s a b a r r e l l e d space. I P N ( n ( E ' , T o ) ) ,n

Wn

by p r o p o s i t i o n 3 . 4 9 ,

i s a n d -decomposition

f o r two b a r r e l l e d

t o p o l o g i e s on H (0) and p r o p o s i t i o n 3 . 1 2 s a y s t h a t t h e s e N t o p o l o g i e s must c o i n c i d e . H e n c e ( H ( E ) , T 0 ) 6' 'L H N ( 0 ) a l g e b r a i c a l l y and t o p o l o g i c a l l y . This completes t h e proof.

Lemma 3.56

L e t E be a q u a s i - c o m p l e t e n u c l e a r and dual

nuclear space.

Then H N ( 0 )

= H(0)

( a l g e b r a i c a Z l y and

topo1ogicaZZ.y). Proof:

'Ce a l w a y s h a v e H N ( 0 ) C H ( 0 ) .

By t h e o r e m 1 . 2 7

pN(nE) = 8 ( n E ) f o r e a c h n o n - n e g a t i v e i n t e g e r n and moreover T

= n

on

w

B(nE)

by p r o p o s i t i o n 1 . 4 4 .

Again by p r o p o s i t i o n

1.44 w e can choose f o r every neighbourhood W o f z e r o a n o t h e r neighbourhood V of zero such t h a t

i=o llpn/Iw 2 W

W

"W(Pn)

50

I &o

llpnll".

lience H (0) = H ( 0 ) a l g e b r a i c a l l y and t o p o l o g i c a l l y . Note a l s o N t h a t a s u b s e t o f H N ( 0 ) i s c o n t a i n e d a n d b o u n d e d i n s o m e HZ(V) i f a n d o n l y i f i t i s c o n t a i n e d a n d b o u n d e d i n some Hm(W), V and W b e i n g neighbourhoods

Corollary 3.57

of 0 i n E.

If E i s a F r d c h e t n u c l e a r s p a c e t h e n

( H ( E ) # T o ) i2 H ( o E ! ) * B

141

Holomorphic functions on balanced sets Corollary 3.58 =

T~

'c6

E i s a Fre'chet n u c l e a r s p a c e t h e n

I f

on H(BJ i f and o n l y if

a

v30, V open

(H"(V)

,I1

!Iv) is a r e g u l a r

i n d u c t i v e l i m i t where O E E '

6'

S i n c e E i s a F r g c h e t s p a c e T~

Proof if

(H(E),.ro)

= T~

o n H(E) i f a n d o n l y

i s an i n f r a b a r r e l l e d l o c a l l y convex space.

A

l o c a l l y convex s p a c e i s i n f r a b a r r e l l e d i f and o n l y i f s t r o n g l y bounded s u b s e t s o f t h e d u a l are e q u i c o n t i n u o u s . (H(E),.,)'@

1 H(0

Now

and t h e e q u i c o n t i n u o u s s u b s e t s of

)

'B ( H ( E ) , T ~ ) 'a r e t h e s u b s e t s o f H(0) w h i c h a r e c o n t a i n e d and b o u n d e d i n H"(V)

f o r some n e i g h b o u r h o o d V o f z e r o i n F'

B

So ( H ( E ) , - r o ) i s

i n f r a b a r r e l l e d i f and o n l y i f e a c h bounded

s u b s e t o f H ( 0 ) i s c o n t a i n e d a n d b o u n d e d i n H"(V) neighbourhood V o f zero, ( H m ( V ) , 11

l i m

__f

OEV,

Ilv)

i.e.

f o r some

i f and o n l y i f

is a regular inductive l i m i t .

V open VCE

E,

Example 3 . 5 9

Example 3 . 4 7 shows t h a t H(0)

inductive l i m i t i f 0 that

T

0

#

on H ( Q

T~

N

E

.

is not a regular

By c o r o l l a r y 3 . 5 8 t h i s s h o w s

We h a v e a l r e a d y o b t a i n e d t h i s r e s u l t

),

by a d i f f e r e n t method proposition

C "1

(example 2 . 5 2 ) .

Example 2 . 5 2 and

3 . 5 8 a l s o show t h a t i f E i s a F r G c h e t n u c l e a r

s p a c e which d o e s n o t a d m i t a c o n t i n u o u s norm t h e n H(oEI

B

1

=

V open VCEi B

is not a regular inductive l i m i t . §

3 . 4 SEMI-REFLEXIVITY

A N D N U C L E A R I T Y F O R SPACES O F

H O L O M O R P H I C FUNCTIONS Our f i r s t r e s u l t

f o l l o w s from a g e n e r a l theorem concern-

i n g s e m i - r e f l e x i v e l o c a l l y convex s p a c e s which have e q u i Schauder decompositions. Proposition 3.60

L e t U be a balanced open s u b s e t o f a

l o c a l l y c o n v e x s p a c e E , Z e t F be a Banach s p a c e and l e t

Chapter 3

142 T

'o,b'

semi-refZexive

T

w'

T

w,b'

~

) on

H(U;F).

i f and onZy i f

(P("E;F)

Then ( H ( U ; F ) , r ) i s

, T I is s e m i - r e f Z e x i u e

f o r e a c h n o n - n e g a t i v e i n t e g e r n and H ( U ; F ) is T . S . r - e o m p Z e t e . ( H ( U ; F ) , T ~ )i s semi-

I n p a r t i c u l a r t h i s means t h a t r e f l e x i v e i f and o n l y i f Corollary

(H(U;F),T,,~) i s reflexive.

I f U i s a baZanced open s u b s e t of a

3.61

FrGchet s p a c e and F i s a Banach s p a c e t h e n ( H ( U ; F ) , r ) , where T ~ } , is s e m i - r e f Z e x i v e i f and onZy i f . ( p ( n ~ ; , ~T)) TE{T T 0' w' i s s e m i - r e f Z e x i u e for e a c h n o n - n e g a t i v e i n t e g e r n . iVe now s h o w t h a t

(H(U),T ) i s a n u c l e a r space i f U i s

an a r b i t r a r y open s u b s e t o f a q u a s i - c o m p l e t e

dual nuclear

Since the projective l i m i t of nuclear spaces i s

space.

n u c l e a r and t h e compact open t o p o l o g y i s a l o c a l t o p o l o g y i t s u f f i c e s t o prove t h i s r e s u l t f o r convex balanced open s e t s . This r e s u l t

can e a s i l y b e proved f o r e n t i r e f u n c t i o n s by u s i n g

T a y l o r s e r i e s e x p a n s i o n s a n d t h e f o l l o w i n g lemma w h o s e p r o o f

We l e t

i s already contained i n t h e proof of theorem 1 . 4 3 . s = sup

Note t h a t s i s f i n i t e by S t i r l i n g ' s

1/n'

(n!) formula.

m

Lemma 3 . 6 2

L e t K a n d ( x ~ ) ~be= r~e s p e c t i v e z y a compact

baZanced s u b s e t and a compact s e q u e n c e i n t h e ZocaZZy c o n v e x s p a c e E. L e t L d e n o t e t h e c Z o s e d c o n v e x baZanced huZZ of m

(Sxn)n=l

*

in E' If ( A n ) n = l E: 2 , and II+IIK 5 ; = l I ~ n I I+(x,)l f o r every t h e n f o r e a c h p o s i t i v e i n t e g e r m t h e r e e x i s t a s e q u e n c e in m

m

i1, ( h m , n ) n = l ) m

such t h a t Proof

I1pIIK

m

z J i t h ;=l

c

m

;=l

b m , J= c=;l

m

I A ~ , ~ IP(X;) I I

IjJ)

m

, and

f o r every P

(x",,,"=l E

C

6""~).

By i n d u c t i o n o n e s e e s ( a s i n t h e o r e m 1 . 4 3 )

f o r every L i n

ofLS(mE).

An a p p l i c a t i o n o f t h e p o l a r i z a t i o n

L

143

Holomoiphic functions on balanced sets formula e a s i l y completes t h e proof. A sequence

-

(x ) i n a l o c a l l y convex space i s s a i d t o be n n o as n m f o r every n

r a p i d l y decreasing i f npx positive integer p.

L e t U b e a c o n v e x b a l a n c e d o p e n s u b s e t of a

Lemma 3.63

q u a s i - c o m p l e t e d u a l n u c l e a r s p a c e E and l e t K b e a ( c o n v e x b a l a n c e d ) compact s u b s e t of U . There e x i s t a f i n i t e dimensional subspace F o f E , compact s u b s e t of F , U o an o p e n s u b s e t of F ,

m

KO a

€tlr

m

( x ~ ) a ~ r=a p~i d l y d e c r e a s i n g s e q u e n c e i n E and W a n e i g h b o u r hood of z e r o i n E s u c h t h a t

Let V d e n o t e a convex b a l a n c e d n e i g h b o u r h o o d o f z e r o

Proof

S i n c e any compact s u b s e t o f a

i n E such t h a t K + V C U . quasi-complete

dual n u c l e a r space i s contained i n t h e convex

h u l l o f a r a p i d l y d e c r e a s i n g s e q u e n c e we c a n c h o o s e ( y n I n a r a p i d l y d e c r e a s i n g s e q u e n c e i n E whose c l o s e d c o n v e x h u l l Choose N a p o s i t i v e i n t e g e r s u c h t h a t contains K. 2 4 l o a n y n € V f o r a l l n > N w h e r e a = II s / b . Let F d e n o t e t h e s u b s p a c e o f E s p a n n e d b y { y l , N

3

a n d l e t K O = {;=l nz m =l

l a n ] 5 1 and

i = la n y n

By o u r c o n s t r u c t i o n K Uo

=

KO

Let An

1

7(VT\F).

+

=

1 2a(N+n)'

integer n.

since

("n);=N+1

m

Uo

E

. . .

,yN}

such t h a t

K}.

i s a compact s u b s e t o f F .

Let

i s an open s u b s e t o f F and K o C Uo.

and x

n

2

= 2a(N+n) yN+n f o r each p o s i t i v e

lhnI

5

F1a - g -= l 1n

= -2 a - -I T= ~- 2 s

and

Chapter 3

144

cKo + 14. v CK

+

4

-

+

V +

1 12

n

ii

4

+

1 3

V +

1 V C K 12

t h i s c o m p l e t e s t h e p r o o f i f we l e t W =

+ V C U 1 12 v*

We s h a l l

We now p r o v e t h e m a i n r e s u l t o f t h i s s e c t i o n .

(H(U)

assume t h a t

, T ~ )i

open s u b s e t o f E n .

s a FrGchet n u c l e a r space i f U i s an

T h i s i s a w e l l known f i n i t e d i m e n s i o n a l

r e s u l t a n d i s g i v e n i n a number o f books on f u n c t i o n a l analysis.

I t i s a l s o a s p e c i a l c a s e o f a r e s u l t which w i l l

be proved,

independently o f t h e following r e s u l t ,

in

chapter 5. Theorem 3 . 6 4

L e t U be a quasi-complete

dual nuclear space.

T h e n ( H ( U ) , . r o ) is a n u c l e a r s p a c e if U is an o p e n s u b s e t o f E . Proof

We may s u p p o s e t h a t U i s c o n v e x a n d b a l a n c e d . m

be a compact s u b s e t o f U and l e t K O ,

Let K

m

( x ~ ) a~n d= ~ Uo’(Xn)n,l, W h a v e t h e same m e a n i n g w i t h r e s p e c t t o K and U a s i n lemma 3 . 6 3 . m

(Bn)n=l (H(Uo),

By t h e n u c l e a r i t y o f

€2,

w e can f i n d

an equicontinuous s u b s e t o f

such t h a t

yo)

llfllK

a n d ($n);=l

(H(Uo),ro)

0

2

~ = ~ l @ n ll $ n ( f ) l m

for

every

in

H(Uo)’

On m u l t i p l y i n g e a c h B n b y a c o n s t a n t i f n e c e s s a r y we may s u p p o s e t h a t t h e r e e x i s t s a r e l a t i v e l y c o m p a c t s u b s e t K1

o f Uo,

145

Holornophic functions on balanced sets KO C K 1 , H(Uo)

m

m

=

ll$lk 2

Since

8Z1

'B=~

*

L e t K~

I$n(f) I

such t h a t

(XnI

enxn; m

T1 s

5

1

IXnI

[en] 5

5 $=1 l X n l

f o r a l l n and e v e r y f i n

5 IlfIjK

for all nI.

[$(xn)[ for every 4 i n

lemma 3 . 6 2

E l

and

implies that

f o r e a c h f d e f i n e d a n d h o l o m o r p h i c on a n e i g h b o u r h o o d o f K 2

RE1 m

where i n C;=l K1

of

m

Bnxn;

;Z1

+ K 3 C Uo + K 3

lon( 5 +

u.

Y E

+

W

KJ

If $

i s a n e l e m e n t o f H(K3+W).

f o r a l l x i n Uo and y i n K3

+

+ W then

-

i s an element o f H ( U o ) . ; Y €K3

1 f o r a l l n ) = K3.

W a n d h e n c e K1 If

Now l e t f E H ( U ) .

T(f)

( x m,n ) mm , n = 1 i s c o n t a i n e d

l a m l n ] 5 ( i ) m f o r a l l m and

E

$(fy)

Since

K3

i s a compact s u b s e t

f

d e f i n e d by f ( x ) = f ( x + y ) Y then the function

H'(Uo) E

C

h=o a,

Note t h a t

Y

^n

f(x)(y) = f(x+y) n!

+ W and t h e s e r i e s converges

u n i f o r m l y w i t h r e s p e c t t o x o v e r t h e compact s u b s e t s o f Uo .it follows t h a t

146

Chapter 3

Since

2

zZl m

Sup

lBnl

4

If(x+y)l

x€K1

IcakJml

Cm

k=l

- $

f o r a l l n,m

-1

5

zm

a l l m and

a n d k we h a v e shown

YEK3

IlfII, where of

;I1

I a

,?j=l

1

I$,(f)

6n

6n

m

<

(H(U),ro)'.

m

and ($n)n=l

i s an equicontinuous subset

This completes t h e proof.

Corollary 3.65

If U is an open s u b s e t of a 3 3 Q s p a c e t h e n

( H ( U ) , r o ) is a F r & c h e t nuclear s p a c e .

3.5

EXERCISES

3.66* -

A convex balanced absorbing subset o f a v e c t o r space

is called a b a r r e l ,

be d-barrelled (resp.

A l o c a l l y convex topology

(resp. d-infrabarrelled)

'I

is said t o

if e v e r y b a r r e l

e v e r y b a r r e l which a b s o r b s bounded s e t s ) which i s a

countable i n t e r s e c t i o n of neighbourhoods of zero i s a neighbourhood o f zero. barrelled Let

T~~

(resp.

and

T

di

Show t h a t t h e c o l l e c t i o n o f a l l d -

d-infrabarrelled)

t o p o l o g i e s form a Q - f a m i l y .

d e n o t e t h e d - b a r r e l l e d and d - i n f r a b a r r e l l e d

topologies associated w i t h r respectively.

If U i s a balanced

open s u b s e t o f a l o c a l l y c o n v e x s p a c e E and F i s a Banach s p a c e show t h a t w,dt

T6 = 3.67 -

Let ( E , T )

Suppose T

and

T ~ , ~ = T

w,di

on H(U;F).

be a l o c a l l y convex space.

Letrl

=

T .

h a s b e e n d e f i n e d for a l l a s t r i c t l y l e s s t h a n t h e

o r d i n a l number

ao.

Let

(E,T

)=+* ( E , T ~ ) . ac a
Let

denote

T~ 0

147

Holomoiphic functions on balanced sets t h e t o p o l o g y on E i n h e r i t e d f r o m ( ( E , T a

)IBJ1@.

t h e r e e x i s t s a n o r d i n a l number y s u c h t h a t yo

Show t h a t

2 y.

(E,

)

T

Show t h a t

0 =

T

Y

for all

T

YO

i s an i n f r a b a r r e l l e d l o c a l l y

YO

convex s p a c e and t h a t associated with

3.68

Let

T

T.

(E,T)

YO

i s t h e i n f r a b a r r e l l e d topology

b e a l o c a l l y convex s p a c e which c o n t a i n s

a c o u n t a b l e fundamental s y s t e m o f bounded s e t s . (E,Tdi)

i s a DF space.

3.69"

L e t qh?-.p,

endowed w i t h t h e

1 5 p

2c l

5 q

norm.

Show t h a t

< + m , denote t h e 2 space P Show t h a t qLp i s a b a r r e l l e d s p a c e

i f and o n l y i f p = q .

3.70*

Let

be a Schauder decomposition f o r t h e

l o c a l l y convex s p a c e E and l e t ( p a ) a E r be a fundamental system o f semi-norms o f E .

let^' d e n o t e t h e l o c a l l y

convex t o p o l o g y g e n e r a t e d by t h e semi-norms

q (x) = sup p (u ( x ) ) a s a ranges over r . Show t h a t n a n (E 3 i s an e q u i - S c h a u d e r d e c o m p o s i t i o n f o r ( E , T ' ) a n d t h a t n n T I i s t h e w e a k e s t l o c a l l y c o n v e x t o p o l o g y on E , f i n e r t h a n T,

which c o i n c i d e s w i t h

T

Schauder decomposition.

3.71

If

on e a c h E

n

and h a s {En}n a s an e q u i -

i s a S c h a u d e r d e c o m p o s i t i o n f o r ( E , T ~ )and

( E , T ~ ) show t h a t

and

'cl

T*

h a v e t h e same a s s o c i a t e d

b a r r e l l e d topology. 3.72 -

Let

3.73*

Let

(EJ)

b e a l o c a l l y convex s p a c e .

i s weak* q u a s i - c o m p l e t e

i f and o n l y i f

Show t h a t

E l

(E,T)' = (E,-rt)l.

CEnIn b e a s h r i n k i n g d e c o m p o s i t i o n f o r t h e l o c a l l y

convex s p a c e E .

Show t h a t { ( E n ) l B I n i s a n e q u i - S c h a u d e r

d e c o m p o s i t i o n f o r E'

6

.

Chapter 3

148

n Show t h a t I @ H y ( E;F)

3.74

and an , ? - a b s o l u t e

i s an$ -decomposition

decomposition f o r (HHY(U;F),~o)

whenever U

i s a b a l a n c e d open s u b s e t o f a l o c a l l y convex s p a c e E and F i s a q u a s i - c o m p l e t e l o c a l l y convex s p a c e . 3.75* -

I f U i s an open b a l a n c e d s u b s e t o f a l o c a l l y convex n s p a c e E show t h a t I @ ( n E ) , ~ o , t I Z = oa n d ( 6 ( E ) , T ~I nW, = ~o ~

a r e & d e c omp o s i t i o n s a n d

4 - a b s o 1u t e

d e compo s i t i o n s f o r

(H(U) , T ~ , a~n d) (H(U) , T ~ , r~e s ~p e )c t i v e l y . I f E i s a DF s p a c e show t h a t T

3.76*

0

,t

=

6

=

T

f o r a n y p o s i t i v e i n t e g e r n and h e n c e d e d u c e t h a t

on/P(nE)

w T

0,t

=

T

6

on H(U) f o r a n y b a l a n c e d o p e n s u b s e t U o f E . I f E i s a Frechet

3.77* T

=

T&

s p a c e and F = ( E ' , T

on H ( F ) .

0

)

show t h a t

I f E i s an i n f i n i t e d i m e n s i o n a l r e f l e x i v e Banach s p a c e 1 a n d F = ( E , o ( E , E ' ) ) show t h a t (@( F ) , T ~ ) ' 2 ( E l ) * a n d t h a t

3.78*

' 0

,ub

#

3.79

T~

on H ( F ) .

Show t h a t H(E) i s T . S .

T~

complete b u t not T . S .

T~

c o m p l e t e when E = t N x C " ) .

3.80*

Let T denote t h e topology of pointwise convergence. P I f E a n d F a r e l o c a l l y c o n v e x s p a c e s s u c h t h a t H ( E ) a n d H(F)

are T.S.

T

P

c o m p l e t e and s e p a r a t e l y c o n t i n u o u s p o l y n o m i a l s

o n E x F a r e c o n t i n u o u s show t h a t s e p a r a t e l y c o n t i n u o u s h o l o m o r p h i c f u n c t i o n s on E x F a r e h o l o m o r p h i c i f a n d o n l y i f H(E x F) i s T . S . 3.81 that

T

P

complete.

I f E i s a quasi-complete (H(E),ro)

is Frschet i f

i n f r a b a r r e l l e d s p a c e show

and o n l y i f E is t h e s t r o n g

d u a l o f a F r e c h e t Montel s p a c e . 3.82

I _

I f E i s an i n f r a b a r r e l l e d c o m p l e t e D F s p a c e show t h a t

( H ( E ) , T ~ )i s b a r r e l l e d i f a n d o n l y i f E i s a M o n t e l s p a c e .

149

Holomotphic functions on balanced sets

~

Show t h a t a s e q u e n t i a l l y c o m p l e t e b a r r e l l e d s p a c e w i t h

3.83*

a b a s i s is complete.

3.84*

I f E i s a quasi-complete n u c l e a r and dual n u c l e a r

s p a c e show t h a t t h e c o m p a c t o p e n t o p o l o g y o n H ( E ) g e n e r a t e d by a l l semi-norms

is

o f t h e form

a s K ranges o v e r t h e compact s u b s e t s o f E . Let 0 b e t h e o r i g i n i n t h e Banach s p a c e E a n d l e t B

3.85*

Show t h a t t h e t o p o l o g y on H(0)

be t h e unit b a l l of E.

is

g e n e r a t e d by t h e semi-norms

where

-

(an)n ranges over a l l sequences of

\anllh

s c a l a r s such t h a t

-.

o as n

3.86”

Let 0 b e t h e o r i g i n i n t“),

of H(0)

i s g e n e r a t e d by a l l

show t h a t t h e t o p o l o g y

semi-norms

o f t h e form

m

PK(f) = c n=o where

n n

s u b s e t s o f CE 3.87 -

and i f ,

Un

in C

If

n

-

r a n g e s o v e r a1

(a )

“1

o as n

.

sequences such t h a t and K r a n g e s o v e r t h e compact

( f n I n i s a s e q u e n c e i n H(O),

f o r each n, such t h a t f

0 the origin in

C (N) ,

t h e r e e x i s t s a neighbourhood of zero

I

‘n

= o all m

’> n s h o w t h a t { f n I n

i s a very s t r o n g l y convergent sequence i n H ( 0 ) .

3.88*

Let U b e a b a l a n c e d o p e n s u b s e t o f a q u a s i - c o m p l e t e

l o c a l l y convex s p a c e E .

For each compact s u b s e t K of E

l e t K = { x E E ; I P ( x ) l 5 IIPIIK f o r e v e r y P i n 6 ( E ) I .

150

=Ku

6

Let

Chapter 3

.

I?

6

Show t h a t

i s a b a l a n c e d open s u b s e t o f E .

K compact

If

f

=

zm

n=o

E

n.

H ( u ) show t h a t

-

d"f0 n!

zm

n=o r

c

l

A

a n d d e f i n e s a h o l o m o r p h i c f u n c t i o n f on U . mapping f

(H(U),T~)

E

isomorphism.

I f T& =

w

( t h e boundary o f U i n E)

(H(U),.ro) i s a l i n e a r

E

a n d ,€

t h e r e e x i s t s a n f i n H(U) 3.89" -

f

-

on H(U) show t h a t f o r e a c h €,

'I

A

Show t h a t t h e

A

%

c o n v e r g e s and

E

n

U,

2 n s u c h t h a t f ( € , )n

_ _ _ f

5 as n

L e t E b e a Banach s p a c e w i t h u n i t b a l l B .

P E P ~ ( ~ E l e )t

IIPIk = a B ( P ) .

show t h a t P - Q

E

If P E P ~ ( ~ E a n)d Q

aU m

as n

m

E

m.

If E

@N(mE)

(?N(n+mE) a n d

I f U i s a b a l a n c e d open s u b s e t o f a m e t r i z a b l e

3.90"

l o c a l l y c o n v e x s p a c e show t h a t

3.91*

T

= T

0

,t

o n H(U).

By u s i n g t h e f a c t t h a t t h e r e e x i s t d i s c o n t i n u o u s

p o l y n o m i a l s on CC

I

,

I uncountable,

show t h a t

(H(C

I

) , T ~ )

i s n o t a semi-Monte1 s p a c e . (H(CC N )

a (H(En),~O). n

=*

Show t h a t

3.93

L e t E and F be l o c a l l y convex s p a c e s and l e t U be a

, T ~ )

b a l a n c e d open s u b s e t o f E . TE{T~, T

T.S.

T

~

T , ~ } ,

Show t h a t

(H(U;F),T),

i s semi-Monte1 i f and o n l y i f

(H(U;F),r)

is

c o m p l e t e a n d ( @ ( n E ; F ) , ~ )i s s e m i - M o n t e 1 f o r e a c h

non-negative 3.94

=

integer n.

Let E b e a l o c a l l y convex s p a c e .

Show t h a t

(H(U),.ro)

i s c o m p l e t e f o r e v e r y open s u b s e t U o f E i f and o n l y i f ( P ( n E ) , ~ o )i s complete f o r each non-negative H(V) i s T . S .

T~

subset V of E.

i n t e g e r n and

c o m p l e t e f o r each convex b a l a n c e d open

15 1

Holomorphic functions on balanced sets L e t {En},

3.95*

that

b e a s e q u e n c e o f Banach s p a c e s .

Show

m

(H(iZ1 E , ) , T ~ ) i s a semi-Monte1 s p a c e i f and o n l y i f e a c h

En i s f i n i t e d i m e n s i o n a l .

3.96* ^n

d f(x)

I f E i s a l o c a l l y c o n v e x s p a c e a n d fEHN(E) show t h a t

n

E @ ~ (

E) f o r e v e r y x i n E a n d e v e r y p o s i t i v e i n t e g e r n .

Show, b y c o u n t e r e x a m p l e , t h a t t h e a b o v e c o n d i t i o n o n fsH(E)

i s n o t s u f f i c i e n t t o i n s u r e t h a t it l i e s i n HN(E).

Show a l s o t h a t H ( E ) N

o f H(E).

3.97*

is a translation invariant subalgebra

L e t E b e a l o c a l l y convex s p a c e and F a normed

l i n e a r space.

A f u n c t i o n f s H(E;F)

i s s a i d t o be o f

e x p o n e n t i a l t y p e i f t h e r e e x i s t a c o n t i n u o u s semi-norm a Ilf(x)II 5 C exp ( c a ( x ) )

on E a n d p o s i t i v e n u m b e r s c , C s u c h t h a t f o r every x i n E .

Let

Exp(E;F) d e n o t e t h e s e t of a l l

holomorphic f u n c t i o n s o f e x p o n e n t i a l t y p e from E i n t o F . Show t h a t f = C n=o

d"f0 n!

E

Exy(E;F)

i f and o n l y i f t h e r e

e x i s t s a c o n t i n u o u s semi-norm a on E s u c h t h a t

3.98 f

E

I f E i s a Banach s p a c e and f

Exp(E;C) =

E

H(E) show t h a t

~ x p ( E )i f a n d o n l y i f t h e r e s t r i c t i o n o f f t o

each one dimensional subspace o f E i s a f u n c t i o n o f expone n t i a l type. 3.99 -

I f E i s a l o c a l l y c o n v e x s p a c e show t h a t t h e

mapping f

=

C

n=o

d"f0 n.

E

Exp(E)

m

nC = o d n f ( o )

E

H(OE)

is a linear bijection. Using t h e above, o r o t h e r w i s e , d e s c r i b e a l o c a l l y convex t o p o l o g y on E x p ( E ) s o t h a t t h e a b o v e b i j e c t i o n i s a l i n e a r

152

Chapter 3

t o p o l o g i c a l isomorphism. Let E b e a Banach s p a c e and l e t f and g b e holomorphic i s an e n t i r e I f h = f, f u n c t i o n s o f e x p o n e n t i a l t y p e on E .

3.100

f u n c t i o n on E show t h a t h

E

Exp(E)

g

.

Let E b e a l o c a l l y convex s p a c e .

3.101 H (E) N

An e l e m e n t f o f

i s s a i d t o b e of n u c l e a r e x p o n e n t i a l t y p e i f t h e r e

e x i s t s a convex b a l a n c e d open s u b s e t V of E s u c h t h a t ^n

l i m s u p a,,(d

f(o))

'/n

<

rn,

Let ExpN(E) d e n o t e t h e s p a c e o f a l l holomorphic f u n c t i o n s o f n u c l e a r e x p o n e n t i a l t y p e on E . f

=

C

E

n=o

ExpN(E)

Show t h a t t h e m a p p i n g

-

rn

nC = o i n f ( o )

E

HN(OE)

is a linear bijection 3.102*

Let V and U b e open s u b s e t s o f t h e l o c a l l y convex

spaces E and F r e s p e c t i v e l y .

Let

-b e TI

a continuous linear

mapping from E i n t o F s u c h t h a t a(V) i s a compact s u b s e t o f U.

L e t R;Hrn(U)

__f

Hrn(V) b e d e f i n e d b y R ( f ) = f o n l v .

Show t h a t R i s a c o m p a c t m a p p i n g .

U s i n g t h i s r e s u l t show

t h a t H(K) i s a 8 3 . 2 s p a c e w h e n e v e r K i s a c o m p a c t s u b s e t o f

a FrEchet-Schwartz 3.103"

space.

I f U i s an open s u b s e t o f a l o c a l l y convex s p a c e E

a n d F i s a s e m i - M o n t e 1 s p a c e show t h a t ( H H y ( U ; F ) , T o ) i s a semi-Monte1 s p a c e . 3.104*

that

I f E i s a q u a s i - c o m p l e t e d u a l S c h w a r t z s p a c e show

(H(U),ro)

U of E.

i s a Schwartz s p a c e f o r any open s u b s e t

153

Holomorphic Jitnctions on balanced sets 53.6

NOTES A N D REMARKS The c o n c e p t o f Q - f a m i l y

J . Schmets

[627]

(definition 3.1)

i s due t o

(see also chapter 2 of the lecture notes

o f J . Schmets [628]

and Ph.

Noverraz

[553,556]) and

d e v e l o p e d n a t u r a l l y f r o m t h e r e s u l t s o f e a r l i e r a u t h o r s on particular associated topologies.

Y.

Komura

first t o discuss associated topologies.

[ 3 9 4 ] was t h e

He was i n t e r e s t e d

o n l y i n t h e a s s o c i a t e d b a r r e l l e d topology and proved proposition 3.5.

An a l t e r n a t i v e p r o o f o f t h i s p r o p o s i t i o n

u s i n g t h e axiom o f c h o i c e i n p l a c e o f t r a n s f i n i t e induction i s due t o M. A.

Roberts

topology.

[608]

Kennedy ( L e c t u r e , D u b l i n ,

1979).

The c o r r e s p o n d i n g r e s u l t s f o r t h e i n f r a b a r r e l l e d

t o p o l o g y and f o r t h e q u a s i - c o m p l e t e complete cases ( p r o p o s i t i o n 3 . 6 )

and s e q u e n t i a l l y

are due t o K .

Noureddine

The a s s o c i a t e d u l t r a - b o r n o l o g i c a l t o p o l o g y i s

[532].

s t u d i e d by H .

Buchwalter i n [108],

t o p o l o g y by K .

Noureddine i n

the barrelled-bornological

[533] and t h e o - b a r r e l l e d

a - i n f r a b a r r e l l e d topologies by K .

Noureddine and J .

and

Schmets

General r e s u l t s f o r Q-families are given i n

[535].

J . Schmets [627,628].

i n t r o d u c e d by H . [53411

December

also studies the associated barrelled

Kelley s p a c e s (example 3.4)

Buchwalter [lo71

(see also K.

were

Noureddine

*

S c h a u d e r d e c o m p o s i t i o n s o f Banach s p a c e s were f i r s t d e f i n e d by M . M .

Grinblyum

[283]

(see B.L.

Sanders

and e x t e n d e d t o l i n e a r t o p o l o g i c a l s p a c e s by C.W. and J . R . proof

R e t h e r f o r d [483].

The o n l y r e s u l t w e u s e w i t h o u t

( i n t h e proof of p r o p o s i t i o n 3.60)

B.L.

S a n d e r s [ 6 2 4 ] a n d T.A.

N.J.

Kalton

[370,371]

[624]) McArthur

Cook

[168].

i s due t o We r e f e r t o

for further details.

T h e c o n c e p t s o f a b s o l u t e d e c ompo s i t i o n ,

/Ir - d e c o m p o s i t i o n

a n d 4 - a b s o l u t e d e c o m p o s i t i o n a r e new a n d a r e i n t r o d u c e d h e r e as a s u i t a b l e t e c h n i q u e f o r t r e a t i n g h o l o m o r p h i c f u n c t i o n s on b a l a n c e d open s e t s .

Propositions 3.10,

3.11 and

Chapter 3

154

3 . 1 3 a r e new w h i l e a s t r o n g e r f o r m o f p r o p o s i t i o n b e f o u n d i n Ph. N o v e r r a z

3 . 1 2 may

[553,556].

Schauder decompositions were i n t r o d u c e d i n t o i n f i n i t e d i m e n s i o n a l holomorphy by S . Dineen and a l l t h e 63.2 and a number o f t h o s e i n § 3 . 3 are t o

results of b e found i n

[185].

These r e s u l t s were motivated by e a r l i e r

r e s u l t s c o n c e r n i n g h o l o m o r p h i c f u n c t i o n s on Banach s p a c e s (S.

Dineen

material in

[177], R.

Aron

[17]!.

The a r r a n g e m e n t o f t h e

i s , h o w e v e r , new a n d m o r e c o h e r e n t t h a n t h a t g i v e n

[185]. The a p p l i c a t i o n o f a s s o c i a t e d t o p o l o g i e s , i n

conjunction with

Schauder decompositions,

t o t h e study of

h o l o m o r p h i c f u n c t i o n s on l o c a l l y convex s p a c e s i s due t o Ph.

Noverraz

proposition i n S. [509].

[553,556] where he p r o v e s theorem 3.26 and 3.29.

Dineen

Propositions 3.29,

3.30,

3.31 a r e given

and l e m m a 3.28 i s due t o L .

[185]

I n v i e w o f t h e o r e m 3 . 2 6 we may a s k i f

Nachbin T

=

,t U a b a l a n c e d open s u b s e t o f a l o c a l l y convex s p a c e . Proposition 3.41,

e x e r c i s e 3.76and

example o f a non-complete To

and S.

Ponte

,t

#

T~

To,ub

= T~

Komura's

Recently J . M .

[395]

Ansemil

+

[ l o ] h a v e s h o w n t h a t -c0

weak t o p o l o g y ,

on H ( U ) ,

Monte1 s p a c e shows t h a t

even on E ' .

E an i n f i n i t e d i m e n s i o n a l

T~

c o r o l l a r y 5.26 a l l give

a p o s i t i v e answer f o r s p e c i a l cases but Y . we may h a v e

0

Tg on P ( ' E ) , 3 ub r e f l e x i v e Banach s p a c e w i t h t h e

a n d h e n c e we d o n o t ,

i n g e n e r a l , have

o n H(U).

The c o m p l e t e n e s s o f

(H(U;F),r

w

) has been i n v e s t i g a t e d

b y many a u t h o r s a n d t h e r e s u l t p r e s e n t e d h e r e 3 . 4 4 ) may b e f o u n d i n S .

Dineen

[ZOO].

i n a s e r i e s o f r e s u l t s which a p p e a r i n S . R.

Aron [ 1 7 ] ,

P.

Aviles

S.B.

Chae

[120],

and 3. Mujica [ 4 1 ] .

(corollary

I t is the latest Dineen

[177,185],

J . M u j i c a [SO31 a n d Aspects o f t h e completeness

question w i l l a r i s e i n each o f t h e remaining c h a p t e r s . Taylor

s e r i e s c o m p l e t e n e s s was i n t r o d u c e d b y S . D i n e e n [ 1 8 5 ] .

155

Holomoiphic functions on balanced sets Lemma 3 . 3 3 i s new a n d a g e n e r a l r e s u l t o f t h e s a m e k i n d f o r $-absolute

d e c o m p o s i t i o n s c a n e a s i l y b e s t a t e d and p r o v e d .

C o r o l l a r y 3.34

and p r o p o s i t i o n 3 . 3 5 a r e new.

i s given i n S.

Dineen

P r o p o s i t i o n 3.36

[ 1 8 5 ] w h e r e o n e may a l s o f i n d a

'I

w

analogue of proposition 3.36. The c l a s s i c a l Montel t h e o r e m s a y s t h a t c l o s e d

( H ( U ) , T ~ ) ( U a n o p e n s u b s e t o f Cn) a r e

bounded s u b s e t s o f compact. result

A number o f d i f f e r e n t g e n e r a l i z a t i o n s o f t h i s

(known c o l l e c t i v e l y a s M o n t e l t h e o r e m s )

holomorphic

for

f u n c t i o n s o f i n f i n i t e l y many v a r i a b l e s h a v e

appeared i n t h e l i t e r a t u r e .

The v a r i e t y o f r e s u l t s a r e

o b t a i n e d by v a r y i n g t h e u n d e r l y i n g l o c a l l y convex s p a c e s , t h e c o n c e p t o f d i f f e r e n t i a b i l i t y a n d t h e t o p o l o g y on t h e corresponding space o f holomorphic

functions.

Most o f t h e

p r o o f s r e q u i r e an a p p l i c a t i o n of A s c o l i ' s theorem.

f i r s t r e s u l t o f t h i s k i n d i s due t o D. $3,8spaces

Pisanelli

[571] f o r

and t h i s i s a p a r t i c u l a r c a s e o f c o r o l l a r y 3.38.

F u r t h e r Montel theorems are t o b e found i n D. [576,578,582], [149]

The

D.

Lazet

[423], J . F .

Pisanelli

Colombeau a n d D .

Lazet

( t h i s a r t i c l e c o n t a i n s p r o p o s i t i o n 3 . 3 7 and

c o r o l l a r i e s 3.38 and 3 . 3 9 ) , S . Dineen

[185,194]

J.F.

Colombeau

[141] and

( s e e a l s o e x e r c i s e s 2.84

and 3 . 1 0 3 ) .

A

number o f t h e above a u t h o r s a l s o p r o v e i n f i n i t e dimensional versions of t h e c l a s s i c a l V i t a l i theorem. [462] d i s c u s s e s l o c a l l y convex spaces which s a t i s f y

Elatos

M.C.

a "Montel" p r o p e r t y and shows t h a t t h e y a r e r e l a t e d t o l o c a l l y convex s p a c e s which s a t i s f y t h e c o n c l u s i o n o f Z o r n ' s theorem (theorem 2 . 2 8 ) .

of

Ascoli s t y l e characterizations

compact s e t s a r e due t o L .

'I

w

R.

[120],

Aron [ 1 7 ] ,

M.C.

Nachbin

Matos [461]

[SO91 ,S.B. C h a e

and J . A .

Barroso

[47,48]. Propositions 3.40, 3.43,

3.44,

3.41,

3.45 and c o r o l l a r i e s 3 . 4 2 ,

and 3.46 a r e due t o S.

Dineen

[ZOO].

3 . 4 7 i s new a n d r e l a t e d t o a n e x a m p l e o f R . i n R. L.

Soraggi

[664]

.

Example

Aron g i v e n

156

Chapter 3 A n a l y t i c f u n c t i o n a l s on l o c a l l y c o n v e x s p a c e s

a r e u s u a l l y r e p r e s e n t e d e i t h e r as f u n c t i o n s o f e x p o n e n t i a l t y p e o r as holomorphic germs a t t h e o r i g i n . natural linear topological

There i s a

(but unfortunately not algebraic)

isomorphism between t h e s e r e p r e s e n t a t i o n s ( s e e e x e r c i s e s 3.93,

3 . 9 4 and 3 . 9 5 ) . The e x p o n e n t i a l t y p e r e p r e s e n t a t i o n i s u s e f u l i n

s t u d y i n g c o n v o l u t i o n o p e r a t o r s ( s e e a p p e n d i x I ) w h i l e we h a v e f o u n d t h e g e r m a p p r o a c h u s e f u l when i n v e s t i g a t i n g topological p r o p e r t i e s of holomorphic f u n c t i o n s .

Since the

r e s u l t s h e r e on a n a l y t i c f u n c t i o n a l s were o r i g i n a l l y p r o v e d u s i n g T a y l o r s e r i e s e x p a n s i o n s a b o u t t h e o r i g i n we a r e e s s e n t i a l l y u s i n g t h e o r i g i n a l method.

Sometimes

however t h e S c h a u d e r d e c o m p o s i t i o n a p p r o a c h c a n b e more e f f i c i e n t - as i n theorem 3.55. Holomorphic f u n c t i o n s o f n u c l e a r t y p e ( d e f i n i t i o n 3 . 4 8 ) were i n t r o d u c e d b y C . P . G u p t a [ 2 9 5 , 2 9 6 ]

L.

Nachbin

[511].

and

The f i r s t i n f i n i t e d i m e n s i o n a l

r e p r e s e n t a t i o n theorem f o r a n a l y t i c f u n c t i o n a l s by holomorphic germs i s due t o P . J .

(HN(U),no)A

2

H(Uo)

Boland

whenever U i s a convex balanced open

subset of a 8 3 Q space. proposition

[ 8 5 ] who p r o v e d t h a t

This i s a stronger r e s u l t than

3.50 f o r ~ 8 3 Q s p a c e s . P r o p o s i t i o n s 3.49,

3 . 5 1 and c o r o l l a r y 3 . 5 2 a r e new. t o P.J.

Boland and S . Dineen

i s given i n 15.4.

3.50,

C o r o l l a r y 3.54 i s due

[go] and an a l t e r n a t i v e p r o o f

Theorem 3 . 5 5 i s due t o R .

Ryan

[620],

L e m m a 3 . 5 6 i s new w h i l e c o r o l l a r i e s 3 . 5 3 a n d 3 . 5 7 a r e proved i n P . J .

Boland and S .

Dineen

[QO]

assumption t h a t E has a Schauder b a s i s . a n d e x a m p l e 3 . 5 9 may b e f o u n d i n P . J . [91] and S . Dineen

[202].

under t h e a d d i t i o n a l Corollary 3.58

Boland and S . Dineen

Further r e p r e s e n t a t i o n theorems

f o r a n a l y t i c f u n c t i o n a l s on a Banach s p a c e a r e due t o J.M.

Isidro

[351] while t h e classical theory f o r functions

o r one complex v a r i a b l e i s due t o A . G.

Kb'the [ 3 9 6 ] a n d C . L .

d a S i l v a Dias

Grothendieck [661].

A.

[285],

Martineau

157

Holomorphic functions on balanced sets [451] i n v e s t i g a t e s t h e case of s e v e r a l v a r i a b l e s . P r o p o s i t i o n 3.60 and c o r o l l a r y 3 . 6 1 are due t o S.

[ 1 8 5 ] . T h e o r e m 3 . 6 4 was f i r s t p r o v e d f o r e n t i r e

Dineen

n u c l e a r s p a c e s by P .

f u n c t i o n s on q u a s i - c o m p l e t e see also E.

Kelimarkka [526],

i n d e p e n d e n t l y , t o a r b i t r a r y open s e t s by P . J . and L . by L .

!Vaelbroeck

Perrot

[83],

Boland

[86]

Our p r o o f i s c l o s e t o t h a t g i v e n

[713].

l a e l b r o e c k , who a l s o p r o v e s

lemma 3.63.

proof o f theorem 3 . 6 4 i s given by J . F . B.

Boland

and a f t e r w a r d s e x t e n d e d ,

A further

Colombeau and

[16@,161] and f o r f u l l y n u c l e a r s p a c e s w i t h a

b a s i s we p r o v i d e a n a l t e r n a t i v e p r o o f i n c h a p t e r 5 (corollary 5 . 2 2 ) .

Applications o f theorem 3.64 t o l i f t i n g

theorems f o r l i n e a r mappingsare t o be found i n W. B.

Kaballo

and t o t h e c l a s s i f i c a t i o n o f S t e i n a l g e b r a s i n

[363]

Kramm [398,399]. E x t e n s i o n s o f theorem 3 . 6 4 t o A and s n u c l e a r t y and

t o nuclear bornologies a r e given i n K.D.

B.

Gramsch and R .

[69,70], [152],

Meise

L . Waelbroeck

J.F.

[67], [713],

Colombeau and B .

Bierstedt,

K.D.

B i e r s t e d t and R .

bleise

J.F.

Colombeau a n d R .

Meise

Perrot

[157,159,160,16

For example t h e f o l l o w i n g r e s u l t i s proved i n

l e t E be a quasi-complete (H(U;F),-r )

Sgtz 1.12 of

(E',-r0)

[152];

l o c a l l y convex space, then

i s an s n u c l e a r

i f and only i f

,1651.

s p a c e f o r any open s u b s e t U o f E

and F a r e b o t h s n u c l e a r s p a c e s ( s e e

[67]).

An a p p r o a c h t o t h e m a t h e m a t i c a l

foundations of

quantum f i e l d t h e o r y u s i n g n u c l e a r i t y and i n f i n i t e dimensional holomorphy i s given i n P . 411, B.

413,414,415,416,4171

Perrot

[158]

and J . F .

e x e r c i s e 3.104.

Colombeau and

Colombeau [ 1 4 5 ] ) .

C o r o l l a r y 3.65 i s due t o P . Schwartz property f o r

Kr6e [ 4 0 6 , 4 @ 7 , 4 @ 8 , 4 0 9 ,

(see also J . F . Boland

[82,83].

The

( H ( U ) , - r ) i s d i s c u s s e d i n o u r n o t e s on

This Page Intentionally Left Blank

Chapter 4

HOLOMORPHIC FUNCTIONS ON BANACH SPACES

Banach s p a c e s a n d n u c l e a r s p a c e s p l a y a n i m p o r t a n t r o l e i n l i n e a r f u n c t i o n a l a n a l y s i s and a l s o i n c l a s s i c a l a n a l y s i s b y way o f a p p l i c a t i o n .

This chapter i s devoted t o t h e study of

h o l o m o r p h i c m a p p i n g s b e t w e e n Banach s p a c e s a n d i n c h a p t e r 5 we d i s c u s s h o l o m o r p h i c f u n c t i o n s on n u c l e a r s p a c e s .

As o n e

w o u l d e x p e c t , s i n c e e v e r y n u c l e a r Banach s p a c e i s f i n i t e dimensional,

t h e s e two t o p i c s p r o c e e d a l o n g q u i t e d i f f e r e n t

l i n e s b u t b o t h c o n f i r m t h a t i n f i n i t e dimensional holomorphy l e a d s t o c o n c e p t s and r e s u l t s which a r e o f i n t e r e s t i n thems e l v e s and q u i t e d i f f e r e n t from what one would e x p e c t from t h e underlying f i e l d s . I n t h i s c h a p t e r we f i n d t h a t t h e r e i s a r i c h i n t e r a c t i o n between t h e t h e o r y o f holomorphic f u n c t i o n s and t h e g e o m e t r y

of Banach s p a c e s .

By t h e g e o m e t r y o f B a n a c h s p a c e s , a t o p i c

t h a t h a s undergone r a p i d development i n t h e l a s t f i f t e e n y e a r s , we mean t h e s t u d y o f g e o m e t r i c p r o p e r t i e s o f t h e u n i t b a l l s u c h a s s m o o t h n e s s , t h e e x i s t e n c e of e x t r e m e p o i n t s , d e n t -

a b i l i t y , uniform c o n v e x i t y , sequentiaZ compactness e t c . I f E i s a Banach s p a c e t h e n t h e c o m p a c t o p e n t o p o l o g y

11 Ik

o n H(E) i s g e n e r a t e d b y subsets of E.

as K r a n g e s o v e r t h e compact

Our m o t i v a t i n g p r o b l e m i s t h e f o l l o w i n g ;

do

t h e r e e x i s t a n y o t h e r s e m i - n o r m s o n H(E) w h i c h h a v e t h e f o r m

I / Ik

f o r some s u b s e t A o f E?.

it w i l l always be

‘c6

semi-norm w i l l n o t b e

I f s u c h a semi-norm

11 IIA

T

w

continuous.

Since

11 IIA

llfla <

0

for e v e r y f i n H ( E ) . 159

the

i s a semi-

norm i f i t i s f i n i t e we a r e l o o k i n g f o r n o n - r e Z a t i v e Z y A such t h a t

exists

c o n t i n u o u s and i f A i s n o t pre-compact

compact This problem has

Chapter 4

160

l e d t o much o f t h e r e s e a r c h we r e p o r t i n t h i s c h a p t e r . I n t h e f i r s t s e c t i o n we d i s c u s s a f e w g e n e r a l p r o p e r t i e s o f h o l o m o r p h i c mappings b e t w e e n Banach s p a c e s . Some o f t h e s e a r e u n r e l a t e d t o t h e t o p o l o g i c a l p r o b l e m b u t are of i n t e r e s t i n themselves. 54.1

ANALYTIC

EQUALITIES

A N D INEQUALITIES

The t h e o r y o f h o l o m o r p h i c f u n c t i o n s o f one o r s e v e r a l complex v a r i a b l e c o n t a i n s a number o f i n t e r e s t i n g and u s e f u l e q u a l i t i e s and i n e q u a l i t i e s and it i s n a t u r a l t o e x t e n d t h e s e t o i n f i n i t e l y many v a r i a b l e s .

Such g e n e r a l i z a t i o n s a r e o f

i n t e r e s t i f t h e y s a t i s f y a t l e a s t one o f t h e f o l l o w i n g criteria; a ) t h e y r e q u i r e new n o n t r i v i a l p r o o f s

(and a s t u d y o f t h e s e

i n t u r n may l e a d t o i m p r o v e d a n d e v e n new f i n i t e d i m e n s i o n a l results), b) t h e y l e a d t o a p p l i c a t i o n s n o t c o v e r e d by t h e c o r r e s p o n d i n g f i n i t e dimensional r e s u l t s , c ) t h e y g i v e r i s e t o a c l a s s i f i c a t i o n problem f o r l o c a l l y convex s p a c e s , d ) t h e y l e n d t h e m s e l v e s t o new i n t e r p r e t a t i o n s w h i c h i n t u r n s u g g e s t new c o n c e p t s a n d p r o b l e m s o r non-existent

( w h i c h may e v e n b e t r i v i a l

i n f i n i t e dimensions).

We p r e s e n t h e r e e x t e n s i o n s o f t h r e e w e l l known r e s u l t s from t h e t h e o r y o f one complex v a r i a b l e ;

S c h w a r z ' s lemma, t h e

maximum m o d u l u s t h e o r e m a n d t h e C a u c h y - H a d a m a r d

formula.

S i n c e t h e s e e x t e n s i o n s w i l l n o t b e r e q u i r e d l a t e r w e do n o t give a comprehensive account.

For both Schwarz's

lemna and

t h e maximum m o d u l u s t h e o r e m we n e e d t h e c o n c e p t o f a n extreme point. Definition 4.1

Banach s p a c e .

L e t K b e a c o n v e x s u b s e t of a c o m p l e x A p o i n t e of

K is

161

Holomorphic functions on Banach spaces -1 < X

( a ) a r e a l e x t r e m e p o i n t i f { e + Ax;

implies x

=

0 ,

( b ) a c o m p l e x e x t r e m e p o i n t i f { e + Ax;

implies x

5 1)CK

o 5

1x1

5 l}CK

= 0.

I t i s c l e a r t h a t e v e r y r e a l extreme p o i n t i s a complex I f e v e r y p o i n t o f norm o n e i s a r e a l e x t r e m e

extreme p o i n t .

p o i n t o f t h e c l o s e d u n i t b a l l o f E then E i s c a l l e d a rotund L

o r a s t r i c t l y corivex Bar.ach s p a c e . convex i f 1 < p <

P (X,O,p)

is s t r i c t l y

f o r any f i n i t e measure s p a c e (X,a,p).

m

If

e v e r y p o i n t o f modulus 1 i s a complex extreme p o i n t o f t h e c l o s e d u n i t b a l l o f E t h e n we s a y E i s a s t r i c t l y c - c o n v e x 1 L ( 0 , l ) i s a s t r i c t l y c - c o n v e x Eanach s p a c e Banach s p a c e . which i s n o t s t r i c t l y convex. Now l e t D = ( z E C ; I z I variable says that If(z)l some z

if f

E

S c h w a r z ' s lemma i n o n e

< 1).

H(D;D)

and f ( o ) = o t h e n

5 I z i f o r a l l z E D and moreover i f I f ( z E D then

)I

lzol

=

for

f ( z ) = cz f o r a l l z i n D where c i s a

c o n s t a n t o f modulus 1.

We u s e t h e f i r s t p a r t o f t h i s r e s u l t

t o p r o v e t h e f o l l o w i n g lemma, w h i c h i s a l s o u s e f u l t h e maximum m o d u l u s t h e o r e m ,

i n extending

and e x t e n d t h e second h a l f t o

mappings b e t w e e n Banach s p a c e s .

Proof

If

If(z)l

= 1 for

some z

E

D t h e n t h e one

d i m e n s i o n a l maximum m o d u l u s t h e o r e m i m p l i e s t h a t f i s a c o n s t a n t mapping i n which c a s e t h e above r e s u l t H e n c e we may a s s u m e f E H ( D ; D ) .

z

-'

Z

-

hence

-az

1

mapping z g(o)

-

-a

=

0 .

If(z)

T h e M'dbius t r a n s f o r m a t i o n

(la1 < 1 ) maps D o n t o D a n d g(z)

=

By S c h w a r z

- f(o)l 5

is trivial.

c1

to

0 .

Hence t h e

f(z)-f(o) belongs t o H(D;D) 1 - f o f (z)

and

162

Chapter 4

L e t E a n d F b e Banach s p a c e s w i t h o p e n u n i t baZZs

Theorem 4 . 3

Let f

U and V z - e s p e c t i v e l y .

i s o m e t r y from E o n t o F .

E

H ( U ; v ) and s u p p o s e d f ( o ) i s a n f ( x ) = d f ( o ) ( x ) for a 2 2

Then

x

i n U and i n p a r t i c u Z a r f i s an i s o m e t r y from U o n t o V . We f i r s t n o t e t h a t b y r e p l a c i n g f b y d f ( o ) - l o f

Proof

we may a s s u m e t h a t E = F a n d d f ( o ) = I , w h e r e I i s t h e

We f i r s t s h o w t h a t f ( o ) =

i d e n t i t y map o n E .

o t h e r w i s e , t h e n b y t h e Hahn-Banach theorem, nJ

$IEE',

II$II =

1, such t h a t $ ( f ( o ) )

for all z ED.

g(z1 = $of( zfO l g (o)I

2

V(0)II )

5 l-lg'(o)12,

and f ( o ) =

0 .

Ilf(o)

=

0 .

Suppose

there exists

11.

Let

S i n c e gsH(D;D)

we g e t

( s e e example 2.31).

Now f i x EED\{o}

and l e t

T o c o m p l e t e t h e p r o o f i t s u f f i c e s t o show w i s i d e n t i c a l l y

5

zero f o r every E.

Our f i r s t s t e p i n p r o v i n g t h i s i s t o s h o w Ib( + Xw ( x ) I I 5 I I x I I for a l l x i n U a n d h e D .

5

x

E

U\{o)and

$I

II$(I I 1 ,

E E l ,

f o r m u l a h ( o ) = $I(-)

X

IIX

II

Let

be given.

and h ( z ) =

1 7

Defjne h by t h e zx +of(-) i f ZED\\CO}.

llxll

S i n c e f ( o ) = o a n d d f ( o ) = I i t f o l l o w s t h a t hEH(D;D). By l e m m a 4 . 2 ,

l e t t i n g z = 511xII, we h a v e

163

Holomorphic functions on Banach spaces

(since the function

5 As

l-t t

i s decreasing)

IIXII.

41 was a n a r b i t r a r y e l e m e n t o f t h e u n i t b a l l o f E’ i t f o l l o w s

by t h e Hahn-Banach t h e o r e m , t h a t

Ilx+Xw5(x)

11

5 1.

(The p r o o f

w o u l d now b e c o m p l e t e i f E was a s t r i c t l y c - c o n v e x s p a c e ) . Now l e t L d e n o t e t h e a l g e b r a o f a l l b o u n d e d l i n e a r m a p p i n g s f r o m

d

Hm(U) i n t o i t s e l f . c ’ 3

mapping f r o m H (U) u n i t b a l l of

i s a Banach s p a c e a n d I ‘

into itself,

= $ ( x + ~ w( x ) )

5

seen t h a t k E H ( D ; D ) .

G i v e n $ e H m ( U ) , I(JIIIu(

Now l e t L : Hm(U)

L(@) = $o(I

11

-

:~1 [ “ ‘ x +1~ w 5 ( x ) ) -$ ( X I 1 I +

1

y

1

1 7

yields

(**)-

Hm(U) b e d e f i n e d b y t h e f o r m u l a By

w5).

5

1,

I t is easily

f o r a l l AED.

A f u r t h e r a p p l i c a t i o n o f lemma 4 . 2 w i t h z =

I$(x)

the identity

is a r e a l extreme p o i n t o f t h e

(see exercise 4.52).

and x E U l e t k ( A )

,

1

I ’ 2 T(L - 1’111 = s u p

(**I 1 1 I + ( x ) 5 T [ + ( X + ~ W 5 ( X ) )- + ( x ) I l

IC.EH~(U)

llJI1lu 5

XEU

1 9

- 1. Hence L = I ’

a n d f o r a n y ~ E E ’ we h a v e

By t h e Hahn-Banach

theorem w

5

0 = L(8)

-

8 = &8ow 2 5

’

i s i d e n t i c a l l y z e r o a n d as we

have a l r e a d y noted t h i s completes t h e p r o o f . O t h e r g e n e r a l i z a t i o n s o f S c h w a r z ’ s lemma a r e a l s o a v a i l a b l e and t h e s e t o g e t h e r w i t h t h e above have a p p l i c a t i o n s t o Banach a l g e b r a t h e o r y .

In p a r t i c u l a r they y i e l d a

g e n e r a l i s e d B a n a c h - S t o n e t h e o r e m f o r J * - a l g e b r a s a n d a new p r o o f o f t h e Russo-Dye t h e o r e m . We now l o o k a t t h e maximum m o d u l u s t h e o r e m .

There a r e a

164

Chapter 4

n u m b e r o f d i f f e r e n t f o r m s o f t h e maximum m o d u l u s t h e o r e m d i s c u s s e d i n t h e l i t e r a t u r e a n d h e r e we c o n f i n e o u r s e l v e s t o the following;

i f f c €i(U), h h e r e U i s a c c n n e c t e d open

s u b s e t o f t, t h e n e i t h e r I f ( z ) l h a s n o maximum o n U o r f i s a c o n s t a n t mapping.

We l o o k a t e x t e n s i o n s o f t h i s r e s u l t

by c o n s i d e r i n g Banach v a l u e d h o l o m o r p h i c m a p p i n g s d e f i n e d on open s u b s e t s o f E .

One c a n e a s i l y s h o w t h a t i f f a H ( U ; F ) ,

U a connected open s u b s e t of

Ilf(z)

11

h a s a maximum t h e n

C and F a Banach s p a c e , and

1IfII i s a c o n s t a n t .

Hence t h e

problem r e d u c e s t o showing t h a t t h e c o n s t a n t mappings are t h e This is not

o n l y holomorphic mappings o f c o n s t a n t modulus.

t r u e i n g e n e r a l , even f o r f i n i t e d i m e n s i o n a l s p a c e s , as t h e f o l l o w i n g example shows. Let f

-zz,

: D

s u p norm t o p o l o g y . but

Ilf(z)

11

Theorem 4 . 4 E-valued

=

f(z) = (l,z),

where

. 2 . i s C with the

z 2m

Note t h a t f i s n o t a c o n s t a n t mapping

1 f o r a l l z a D. L e t E b e a B a n a c h space.

Each ho2omorphic

m a p p i n g f d e f i n e d on a c o n n e c t e d o p e n s u b s e t of

C

f o r u h i e h l l f ( z ) l l h a s a rncz-immz ic c o n s t a n t if a n d o n l y i f E

is a s t r i c t Z y c - e o n v e x B a n a c h s p a c e . Proof

F i r s t suppose

e E

IleII = 1 , i s n o t a c o m p l e x

E,

extreme p o i n t o f t h e u n i t b a l l . Ile+zxll 5 1 f o r a l l z some l z o ]

E C ,

1 we h a v e

Choose

IzI 5 1 .

If

IleII 5 y1( I l e + z o x I I

since t h i s is impossible

X E

+

Ile-zoxI1)

Ile+zxII = 1 f o r a l l

The f u n c t i o n f ( z ) = e + z x , z

E

D,

E such t h a t

Ile+zoxII < 1 f o r

Z E

i s non-constant

< 1 and

E,

121

but

5 1.

Ilf(z)

11

i s a c o n s t a n t f u n c t i o n o f z r e s t r i c t e d t o D and hence w e

have proved t h e theorem i n one d i r e c t i o n . Now s u p p o s e E i s a s t r i c t l y c - c o n v e x B a n a c h s p a c e .

-

L e t U b e a c o n n e c t e d o p e n s u b s e t o f E. suppose t h e mapping z

E

U

Let

f

E

H(U;E)

and

Ilf(z)II i s a c o n s t a n t m a p p i n g .

By u s i n g t r a n s l a t i o n s i f n e c e s s a r y w e may s u p p o s e D C U a n d s u p I l l f ( z ) \ \ ; z E D ) 5 1.

By lemma 4 . 2

and t h e Hahn-Banach

165

Holomorphic functions on Banach spaces theorem

(If(o) + A ( f ( z ) - f ( o ) ) ( l i l i f

Z E 1x1 I#, 1- z

D\(o).

S i n c e t h e u n i t b a l l o f E c o n t a i n s no complex extreme p o i n t s it follows t h a t

for a l l z near zero.

f(z) = f(o)

This

completes the proof.

F o r a r b i t r a r y f u n c t i o n s we a l s o h a v e t h e f o l l o w i n g method f o r r e c o g n i s i n g n o n - c o n s t a n t which a r e of

holomorphic functions

constant modulus.

L e t E be a c o m p l e x Banach s p a c e and l e t f ( z ) non c o n s t a n t hoZomorphic mapping from D

=

1"

a n z n be a

n=o

{ Z E( c ; l z l

=

11

<

.

.. . I i f and o n l y o n E and an o p e n s u b s e t U o f D s u c h t h a t ( ( l f ( z ) ( ( Iis c o n s t a n t on U. {

Then a.

into E.

C l o s e d Span { a l , a Z ,

i f t h e r e e z i s t s a n e q u i v a Z e n t norm111

1))

We now l o o k a t t h e C a u c h y - H a d a m a r d

formula.

formula i n one dimension s t a t e s t h e f o l l o w i n g ;

This m

if

i s a sequence o f complex numbers and r = ( l i m s u p l a ll'n)-l n - - - + m n d n a z then the series c o n v e r g e s u n i f o r m l y on I z ; l z l 5 r o ) E=o n r. f o r any r 0

T h e s i t u a t i o n i n i n f i n i t e l y many v a r i a b l e s

is quite

different. Lemma 4 . 5

then

1"

n=o

I f

$nn

-

i f and onZy i f $ , ( x )

for e v e r y x i n E ( i . e . $ n Proof

If

1" e:

n =1

E

HIE)

then

f o r e v e r y x i n E a n d z i n C.

-

o in t h e

1"

n=l

n - m

+n ( x ) 1" en H G ( E ) .

Conversely if

n =1

IT

E

0

W*

o a s n-

m

topology).

(Gn(x))"zn

converges

By t h e C a u c h y - H a d a m a r d

-

i n one v a r i a b l e l i m sup I @ , ( X ) ~ ~

then f =

-

E i s a Banach s p a c e and $ n E' ~ for a l l n

H(E)

E

as n

=

n-l- "imlm ~5

Since t h e nth

$n ( x )

I

formula

= 0.

f o r every x i n E derivative o f f a t 0

Chapter 4

166

i s $" a n d t h i s i s c o n t i n u o u s we may a p p l y t h e o r e m 2 . 2 8 t o n complete t h e proof.

Let E be a s e p a r a b l e H i l b e r t s p a c e w i t h

Example 4 . 6 orthonormal

cm z n e n

E.

lm

-

$n w h e r e $n i s e v a l u a t i o n n n=l S i n c e $no as n m a t the nth coordinate of E.

all

n=l

E

i n (E',o(E',E))

n

Hence f =

lemma 4 . 5 i m p l i e s t h a t f

= 1 and l i m s u p

n - -

II

d"fj0) n.

p

E

H(E).

However

= 1.

Example 4 . 6 shows t h a t i n i n f i n i t e d i m e n s i o n s we h a v e t o d i s t i n g u i s h between t h e " r a d i u s o f p o i n t w i s e convergence" and t h e " r a d i u s o f u n i f o r m convergence".

A f u r t h e r concept

i s t h e r a d i u s o f b o u n d e d n e s s w h i c h e n t e r s i n a n a t u r a l way and p l a y s an i m p o r t a n t r o l e i n l a t e r developments.

Let U

b e an o p e n s u b s e t o f a l o c a l l y c o n v e x s p a c e E a n d l e t B b e a balanced closed subset of E .

We l e t If E is

d B ( 5 , U ) = S U ~ I ~ A I ; A E E , ~ + A B C Uf o) r e v e r y 5 i n U .

a normed l i n e a r s p a c e a n d B i s t h e u n i t b a l l o f E t h e n d B ( 5 , U ) i s t h e u s u a l d i s t a n c e o f 5 t o t h e complement o f U i n E.

Now l e t F b e a Banach s p a c e a n d l e t f € H ( U ; F ) .

B r a d i u s of b o u n d e d n e s s o f f a t 5 , r f ( t , B ) ,

s u p C ~ A ) ; A E ~ : , ~ + X B C IUl f 1,1 5 + A B The B r a d i u s of defined a s sup

<

The

i s defined as

a).

uniform convergence of f a t 5 , Rf(c,B),

I1X(

is

; X ~ C , c + h B c U a n dt h e T a y l o r s e r i e s o f f a t

5 c o n v e r g e s t o f u n i f o r m l y on c + A B ) . Proposition 4 . 7

L e t U be an o p e n s u b s e t of a l o c a l l y c o n v e x

s p a c e E, l e t F b e a Banach s p a c e and s u p p o s e f

5

EU,

B

E

H(U;F).

is a c l o s e d baZanced s u b s e t of E and r f ( E , B )

If

o then

167

Holomoiphic functions on Banach spaces Proof

We f i r s t n o t e t h a t i f E = U t h e n d B ( S J U ) = +

a n d t h e a b o v e may r e d u c e t o

m

=

m

=

m

T h i s however s a y s

m.

t h a t f i s bounded and t h e T a y l o r s e r i e s c o n v e r g e s u n i f o r m l y on 5 + XB f o r e v e r y A E C i f a n d o n l y i f l i m s u p I ( i " f ( 5 ) n - m n! If o <

la1

< rf(S,B)

then

(by t h e Cauchy i n e q u a l i t i e s ) .

Since rf(c,B)

5 d B ( c , U ) we h a v e shown t h a t

The a b o v e a l s o shows o n t a k i n g la1 <

l a q \ <

rf(S,B),

that

(Note t h a t s i n c e r f ( S J B ) 7 every n).

0

we h a v e

Hence i f 5 + (1-E)BBCU

, 2l ft (dSl l B <

for

B

=O;

168

Since

Chapter 4

E

was a r b i t r a r y i t f o l l o w s t h a t

Now s u p p o s e y < R f ( < , B ) .

Hence f o r any

2,

o < E

B).

< Corollary 4.8

Since

4

Then

y,

2

we h a v e

and y were a r b i t r a r y t h i s i m p l i e s

r f ( < , B ) and c o m p l e t e s the p r o o f .

If E i s a L o c a l l y c o n v e x s p a c e , F is a

Banach s p a c e and K i s a compact b a l a n c e d s u b s e t of E t h e n wl$";;o) Proof

~l:/~=

o for e v e r y f s H ( E ; F ) .

Since a holomorphic function i s continuous it i s

bounded on e a c h compact s u b s e t o f E and t h e r e s u l t f o l l o w s from p r o p o s i t i o n 4 . 7 . I f E i s a f i n i t e dimensional space then rf(<,B)=dB(S,U) f o r any bounded s u b s e t B o f E , any f E H ( U ) .

any open s u b s e t U o f E and

F o r t h i s r e a s o n t h e c o n c e p t of r a d i u s of

boundedness i s n o t i n t e r e s t i n g i n f i n i t e dimensions. The r e m a i n d e r o f t h i s s e c t i o n i s d e v o t e d t o v a r i o u s p r o p e r t i e s o f t h e r a d i u s of boundedness.

These r e s u l t s

w e r e a l l m o t i v a t e d b y t o p o l o g i c a l c o n s i d e r a t i o n s , w h i c h we discuss i n the next section, but are also of independent interest.

We r e s t r i c t o u r s e l v e s t o e n t i r e f u n c t i o n s o n a

169

Holornophic functions on Banach spaces Banach s p a c e E w i t h c l o s e d u n i t b a l l B .

I n t h i s c a s e we

write r (5) i n place of rf(C,B). N o t e t h a t r (C) i s a n f f i s o m e t r i c p r o p e r t y o f t h e Banach s p a c e E and w i l l change i f t h e Banach s p a c e i s r e n o r m e d e v e n w i t h a n e q u i v a l e n t norm. If f i s t h e f u n c t i o n c o n s i d e r e d i n example 4 . 6 t h e n

p r o p o s i t i o n 4 . 7 shows t h a t r f ( o ) f o r a l l E E E)

1 + ~ ,E >

0 ,

= 1 (and hence rf(c)

m

a n d s o f i s unbounded on e v e r y b a l l o f r a d i u s

centered a t the origin.

T h i s a l s o shows t h a t

t h e Taylor series expansion a t zero converges at a l l points

o f E b u t d o e s n o t c o n v e r g e u n i f o r m l y on a n y b a l l o f r a d i u s greater than 1 centered at zero. Our n e x t r e s u l t on t h e r a d i u s o f b o u n d e d n e s s s a y s t h a t e v e r y i n f i n i t e d i m e n s i o n a l Banach s p a c e s u p p o r t s an e n t i r e function with non-trivial

o f boundedness.

(i.e.

n o t i d e n t i c a l l y + -)

radius

This i s a consequence o f t h e following

deep r e s u l t . Proposition 4.9

E i s a n i n f i n i t e d i m e n s i o n a 2 Banach

If

space t h e n t h e r e e x i s t s a sequence i n E ' ,

II$nll

= 1 f o r a22

n and $ , ( x )

---+

o as n

x i n E.

-

( $ n ) n J such t h a t

f o r every

C o r o l l a r y 4.10 I f E i s a n i n f i n i t e d i m e n s i o n a 2 Banach s p a c e t h e n t h e r e e x i s t s a C-valued e n t i r e f u n c t i o n o n E , f ,

-

suah t h a t r f ( c ) < Proof

Let f

=

1"

for e v e r y 5 i n E .

n=o

9:

where

given by p r o p o s i t i o n 4 . 9 . e x a m p l e 4.6, r f ( o )

= 1.

($n)n is t h e sequence i n E '

By lemma 4 . 5 f e H ( E ) a n d , a s i n Hence r f ( c )

<

00

for all S E E.

This completes t h e proof. O u r n e x t r e s u l t s t a t e s t h a t r f may t a k e a r b i t r a r i l y

small v a l u e s even o v e r a bounded s e t .

Chapter 4

170

Proposition 4.11 I f E i s an i n f i n i t e d i m e n s i o n a Z Banach s p a c e w i t h u n i t baZZ B t h e n t h e r e e x i s t s an e n t i r e f u n c t i o n f on E s u c h t h a t inf {rf(x);

xeB1 = o

However rf does have regularity properties as the following proposition shows. I f

Proposition 4.12 rf(o)

<

m,

where

then

(a) Irf(x) (b)

E i s a Banach s p a c e and fEH(E),

- rf(y)

I

5

IIx-rll f o r

aZZ X,Y

E

E,

- log rf i s a p Z u r i s u b h a r r n o n i c f u n c t i o n on E. I n v i e w of (a), ( b ) s a y s t h a t 1 - log If(') 5 J2"( - l o g rf (x+eif3y))de

f o r e v e r y x,y i n E

The above proposition is not difficult to prove (indeed (a) i s obvious) but we shall omit the proof since it would first require a discussion o f plurisubharmonic functions. It is rather remarkable that conditions (a) and (b) of proposition 4.12 characterize radii of boundedness of holomorphic functions o n certain spaces. A proof of this type of result involves, as condition ( b ) might suggest, methods used in solving the Levi problem. We,therefore, only state some o f the important results in this area.

It is

also worth noting that this result have no analogue in the theory o f several complex variables. Theorem 4.13

Banach s p a c e and l e t g:E conditions :

-

L e t E b e an i n f i n i t e dirnensionaZ s e p a r a b l e R+ s a t i s f y t h e folZowing

(a) Ig(x) - g(y)l 2 Ib-yll f o r X,Y i n E , (b) - log g i s a p Z u r i s u b h a r m o n i c f u n c t i o n on E .

171

Holomolphic functions on Banach spaces

Then ( 1 ) if E = 2

1

t h e r e e x i s t s f s H ( E ) such t h a t g = r f ,

( 2 ) if E has a Schauder b a s i s t h e n t h e r e e z i s t s f

such t h a t

qn 5

E

H(E)

r f 5 g.

E x a m p l e s h a v e b e e n c o n s t r u c t e d w h i c h show t h a t we c a n n o t r e p l a c e l l by lp, 1 < p <

m,

Whether

i n (1) above.

o r n o t t h e o r e m 4 . 1 3 ( 2 ) h o l d s f o r a r b i t r a r y s e p a r a b l e Banach s p a c e s i s s t i l l an open p r o b l e m . Condition

(a) of p r o p o s i t i o n 4.12 says t h a t rf i s a

Lipschitz function with Lipschitz constant l e s s than o r e q u a l t o 1.

Example 4 . 1 4 shows t h a t t h i s c o n s t a n t i s t h e

b e s t p o s s i b l e i n g e n e r a l b u t f o r f u n c t i o n s which s a t i s f y s p e c i a l growth c o n d i t i o n s one can f i n d a s m a l l e r c o n s t a n t . On t h e o t h e r h a n d i n B a n a c h s p a c e s w i t h a s u i t a b l e g e o m e t r y we may r e p l a c e t h e i n e q u a l i t y o f p r o p o s i t i o n 4 . 1 2 b y a s t r i c t inequality. Example 4 . 1 4

If f

:zl

-

t i s given by

T h i s c a n n o t h a p p e n on u n i f o r m l y c o n v e x Banach spaces. A Banach s p a c e E is s a i d t o be uniformly Definition 4.15. c o n v e x if f o r e v e r y E > o t h e r e e x i s t s a 6 > o s u c h t h a t for X,Y E

E,

I(x(I =

IIYII

= 1,

IIx+YII

'> 2 - 6

we have

Ilx-yII

i

E

172

Chapter 4

i s a u n i f o r m l y convex s p a c e f o r l < p < m b u t Z1 i s

1,

P

n o t uniformly convex. Proposition 4.16

If E i s a u n i f o r r n z y c o n v e x i n f i n i t e

dimensionaZ Banach s p a c e , f F H @ ) and r f ( o ) <

-

lrf(x)

rf(y)

1

IIx-rll f o r aZZ x , y

<

then

i n E.

Suppose t h e c o n c l u s i o n i s f a l s e .

Proof

m,

Then t h e r e e x i s t

a n e n t i r e f u n c t i o n f on E a n d a u n i t v e c t o r x i n E s u c h t h a t = 1 and r

rf(o)

f

(Ax) = 1 - A

f o r each i n t e g e r n If(xn) 1-

1 n

I

> n,

1

I

llXnII

I 1+

- A

2 we c a n c h o o s e x

1

-

1.

n

T h e n Ilynll

xn = ( l - X ) y n + Ax. @ nE E '

<

-

X

Hence

< 1.

such t h a t

n

1 n

llxn-Xxll 5 1 - X +

1 L e t yn = = ( x n - X x ) .

theorem,

for some A , o

and

1 as n

--

and

F o r e a c h n c h o o s e , by t h e H a h n - B a n a c h

such t h a t

Il@nII

= 1 and @ n ( x n ) =

llxnII.

For

a l l n we h a v e @,(Xn) =

-

=

llXnII

-

(l-Al$n(Yn)

S i n c e l $ n ( y n ) l 5 IlynII $,(x)

Hence n

1.

+

m

1 as n

xnII

L

-

and l i m n - +

$n(x

-. +

+

1 as n

xn)

=

-

A q X ) .

@,(XI

+

m

it follows t h a t

IIxnII

IIx+xnII = 2 .

By u n i f o r m c o n v e x i t y i t f o l l o w s t h a t This contradicts t h e f a c t t h a t

If(xn)

I(x-xn(I

I

-

2 as

o as n

> n f o r a l l n and

completes t h e proof. 14.2

B O U N D I N G SUBSETS O F A B A N A C H SPACE

I n t h e p r e v i o u s s e c t i o n we c o n s i d e r e d s e t s a n d r e g i o n s w h e r e a s i n g l e f u n c t i o n was b o u n d e d .

We now l o o k a t s e t s o n

which e v e r y holomorphic f u n c t i o n i s bounded.

-.

173

Holomorphic functions on Banach spaces Definition 4.17

space E .

L e t U b e a n o p e n s u b s e t o f a ZocaZZy c o n v e x

A s u b s e t .4 o f U i s s a i d t o b e b o u n d i n g f o r U i f

We s h a l l u s e t h e t e r m b o u n d i n g s e t when t h e d o m a i n

s p a c e U i s e a s i l y u n d e r s t a n d a b l e from t h e c o n t e x t .

Bounding

s e t s arise n a t u r a l l y i n problems o f a n a l y t i c continuation, c o n s t r u c t i o n o f t h e envelope o f holomorphy and i n problems c o n c e r n i n g t o p o l o g i e s on H ( U )

.

We b e g i n b y c o l l e c t i r g s o m e s i m p l e p r o p e r t i e s o f b o u n d i n g s e t s .

L e t U b e a n o p e n s u b s e t of a l o c a l l y c o n v e x

Lemma 4 . 1 8

s p a c e E and Z e t F b e a Z o c a Z l y c o n v e x s p a c e .

Then

( a ) e v e r y compact s u b s e t o f U i s b o u n d i n g , ( b ) t h e cZosure o f a bounding s e t i s bounding, (c) i f f cH(U;F)

and A i s a b o u n d i n g s u b s e t o f U t h e n f ( A )

i s a b o u n d i n g s u b s e t of P

\\fllA

( d ) i f A i s a bounding s u b s e t o f U t h e n

i s a

T*

c o n t i n u o u s semi-norm on H ( U ) .

Proof

( a ) , (b) and ( c ) a r e obvious.

Let V = ( f

E

H(U)

T*

closed.

fl

XEA

{f EH(U);

Since (H(U),r6)

neighbourhood of z e r o i n (H(U),T*)

is a

T*

XEA

)f(x)

I

We p r o v e ( d ) .

; Ilf{lA 5 1 1 . V i s c o n v e x b a l a n c e d a n d

a b s o r b i n g and s i n c e V = also

= sup

I f ( x ) l 5 1) i t i s

is barrelled V is a and h e n c e

11 1,

continuous semi-norm.

Corollary 4.19

L e t U b e a balanced open s u b s e t o f a ZocalZy

convex space E t h e n (a) A C U

i s b o u n d i n g i f and onZy i f

(b) A C E

i s b o u n d i n g i f and onZy i f

Chapter 4

174

Proof

The c o n d i t i o n s g i v e n i n ( a ) and ( b ) a r e o b v i o u s l y If A i s

s u f f i c i e n t a n d w e now s h o w t h a t t h e y a r e n e c e s s a r y . b o u n d i n g t h e n p ( f ) = IlfIIA i s a

on H(U). -r6

s e m i - n o r m o n H(U)

irn ll*]lA

1

<

=o

m

If U = E then

number B and e v e r y *n

1"

gn

Em

=0

1"

l l ~ l l A<

q

) is also a

E

H(U).

(a).

m

f o r e v e r y complex

E

m

f

n!

n=o

i=o B n dnfo n! H(E) l m 4H ( E ) . n=o anf n.

p

(proposition 3.17).

f o r every

This completes t h e proof of

11 = 0

c o n t i n u o u s semi-norm

Since U i s balanced y ( f ) =

continuous

Hence

T~

Hence, by ( a ) ,

0

f o r e v e r y p o s i t i v e number 8 and s o

If U is a baZanced o p e n s u b s e t o f a

Corollary 4.20

l o c a l l y c o n v e x s p a c e E t h e n t h e b a l a n c e d h u Z l of e v e r y b o u n d i n g s u b s e t o f U is b o u n d i n g . I t i s n o t known i f t h e c o n v e x h u l l o f a b o u n d i n g s u b s e t of E i s s t i l l bounding.

T h i s would amount t o showing t h a t

t h e bounding s e t s form a bornology. Corollary 4.21

t h e locally c o n v e x s p a c e E c o n t a i n s a non-precornpact b o u n d i n g s e t t h e n T # T~ o n H ( E ) . Proof

I f

Suppose A i s a non-precompact bounding s e t .

p r o v e o u r r e s u l t we show t h a t t h e s e m i - n o r m

on H(E) i s n o t

T

w

continuous.

Suppose

pA were T

and p o r t e d by t h e compact s u b s e t K o f E . (IfllA 5 C ( V )

IlfIIV

5

IlfnIIA

w

To

I/fllA

continuous

For every

neighbourhood V o f K t h e r e would e x i s t C(V) Hence

pA(f) =

> o such t h a t

for every f i n H(E). C(V) IlfnI/V f o r a l l f i n H ( E ) a n d e v e r y p o s i t i v e

i n t e g e r n and t h u s

-

Letting n

m

K we h a v e

V-

175

Holomolphic functions on Banach spaces we s e e t h a t

Ilfl(A

Ilflh

I l f I I v.

5

On l e t t i n g

IIfIIK f o r e v e r y f i n H ( E ) .

By t h e Hahn-Banach t h e o r e m i t f o l l o w s t h a t A l i e s i n t h e S i n c e K i s compact i t s c l o s e d

c l o s e d convex h u l l o f K .

convex h u l l i s precompact and t h i s c o n t r a d i c t i o n completes the proof. Our n e x t p r o p o s i t i o n , and i t s c o r o l l a r i e s ,

show t h a t

b o u n d i n g s e t s b e h a v e more l i k e compact t h a n bounded s e t s . I f E i s a l o c a l l y convex s p a c e f E H ( E ) fx : E

C be d e f i n e d by f x ( y )

I

that fx EH(E)

and

X E

f(x+y).

=

E let I t i s immediate

f o r every x i n E.

Proposition 4 . 2 2

If E

is a m e t r i z a b l e l o c a l l y c o n v e x s p a c e and A i s a b o u n d i n g s u b s e t o f E t h e n ( f x ) x A i s a T

0

bounded s u b s e t of H ( E )

Proof

for e a c h f i n H ( E ) .

F i r s t l e t K be a f i x e d compact s u b s e t o f E .

I f K1

i s any o t h e r compact s u b s e t o f E t h e n , f o r a given f i n H(E)

3

s i n c e K+K1

i s a compact s u b s e t o f E .

Hence ( f x ) x E K i s a metrizable H(E)

T

and

T

T&

Since E is

bounded s u b s e t o f E . define

t h e same b o u n d e d s u b s e t s o f

and hence

YEA

S i n c e K was a r b i t r a r y ( f )

Y

YEA

is a

a n d we h a v e c o m p l e t e d t h e p r o o f .

T

0

bounded s u b s e t o f H ( E )

Corollary 4 . 2 3

If E i s a m e t r i z a b z e l o c a l l y c o n v e x s p a c e , f E H ( E ) and A i s a b o u n d i n g s u b s e t o f E t h e n t h e r e e x i s t s a neighbourhood V o f z e r o i n E such thatIlfIIA+V < . Proof

By p r o p o s i t i o n 4 . 2 2 ( f x ) X E A i s a

T

bounded s u b s e t

Chapter 4

176 of H(E).

S i n c e E i s m e t r i z a b l e i t i s a l s o a l o c a l l y bounded and hence t h e r e e x i s t s a neighbourhood V o f

subset of H(E)

zero such t h a t sup XEA

llfxIIV

<

Hence s u p If(x+Y)l XEA

m .

=IlfllA+V

<

YEV

Corollary 4.24

I f E is a m e t r i z a b l e l o c a l l y c o n v e x s p a c e

t h e n t h e v e c t o r sum of f i n i t e l y many b o u n d i n g s u b s e t s of E

i s a l s o a bounding s u b s e t o f E . Proof

L e t A1

and A2 b e b o u n d i n g s u b s e t s o f E and l e t

f EH(E).

By p r o p o s i t i o n 4 . 1 8 t h e s e m i - n o r m g~ H ( E )

_ _ _ f

c o n t i n u o u s a n d p r o p o s i t i o n 4 . 2 2 imp i e s t h a t bounded s u b s e t o f H ( E ) .

~-

T

6

(fx)xEA2 i s a

T~

Hence +

and t h u s A1+A2

llgl~lis

A2

<

m

i s a bounding s u b s e t of H ( E ) .

The e x t e n s i o n

t o f i n i t e l y many b o u n d i n g s e t s i s c a r r i e d o u t i n a n o b v i o u s manner and t h i s completes t h e p r o o f . We now i n v e s t i g a t e t h e s i z e o f b o u n d i n g s e t s .

Since

a n y t w o l o c a l l y c o n v e x t o p o l o g i e s w h i c h d e f i n e t h e same c o n t i n u o u s d u a l h a v e t h e same b o u n d e d s e t s , b o u n d i n g s e t s a r e bounded.

On t h e o t h e r h a n d s i n c e e v e r y h o l o m o r p h i c

f u n c t i o n i s c o n t i n u o u s it f o l l o w s t h a t r e l a t i v e l y compact s e t s are always bounding. o f t h e above,

We h a v e a l r e a d y g i v e n a r e f i n e m e n t

c o r o l l a r y 4.10,

w h i c h we may r e s t a t e a s f o l l o w s .

Each b o u n d i n g s u b s e t of a n i n f i n i t e Proposition 4.25 d i m e n s i o n a l l o c a l l y c o n v e x s p a c e E i s a nowhere d e n s e s u b s e t of

E.

Our n e x t t w o r e s u l t s show t h a t f o r a l a r g e c l a s s o f s p a c e s t h e bounding s e t s c o i n c i d e w i t h r e l a t i v e l y compact s e t s . I f t h i s were t r u e o f a l l l o c a l l y c o n v e x s p a c e s t h e n

bounding s e t s would n o t b e v e r y i n t e r e s t i n g .

We p r o v e s o m e

o f o u r r e s u l t s f o r b o u n d i n g s u b s e t s o f Banach s p a c e s .

These

177

Holomorphic functions on Banach spaces can be extended t o a r b i t r a r y l o c a l l y convex spaces q u i t e e a s i l y a n d t o a r b i t r a r y o p e n s e t s i f we r e p l a c e r e l a t i v e l y compact sets by precompact s e t s .

The d i s t i n c t i o n

between r e l a t i v e l y compact and precompact s e t s i n v o l v e s p r o b l e m s o f a n a l y t i c c o n t i n u a t i o n w h i c h we b r i e f l y d i s c u s s

later. Proposition 4.26

Every bounding s u b s e t o f a separabZe

ZocaZZy e o n v e x s p a c e is r e t a t i v e l y c o m p a c t .

quasi-compZete

Irn b e a d e n s e s e q u e n c e i n t h e l o c a l l y n n=l convex space E and l e t A b e a bounding s u b s e t o f E . Let Let cx

Proof

p

E

Let

cs(E) and

vm

=

f J n=1

E

> o be arbitrary.

{xn+x ; p(x)

<

m

( v ~ ) ~ i s =a n~ i n c r e a s i n g

€1. Since

c o u n t a b l e open c o v e r o f E .

11 1,

i s a rC6c o n t i n u o u s

s e m i - n o r m on H(E) i t f o l l o w s t h a t t h e r e e x i s t C > o a n d N a p o s i t i v e i n t e g e r such t h a t H(E).

n

Ilf(/A

5

IlfllVN

C

On r e p l a c i n g f b y f n , t a k i n g n t h

+m

Ilfla 5

it follows t h a t

By t h e H a h n - B a n a c h

roots,

and l e t t i n g

f o r every f i n H(E).

theorem A l i e s i n t h e closed convex

S i n c e p and

h u l l o f VN,

IlfllVN

for every f i n

E

were a r b i t r a r y it f o l l o w s t h a t

A i s a precompact subset o f E .

S i n c e E is quasi-complete

A i s i n f a c t r e l a t i v e l y compact and t h i s completes

the

proof. Using t h i s r e s u l t w e o b t a i n a f u r t h e r class o f spaces f o r w h i c h t h e same r e s u l t h o l d s . Theorem 4 . 2 7

L e t E be a Banach s p a c e w h i c h is i s o m o r p h i c

t o a s u b s p a c e of C(T),

T a s e q u e n t i a l l y compact H a u s d o r f f

s p a c e , where C ( T ) i s endowed w i t h t h e s u p norm t o p o l o g y . Then t h e b o u n d i n g s u b s e t s of E a r e r e l a t i v e l y c o m p a c t . Proof

Let

CxnIn b e a bounded s e q u e n c e i n subsequence. Then t h e r e e x i s t

E which h a s no c o n v e r g e n t E

> o and t

n,m

E

T with

I X , ( ~ , , ~ )- xm(t - xE m ~ ~ n,m ) ~ = ~ ~ x n 2

178

Chapter 4 CO

f o r a l l n,m.

Now c h o o s e a s u b s e q u e n c e o f ( t n , m ) , ( t k ) k , l ,

which i s d i s c r e t e and c o n v e r g e s t o t e T and a s u b s e q u e n c e m

m

E if of (XnIn=l> (YnIn=lI such t h a t l y n ( t k ) - y m ( t k ) l m S i s a compact s u b s e t o f n,m>k. Let S = I t k l k = l u { t ] .

T a n d (y,

m

i s n o t a r e l a t i v e l y compact s u b s e t o f C(S).

1s) k = l

S i n c e C(S) i s s e p a r a b l e t h e r e e x i s t s by p r o p o s i t i o n 4 . 2 6

f EH(C(S)) such t h a t sup If(yk)

k

I

=

m.

I f RS d e n o t e s t h e

r e s t r i c t i o n m a p p i n g f r o m C(T) t o C(S) t h e n g = f0RSlE

E

H(E)

pro0 f .

and s u p l g ( x n ) l = n

-.

This completes t h e

I f E i s a Banach s p a c e a n d t h e c l o s e d u n i t b a l l B of E'

i s weak* s e q u e n t i a l l y c o m p a c t t h e n we may embed E

i n C(B) a n d h e n c e t h e b o u n d i n g s u b s e t s o f E a r e r e l a t i v e l y compact.

Thus t h e b o u n d i n g s e t s a r e r e l a t i v e l y compact i n

E whenever E '

h a s t h e Radon-Nikodym p r o p e r t y a n d i n

p a r t i c u l a r whenever E i s r e f l e x i v e o r weakly compactly generated.

A complete geometric o r l i n e a r c h a r a c t e r i z a t i o n

o f Banach s p a c e s i n w h i c h a l l b o u n d i n g s e t s a r e r e l a t i v e l y c o m p a c t i s s t i l l n o t known. We now c o n s t r u c t a n e x a m p l e o f a n o n - r e l a t i v e l y c o m p a c t b o u n d i n g s u b s e t o f a Banach s p a c e .

Here,

once

a g a i n , t h e g e o m e t r y o f t h e Banach s p a c e p l a y s an i m p o r t a n t role.

S i n c e i t c a n e a s i l y b e shown t h a t a n o n - r e l a t i v e l y

c o m p a c t b o u n d i n g s u b s e t o f a Banach s p a c e , i f s u c h e x i s t s , c a n b e c o n t i n u o u s l y mapped o n t o a n o n - r e l a t i v e l y c o m p a c t b o u n d i n g s u b s e t o f 1- i t i s n a t u r a l t h a t we i n v e s t i g a t e bounding s u b s e t s o f 1-.

B o u n d i n g s u b s e t s o f 1, h a v e ,

i n f a c t , b e e n c o m p l e t e l y c h a r a c t e r i z e d and we now q u o t e , without proof, t h i s characterization. Theorem 4 . 2 8

( a ) If A i s a bounded s u b s e t o f Z,

following conditions are equivalent; ( i ) A is a bounding s u b s e t o f I,.

then the

179

Holomorphic functions on Banach spaces

(ii)

every sequence ($,),cZL

which converges pointwise

t o z e r o c o n v e r g e s u n i f o r m l y t o z e r o on A, ( i i i ) t h e r e is no s e q u e n c e (a

)

i n A w h i c h is e q u i v a t e n t

n n t o t h e u n i t v e c t o r b a s i s in l 1

(iv)

-

t h e r e is no c o n t i n u o u s l i n e a r mapping T: w i t h c o n t i n u o u s i n v e r s e , T - ~ : T ( Z 1)

\

2

+

zl,

m

such

t h a t T ( B ) C C o n v e x H u l l o f A uhere B is t h e u n i t b a l t

o f 11’ (v)

A i s weakly c o n d i t i o n a l t y compact

(i.e.

each sequence

i n A c o n t a i n s a weak Cauchy s u b s e q u e n c e ) . (b)

The c o n v e x h u l l of a b o u n d i n g s u b s e t of 2 , is b o u n d i n g .

(c)

E v e r y bounded s u b s e t of c o is a b o u n d i n g s u b s e t of 1 ,

We now s h o w t h a t bounding sets.

z m

contains

T h i s , of c o u r s e ,

c l o s e d non-compact

is a l s o a c o r o l l a r y

of

theorem 4.28. We f i r s t n e e d s o m e p r e l i m i n a r y r e s u l t s n I f p(z) = 1 + aiZ1 + z is a p o l y n o m i a l

i:il

Lemma 4 . 2 9

of one c o m p t e x v a r i a b l e t h e n s u

IlzTi=1

By P a r s e v a l ‘ s f o r m u l a

Proof

2 + 1 = 1 ‘/2 n

Ip(z)(

1

C(xnIncl

E

zm;

defined on 2 ,

Lemma 4.30

xn = o i f n let

Let p

L

S)

1

and

(S)

2

=

and if f i s a f u n c t i o n

IIfIIS = s u p { f ( x )

B(nzm)

Ip(eie) I2d0

-71 TI

If S i s a s u b s e t of N l e t 2 , m

fi

E

I; 7

x

E

Z,(S),

IIxII 5 1 1 .

o be a r b i t r a r y .

Then

t h e r e e x i s t s an i n f i n i t e s u b s e t S of N s u c h t h a t IIplls 5

E.

180

Chapter 4 Let P =

Proof

where A

E

Ls(("Im).

T h e n we c a n c h o o s e a s e q u e n c e o f m u t u a l l y d i s j o i n t

i s false.

(Sn)n, such t h a t

infinite subsets of N,

By h o m o g e n e i t y w e c a n f i n d x1

Iblll(

and P ( x l ) I(x211 ~ , 1

1,

B Y lemma 4 . 2 9

sup Ixl.1

we h a v e s h o w n IIpII S i m i l a r l y IIp

IPII

L

since

1 1s g u

IF 11p Ilf' ( 1

Suppose t h e c o n c l u s i o n

>

0 si i=l <

-

Now l e t u

=

P(x2) =

s 1 u s2 s4

->

n

=

( 0 ,

. . .

1, o nt'1

.o,

7

=u

. . . .

with

C then

s l n s2

=

n=l

Theorem 4 . 3 1

a b o u n d i n g s u b s e t of

A i s

Suppose A i s n o t bounding.

e x i s t s an e n t i r e f u n c t i o n f on 2,

f o r every x i n

I,,

)

f o r each

pos it i o n A is a closed

fun}.

s u b s e t o f 1-

00

Since

E

(S2)

and h e n c e w e h a v e completed t h e p r o o f

bounded non-compact

-

2,

This is impossible f o r a l l n

m

as n

E

GE a n d a p p l y i n g t h e s a m e m e t h o d

p o s i t i v e i n t e g e r n and l e t A

Proof

E

> ATE. -

(JT)".

E

for all n.

>

If A

E.

JTE.

I P ( x ~ + xX I ) /

Sn

and x2

Zm(S1)

E

IlPII

2.,

By c o r o l l a r y 4 . 1 9 t h e r e

such t h a t

we c a n c h o o s e ( i f n e c e s s a r y

181

Holomorphic functions on Banach spaces

z l m ]a n

b y r e s t r i c t i n g f t o z,(S) Q1

positive integers,

(nj)j=l,

( n j

An. J

u

E

j

zm)

An,

where

I

S1 i n f i n i t e s u c h t h a t k

c IxTz1 o < r L n l

(,1) n

su

for all j .

1"

) = 1 for all j .

Ls ( " j

0

belongs t o H(Z-1 'n'f(o) j=1 P f ( o ) (uj)

The f u n c t i o n g =

h

such t h a t

-> 25 >

l-(uj)\l'nj n.! I

increasing sequence of

1

4

For each i n t e g e r j

- dnJn(o) n ' j' S1 a n d

.

Cet k l

( X U ~ ) ~ ~ -5 ' ;l!. 1~ ~

]]An

and

let = 1.

Choose

This i s possible

s1

1

b y lemma 4 . 3 0 . Now s u p p o s e k i km

E

and Si h a v e b e e n c h o s e n f o r

Sm-1 a n d l e t C,

4

s u c h t h a t k,

S,

=(kl,

. .

.,km}.

1ziLm-1.

Choose

Choose S m ~ S m - l

, Sm i n f i n i t e a n d

S i n c e C m i s a f i n i t e s e t t h e c l o s e d u n i t b a l l o f loo ( C m )

i s c o m p a c t a n d w e a r e t a k i n g t h e supremum o v e r a c o m p a c t s e t o f a f i n i t e sum o f c o n t i n u o u s f u n c t i o n s e a c h o f w h i c h c a n b e made a r b i t r a r i l y s m a l l b y a n a p p r o p r i a t e c h o i c e o f S m (lemma 4 . 3 0 ) . By i n d u c t i o n we o b t a i n a n i n c r e a s i n g s e q u e n c e o f p o s i t i v e integers

m

(km)m,l.

By r e s t r i c t i n g e v e r y t h i n g t o l , ( S )

where

m

S = ul{km}

l e t CA

For each m

we may s u p p o s e k m = m f o r a l l m .

= S\Cm

= (n;n>m}.

Each z

i n a u n i q u e manner a s x+y where x

E

E

Z,(S)

lm(C,,,),

can be w r i t t e n y

E

lm(Clm)

Chapter 4

182

II.II

and = sup (nxiiSiiyii). U s i n g t h e a b o v e n o t a t i o n we d e f i n e f o r e a c h p o s i t i v e i n t e g e r

m , T,

@ ( n k m Z,(S)),

E

by

fink

Tm(z) =

nkm

1,

t h i s implies that

m= 1

IAj I

i s a polynomial

(x) km*

m

Tm(x) converges f o r each x i n t,(S)

and hence, by theorem 2.28,

been chosen,

n

(x) = Ank

1, and

1"

m= 1

Trnc H ( l w ( S ) ) .

i Amurn)! 2 1 f o r i = 1 ITi(l m= 1

of degree s t r i c t l y l e s s than n

( i n A)

By C a u c h y ' s i n e q u a l i t i e s t h e r e e x i s t s A . J+1

. .

Let a = ( A 1 J A 2 , .we

.,AnJ

have ITj(a)l

=

lTj(A1,

By c o n s t r u c t i o n

. .

. .

.)

.,Aj,o,

E

z,(S).

. .

E

CC,

kj+l'

IAj+ll

. ) [ = I T j ( cj

m= 1

Amurn)[ 2 1 .

2 1 and t h i s c o n t r a d i c t s t h e

1"

and completes t h e p r o o f .

fact that

d

m

j=l Tj

J

E

H(Z,(S))

5 1,

For each i n t e g e r j

Hence l i m s u p IT.(a)ll'kj j

,..., j .

183

Holomoiphic functions on Banach spaces

We d o n o t know o f a n y B a n a c h s p a c e E f o r w h i c h T

#

'c6

o n H(E) a n d i n w h i c h t h e c l o s e d b o u n d i n g s e t s

a r e compact. § 4 . 3 H O L O M O R P H I C FUNCTIONS O N B A N A C H SPACES WITH A N

BASIS

UNCONDITIONAL -

I n t h i s s e c t i o n we l e a v e c o u n t e r e x a m p l e s a s i d e a n d o n H(U) i f U i s a b a l a n c e d o p e n s u b s e t 6 o f a Banach s p a c e w i t h a n u n c o n d i t i o n a l b a s i s . The p r o o f show t h a t

T~

involves,

as i n t h e previous section, geometric properties

= T

o f Banach s p a c e s .

A l s o we s e e t h e u s e f u l n e s s o f a b a s i s

o r a coordinate system

-

used h e r e re-appear

i n studying holomorphic functions

-

and v a r i a t i o n s o f t h e t e c h n i q u e s

on f u l l y n u c l e a r s p a c e s i n t h e n e x t c h a p t e r . We o r d e r t h e f i n i t e s u b s e t s o f N ,

t h e n a t u r a l numbers,

by s e t i n c l u s i o n . Definition 4.33

i n a Banach s p a c e E i s

(en);=1

A basis

c a l l e d an u n c o n d i t i o n a l b a s i s i f f o r any x =

1"

n =1

xn en

E

E

-

l i m J+

JCN,J

( i . e . given

E>O

IIx

-

1

i E J

finite

x

j

'

e.11 = o

t h e r e e x i s t s a f i n i t e s u b s e t J E of N such

t h a t for any f i n i t e s u b s e t J o f N w h i c h c o n t a i n s J c we have

IIx -

1

j EJ

xjej

11

5

E)

.

l p , 1 5 p < -, a n d c o a l l h a v e u n c o n d i t i o n a l b a s e s and t h e f i n i t e p r o d u c t o f s p a c e s w i t h an u n c o n d i t i o n a l b a s i s a l s o h a s an u n c o n d i t i o n a l b a s i s .

The s p a c e o f a l l

c o n v e r g e n t s e r i e s i s an example o f a Banach s p a c e ( w i t h a b a s i s ) which h a s n o t a n u n c o n d i t i o n a l b a s i s . The f o l l o w i n g r e s u l t i s w e l l known a n d c o n s e q u e n t l y we d o not include a proof.

Chapter 4

184

Lemma 4 . 3 4

g i v e n by

-

I f E is a Banach s p a c e with an u n c o n d i t i o n a 2 m

-

( e n ) n = l b t h e n t h e b i l i n e a r mapping from l m x E

basis,

1"

((Bn)n=19

il=l

xnen)

1"

n=l

BnXnen

E

is w e l l d e f i n e d and c o n t i n u o u s . The a b o v e p r o p e r t y i n f a c t c h a r a c t e r i s e s B a n a c h s p a c e s with an unconditional b a s i s . Lemma 4.34

a l l o w s us t o renorm E w i t h an e q u i v a l e n t

b u t more u s e f u l norm.

L e t ( E , \ ] 11)

Lemma 4 . 3 5

b e a Banach s p a c e with an

CO

( e n ) n = l , t h e n t h e norm

unconditional basis,

i s e q u i v a l e n t to t h e o r i g i n a l norm o n E . H e n c e f o r t h we s h a l l a s s u m e t h a t t h e g i v e n n o r m o n E satisfies

lm x e n=l n n

E

IIC" n =1

111

xnen[[ = sup XnXn enII f o r a l l ~cl\r,J finite ~ E J 1 E and i n t h i s c a s e t h e b i l i n e a r mapping o f

bj15

lemma 4 . 3 4 h a s n o r m 1 . We now i n t r o d u c e s o m e n o t a t i o n f o r t h e B a n a c h s p a c e E

with unconditional basis If o 5

m i n 5 "

generated by e

j'

n =

(e )" n n=l'

w i l l

E:

5 n.

rn < j

.

denote t h e closed subspace of E I f m=o we w r i t e E

n

and i f

m

m we write E Note t h a t E o = E . We l e t B d e n o t e m t h e u n i t b a l l o f E a n d l e t B m d e n o t e t h e u n i t b a l l o f 2,.

n Let n m ,

0

5 m 5 n 5

n o n t o Em w h e r e

IT

n

and

denote the natural projection of E

m, 71

m

are given t h e i r obvious meanings.

'The f o l l o w i n g s i m p l e f a c t s a r e e a s i l y v e r i f i e d ,

185

Holomorphic functions on Banach spaces (a)

T;(B)

(b)

(B,

x

Now l e t

= B

n

E*m

B)n

E:

=

m

(Bn)n=l

E

B f l E.:

lm

Now suppose S1,

then

. .,

.

Sm-l

is a finite increasing

s e q u e n c e o f p o s i t i v e i n t e g e r s a n d B1,

. . .,

B,

are non-

We d e f i n e t h e s e q u e n c e ( a n ) n a s

n e g a t i v e r e a l numbers. f o 11o w s

B1' a

=

n

n 5 'i-1

s1 < n i S

i'

25izm-1

i f n > Sm-l

'm a n d we l e t

lAil

5 1,

z.

E

E

'i

si-1

w h e r e S o = o a n d Sm =

Lemma 4 . 3 6

basis,

L e t E be a Banach s p a c e w i t h an u n c o n d i t i o n a Z (en)n=l and norm s a t i s f y i n g t h e c o n d i t i o n o f m

Zemrna 4 . 3 5 . (a) i f

Then m

(pn)n=l

E

co,

(Bn)n

x B

i s a r e l a t i v e l y compact

subset of E , (b) i f

m}

(Bi)y=l i s any f i n i t e sequence

o f p o s i t i v e reaZ

186

B

Chapfer 4

. .,

*

S1'

B1'

3

6,

* ,

*

f

Sm-1

yB f o r any f i n i t e i n c r e a s i n g s e q u e n c e

. . .,

of p o s i t i v e i n t e g e r s S1, Proof

(a)

Sm-l.

(Bn)n=l x B i s a bounded s u b s e t of E .

By lemma 4 . 3 4

Hence i t s u f f i c e s t o n o t e t h a t s u p

1-

;

(b)

x e n n

n=l

If z

E

yB t h e n

E

z

i=l

1' Bi 5

implies z

E

1 we h a v e

B

=

m z

B~

S1'

*

*

.,

B,,

*

*

.,Bm

E

Hence z =

si

E ~ i - l

Sm-1

o as j

z i where z i

-.

i

Bnxnenll

n=j

T

y-lm

=

a l l i where So = 1 and S since

lBnl

B} 5 s u p nlj

{Ill"

.

-

and

03

i '

B n ES

1m

i=l

for

i-1

Bi(;,zi)

and

1

f o r a l l i and t h i s

This completes t h e proof.

We now d e c o m p o s e h o m o g e n e o u s p o l y n o m i a l s b y u s i n g t h e d e c o m p o s i t i o n g i v e n by t h e b a s i s . Let P

E

@("E),

A e &S(nE) k

i n t e g e r then E = E +Ek

A

and A = P .

and h e n c e each x

i n a u n i q u e f a s h i o n a s x + x 2 where x1 1

= P(x1+x2)

P(x)

continuous k

by A j .

-

= A(x1+x21n

The mapping x

E

If k is a positive

E

=

In

j =o

(y)

E

E

Ek

E can be w r i t t e n

and x 2

A(xl)j(x2)

E

Ek.

Hence

n- j

~ ( x ~ ) j ( nx- j~ ) d e f i n e s

a

n-homogeneous poZynomiaZ w h i c h we shaZZ d e n o t e

We t h e n h a v e P

=

1"

j=o

k

A.. J

W i t h t h e a b o v e n o t a t i o n we o b t a i n t h e f o l l o w i n g lemma.

187

Holornorphic functions on Banach spaces

If K i s a c o m p a c t s u b s e t of E ' m - 1

Lemma 4 . 3 7

f o r a22 j , Proof

5 j 'n

(a) Let

x =

Irn

xm

Es m-1

i=l

Y E

K and

(y

=

1 2ri

. +

Hence,

l:i

Bixi)

B be arbitrary.

Then

i E s i - l, 2 5 i Z r n - 1 , x 1 € E S 1

and

by t h e Cauchy i n t e g r a l f o r m u l a ,

=

(7)A(y+Irn-' i =1

P(y

lAl=

X E

S

x i ' where xi

A 'm-1

j

0

then

+

i=l

Pixi)'

Pixi

+

n-j (Bmxm)

A Bmxm)

dX

A j+ 1

1

I t now s u f f i c e s t o n o t e t h a t

Im-' Bixi i=l

+ A Bmxm

E:

and t o take t h e supremum o f b o t h s i d e s ( b ) U s i n g t h e a b o v e n o t a t i o n we h a v e

Chapter 4

188 Hence y +

1"

$ i ~ i + 'mxm+ 1 = y i=l

+

iy = - l l! 3 i x i

B m ( x m + mx t l 1

+

and t h i s p r o v e s t h e d e s i r e d e q u a l i t y In order t o avoid unnecessary subscripts i n t h e f o l l o w i n g p r o p o s i t i o n we s h a l l a d o p t t h e f o l l o w i n g c o n v e n t i o n ; when w e s a y I t b y t a k i n g s u b s e q u e n c e s i f n e c e s s a r y " t h e n we s h a l l assume w i t h o u t l o s s o f g e n e r a l i t y t h a t t h e o r i g i n a l sequence has t h e d e s i r e d p r o p e r t y .

If U i s a baZanced o p e n s u b s e t of a Banach

Theorem 4 . 3 8

on H(U).

s p a c e E w i t h an u n c o n d i t i o n a Z b a s i s t h e n -ccw = -c6

is the

S i n c e E i s a n o r m e d l i n e a r s p a c e -c6

Proof

b o r n o l o g i c a l t o p o l o g y a s s o c i a t e d w i t h -ccw a n d h e n c e i t s u f f i c e s t o show t h a t a n y Banach v a l u e d

bounded

T~

h e n c e T6 b o u n d e d ) l i n e a r f u n c t i o n o n H(U)

is

Tcw

(and

continuous.

The i d e a o f t h e p r o o f i s t o f i r s t

Let T b e s u c h a f u n c t i o n .

show, u s i n g t h e f a c t t h a t T i s

T~

continuous,

i s s u p p o r t e d b y a c e r t a i n open s e t .

that T

Then u s i n g i n d u c t i o n

we c h i p a w a y a t t h i s o p e n s e t a n d s h o w t h a t T i s s u p p o r t e d by a s e q u e n c e o f open s e t s which t e n d t o a compact s e t . Let B b e t h e open u n i t b a l l o f E . (nB);,l

i s an open c o v e r i n g o f E and h e n c e ,

confinuous, such t h a t U

If U = E,

there exist a positive integer n

llT(f)

11

C

IlfIln

then

since T is 0

T~

and C > o

f o r e v e r y f i n H(E)

(*).

If

0

# E l e t k d e n o t e t h e s e t o f a l l compact b a l a n c e d s u b s e t s

o f U which l i e i n En each K

u

E

3?

f o r some p o s i t i v e i n t e g e r n . 1

For

l e t V K = K + Td(K,,&U)B w h e r e d(K,,&U) i s t h e

VK i s open and d i s t a n c e from K t o t h e complement o f U . VK = u . S i n c e E i s s e p a r a b l e we c a n c h o o s e f r o m t h i s K C k

189

Holomophic functions on Banach spaces covering a countable subcover o f U,

is

m

(VK ) i = l . i

Since T

r and

continuous there exist a positive integer

T~

C ' > o such t h a t

llT(f)

11

L e t ai

5

sup

C' 1

=

4

. ,r

i=l,.

d(Ki,AU)

llfllv

f o r e v e r y f i n H(U)

(**).

Ki f o r i = 1 , . . . , r and suppose

f o r i = l , , . . rif U # E .

S

K i C E

I f U=E l e t r = l , K = { o ) and al=n 1

1

:

Then ( * ) a n d ( * * ) i m p l y t h a t t h e r e e x i s t s C > o s u c h t h a t 1

f o r a l l Pn

E

L?(nE) 2 n d f o r a i l n

Now s u p p o s e we a r e g i v e n m + l p o s i t i v e n u m b e r s C , Y @ 1 J ' * . J 8 , Y a s t r i c t l y i n c r e a s i n g sequence o f p o s i t i v e i n t e g e r s S

3

and y > 1 such t h a t

* * * 'Sm-l

for all ?ll+l

P

n

E

@(nE) and a l l n .

> o a n d y1

T h e n we c l a i m t h a t g i v e n

> y t h e r e e x i s t Cm+l

> o and sm > s

~

-

such t h a t

for all

pn

E

(nE) and f o r a l l n .

Suppose o t h e r w i s e . Then f o r e v e r y p o s i t i v e i n t e g e r n t h e r e a homogeneous p o l y n o m i a l o f d e g r e e k n , s u c h t h a t e x i s t s P,,

~

Chapter 4

we

f i r s t show sup kn =

n

-.

Otherwise, by t a k i n g a subsequence

i f n e c e s s a r y , w e may s u p p o s e k n = M f o r a l l n . By lemma 4 . 3 6 ( b )

and hence t h e sequence

m

'n SUP i = lJ

. .. , T

(Y 1a i l k n IIPnlb K i + ~sB1 a. 1""

I .

1

- - J sm - l ' s m - l

+n

9Bm+l

Y

i s a l o c a l l y bounded s u b s e t o f @ ( - E ) and a subset of H(U).

T

6

n=l

bounded

f o r a l l n t h i s i s i m p o s s i b l e a n d h e n c e s ~ kpn =

m.

t a k i n g a s u b s e q u e n c e i f n e c e s s a r y we may a s s u m e t h a t

By (knIn

i s s t r i c t l y i n c r e a s i n g sequence o f p o s i t i v e i n t e g e r s . Asm-l+n a n d h e n c e , b y (4.4), t h e r e e x i s t s pn = l k n J j =o

Now

f o r each i n t e g e r n ,

jn,

0

2 jn I kn, such t h a t

,

Now s u p p o s e l i m s u p n - m

kn

-

jn

kn

i f n e c e s s a r y we may s u p p o s e

=

E

k -j

>

0 .

By t a k i n g a s u b s e q u e n c e

n n +E 7 n

as n

-

3

2

K

F a II

w

(0

v

K

n

3

.4

c,

F

cd a,

m 0 0

u

c x k

cd

k c, .ti

P

a,

3

0

z

W

vl

x II

z

-4

a,

.4

-

8 w.4 -

cd

w

F

w

c

W

I

-

F

.ri

F

m

A

E

I

d

+

.ri

x

.4

a,

.4

w.4 V

x

a,

K

'n Y

K

+ 3 I

F

v

cl

n

K

F

E d m II w -4

Y

V

II

-

n

x

V

>

F

-n

4 -

I

3

w

.4

E m

0

c4

111

.4

a,

F:

c,

+F

.rl

k a, M a,

z

7 m

V

a

A

0

P

<+I k cd

Y

rd

F

.rd w .4

W

a,

II

d

. t i

x

a, eII

w

x

II

w .4

8

w K el

c, a,

F

rl

V

c cd a, k

0 Cr,

K

a,

n .4

K

.4

x

+ E

,-I I

VI A W.4

II N

x

a,

F

K

\o

M

d

E E

d

: a, v)

0

a

c4 3 ln

3 0

z

a,

3:

K a V,

P

x

m

E

I

+3

A

z

+

a

x

d

.4

a,

.4

F

F

II

cd

I

3

+

E m

w.4 II rl

x

3

a, k a, F V

K

+

E

I

d

m

E

I

E

-

m

d r l I +

m

.

. . . .

'

3

3

%

. . *

m

4

v ) m

+

3

II W.4

E

v

u

+

+ x x

V

x

n

F

m

4

.r

EE

3 I

+F

Y

d '

a

n

M

r .

V

a

d

E

192

Chapter 4

S i n c e 6 > o was a r b i t r a r y a n d lirn s u p i s z e r o .

s

s e q u e n c e a n d A m-l jn

i s p o s i t i v e it follows t h a t t h e

E

Hence,

since k

+n

k

E

p(

s

i s a s t r i c t l y increasing

n

the series

nE),

+n

A.m-l

Jn

1"

n=o

- , sm..l'sm-l+" 1' ~ * i + ( a i Y 1 k n ~ ~ p n B~ s ~ i = l , . ,r B1' * * *,Bm+l

SUP

.

d e f i n e s an e n t i r e f u n c t i o n on E More o v e r s

A m-1

+n

1

jn

l i m sup

k

sl"

UP i=l,

. . ,r

S

> -

-

m

However i f f = is a

$,,...

l i m sup

n

T

> -

n

1"

(and hence a

consequently l i m n - +

(1

m - 1 ' sm - 1 + n

2 s

Bm+1

1.

E

n.

n=o

* .

H(E)

) bounded

then {gn

~

T(

dnf(o) 'In n! 111

m

g i v e s a c o n t r a d i c t i o n and s o l i m n-+

dnfo n! 'n=o

s u b s e t o f H(E) a n d

T

-

=

k

n

-j

kn

-

Thus t h e above

0 .

n

o as n

__f

m .

We now c o n s i d e r t h i s c a s e . S i n c e r i s f i n i t e and f i x e d and t h e sequence i n f i n i t e we may s u p p o s e ,

(kn)nmZ1 i s

from ( 4 . 2 ) and t a k i n g a subsequence

i f n e c e s s a r y , t h a t t h e r e e x i s t s i,l2izr, s u c h t h a t

for all n.

'

I

kn

193

Holomoiphic functions on Banach spaces

On t h e o t h e r h a n d

(lemma 4 . 3 7 ( a ) )

2

Bm+l

n Y 1 (m (-1 n

II

k

k -j

n

8,

s

n

1

-+n

'/k,

( b y lemma 4 . 3 7 ( b ) ) .

'

II

194

Chapter 4

T h i s i s i m p o s s i b l e s i n c e y 1 > y a n d t h u s we h a v e p r o v e d t h e r e q u i r e d s t e p i n o u r induction argument. Aside

A s i m p l i f i e d v e r s i o n o f t h e above goes a s f o l l o w s ;

i f t h e i n d u c t i o n s t e p d i d n o t w o r k t h e n we c o u l d f i n d jn kn - j n , where is evaluation a t the nth fn= c o o r d i n a t e , such t h a t t h e sequence ( f n ) n did n o t s a t i s f y

4,

If

(4.4).

nth

4,

hn

-

k -j

n n ___ kn

E

> o then the rapid decrease of the

c o o r d i n a t e overcomes t h e g e o m e t r i c growth

first coordinate so that

kn-jn/k

n

--+

1,

fn

E

H(E).

of the

Otherwise

o so t h a t t h e e f f e c t of t h e nth coordinate i s

n e g l i g i b l e and

1n

fn behaves

like

1, 4,j n .

In both cases

we s a w t h a t t h i s l e d t o a c o n t r a d i c t i o n .

We now c o m p l e t e t h e p r o o f o f t h e t h e o r e m . Let

(yn)n d e n o t e a s e q u e n c e o f r e a l numbers,yn

such t h a t

T:=~

Now u s i n g ( 4 . 1 )

'n

= y

< 2'

> 1,

and i f U # E such t h a t

a s t h e f i r s t s t e p i n t h e i n d u c t i o n and

s i n c e ( 4 . 2 ) --J ( 4 . 3 ) we c a n f i n d a s t r i c t l y i n c r e a s i n g sequence of p o s i t i v e integers,

m

( s ~ ) ~ a=n d~ ,(Cn):=l

a

195

Holomorphic functions on Banach spaces s e q u e n c e o f p o s i t i v e numbers s u c h t h a t llTIPn)

II

f o r a l l Pn Let K =

n

F cIT,+l h l . .-Ym) SUP

i=l,

...

@ ( n E ) and a l l n .

E

n i Ki+B , r a i l I p n l I"i

sl,. .,s m + l l J I J * * J

1 m'

where % = ( % , ) , and 1 i f n < s2

< n < s i i+l Since

E

c0 3 K i s a compact s u b s e t o f E .

F o r e a c h i , liicr, l e t L i

= yKi+ya

i

K.

Li

is a

compact s u b s e t o f E and moreover f o r e a c h i

Hence L =

cfj i=l

Li

i s a compact s u b s e t o f U . Moreover i f V i s

any open s e t which c o n t a i n s L t h e r e e x i s t s a p o s i t i v e i n t e g e r nv such t h a t

S i n c e U i s b a l a n c e d we c a n c h o o s e A > 1 s u c h t h a t XL i s a g a i n a compact s u b s e t o f U .

I f W i s any open s u b s e t o f U

w h i c h c o n t a i n s XL t h e n t h e r e e x i s t s a n e i g h b o u r h o o d V o f K s u c h t h a t XLCXVCW. Hence, f o r any f = proposition 3.16

1" dllfo n!

n=o

E

H(U) we h a v e , b y

196

Chapter 4

H e n c e T i s p o r t e d b y t h e c o m p a c t s u b s e t XL o f U . This completes the proof. By m o d i f y i n g t h e a b o v e p r o o f p l a c e of t h e b a s i s )

o n e c a n s h o w t h a t T~ =

whenever E i s a s u b s p a c e o f sense of Shilov.

( u s i n g C e s r . r o sums i n T~

on H(E)

1

L [ o , ~ I Th]o m o g e n e o u s i n t h e

The p r o o f however i s j u s t as d i f f i c u l t

a s t h e a b o v e a n d we d o n o t i n c l u d e i t . §

4.4

F U R T H E R RESULTS A N D EXAMPLES CONCERNING HOLOMORPHIC

FUNCTIONS O N B A N A C H SPACES

Iie c o m m e n c e t h i s s e c t i o n b y e x h i b i t i n g a g e n e r a t i n g f a m i l y o f semi-norms

f o r (H(U),ru),

s u b s e t o f a Banach s p a c e .

U a b a l a n c e d open

We t h e n g i v e a n u m b e r o f e x a m p l e s

a l l o f which i n v o l v e bounding s e t s . Proposition 4.39

L e t U be a baZanced open s u b s e t o f a

Banach s p a c e E .

The

T~

topoZogy on H ( U )

i s generated

b y t h e semi-norms

where

B i s t h e u n i t baZZ o f

C J ( a n ) ~ z Or a n g e s o v e r c

K r a n g e s o v e r t h e compact s u b s e t s o f U .

and

197

Holomorphic functions on Banach spaces Let K b e a c o m p a c t b a l a n c e d s u b s e t o f U a n d l e t

Proof (IJ

E C ~ .

I f V i s any balanced neighbourhood o f K

t h e n t h e r e e x i s t A < 1 and n o , a p o s i t i v e i n t e g e r ,

such t h a t

AV i s a neighboushood o f K and K + a n B C V f o r a l l n 2 n 0 .

By u s i n g t h e C a u c h y i n e q u a l i t i e s w e c a n f i n d c l > o s u c h t h a t

Thus p

is a

K,

T

W

c o n t i n u o u s semi-norm on H(U).

Conversely suppose p is a

Tw

c o n t i n u o u s s e m i - n o r m o n H(U)

p o r t e d by t h e compact b a l a n c e d s u b s e t K o f U .

If X > 1 is

such t h a t A K C U then proposition 3.24 implies t h a t t h e s e m i -norm $(f) =

1” n=o

A”p(-) d n f ( 0 n.

is

c o n t i n u o u s and p o r t e d by

T W

AK.

Moreover p

5 $.

c ( n > > o such t h a t

Fo? e a c h p o s i t i v e - i n t e g e r n t h e r e e x i s t s mf 0 2 c(n) d m f o

IlnK+3

$*(I

f o r e v e r y f i n H(U) a n d a l l m .

f o r e v e r y f i n H(U) a n d a l l m . For each i n t e g e r n choose a p o s i t i v e i n t e g e r jn such t h a t

‘(”)/,j

2

1 for all j

2 jn.

We may a s s u m e , w i t h o u t l o s s o f

Chapter 4

198

m

generality,

t h a t t h e sequence (jn)n=l

increasing.

1'

is s t r i c t l y

1 for j A j2

Now l e t

a. =

-

J The s e q u e n c e

(aj)Yzo

for j

n

< j c j n + l , nL2

lies i n c

and,

for e a c h f i n H ( U ) ,

we h a v e

Hence any

T~

c o n t i n u o u s s e m i - n o r m on H ( U )

i s dominated by

a semi-norm o f t h e r e q u i r e d t y p e and t h i s c o m p l e t e s t h e proof. We now l o o k a t h o l o m o r p h i c f u n c t i o n s o n a c o u n t a b l e d i r e c t sum o f B a n a c h s p a c e s .

P r o p e r t i e s of bounding sets

enable us t o s e t t l e t h e "completion

problem" f o r such

spaces and t h e techniques developed prove useful i n d i s c u s s i n g h o l o m o r p h i c f u n c t i o n s on s p a c e s o f d i s t r i b u t i o n s (chapter 5). E i where e a c h E i is a Let E = 1=1 On H ( E ) , T and ~ T~ d e f i n e t h e same bounded

Proposition 4.40

Banach s p a c e . sets. Proof let

L e t Fn =

p be a

T~

In

i = l Ei

f o r e a c h p o s i t i v e i n t e g e r n and

c o n t i n u o u s s e m i - n o r m on H ( E ) .

without loss of generality t h a t

We may a s s u m e

199

Holomoiphic functions on Banach spaces

p(f)

1-

=

pc-1

n=o

n

f o r every f =

1

n=o

dnf(o) 7

We f i r s t c l a i m t h a t t h e r e e x i s t s a p o s i t i v e

i n H(E). integer n

such t h a t i f f E H ( E )

P

Suppose o t h e r w i s e . can choose P

n’

and f I F n

= o then p ( f )

=

0.

P

T h e n f o r e v e r y p o s i t i v e i n t e g e r n we

a homogeneous p o l y n o m i a l ,

s u c h t h a t pn

= 0

I Fn

Vie now s h o w t h a t t h e s e q u e n c e (Pn):=l is and p(Pn) # 0. l o c a l l y bounded. For each n l e t B denote t h e u n i t b a l l n of En. I f x E E t h e n X E Fn f o r s o me i n t e g e r n . Hence t h e r e

. . .n ,

e x i s t A.>o,

i=l,

such t h a t

By u s i n g t h e b i n o m i a l e x p a n s i o n we c a n f i n d X n + l > o such t h a t

’ [I

l n * l XiBif

IIP.

M +

X+iZl

1.

2n+1

f o r i=l,.

..,

n+l

and by p r o c e e d i n g i n t h i s manner, s i n c e e a c h s t e p o n l y i n v o l v e s a f i n i t e n u m b e r o f p o l y n o m i a l s , we c a n f i n d a sequence o f p o s i t i v e numbers, I I p j IIx+Cm n=l

n Bn

<

m

Hence {Pn,)n,l polynomials.

M+l

m

(Xn)n=1’ such t h a t

for all j.

i s a l o c a l l y bounded f a m i l y o f

S i n c e we o n l y u s e d t h e p r o p e r t y P m

= o

n

f o r each n it follows t h a t {a p 1 is also a locally n n n=l bounded f a m i l y o f p o l y n o m i a l s f o r any s e q u e n c e o f s c a l a r s co

(an)n=l

*

n L e t a =-p ( p n )

f o r each n .

T h e n {anPnl:=l

i s a l o c a l l y bounded and h e n c e a r 6 bounded s e q u e n c e of

Chapter 4

200

S i n c e p ( a P ) 2 n f o r a l l n and p n n continuous t h i s i s impossible and e s t a b l i s h e s o u r c l a i m .

holomorphic f u n c t i o n s .

is

T~

Let

( f a ) a E A b e an a r b i t r a r y

l e t p be a

T

b o u n d e d s u b s e t o f HIE) a n d

0

c o n t i n u o u s s e m i - n o r m on H ( E ) .

T&

To c o m p l e t e

t h e p r o o f we m u s t s h o w s u p p ( f ) m . By t h e a b o v e we may a = 0 choose a p o s i t i v e i n t e g e r N such t h a t i f f E H ( E ) and f FN

I

then p(f) = ?,(x+y)

'L

F o r e a c h C(EA l e t f a

0 .

f a ( x ) where x E F

=

and

N

YE

H(E) b e d e f i n e d by

E

lm

n=N+1

Since

En.

lac* i s a l o c a l l y bounded f a m i l y i n H(FN)

'fa[

follows

it

FN %

( f a ) a ~ Ai s a l o c a l l y b o u n d e d a n d h e n c e a

that

subset of H(E).

S i n c e fa

'L

I

2.

we h a v e p ( f a ) = p ( f , )

= fa, I

FN

f o r a l l a a n d s o s ~ pp( f a )

pN 'L

= s;p

bounded

T~

p(fa)

m.

This completes

the proof. Proposition 4.41

T~

if a n d only if each E i

Proof

1-

i=l

i =1

Ei

where each E i

Then ( H ( E ) , r ) is complete f o r

space. OT

1"

Let E =

E i s

(H(c"))

I f each Ei (C")

,TI

T

= T ~ , T

~

,

~

,

~

,

is a finite dimensional space.

a n d we h a v e a l r e a d y s e e n ( e x a m p l e 2 . 4 7 )

i s complete.

ow

s u p p o s e a t l e a s t o n e E~

an i n f i n i t e d i m e n s i o n a l Banach s p a c e .

ll$nII

= 1.

an e n t i r e f u n c t i o n f on El

that is

Without l o s s o f

is i n f i n i t e dimensional.

.rn d e n o t e t h e n a t u r a l p r o j e c t i o n f r o m E o n t o E n .

nE A~ w i t h

T

is a f i n i t e dimensional space then

g e n e r a l i t y we may s u p p o s e E l

n let @

is a Banach

Let

For each

By c o r o l l a r y 4 . 1 0 t h e r e e x i s t s with r f ( o ) = 1.

T

~

,

~

20 1

Holomorphic functions on Banach spaces

1"

En and e a c h compact s u b s e t o f E i s c o n t a i n e d a n d n=l c o m p a c t i n some Fm i t f o l l o w s t h a t t h e p a r t i a l sums o f g

Fm =

form a Cauchy s e q u e n c e i n ( H ( E ) , T ~ ) . g

I

We now s h o w t h a t

H(E).

Suppose o t h e r w i s e .

Then t h e r e would e x i s t a convex b a l a n c e d Ilgllv = M <

neighbourhood V o f zero i n E such t h a t

For

03.

e a c h p o s i t i v e i n t e g e r n l e t Bn b e t h e u n i t b a l l o f E n . e x i s t s f o r e a c h n a p o s i t i v e number 6

There

such t h a t &,B,CV.

n

4

Choose n a p o s i t i v e i n t e g e r such t h a t n >

and choose

7

O1

6n x E-B f o r which @ (x) 2 n n

#

- -

e x i s t s a sequence i n El,

I f ( y m )I

m

(ym m = l '

as m

m

By o u r c o n s t r u c t i o n t h e r e

0 .

IlymII 5 2 ,

m

1

T(2X) =

as m

--

4

t h i s shows t h a t g

not complete.

By e x a m p l e

such t h a t

1.24

Ym

7

X E

V for a l l m.

H e n c e ( H ( E ) , T ~ )i s

H(E).

( p (nE)

t

, T ~ )and

( 6( n E ) , 8 )

a r e complete 1,ocally convex spaces f o r each p o s i t i v e i n t e g e r n.

Hence H(E) i s n o t T . S . T o

H(E) i s

n o t T.S.T

0

complete.

complete and hence

,b

a complete l o c a l l y convex s p a c e . TO

,b

=

T~

T

=

To,

T

~

,

~

,

T

~

(H(E),T

0

,b

)

is not

By p r o p o s i t i o n 4 . 4 0

a n d t h u s we h a v e s h o w n t h a t

complete f o r

By c o r o l l a r y 3 . 3 4

,

oTr

(H(E),T) i s n o t

~T 6, . ~

This completes

the proof. Note t h a t t h e above a l s o shows t h a t t h e r e e x i s t

'c6

Chapter 4

202

b o u n d e d s u b s e t s o f H(Cm En) w h i c h a r e n o t l o c a l l y b o u n d e d n =1 whenever a t l e a s t one E space.

n

i s an i n f i n i t e d i m e n s i o n a l Banach

We now b r i e f l y c o n s i d e r a n e x t e n s i o n p r o b l e m w h i c h a r i s e s o n l y in i n f i n i t e d i m e n s i o n a l a n a l y s i s .

If

F is a

s u b s p a c e o f a l o c a l l y c o n v e x s p a c e E when c a n e v e r y h o l o m o r p h i c f u n c t i o n on F b e e x t e n d e d t o a h o l o m o r p h i c f u n c t i o n on E ?

Two i n t e r e s t i n g d i s t i n c t c a s e s o f t h i s

p r o b l e m a r i s e when ( a ) F i s a c l o s e d s u b s p a c e o f E a n d ( b ) when F i s a d e n s e s u b s p a c e o f E .

Problem ( a ) c o n c e r n s

a n a t t e m p t t o f i n d a h o l o m o r p h i c Hahn-Banach

t h e o r e m and

w i l l r e a p p e a r i n o u r d i s c u s s i o n on h o l o m o r p h i c f u n c t i o n s

Example 4 . 4 2 , w h i c h u s e s p r o p e r t i e s

on n u c l e a r s p a c e s .

o f b o u n d i n g s e t s , shows t h a t i n g e n e r a l we d o n o t o b t a i n a Problem (b) i s t h e

p o s i t i v e s o l u t i o n t o t h i s problem.

holomorphic analogue o f f i n d i n g t h e completion of a E x e r c i s e s 1 . 8 9 and 2 . 9 4 a r e r e l a t e d

l o c a l l y convex s p a c e .

t o problems ( a ) and (b) r e s p e c t i v e l y . This example i s devoted t o showing t h a t

Example 4 . 4 2

n o t e v e r y holomorphic f u n c t i o n on c h o l o m o r p h i c f u n c t i o n on l m . m

Let A = (u ) where u = (o,.. n n=l n positive integer n. c

0

. , 1,o

can be extended t o a

. . .)

f o r each

T nth place

A i s a c l o s e d non-compact

subset of

By p r o p o s i t i o n 4 . 2 6 A i s n o t a b o u n d i n g

and o f lm.

s u b s e t o f c o and by t h e o r e m 4 . 3 1 A i s a b o u n d i n g s u b s e t o f NOW suppose each holomorphic function o f co has a

1m .

h o l o m o r p h i c e x t e n s i o n t o lm. f E H ( c ~ )s u c h t h a t

IlfIIA =

a.

By t h e a b o v e t h e r e e x i s t s %

I f f c H( 2,)

%

and f

=

f then

IC 0

11?11,

=

llflk =

a

and t h i s c o n t r a d i c t s t h e f a c t t h a t A i s

a bounding s u b s e t of

lm.

H e n c e we h a v e shown t h a t t h e r e

e x i s t h o l o m o r p h i c f u n c t i o n s on c o w h i c h c a n n o t b e e x t e n d e d h o l o m o r p h i c a l l y t o tm.

203

Holomoiphic functions on Banach spaces

I f E i s a l o c a l l y convex space w i t h completion E t h e n

t h e r e e x i s t s a subspace of E , E u J

which i s c h a r a c t e r i z e d by

the following properties (1) E (2)

r.

C E , C

E,

each holomorphic f u n c t i o n on E can b e extended t o a h o l o m o r p h i c f u n c t i o n on Eo,

(3)

fi

A

F C E , F a s u b s p a c e of E , and e a c h h o l o m o r p h i c f u n c t i o n on E c a n b e e x t e n d e d t o a h o l o m o r p h i c f u n c t i o n on F t h e n F C E L o .

if E C

E & i s c a l l e d t h e h o l o m o r p h i c c o m p l e t . i o n of E .

Proposition

4.43

space t h e n E6

=

u

If E

i s a metrizable l o c a l l y convex

A

i s t h e c l o s u r e of

where

ACE, A bounding

fi

in E

A

Let E B =

Proof

(cn)nC E

Ux

-

I f 5 E ECp t h e n t h e r e e x i s t s

ACE A bounding

s u c h t h a t 5,

-

.5 a s n

m.

Since 5

EB.

E

Ls)

n

n

and 5

E

and h e n c e s u p [ f ( S n ) l <

l i m f ( s ) e x i s t s f o r e v e r y f i n H(E)

- - + n

f o r every f i n H(E).

E

i s a bounding subset o f E

Thus {En),

On t h e o t h e r h a n d i f

T h i s shows t h a t E @ C E B .

A i s a bounding s u b s e t o f E and f

E

H(E) t h e n b y c o r o l l a r y 4 . 2 3

t h e r e e x i s t s a convex balanced neighbourhood V of z e r o i n

E such t h a t about

Ilf[(A+V <

a.

By u s i n g T a y l o r s e r i e s e x p a n s i o n s

p o i n t s o f A w e f i n d t h a t t h e r e e x i s t s a holomorphic

function

1

%

o n A+W s u c h t h a t f

IA + V A

i n t e r i o r o f t h e closure of V i n E. t h a t rCEOa n d h e n c e Eu

= EB.

-

= f

'

A+V

where

W is

the

S i n c e A + W 3 K t h i s shows

This completes t h e proof.

204

Chapter 4 P r o p o s i t i o n 4.43 and t h e s o l u t i o n t o t h e Levi problem

may b e u s e d t o p r o v e t h e f o l l o w i n g r e s u l t . Proposition 4.44

If E i s an i n f i n i t e d i m e n s i o n a Z

m e t r i z a b l e ZocaZZy c o n v e x s p a c e of c o u n t a b l e a l g e b r a i c d i m e n s i o n t h e n E i s hoZomorphicaZZy c o m p l e t e if and o n l y

if E a d m i t s a c o n t i n u o u s norm. I n p a r t i c u l a r p r o p o s i t i o n 4 . 4 4 s a y s t h a t an i n f i n i t e countable a l g e b r a i c dimension

d i m e n s i o n a l normed s p a c e o f

i s holomorphically complete. 1 4 . 5 EXERCISES

Let X b e an H a u s d o r f f

4.45

t o p o l o g i c a l s p a c e and l e t

b e t h e s p a c e o f bounded c o n t i n u o u s complex v a l u e d

,fb(X)

f u n c t i o n s on X w i t h t h e s u p norm t o p o l o g y .

Show t h a t f

i s a r e a l o r complex extreme p o i n t o f t h e u n i t s p h e r e o f g b ( X ) i f and o n l y i f 4.46"

( f ( x )\

= 1 f o r every x i n X.

Let E and F b e complex Banach s p a c e s w i t h o p e n u n i t

b a l l s U and V r e s p e c t i v e l y .

Show t h a t e v e r y f

E

H(U;V)

w i t h d f ( o ) = L and f ( o ) =o i s l i n e a r i f and o n l y i f L i s m

a complex e x t r e m e p o i n t o f t h e u n i t b a l l o f H (U,v)

4.47*

Let f b e a c o n t i n u o u s f u n c t i o n mapping t h e c l o s e d

u n i t d i s c o f t h e complex p l a n e i n t o a complex Banach Suppose f i s holomorphic on t h e open d i s c .

algebra B. If

1x1

( f(h)

1,

5 1

(1 la

= 1 show t h a t

4.48"

d e n o t e s t h e s p e c t r a l r a d i u s ) whenever

(f(x)l,

5 1 for a l l h,IXI

2

Let B b e a Banach a l g e b r a a n d l e t f : D = { z c

b e a n a n a l y t i c f u n c t i o n s u c h t h a t f ( o ) = o and for all z

z

E

D\{o)

4.49 -

1.

E

D.

or

Show t h a t e i t h e r I f ( z ) If(z)la

5

IzI

lo

<

c

:lzl
(f(z)

I,

5 1

( z ( for all

f o r a l l z i n D.

L e t U b e an o p e n s u b s e t o f a Banach s p a c e E which

B

205

Holomophic functions on Banach spaces c o n t a i n s t h e o r i g i n and l e t f

-

spectrum of f t h a t f-XI

as I X E C ;

,a(f),

: V

We d e f i n e t h e

H(U;E).

E

-fi

V,W

open, OEV, W C U such

W i s a b i h o l o m o r p h i c mapping}.

Show t h a t

o(f) = o(df(o)).

Let E b e a B a n a c h s p a c e w i t h o p e n u n i t b a l l B ,

4.50*

For

each x i n E l e t

-

K(x) = : B

If f

f4

E

II+II

E';

= sup

= 1)

E then the numerical range of f,W(f),

4

d e f i n e d as { d ( f ( x ) ) ; IW(f)l

= 4(x)

(1x1

; A

E K ( x ) , IIxI( = 1 1 .

is

Let

W(f)).

E

I f f I B i s h o l o m o r p h i c show t h a t

m w h e r e k = 1, k 1= e a n d k m = m / m - 1

If f ( o )

for m 2 2.

= o

a n d W(f) i s r e a l show t h a t f i s a l i n e a r m a p p i n g . 4.51*

-

L e t E and F b e Banach s p a c e s w i t h open u n i t b a l l s

U and V r e s p e c t i v e l y .

Let f

: U

V be a biholomorphic

mapping from U t o V ( i . e .

f i s holomorphic and b i j e c t i v e

and f - l

I f f ( o ) = o show t h a t f i s t h e

i s holomorphic).

restriction t o U of a linear isometry of E onto F. 4.52* -

6

h

Let B b e t h e open u n i t b a l l o f a Banach s p a c e .

o and suppose f

E

C ,

[hi

K O = 1 , K1

E

H(B;g)

5 1 and x E B . = e a n d Km

=

s a t i s f i e s I(x+Af(x) Show t h a t

mm/m-l

for m

1 d n f1 2.

( 0 )

7 (1

Let

11

2 1+6 for all 2 Km6 w h e r e

By c o n s i d e r i n g

t h e c a s e 6 = o deduce t h a t t h e i d e n t i t y mapping on E i s a r e a l e x t r e m e p o i n t o f t h e u n i t b a l l o f Hm(B;E) endowed w i t h t h e s u p norm t o p o l o g y . 4.53*

L e t L(H) b e t h e a l g e b r a o f a l l b o u n d e d l i n e a r

o p e r a t o r s from t h e H i l b e r t s p a c e H i n t o i t s e l f and l e t U b e t h e open u n i t b a l l o f L(H). t h a t t h e mapping

I f S E U and A = 1-S*S

show

206 2

E

Chapter 4 L(H)

___f

(s-z)( I - ~ * z ) - ~ A ' / ~

A*-'/2

i s a b i h o l o m o r p h i c mapping of U i n t o i t s e l f which h a s S(1+A1/q-' 4.54*

a s a unique f i x e d p o i n t .

L e t U b e a b o u n d e d o p e n s u b s e t o f a Banach s p a c e

and l e t f EH(U;U).

I f d ( f ( U ) , % U ) > o show t h a t f h a s a

unique f i x e d p o i n t . 4.55 -

If f

-

: c

C i s d e f i n e d by

1"

m

f((xn}n=l ) =

show t h a t f

n=2

f =

I f E i s a B a n a c h s p a c e , (bne

1" 0:

n=1

E

H(co)

and

-1

m

4.56

E

E l

all n,

and

H(E) show t h a t r f ( x ) i s a c o n s t a n t and

find t h i s constant

4.57*

L e t E a n d F b e Banach s p a c e s a n d l e t f e H ( E ; F ) .

that A =

EF';

f

( 0 ) )

i s a s e t of f i r s t category

I f E i s s e p a r a b l e show t h a t t h e r e e x i s t s g

i n E.

such t h a t rf = r

4.58

r + o f( 0 ) > r

Show

. g

E

H(E)

G i v e an e x a m p l e o f a n i n f i n i t e d i m e n s i o n a l Banach

s p a c e E and an f i n H(E) many c o o r d i n a t e s b u t r variables. 4.59* -

f

such t h a t f "depends"

I f T i s an i n f i n i t e d i s c r e t e s e t and f

i s such t h a t r

show t h a t

$

on i n f i n i t e l y

" d e p e n d s " o n l y on f i n i t e l y many

Ir,(x)

-

rf(y)

E

I

H(c

0

(T))

< 1lx-y

11

f o r a l l x, y i n co(T). 4.60

L e t E b e a Banach s p a c e .

f EH(E) Frgchet

f o r which r f topology

T

+

0~

Show t h a t t h e s e t o f a l l

can be g i v e n a unique

which i s f i n e r t h a n t h e compact open

207

Holomorphic functions on Banach spaces topology

.

Show t h a t

((fEH(E); rf

E

- 1 , ~ ) is a locally m

+

convex

Fr6chet algebra 4.61 -

I f e a c h c o m p a c t s u b s e t o f a Banach s p a c e E l i e s i n a

s e p a r a b l e c o m p l e m e n t e d s u b s p a c e show t h a t t h e c l o s e d

bounding s u b s e t s o f E a r e compact.Using t h i s r e s u l t g i v e an e x a m p l e o f a Banach s p a c e whose c l o s e d b o u n d i n g s e t s a r e a l l compact b u t which i s n o t a weakly c o m p a c t l y g e n e r a t e d Banach s p a c e .

4.62

By u s i n g b o u n d i n g s e t s show t h a t l m d o e s n o t c o n t a i n

any i n f i n i t e d i m e n s i o n a l s e p a r a b l e complemented s u b s p a c e s . 4.63* -

Let f

E

H

G

(U;F)

where U i s an open s u b s e t o f a l o c a l l y

c o n v e x s p a c e E a n d F i s a Banach s p a c e whose d u a l b a l l i s

weak* s e q u e n t i a l l y c o m p a c t .

Show t h a t f

E

HHY(U;F) i f

gof EH(U) f o r e v e r y g i n H(F). 4.64*

gl

I f f E H ( c o ) show t h a t t h e r e e x i s t s g e H(1,) i f and o n l y i f r = + m . f -

CO

4.65

L e t E b e a Banach s p a c e a n d l e t td,);=,

n u l l sequence i n E ' .

Let

m

b e a weak*

(kn)n=l be a s t r i c t l y i n c r e a s i n g

s e q u e n c e o f p o s i t i v e i n t e g e r s and f o r e a c h n l e t j negative i n t e g e r with o 5 j n 2 kn. jn

f

such t h a t

= f

=

kn-jn

1" 6, 4, n=l

be a non-

Show t h a t

H(E) i f a n d o n l y i f l i m i n f nm

E

n

kn-jn

kn

is positive. 4.66

If E =

1"

n=1

E

n

w h e r e e a c h E n i s a Banach s p a c e w i t h

a n u n c o n d i t i o n a l b a s i s show t h a t 4.67 E,

T

w

= T~

on H(E).

I f K i s a c o m p a c t b a l a n c e d s u b s e t o f a Banach s p a c e

F i s a Banach s p a c e a n d B i s t h e u n i t b a l l o f E show t h a t

t h e t o p o l o g y o f H(K;F)

i s g e n e r a t e d b y t h e semi-norms

208

Chapter 4

m

where

(an)n=o ranges over co

=*

If f

:

show t h a t f

E

z2

-

C i s d e f i n e d by f ( { x

Z2,

p(L(Z2)).

}m

n n=l

)=I"

2

xn,

n=l

s i n c e it i s s e p a r a b l e , can be

i d e n t i f i e d with a closed subspace of &[o,l]

(say n ( 2 ) ) . 2

Show t h a t t h e r e e x i s t s n o h o l o m o r p h i c f u n c t i o n o n , g [ o . 11 w h o s e r e s t r i c t i o n t o ~ ( 2 ~i s) e q u a l t o f . /

4.69"

Let

fi

b e a c o n t i n u o u s s u r j e c t i o n from 2,

onto co.

Show t h a t t h e i d e n t i t y m a p p i n g from c o t o c o c a n n o t b e l i f t e d t o Z1 t o Zl

c

i.e.

show t h a t no h o l o m o r p h i c m a p p i n g , %

e x i s t s s u c h t h a t n o f = I d on c

.

%

from

f,

I f U i s a b a l a n c e d open s u b s e t o f a Banach s p a c e

4.70*

show t h a t e v e r y n u l l s e q u e n c e i n ( H ( U ) , T ) sequence where T =

T

0'

T

w

i s a Mackey n u l l

or T ~ .

Let U b e an o p e n s u b s e t o f a Banach s p a c e E and l e t

4.71* -

F b e a Banach s p a c e .

Let

T~

of

b e t h e t o p o l o g y on H ( U ; F )

uniform convergence o f f u n c t i o n s and t h e i r f i r s t n d e r i v a t i v e s on t h e compact s u b s e t s o f U where n = o , l ,

. . . ,m .

I f E i s i n f i n i t e d i m e n s i o n a l show t h a t

T

~

, n

=o,l,

m , W

Show t h a t

H(U;F). 4.72*

...,

Let

a l l d e f i n e t h e same bounded s u b s e t s o f ( H ( U ; F ) , T ~ )i s c o m p l e t e f o r n = o , l ,

( P a ) @ b e a f a m i l y o f s c a l a r v a l u e d homogeneous

p o l y n o m i a l s on t h e Banach s p a c e E , degree n

.

...,-.

If

sup

I Pa(~)ll/n.

<

m

p a b e i n g homogeneous o f

f o r e v e r y x i n E show t h a t

t h e r e e x i s t s a neighbourhood V o f zero i n E such t h a t

209

Holomorphic functions on Banach spaces

*

4.73

I f E i s a Banach s p a c e show t h a t t h e f o l l o w i n g

conditions are equivalent a) E has t h e approximation property, b ) ( H ( E ) , T ~ )h a s t h e a p p r o x i m a t i o n p r o p e r t y c ) IH(E) @ E i s d e n s e i n ( H ( E ; E ) , T ~ ) .

F o r e a c h p o s i t i v e i n t e g e r n show t h a t

4.74

and hence deduce t h a t

(6' ( n l l ) , @ ) z 2,

( H ( l l ) , ~ u )has t h e approximation

property.

4.75*

I f E i s a s e p a r a b l e m e t r i z a b l e l o c a l l y convex space

show t h a t

(H(U),.ru) i s q u a s i - c o m p l e t e f o r any open s u b s e t

U of E.

4.76"

x =

L e t E b e a Banach s p a c e w i t h a monotone b a s i s

1"

n=l

x e n n

E

E.

Let B b e t h e open u n i t b a l l o f E

If

e v e r y automorphism o f B h a s t h e form

where

IT

i s a permutation of t h e positive integers,

for all n and

SLIP

i s m o r p h i c t o .c 4.77* -

Let

r

lanl

<

1 , show t h a t E i s i s o m e t r i c a l l y

.

b e an u n c o u n t a b l e d i s c r e t e s e t .

H ( ( c o ( r ) ) , T ~ )= ( H ( C ~ , ~, (T ~~ ) ). s u b s e t s o f H ( c o ( r ) ) a n d t h e -c0 H ( c ~ , ~ ( ~a r)e ) l o c a l l y b o u n d e d . shows t h a t t h e l o c a l l y bounded

=

T~

Show t h a t

Show t h a t t h e r o - b o u n d e d

bounded s e q u e n c e s i n Give an example which

bounded s u b s e t s o f H(c

0,P

(r))

are not

1

Chapter 4

210

4.78*

L e t E b e a s e p a r a b l e Banach s p a c e w i t h o p e n u n i t

b a l l B and l e t D b e t h e open u n i t d i s c i n C . t h e r e e x i s t s an f i n H ( D ; B ) 4.79

Show t h a t

-

such t h a t f(D) 3 8 .

L e t E b e a Banach s p a c e a n d s u p p o s e t h e r e e x i s t s a

sequence i n

I

m

(4n)n=1,such t h a t

E l ,

I1xII = S X P ~ + ~ ( Xf o) r

By u s i n g e x e r c i s e 4 . 7 8 show t h a t E c a n b e

every x i n E .

embedded i s o m e t r i c a l l y i n H m ( D ) , D = E Z E C ; I z \ < l ) . 4.80* -

m

Let B be t h e open u n i t b a l l o f co and l e t

b e a s e q u e n c e i n CC s u c h t h a t

( a n ( < 1 and

Im ( l - ( a n l )<

n=l

m.

Show t h a t t h e r e e x i s t s an f i n A ( B ) s u c h t h a t

-

f l B E H ( B ) a n d { z c B ; f ( z ) = o ) = {(B,),EB;

B m = a m f o r some m}.

A s u b s e t A o f a l o c a l l y convex s p a c e E i s c a l l e d

4.81*

a (holomorphic o r a n a l y t i c ) f cH(E)

determining s e t i f

and f l A = o i m p l i e s f E

0 .

Show t h a t E c o n t a i n s a

compact d e t e r m i n i n g s e t i f and o n l y i f

( E ' , T ~ )admits a

Show t h a t a m e t r i z a b l e l o c a l l y c o n v e x

c o n t i n u o u s norm.

s p a c e c o n t a i n s a compact d e t e r m i n i n g s e t i f and o n l y i f i t is separable. 4.82

I _

L e t E and F b e Banach s p a c e s and l e t K b e a compact

-

d e t e r m i n i n g s e t f o r h o l o m o r p h i c f u n c t i o n s on E .

f : K

F i s a mapping s u c h t h a t f o r each $ i n F '

e x i s t an open neighbourhood V that

+

o f K and an f

+

E

+

there

H(V ) s u c h

= + o f show t h a t t h e r e e x i s t s an o p e n n e i g h b o u r h o o d

'1K V o f K and a n

14.6

If

2

E

H(V)

%

such t h a t f l K = f.

NOTES A N D R E M A R K S The i m p o r t a n c e o f c o m p l e x e x t r e m e p o i n t s

( d e f i n i t i o n 4.1)

i n t h e t h e o r y o f v e c t o r v a l u e d maximum m o d u l u s t h e o r e m s w a s f i r s t n o t e d by E . prove theorem 4 . 4 .

Thorp and R .

I " h i t 1 e y i n [685] w h e r e t h e y

They a l s o g i v e e x a m p l e s

-

e.g L(o,l)

-

Of

21 1

Holomoiphic functions on Banach spaces s p a c e s which a r e s t r i c t l y c-convex b u t n o t r o t u n d .

The

proof o f theorem 4 . 4 given h e r e i s due t o L . Harris

[305].

Many o f t h e o t h e r known i n f i n i t e d i m e n s i o n a l maximum modulus theorems a l s o i n v o l v e complex extreme p o i n t s .

L.

H a r r i s [304] shows t h a t t h e complex e x t r e m e p o i n t s o f a

convex b a l a n c e d compact s e t K form a boundary f o r t h e a l g e b r a o f a l l c o n t i n u o u s complex v a l u e d f u n c t i o n s on K which a r e "holomorphic" on t h e a n a l y t i c d i s c s i n K .

This r e s u l t

may b e r e g a r d e d a s a h o l o m o r p h i c K r e i n - M i l m a n t h e o r e m . J.

Globevnik [257] proves t h e r e s u l t quoted a f t e r theorem 4.4

and a p p l i e s it i n s t u d y i n g t h e r e s o l v e n t f u n c t i o n o f an operator.

Further p e r t i n e n t r e s u l t s a r e t o be found i n For i n s t a n c e J . Globevnik

[32,258,260,276,631,693,694].

c h a r a c t e r i s e s h o l o m o r p h i c f u n c t i o n s o f c o n s t a n t norm i n [260],

J . Globevnik and 1.Vidav d i s c u s s o p e r a t o r v a l u e d

holomorphic f u n c t i o n s i n [276] and g e n e r a l i s e e a r l i e r r e s u l t s

of A.

Brown a n d R . G .

Douglas

[lo61 and i n [258] 3. Globevnik

p r o v e s a maximum m o d u l u s t h e o r e m f o r B a n a c h a l g e b r a s i n w h i c h t h e s u p norm i s r e p l a c e d b y t h e s p e c t r a l r a d i u s . Schwarz's v a r i a b l e s by C .

l e m m a was e x t e n d e d t o f u n c t i o n s o f t w o C a r a t h e o d o r y [lll],

dimensional algebras by C . L .

t o some f i n i t e

S i e g e 1 [658] and t o c e r t a i n

i n f i n i t e d i m e n s i o n a l o p e r a t o r a l g e b r a s by R . S. [568]. t o L. H.

Phillips

i s due

The g e n e r a l r e s u l t g i v e n h e r e , t h e o r e m 4 . 3 , Harris

[304,305].

An a l t e r n a t i v e p r o o f u s i n g

Cartan's uniqueness r e s u l t

[113]

(see I .

Shimoda [656]

f o r a normed l i n e a r s p a c e v e r s i o n o f t h e u n i q u e n e s s theorem) papers

i s giv'en i n L .

[306,307,308]

Harris

In a s e r i e s o f

L . H a r r i s p r o v e s a number o f o t h e r

g e n e r a l i z a t i o n s o f Schwarz's t o J*algebras,

[304,313].

l e m m a and g i v e s a p p l i c a t i o n s

Mzbius t r a n s f o r m a t i o n s ,

t h e numerical

range and t h e c h a r a c t e r i s a t i o n o f extreme p o i n t s , [327, c o r o l l a r y 6.81, A.

Renaud

S.J.

[604] and J . P .

G r e e n f i e l d and N . R .

M.Herv6

Wallach

[282],

Vigug [696, p r o p o s i t i o n 1 . 2 . 1 1

a l s o obtain i n f i n i t e dimensional generalizations o f t h i s

lemma.

212

Chapter 4 A.E.

Taylor [678, p.474-4751

was t h e f i r s t t o n o t e

t h a t e n t i r e f u n c t i o n s o n c e r t a i n B a n a c h s p a c e s may h a v e f i n i t e r a d i i of uniform convergence.

Subsequently L.

Nachbin

[SO91

g a v e t h e d e f i n i t i o n o f r a d i u s of b o u n d e d n e s s a n d p r o v e d Around t h e same t i m e H .

proposition 4.7.

Alexander

[5]

w r o t e h i s d i s s e r t a t i o n on a n a l y t i c c o n t i n u a t i o n i n Banach spaces.

H i s i n v e s t i g a t i o n s l e d t o t h e concept of bounding

s e t ( h e w i s h e d t o know i f

( H ( E ) , T ~ ) was b a r r e l l e d )

a n d showed

t h a t t h e bounding s u b s e t s o f a H i l b e r t s p a c e were precompact. T h i s r e s u l t was p r o v e d i n d e p e n d e n t l y b y S . who was n o t i v a t e d b y L .

Dineen

[177]

Nachbin's proof of proposition 4.7.

'The n e x t d e v e l o p m e n t was t h e o r e m 4 . 2 7 w h i c h was p r o v e d i n d e p e n d e n t l y by S . Dineen [336,339].

[181] and A .

The p r o o f o f t h e o r e m 4 . 2 7

preliminary r e s u l t , proposition 4.26) t o M.

S c h o t t e n l o h e r [531,638]

t h e o r e m may b e f o u n d i n K .

Hirschowitz

(including the g i v e n h e r e i s due

a n d f u r t h e r comments on t h i s

Rusek

[616] and M.Bianchini

[64].

6

Complete l i n e a r c h a r a c t e r i z a t i o n s o f t h e Banach s p a c e s

i n which a l l bounding s e t s a r e r e l a t i v e l y compact and t h e

class

o f Banach s p a c e s w i t h weak* s e q u e n t i a l l y compact

d u a l b a l l s a r e s t i l l unknown. M.

By t h e o r e m 4 . 2 7

6c a.

S c h o t t e n l o h e r [642] n o t e s t h a t e v e r y we5kly compactly

generated

(WCG) s p a c e ( s e e J . D i e s t e l

[171])

lies in

and

t h a t & i s closed under t h e operation of taking closed subspaces.

S i n c e t h e r e e x i s t c l o s e d s u b s p a c e s o f WCG

s p a c e s w h i c h a r e n o t W C G we s e e t h a t W C G # and F .

J.

llagler

S u l l i v a n [ 3 0 1 ] p r o v e t h a t smooth Banach s p a c e s l i e

i n @I a n d

C.Stegal1

Asplund s p a c e . i n J.

&.

F u r t h e r r e s u l t s on t h e c l a s s

H a g l e r and W . B .

[300] and R .

[ 6 7 1 ] s h o w s t h a t 03 c o n t a i n s e v e r y w e a k

Haydon

Johnson

63

are given

[299], J . Hagler and E.Odel1

[320].

A s u b s e t A o f a Banach s p a c e E i s l i m i t e d i f e v e r y

weak* n u l l s e q u e n c e i n E '

t e n d s t o z e r o u n i f o r m l y on A .

L i m i t e d s e t s were i n t r o d u c e d b y J . D i e u d o n n 6 [ 1 7 3 ] lemma 4 . 2 s h o w s t h a t b o u n d i n g s e t s a r e l i m i t e d .

and

Limited

s e t s may b e r e g a r d e d a s t h e l i n e a r a n a l o g u e o f b o u n d i n g s e t s .

213

Holornorphic functions on Banach spaces We d o n o t know o f a n y l i m i t e d s e t w h i c h i s n o t b o u n d i n g . J.

Bourgain, J . D i e s t e l and D .

IVeintraub

[98] summarise

m o s t o f t h e known r e s u l t s o n l i m i t e d s e t s a n d p r o v e s o m e new r e s u l t s ,

They show,

s u b s e t s o f L 1 (X,F,v) e x i s t s a n L1(X,F,v) e x a m p l e s show t h a t

for instance, that the limited

a r e r e l a t i v e l y compact and s i n c e t h e r e which does n o t belong t o

a#

63.

Bayoumi

A.

this

[55] s t u d i e s

bounding s e t s i n complete separable metrizable topological vector spaces.

He proves f o r c e r t a i n non-locally

convex

s p a c e s t h a t t h e bounding s e t s are r e l a t i v e l y compact i f t h e open b a l l s d e f i n e d by t h e m e t r i c are p o l y n o m i a l l y convex and deduces t h a t t h e bounding s u b s e t s o f I p , o < p < l , a r e r e l a t i v e l y compact. The p r o o f o f t h e o r e m 4 . 2 7 m o t i v a t e d a number o f t h e

In t h i s

o t h e r r e s u l t s p r e s e n t e d i n s e c t i o n 4 . 1 and 4 . 2 . c a t e g o r y we may i n c l u d e lemma 4 . 5 , 4.18,

c o r o l l a r i e s 4.19.4.20

and c o r o l l a r i e s 4 . 2 3 and 4 . 2 4 . A.

Hirschowitz

corollary 4.8,

and 4 . 2 1 ,

lemma

proposition 4.22

T h e s e may b e f o u n d i n

[339] and S . Dineen

[181,184].

N e x t came t h e p r o o f o f t h e e x i s t e n c e o f a c l o s e d non-compact bounding s u b s e t o f Zm,

theorem 4.31,

and t h i s

i s due t o S . Dineen [178].

This motivated t h e deep

a n a l y s i s o f t h e geometry o f

Lm

undertaken by B.

i n [361] and l e d t o theorem 4 . 2 8 .

Josefson

Josefson only gives the

first f o u r conditions o f theorem 4.28(a) but we have included condition

( v ) which i s e q u i v a l e n t t o ( i i i ) by

t h e celebrated' theorem o f H . P .

Rosenthal

Tzafriri

[447]).

[611] ( s e e

J.

L i n d e n s t r a u s s and L .

Recently

R.

Haydon [ 3 2 0 , 3 2 1 ] o b t a i n e d e x a m p l e s o f B a n a c h s p a c e s

w h i c h d o n o t c o n t a i n lm b u t c o n t a i n c l o s e d non-compact bounding s e t s D.

( s e e a l s o J . Bourgain, J . D i e s t e l and

Weintraub [ 9 8 ] ) . With t h e o r e m 4 . 2 7 as m o t i v a t i o n B .

Josefson

[359]

proved p r o p o s i t i o n 4 . 9 ( a l s o due i n d e p e n d e n t l y and w i t h a

214

Chapter 4

different proof t o A.

Nissenweig [529]) and a p p l i e d it t o

o b t a i n c o r o l l a r y 4.10 and p r o p o s i t i o n 4.25.

In c o n s t r u c t i n g

a counterexample t o t h e Levi problem B . J o s e f s o n [358] showed t h e e x i s t e n c e o f a c l o s e d p r e c o m p a c t non-compact b o u n d i n g s u b s e t o f a p o l y n o m i a l l y convex domain i n c o ( r ) , r uncountable. P r o p o s i t i o n s 4 . 1 1 and 4.12 a r e due t o R. and P . C.O.

Aron

[20]

Lelong [425] r e s p e c t i v e l y and t h e s e m o t i v a t e d

Kiselman [385,386,387]

t o undertake a detailed

i n v e s t i g a t i o n o f t h e r a d i u s of convergence.

Kiselman

f i r s t posed t h e problem of p r e s c r i b i n g t h e r a d i u s of uniform c o n v e r g e n c e a n d p r o v e d t h e o r e m 4 . 1 3 when E=co o r

l s <

P’ and t h e r a d i u s o f c o n v e r g e n c e depended o n l y on a f i n i t e

number o f v a r i a b l e s .

m,

Subsequently t h e f i n i t e n e s s

r e q u i r e m e n t was r e m o v e d a n d t h e o r e m 4 . 1 3 p r o v e d b y G . C o e u r 6 A more a c c e s s i b l e p r o o f o f t h i s t h e o r e m h a s b e e n

[136].

given by M .

Schottenloher i n [641].

N.

Cherfaoui [122]

s t u d i e s t h e r a d i u s of boundedness of plurisubharmonic f u n c t i o n s on O r l i c z s p a c e s . are due t o C . O .

Example 4 . 1 4 and p r o p o s i t i o n 4 . 1 6

Kiselman [385].

Many f u r t h e r e x a m p l e s

a n d r e s u l t s o n t h e r a d i u s o f b o u n d e d n e s s Lor t h e r a d i u s o f uniform convergence) are contained i n t h e paper of Kiselman quoted above.

M. S c h o t t e n l o h e r [ 6 4 2 ] h a s w r i t t e n a v e r y r e a d a b l e s u r v e y a r t i c l e on b o u n d i n g s e t s and t h e r a d i u s of boundedness. If

T~

=

T&

o n H(U)

and U i s a holomorphically

convex open s u b s e t of a l o c a l l y convex s p a c e t h e n U i s a domain o f holomorphy.

This provided the motivation f o r

theorem 4.38 proved i n S . Dineen [180].

G. Coeur6 [130] 1 g r o v e d t h e same r e s u l t f o r s u b s p a c e s o f L [ o , 2 1 ~ 1 w h i c h a r e

homogeneous i n t h e s e n s e o f S h i l o v . An i n i t i a l d i f f i c u l t y i s d e a l i n g w i t h t h e

T~

topology

i s t h a t a g e n e r a t i n g s e t o f semi-norms i s n o t p r e s c r i b e d . T h i s d i f f i c u l t y i s overcome by p r o p o s i t i o n 4.39

215

Holomoiphic functions on Banach spaces w h i c h was f i r s t p r o v e d f o r e n t i r e f u n c t i o n s b y S . D i n e e n [177] and a f t e r w a r d s extended t o holomorphic f u n c t i o n s

on b a l a n c e d o p e n s e t s b y R .

Aron [ 1 7 ] .

Propositions 4.40

and 4 . 4 1 a r e due t o S . Dineen [185] and show, i n a s i m p l e fashion,

t y p i c a l phenomena which a r i s e i n t h e s t u d y o f

h o l o m o r p h i c f u n c t i o n s on n o n - m e t r i z a b l e l o c a l l y c o n v e x spaces.

E x a m p l e 4 . 4 2 may b e f o u n d i n S . D i n e e n [ 1 8 4 ] a n d

a g e n e r a l i n v e s t i g a t i o n o f Hahn-Banach t y p e e x t e n s i o n

t h e o r e m s f o r h o l o m o r p h i c f u n c t i o n s on Banach s p a c e s i s undertaken i n R.

Aron a n d P .

Berner [26] and R.

Aron [ 2 5 ] .

They show i n [ 2 6 ] t h a t f & H ( c O ) h a s a h o l o m o r p h i c e x t e n s i o n

t o Zm i f and o n l y i f rf

=

Hahn-Banach e x t e n s i o n

+m.

t h e o r e m s f o r h o l o m o r p h i c f u n c t i o n s on f . u l l y n u c l e a r s p a c e s a r e discussed i n 15.4. The c o n c e p t o f h o l o m o r p h i c c o m p l e t i o n i s d u e t o Proposition 4.43 i s due t o

A.

Hirschowitz

[339].

A.

Hirschowitz

[339] f o r normed l i n e a r s p a c e s and t o

S.

Dineen [184]

f o r metrizable l o c a l l y convex spaces.

Proposition 4.44

is given i n

[184].

F u r t h e r r e s u l t s on

h o l o m o r p h i c c o m p l e t i o n may b e f o u n d i n A . H i r s c h o w i t z [337,339,343], R.R.

G.

Baldino [43],

Ph. N o v e r r a z

CoeurB [ 1 3 5 ] , Ph. Noverraz

[206] and M .

S.

Dineen [ 1 9 0 ] ,

[542,547],

S . Dineen and

Schottenloher [633,642,645].

For

example G . Coeur6 [135] shows t h a t a h o l o m o r p h i c a l l y complete p r o p e r s u b s p a c e o f a Banach s p a c e i s a p o l a r s e t M.

S c h o t t e n l o h e r [633] shows t h a t

(ExF)"

= E m x FG

f o r any

m e t r i z a b l e l o c a l l y convex s p a c e s E and F and S . Dineen and Ph. N o v e r r a z [206] show, u s i n g G a u s s i a n m e a s u r e s ,

that

" a l m o s t a l l " a l g e b r a i c h y p e r p l a n e s i n a Banach s p a c e a r e not holomorphically complete.

This Page Intentionally Left Blank

Chapter 5 HOLOMORPHIC FUNCTIONS ON NUCLEAR SPACES WITH A BASIS

We h a v e a l r e a d y s e e n t h a t n u c l e a r h o l o m o r p h i c f u n c t i o n s a n d n u c l e a r s p a c e s e n t e r i n t o t h e g e n e r a l t h e o r y o f i n f i n i t e dimens i o n a l holomorphy i n a n a t u r a l way.

For i n s t a n c e ,

i n 53.3,

we

saw t h a t t h e d u a l o f t h e s p a c e o f h o l o m o r p h i c f u n c t i o n s o n a Frgchet space

E

with t h e approximation property,

endowed w i t h

t h e c o m p a c t o p e n t o p o l o g y , may b e i d e n t i f i e d v i a t h e B o r e 1 t r a n s f o r m w i t h t h e s p a c e o f n u c l e a r germs a t t h e o r i g i n i n Moreover,

E

if

i s nuclear,

isomorphism between

(H(E)

we o b t a i n a l i n e a r t o p o l o g i c a l

, T ~ )

and

E' B'

H(OEl). B

I n t h i s c h a p t e r , w e i n v e s t i g a t e holomorphic f u n c t i o n s on f u l l y nuclear spaces basis.

( d e f i n i t i o n 1.49) which have a Schauder

T h e b a s i s p r o v i d e s u s w i t h a coordinate s y s t e m a n d t h u s

i t i s n o t s u r p r i s i n g t h a t we u s e r e f i n e m e n t s o f a n u m b e r o f t h e

t e c h n i q u e s d e v e l o p e d i n t h e l a s t two c h a p t e r s .

The f i n a l

r e s u l t s we o b t a i n c o n c e r n i n g t h e d i f f e r e n t t o p o l o g i e s d o n o t i n t r i n s i c a l l y depend on t h e e x i s t e n c e o f a b a s i s and h a v e i n s p i r e d a number o f a n a l o g o u s r e s u l t s o n s p a c e s n o t n e c e s s a r i l y having a basis, section 5.4,

w h i c h we p r e s e n t e d i n c h a p t e r s 1 a n d 3 .

c e r t a i n cases.

The e x i s t e n c e o f a b a s i s a l s o a l l o w s u s t o

d e f i n e i n f i n i t e d i m e n s i o n a l a n a l o g u e s o f Reinhardt domains

poZydiscs.

In

w e a l s o s h o w how t o r e m o v e t h e b a s i s h y p o t h e s i s i n

and

When we d o n o t h a v e a b a s i s , we a r e o b l i g e d t o

confine ourselves t o entire functions. The f i r s t s e c t i o n o f

t h i s chapter i s devoted t o

l i n e a r and

geometric properties of c e r t a i n c l a s s e s of nuclear spaces with a basis.

We o n l y p r e s e n t r e s u l t s n o t a l r e a d y c o n t a i n e d i n t h e

s t a n d a r d t e x t s on n u c l e a r spaces.

217

I n 5 5 . 2 w e show t h a t

218

Chapter 5

(H(U),-ro)

i s an open poly-

U

h a s a n a b s o l u t e b a s i s whenever

disc i n a f u l l y nuclear space with a basis.

T h i s b a s i s problem

a n d t h e f o r m o f i t s s o l u t i o n m o t i v a t e d much o f t h e r e s e a r c h reported i n t h i s chapter. S o l u t i o n s t o similar b a s i s problems l e d t o a c h a r a c t e r ization of

set of

(H(U),ro)'

as a s p a c e o f germs on a compact sub-

This duality theory clarifies the relationship

Ei.

between t n e d i f f e r e n t t o p o l o g i e s on

H(U)

and p l a c e s i n p e r -

s p e c t i v e a number o f t h e c o u n t e r e x a m p l e s from p r e v i o u s c h a p t e r s

I n 9 5 . 3 , we s h o w t h a t a nuclear

DN

chapter 6 . t h o s e of

=

T~

T

space with a basis.

6

on

H(E)

A converse

whenever

E

is

i s given i n

The methods u s e d h e r e w e r e p a r t i a l l y m o t i v a t e d b y

94.3

w h e r e we d i s c u s s e d h o l o m o r p h i c f u n c t i o n s o n

Banach s p a c e s w i t h a n u n c o n d i t i o n a l b a s i s .

In 55.4,

we d i s c u s s

holomorphic f u n c t i o n s on c e r t a i n n u c l e a r s p a c e s which do n o t n e c e s s a r i l y have a b a s i s and d e s c r i b e a r e l a t i o n s h i p between e x t e n s i o n theorems f o r holomorphic f u n c t i o n s and p r o p e r t i e s o f t h e compact open t o p o l o g y . The r e s u l t s o f t h i s c h a p t e r a r e proved i n d e p e n d e n t l y o f o u r p r e v i o u s r e s u l t s on n u c l e a r holomorphic f u n c t i o n s and h o l o morphic f u n c t i o n s on n u c l e a r s p a c e s .

55.1

NUCLEAR S P A C E S WITH A B A S I S We f i r s t d i s c u s s a b s o l u t e b a s e s i n l o c a l l y c o n v e x s p a c e s .

This i s a p a r t i c u l a r c a s e of a b s o l u t e Schauder decompositions previously discussed i n §3.1

-

one t a k e s each o f t h e sub-

spaces t o b e one dimensional

-

and a number o f t h e t e c h n i q u e s

used t h e r e are a l s o a p p l i e d h e r e .

Let space

E.

p E cs(E)

m

(en)n=l

b e a S c h a u d e r b a s i s i n t h e LocaZZy c o n v e z

T h e b a s i s i s s a i d t o b e a b s o l u t e i f for a n y

there exists

q

E

cs(E)

such t h a t

219

Holomorphic functions on nuclear spaces with a basis

p

E

This is obviously equivalent to saying that for any cs(E) the mapping

defines a continuous semi-norm on

E

We now give two elementary but useful results concerning the uniqueness o f topologies on locally convex spaces with an absolute basis. Let

Lemma 5 . 1

and

T~

T~

o g i e s on t h e v e c t o r space

b e two ZocaZZy c o n v e x t o p o Z and suppose

E

absoZute b a s i s f o r both topoZogies.

if

i s an

(en);,l

Then

T~

if a n d onZy

= T~

( E , T ~ )= ~ ( E , . r 2 ) l .

Proof

If

=

T~

then, trivially,

T~

Suppose conversely that continupus semi-norm on generality, that

Let

$:

Then

+

E

E

-+

C

(E,T~)!

( E , T ~ ) ! = (E,.r2)!.

E.

(E,r2)!.

=

Let

p

be a

T~

We may assume, without l o s s o f

be given by

( E , T ~ ) !=

( E , T ~ ) !

continuous semi-norm every x in E .

q

x

=

Hence for all

on

m

and hence there exists a E

such that

xnen

in

E

I$(x)l

we have

d

T1

q(x)

for

220

Chapter 5 ?d

By the definition of absolute basis q is a T~ continuous semi-norm on E. Hence p is T~ continuous and T~ T ~ . Interchanging T~ and T~ we find that This completes the proof. T1 - T 2 .

T~

and hence

T~

Our next result is similar to proposition 3.12 Let T~ and T~ Lemma 5.2 topoZogies o n t h e v e c t o r s p a c e

basis for both

( E , T ~ ) and

b e two l o c a l l y c o n v e x b a r r e l l e d

E.

If

(en)i=lis

( E , T ~ ) then

T~

=

T

~

Proof Let p be a T~ continuous semi-norm on may suppose, without loss of generality, that

W

Since

,

absolute

CIY! .

E.

We

is an absolute and hence a Schauder basis f o r

( E , T ~ ) the semi-norms

are

T~

continuous for each positive integer

m.

Let

V is a T~ closed convex balanced absorbing subset of E and hence, since ( E , T ~ ) is barrelled, a T~ neighbourhood of continuous semi-norm and T 2 < Ti. zero. Thus p is a T~ and T~ in the above, we see that T~ 6 T * Interchanging T~ and hence T~ = T ~ . This completes the proof.

22 1

Holornotphic functions on nuclear spaces with a basis L o c a l l y convex spaces w i t h an a b s o l u t e b a s i s can b e ident i f i e d with dense subspaces of sequence spaces. Definition 5.3

Let

b e a c o l l e c t i o n o f s e q u e n c e s o f non-

P

r>O

n e g a t i v e r e a l numbers s u c h t h a t f o r e a c h p o s i t i v e i n t e g e r there exists space

m

in

P.

P

E

with

ar

The sequence

> 0.

i s t h e s e t of all s e q u e n c e s o f c o m p l e x n u m b e r s ,

A(P)

(Xn) n = 1'

m

(an)n=l

=

01

m

ln=l/xnlan <

s u c h that We endow

pa,

semi-norms

w i t h t h e t o p o l o g y g e n e r a t e d by t h e

A(P) Q

The e l e m e n t s o f

=

m

a = (an)n=l

f o r every

m

m

("n)n=l

P

where

P,

E

are called weights.

If

m

is

(4n)n=1

any s e q u e n c e o f n o n - n e g a t i v e numbers and

i s a neighbourhood of zero then t h e sequence c a l l e d a c o n t i n u o u s weight on

Hausdorff

The c o n d i t i o n

P.

Q

l o c a l l y convex space.

forms an a b s o l u t e b a s i s f o r

Now l e t (en);=1

9

p

=

l i n e a r mapping from in

E

is a

A(P)

i s a complete l o c a l l y

m

)

n(P).

+

We r e f e r t o t h i s b a s i s a s

h(P).

b e a l o c a l l y convex space w i t h a b s o l u t e b a s i s

E

Let

A(P)

( u ~ ) ~ =u ~ =, ( 0 , . . . 0 , 1 , 0 , . . . ) nth position

convex s p a c e and t h e s e q u e n c e

the unit vector basis of

(Bn)n=l

a d m i t s a c o n t i n u o u s norm

i f and o n l y i f t h e r e e x i s t s a c o n t i n u o u s w e i g h t on

A(P)

m

that

ensures t h a t

> 0

A(P)

c o n s i s t i n g o f p o s i t i v e numbers.

is

We may t h e n a s s u m e , w h e n -

A(P).

e v e r n e c e s s a r y and w i t h o u t l o s s o f g e n e r a l i t y , belongs t o

m

(Bn)n=l

{ (P(en)

E

m

I n = 1 3 pEcs ( E ) .

into

A(P)

There is a natural

which t a k e s t h e b a s i s

onto the u n i t vector basis of

(en =: 1 mapping i s g i v e n by

A(P).

This

Chapter 5

222

and

E

is then linearly isomorphic with its image in

E

is isomorphic to h(P) locally convex space.

A(P).

if and only if it is a complete

When we identify a locally convex space containing an absolute basis with a subspace of a sequence space, we shall always assume that the above identification is used. Nuclear sequence spaces have a particularly nice and practical characterization as the following fundamental result shows.

This is the Grothendieck-Pietsch criterion.

Proposition 5 . 4 The s e q u e n c e s p a c e A(P) i s n u c Z e a r if m m and onZy i f f o r e a c h E P there e x i s t ( u ~ ) ~E = II'~ m

and

E

such t h a t

P

an 6 u

0.1

n n

f o r all

This criterion can be rephrased as follows: n u c Z e a r if and onZy i f f o r e a c h s e q u e n c e of n o n - n e g a t i v e

I:=,

n

reaZ numbers, m

and

<

m

(6nan)n=1

E

1

n. A(P)

i s

there exists a

P m

(6n)n,l,

such t h a t

i s a c o n t i n u o u s w e i g h t on

A(P).

Since a locally convex space is nuclear if and only if its completion i s nuclear, this criterion can also be applied to locally convex spaces with an absolute basis.

We obtain the

following: i f is an absolute basis for the locally convex space E then E is nuclear if and only if for each q 3 p, such that p E cs E) there exists q E cs(E),

U S ng proposition 5 . 4 it is possible to obtain a further representation of nuclear sequence spaces. If A(P) is nuclear, then

A(p)

=

m

{(XnInZl;

SUP

n

223

Holomorphic functions on nuclear spaces with a basis

Furthermore,

the topology of

a l l semi-norms

o f t h e form

where

m

a =

ranges over

i s a l s o g e n e r a t e d by

A(P)

The neighbourhood s y s t e m s

P.

generated by t h e d i f f e r e n t systems o f semi-norms d e s c r i b e d above have d i f f e r e n t geometric p r o p e r t i e s - one being o f t h e Q1

t y p e and t h e o t h e r being o f t h e

system has i t s advantages.

-

type

Q~

and each

Since they are equivalent systems

we may u s e w h i c h e v e r i s m o r e s u i t a b l e . D e f i n i t i o n 5.5 IJ

B,

B

T h e m o d u l a r h u l l of a s u b s e t

A(P)

of

,

i s defined as

The modular h u l l i s a l s o c a l l e d t h e s o l i d h u l l . Lemma 5 . 6 A(P)

I f

E v e r y bounded s u b s e t

of

B

t h e nuclear space

E

i s a n u c l e a r s p a c e w i t h an a b s o l u t e b a s i s , h

E

bounded s u b s e t of

Proof i

i

Let

B = (x ) .

m

f o r each

x i - (Xn)n=l

E)

( t h e c o m p l e t i o n of

t h e c l o s u r e of a bounded s u b s e t of

1 E

I

i.

then every

i s contained i n

E.

b e a bounded s u b s e t o f A(P), i Let xn = s y p l x n I f o r each

T o c o m p l e t e t h e p r o o f o f t h e f i r s t h a l f o f t h e lemma,

i c e s t o show

m

( x ~ ) E~ A(P1. = ~

Let

m

(an)n=l

Grothendieck-Pietsch c r i t e r i o n , there e x i s t m

(aA)n=1 bounded and

A(P).

i s c o n t a i n e d i n t h e m o d u l a r h u l l of a n e l e m e n t of

E

P.

E

where n.

it s u f f -

By t h e m

( u ~ ) ~ = ~ and E II;

such t h a t a n < u a' for all n. n n i s u p suplaAxnl = M < m. Hence Ia;x,I i n

Since

P

$

M

B

is

for all

n

m

Thus

(Xn)n=l

point

and

A(P)

E

B

l i e s i n t h e modular h u l l o f t h e

(Xn);=l.

Now s u p p o s e m

,(en)n=l.

i s a nuclear space with absolute basis

E

Identify If

E S A(P).

with a dense subspace of

E

i s a bounded s u b s e t of

B

Then

A(P).

then

h(P),

c o n t a i n e d i n t h e modular h u l l of

( x ~ ) T =E ~A ( P ) .

i s o b v i o u s l y a bounded s u b s e t o f

E

is

B

The s e t

h

and i t s c l o s u r e i n

E

can b e i d e n t i f i e d w i t h t h e modular h u l l of an element of

A(P).

This completes t h e proof. Corollary 5.7

basis,

then

i s a n u c l e a r space w i t h an a b s o l u t e

A

(E);.

=

E

is a n u c l e a r s p a c e w i t h a n a b s o l u t e A is a n i n f r a b a r r e l l e d s p a c e if and o n l y if E

Corollary 5.8

basis, then

E

If

EA

E

I f

is b a r r e Z l e d . We now l o o k a t t h e s t r o n g d u a l o f a n u c l e a r s p a c e w i t h a n

We m a y ,

absolute basis. E

i f we w i s h

by t h e above,

assume t h a t

i s complete.

Proposition 5.9

ute basis

m

o v e r , if (e;)n=l Proof

If m

ordinate then m

Let m

(en)n=l. (e;)n,l

(E,.r) e;

be a n u c l e a r space w i t h a b s o l -

is e v a l u a t i o n a t t h e

is a n a b s o l u t e b a s i s f o r

is a l s o a S c h a u d e r b a s i s f o r

i s a n a b s o l u t e b a s i s for Let

T

nth E;I.

coMore-

(E,.rb)

(E,T~);I.

b e a l i n e a r f u n c t i o n a l on

E

which i s

then

Holornorphic functions on nuclear spaces with a basis bounded on t h e

bounded s u b s e t s o f

b e a bounded s u b s e t of

B

r.

in

[B]

=

E,

R;

en

x n en '. J C N f i n i t e ,

i s a bounded s u b s e t of

M = sup

m

x* =

i o n B,

E.

by t h e Grothendieck-Pietsch E

B = (x B )

and l e t

en

x:

BEr

f o r each

'

We c l a i m t h a t

{I n d e

(u,,):=~

where

E

225

such t h a t

q ( x Bn e n ) .

If

p

E

E

[O,ZII],

p(en) 6

E

r

all

n}.

then w e can choose,

cs(E)

q

criterion,

B,

E

unq(en)

cs(E)

for all

and

n.

Let

Then

Rsr,nEN

Hence

Hence i f basis for

T

[B]

is

i s a bounded s u b s e t o f

T

continuous o r

( E , T ~ ) then,

E

m

(en)n=l

and

-

B C [B].

i s a Schauder

f o r any p o s i t i v e i n t e g e r

m,

we h a v e

226 and

Chapter 5 m

i s a Schauder b a s i s f o r t h e dual space i n both

(e;)n=l

cases.

Moreover,

m

and hence

i s an absolute b a s i s .

(eA)n,l

This completes t h e

proof. On c o m b i n i n g lemma 5 . 6 a n d p r o p o s i t i o n 5 . 9 ,

we i m m e d i a t e l y

obtain the following r e s u l t . C o r o l l a r y 5.10

basis.

E C A(P)

IS

E

Let

h

be a n u c l e a r space w i t h an a b s o l u t e E' B

then

A(P')

where

C o r o l l a r y 5 . 1 0 a n d t h e G r o t h e n d i e c k - P i e t s c h c r i t e r i o n may b e u s e d t o d e c i d e when a g i v e n n u c l e a r s p a c e w i t h a n a b s o l u t e b a s i s i s a dual nuclear space.

We p r e f e r , h o w e v e r ,

t o r e l y on

t h e f o l l o w i n g more p r a c t i c a l c r i t e r i o n which c o v e r s most n o t a l l ) cases i n which

is nuclear,

E

(if

d u a l n u c l e a r and h a s

an a b s o l u t e b a s i s . Definition 5.11

A

l o c a l l y convex space

n u c l e a r s p a c e is i t h a s a n a b s o l u t e b a s i s e x i s t s a s e q u e n c e of p o s i t i v e r e a l numbers

1;=1

6 n

<

m

E

A-

i s an m

(en)n=l

and t h e r e

(Sn)ZZ1

such t h a t

and t h e s e m i - n o r m

is continuous w h e n e v e r

p

E

cs(E).

T h i s d e f i n i t i o n i s e a s i l y compared w i t h t h e G r o t h e n d i e c k Pietsch criterion for nuclearity - for sequence

m

(6n)n=1

shows t h a t e v e r y

A - nuclearity the

i s i n d e p e n d e n t o f t h e semi-norm A-nuclear

p

-

and

space i s also a nuclear space.

Holomotphic functions on nuclear spaces with a basis T h e s t r o n g d u a l of

Proposition 5.12 A-nuclear

an

Proof

a n A-nuclear

s p a c e is

space.

Let

03

b e a n A-nuclear

E

We i d e n t i f y

(en)n=l.

221

t h e u s u a l manner.

space with absolute b a s i s

w i t h a dense subspace of

E

A(P)

By t h e d e f i n i t i o n o f A - n u c l e a r i t y

in

the

mapping m

6 : (Xn)n=l

E

A(P)

__f

(Gnxn)n=l

i s a l i n e a r t o p o l o g i c a l isomorphism from m

Hence

(xnln=l,

Since

E'

P'

6 -

P A(Pl)

' . ( I x n / ) n" = l 1

=

i f and o n l y i f

A(P)

E

(E);

w e h a v e shown t h a t

m

El

B

A(P)

E

onto

(Gnxn)n=l

E

A(P).

A(€').

( c o r o l l a r i e s 5 . 7 and 5.10)

is an A-nuclear space.

The c o m p l e t i o n o f a n A-nuclear space.

A(P)

where

(Xn)n=l

the proof.

A(p)

E

This completes

space i s an A-nuclear

C o u n t a b l e p r o d u c t s a n d c o u n t a b l e d i r e c t sums o f A -

n u c l e a r spaces a r e A-nuclear

spaces.

space with a b a s i s i s an A-nuclear

Every Frcchet n u c l e a r

space and hence everyB3'LL

space w i t h a b a s i s i s A-nuclear. Let CD

absolute basis

(en)n=l.

b a s i s for t h e A - n u c l e a r Proof 5.11

Let

for

E.

m

( q n = 1 Since

(E,T)

be an A-nuclear space with m

Then space

(en)n=l

is also an absolute

(E,.rb).

be t h e sequence s a t i s f y i n g d e f i n i t i o n

I;=, 6n

<

we may

w

assume, by

reordering t h e b a s i s i f necessary, t h a t t h e sequence i s i n c r e a s i n g and t h a t

i t y implies that

>1.

If

p

E

cs(E)

m

(6n)n=1

then A-nuclear-

Chapter 5

228

d e f i n e s a c o n t i n u o u s semi-norm on

m- 1 In=1 ~

t h e sequence ISm(x -

of

E.

Since

6-

E.

Let

rb

as

e

n+m

~

i) s }a

{6;x;

1

Er

~r b - = b o u~n d e d

c o n t i n u o u s semi-norm on

, n = l *, ,

C = s u p q(6:xg)

subset

it f o l l o w s t h a t

i s a bounded s u b s e t of

x:en}Bsr implies that of

+ m

is a

q

Now, i f

+

m

~

E.

...

E

E

and

then A-nuclearity

i s a l s o a bounded s u b s e t

and l e t

n,B

it follows t h a t

Hence since

m

(en)n=l

is a

rb

c o n t i n u o u s semi-norm on

i s an a b s o l u t e b a s i s f o r

(E,rb).

E.

Moreover,

229

Holomorphic functions on nuclear spaces with a basis

(E,rb)

i s an A-nuclear

space.

This completes t h e proof

An i n f r a b a r r e l l e d A - n u c l e a r s p a c e

Corollary 5.14

(E,T)

is b o r n o l o g i c a l . Proof

p

Let

be a

p

proposition 5.13, norm

Let

q

c o n t i n u o u s s e m i - n o r m on

T~

i s dominated by a

By

which h a s t h e form

U = {xeE;q(x)$l}.

is ?-closed

U

bounded s e t s . S i n c e ( E , r ) bourhood o f z e r o i n norm on

E.

c o n t i n u o u s semi-

T~

E.

Since

and h e n c e

E

q>p

c o n t i n u o u s and h e n c e

and a b s o r b s a l l

is infrabarreled, q

is a

T

p

t h i s shows t h a t

U

T-

i s a neighcontinuous semi-

is also

T

This completes t h e proof.

T = T ~ .

We r e c a l l f r o m c h a p t e r 1 t h a t a l o c a l l y c o n v e x s p a c e i s a f u l l y nuclear space i f nuclear spaces. Schauder b a s i s

and

E'

a

E

are both reflexive

i s a f u l l y n u c l e a r s p a c e and h a s a

E

If

E

m

(en)n=l

( h e n c e f o r t h we u s e t h e t e r m f u l l y

nuclear space with a b a s i s ) then t h e b a s i s equicontinuous b a s i s s i n c e an a b s o l u t e b a s i s s i n c e

E

E

co

(en)n=l

i s an

i s b a r r e l l e d and h e n c e it i s

is nuclear.

By p r o p o s i t i o n 5 . 9

t h e strong dual of a f u l l y nuclear space with a b a s i s i s a l s o a f u l l y nuclear space with a b a s i s .

Every r e f l e x i v e A-nuclear

s p a c e i s a f u l l y n u c l e a r s p a c e w i t h a b a s i s a n d we d o n o t know o f any f u l l y n u c l e a r s p a c e w i t h a b a s i s which i s n o t an Anuclear space.

C o u n t a b l e p r o d u c t s a n d c o u n t a b l e d i r e c t sums

of f u l l y nuclear spaces with a b a s i s a r e a l s o f u l l y nuclear

spaces with a b a s i s .

We i n t r o d u c e f u r t h e r c l a s s e s o f n u c l e a r

s p a c e s i n l a t e r s e c t i o n s o f t h i s c h a p t e r a n d a l s o g i v e a number of examples.

Most o f t h e c l a s s i c a l n u c l e a r s p a c e s e n c o u n t e r e d

i n a n a l y s i s a r e r e f l e x i v e A-nuclear nuclear space with a b a s i s ,

m

spaces.

(en)n=l,

If

E

is a fully

we f i x o n c e a n d f o r a l l

Chapter 5

230

a representation of

and

E

EA

as sequence spaces

such t h a t t h e cadonical d u a l i t y between

A(P')

n a t u r a l l y t r a n s f e r r e d t o t h e d u a l i t y between

and

A(P)

and

E

and

A(P)

is

E'

A(P')

We t h u s h a v e

m

Cn=1 W n Z n

=

Z E E

where

and

W E E ' .

Definition 5.15

Ic)

A

be a nucZear sequence

is said t o b e R e i n h a r d t if w h e n e v e r

A

z

A(P)

A C A ( P ) .

s p a c e and l e t

(a)

E

Let

=

m

(ZnIn=1

A

(en)n

and

is a p o l y d i s c if

A

R~

then

h a s e i t h e r of

ie (e

n

m

ZnIn=1

E

A.

the

foZZowing forms

where

if

B,

a >0

E

f o r a22

[O,+m]

and

0

.

(+-)

n,

a

.

(+a)

+m

= 0.

The R e i n h a r d t h u l l o f a r b i t r a r y s u b s e t s of d e f i n e d i n a n o b v i o u s way.

=

E

is

A p o l y d i s c which h a s t h e form

i s open i f and o n l y i f ( R ~ E ) P ~ and a p o l y d i s c which has t h e form (**) is always closed.

(*)

Holomorphic functions on nuclear spaces with a basis

23 I

The origin is a compact polydisc and the whole space is a n open polydisc.

By the Grothendieck-Pietsch criterion for

nuclearity every nuclear space with an absolute basis has a fundamental neighbourhood system consisting o f open polydiscs. It is also immediate that every polydisc is modularly decreasing and every modularly decreasing set is Reinhardt. In studying holomorphic functions on fully nuclear spaces with a basis, we find that the multiplicative polar, which we now introduce, is more useful than the usual linear polar o f functional analysis.

~f E 2 A ( P ) is a fuzzy nuclear space A C E w e define AM (the muZtiplicatiue

Definition 5.16

with a basis and poZar of

A)

as

It i s immediate that subset o f

EE,

AM

is a closed modularly decreasing

and

is a closed subset o f

E

which contains

A.

The following two simple, but technical, results play a crucial role in the sequel.

Let

U

space with a basis

E

Lemma 5.17 4

E ' = A(PI). B

b e a n open polydisc in a fuZZy nucZear A(P).

Then

UM

is a compact poZydisc

U contains a fundamental system of compact sets consisting of compact poZydiscs and UM has a fundamentaz neighbourhood system consisting of open polydiscs. K Interior(K M ) establishes a one-to-one The mapping correspondence between compact poZydiscs in U and open polyM disc neighbourhoods of U . in

Furthermore

-

232

Chapter 5 u)

Proof m (nn)n= 1

E

Let U = { ( z ~ ) ~ =E ~A(P); P and let

v

m

{(zn)n=l

=

E

U M = V o = {(w,):,~

Then

*(PI; E

suplznan/ < 1) n

m

where

Cn=l/zn"n/h 11.

A(P');

sup1 w n n

,<

11.

is a compact Since E is complete and dual nuclear V o subset of E; and hence UM is a compact polydisc in A(P'). K

Let

be a compact subset of AK

such that

X

m

m

(Bn)n=l

E'

U.

for all

(Bn)n=l

Hence

E

U

this shows that

then for all

n

and

and

Since the modular hull of any element of U,

n

By lemma 5 . 6 ,

m

= Modular Hull o f

polydisc in

U.

hBnan 6 1 99

I

There exists

is also a compact subset o f

x

if B n = suP{lznl; (ZnIn=l E K } m ('n)n= 1 E A(P). Since A K C U, hence

U.

U

U

is a compact

contains a fundamental

system of compact sets consisting o f compact polydiscs. Now let W be a neighbourhood o f U M in We can choose

Ei.

a,

(aA)n=l

where

Since

E

Bn

P'

such that

1

0

=

m

(Bn)n=l

if

a' = 0

n

I/(an+l/a

E

P'

'1

if

a,:

# 0.

n

this implies that

Ui4

has a fundamental

neighbourhood system consisting o f open polydiscs.

Holomorphic functions on nuclear spaces with a basis i s any open p o l y d i s c neighbourhood of

V

If

may c h o o s e

a,

O<5<1,

such t h a t

B

and,

since

vM c

Ei

(UM)M =

rr

E

K

KM

i s a n e i g h b o u r h o o d of z e r o i n

and hence

i s a neighbourhood of

KM

m

M (K )

interior

and h e n c e i n

E

(B)" A m

(6n)n,l

Lemma 5.18

sup/ n

<1 W

such

:: 11

and

UM

c o r r e s p o n d e n c e i s now e v i d e n t i f o n e n o t e s t h a t

for a n y c l o s e d p o l y d i s c B,

A(P');

U.

Ei.

in

UM

is a

and hence

KM

KOC

i s a n open p o l y d i s c neighbourhood of

The one-to-one

6 =

then

U,

VM

E' Choose X > 1 B' i s a l s o a compact s u b s e t of U . Then

XK

If

OF

which l i e s i n

i s a compact p o l y d i s c i n

If

disc

Hence

UM.

is also f u l l y nuclear with a basis,

compact p o l y d i s c i n

that

t h e n we

UM

i s a l s o an open poly-

BV

d i s c n e i g h b o u r h o o d of t h e compact s e t 1 (BLqM = -

233

=

A,

(AM)M = A ,

and for a n y o p e n p o l y -

This completes t h e proof.

BM.

i s a subset of a sequence space

A(P)

and

i s a s e q u e n c e of s c a l a r s , w e l e t

Let

space w i t h a b a s i s ,

b e an open p o l y d i s c i n a f u l l y n u c l e a r

U

E

A(P),

and l e t

K

be a compact

234

Chapter 5 U.

poZydisc i n m

T h e r e e x i s t a s e q u e n c e o f reaZ numbers

6n > 1

6 = (6n)n=1,

hood o f z e r o i n U

s u b s e t of

Since

Proof

m

<

such t h a t

A(P)

and

In=,

and

m,

6K

and

.= A ( P t )

E;

i s a f u l l y n u c l e a r space, w e can

m

( B ~ ) ~i n = P~ '

such t h a t

, i s a r e l a t i v e l y compact s u b s e t o f E

2"

Bn

if

1 +-

Then

U.

= 0

n

and l e t E~

E

=

> 1

( E

a neighbour-

G(K+V)CU.

choose a sequence

Let

V

is a r e Z a t i v e Z y compact

if

)"

n n=l'

for all

n

and

B,

f 0

235

Holomorphic functions on nuclear spaces with a basis Moreover,

if

(

then

z ~ ) L = K~ E

and h e n c e

EK

Bn(l

6

IEnznl

+

Bn # 0

and

B:, - ) Bn

=

B,

(otherwise +

c K1.

Now c h o o s e a n e i g h b o u r h o o d o f z e r o , EK + V' C U .

Let

Next c h o o s e

(an);=1

an d u l a ' n n

Let

{ ( z ~ ) ; = ~ A(P);

=

hood o f z e r o i n

Let

for a l l

A(P).

if

2"

Moreover,

cV ' .

= inf(En,sA)

We h a v e

6,,

>

3

1

6n

all all

W

i s a neighbour-

E '

=

n n,

and moreover,

This completes t h e proof.

W

( E : ) ~ = ~ .

an#O

( z ~ ) E~ W= ~ then

6n

U

< 1).

n,

all

< 1

and l e t

m

if

E'W

E~

suplzna;l n

u,!,

an=0

Now l e t

Since

n.

=

E:

if

hence

,ti,

W

(u;),,=~

P,

(a;),"=1

such t h a t

V',

be chosen so t h a t

P

E

such t h a t W

zn = 0)

for all and 6K

Izn~;anI

n

I",l q 1

s

Izna;I

and l e t

6

I",l

6 =

1

En

+

6

and

1

m

(6n)n=1.

I;=,

l' <

-

'n

m.

i s a r e l a t i v e l y compact s u b s e t o f

236

Chapter 5

If

E

happens to be a reflexive A-nuclear space, then the

proof o f the above results f o r the particular case much simpler. 00

(6nzn)n=l

m

A(P),

E

U=E

are

m

( z ~ ) ~E =A ( P~ ) +

This is because the mapping

the sequence defining A-nuclear-

ity.is a linear topological isomorphism. T h i s observation also leads to the following useful lemma. A(P)

Let

Lemma 5.19 let

m

6 = (6nn)n=1

b e a r e f Z e x i v e A - n u c l e a r s p a c e and

be t h e s e q u e n c e g i v e n by t h e A - n u c Z e a r i t y

of

E.

(a) If

f o r m s a f u n d a m e n t a l s y s t e m of c o m p a c t

(K)KE'k

s u b s e t s of

(6K)KEk

then

A(P)

i s also a

f u n d a m e n t a l s y s t e m of c o m p a c t s e t s for

ibi

A(P).

If

9

i s a f u n d a m e n t a l s y s t e m of n e i g h b o u r h o o d s

of

0

in

A(P)

(K+6V)VEv

then

i s a fundamental

s y s t e m of n e i g h b o u r h o o d s o f t h e c o m p a c t s u b s e t

ic)

If

K

A(P).

Of

V

i s a SundamentaZ

s y s t e m of i n c r e a s i n g

c o u n t a b l e o p e n c o v e r s of

R(P)

then so also

i s {

§5.2

m

(6vn)n=l;

m

(VnIn=l

E P1.

HOLOMORPHIC FUNCTIONS ON FULLY NUCLEAR SPACES WITH A BASIS

In this section, we show that the monomials form a basis for certain spaces o f holomorphic functions and analytic functionals on fully nuclear spaces. Definition 5 . 2 0 integer for a l l large).

I f

zm

m

n E

=

N

Let

N")

=

and

mn

= 0

and

m mn r! n = l 'n .

z

=

m

{(mn)n=l;

mn

m

(zn)n=l

a non-negatioe

n sufficiently

for a l l E

A(P)

we l e t

237

Holomotphic functions on nuclear spaces with a basis

The function

z

m

N").

each

in

A monomial

Iml

m

A(P)

E

-

m

z

E

i s c a l l e d a monomial f o r

CC

i s a Im(-homogeneous p o l y n o m i a l on

A(P)

where

mn.

=

Theorem 5 . 2 1

Let U

b a s i s and l e t

E

A(P)

--J

be a f u l l y n u c l e a r s p a c e w i t h a

be an open p o l y d i s c i n

E.

The m o n o m i a l s

form a n a b s o l u t e b a s i s f o r t h e c o m p l e t e n u c l e a r s p a c e (HHy(U) ,

Moreouer,

T ~ ) .

i f

E

is A - n u c l e a r

(H(E)

then

, T ~ )

i s a l s o A-nuclear. Proof

Let

be a compact s u b s e t o f

K

By lemma 5 . 1 8 ,

U.

t h e r e e x i s t s a sequence of p o s i t i v e r e a l numbers, m 1 6 n ' 1 and 6 < m , such t h a t 6K i s a ('n)n=l> m r e l a t i v e l y compact s u b s e t o f n U . Let 5 = (En)n=l E U .

I;=,

each p o s i t i v e i n t e g e r

r

Izn( c ( E n [

and

nsr

let

[tlr

= { ( z ~ ) E~ E=; ~

f o r n>r}. n f i n i t e d i m e n s i o n a l compact p o l y d i s c i n U . If and

m

E

5

=

for

z

For

m

= 0

[Elr f

E

is a HHy(U)

let

N

where

and

Support

m

(5n)n=1

6, # 0

i s chosen s o t h a t

C S u p p o r t (m) 1 .

ts)

if

# 0, ( i . e .

mn

The Cauchy i n t e g r a l f o r m u l a i n s e v e r a l v a r i a b l e s i m p l i e s that

am

f(z) all

d o e s n o t depend on t h e c h o i c e o f

1

=

a zm

for all

z

mENr

E

[t],,

6 all

and

r

E

N

and

6 i n U.

Now l e t

6

E

K.

If

m

E

N

and

Support (5)

S u p p o r t (m)

238 then

Chapter 5

Em

and

= 0

S u p p o r t (m) (z { 1 ,

lamEml 6

. . . ,r 1

In b o t h cases, w e have

m I/amz

/IK

i

I/f

11 .

and

l a m (m

I

i

I I f / ( K and hence

I / f l I K . Applying t h i s r e s u l t t o

Hence

Since

m

Otherwise, S u p p o r t ( c ) C

1 6

n

<

-

t h i s means t h a t

6K

we o b t a i n

239

Holornorphic functions on nuclear spaces with a basis Hence U,

f(z) =

I

4

and s i n c e

1

(N)

mcN f and

a z m , u

m

d e f i n e s a h y p o a n a l y t i c f u n c t i o n on a r e b o t h h y p o c o n t i n u o u s on

f

a g r e e on a s e t whose s e q u e n t i a l c l o s u r e i s f(z) f(z) = N

f o r every

z

in

U.

N")

Let E>O be a r b i t r a r y . Choose J 1 such t h a t f 6K 6 mEN(N)\ J 6 m

1 1 11

subset of

N")

1

which c o n t a i n s

J

and

it follows t h a t a ,mcN (N) ,

U,

The c o e f f i c i e n t s

are obviously u n i q u e l y determined by

U

f.

m

a f i n i t e subset of E .

If

J'

i s any f i n i t e

then

and hence

Since

and

6K

i s a r e l a t i v e l y compact s u b s e t o f

U

w e h a v e shown

t h a t t h e monomials form a n a b s o l u t e b a s i s f o r t h e s p a c e o f h y p o a n a l y t i c f u n c t i o n s on As t h e s p a c e A(Q)

U

w i t h t h e compact open t o p o l o g y .

( H H y ( U ) , ~ O ) i s c o m p l e t e , we may i d e n t i f y i t w i t h

where

and

&(U)

i s t h e s e t o f a l l compact s u b s e t s o f

Since

the Grothendieck-Pietsch c r i t e r i o n implies t h a t

U.

240

Chapter 5

(HHy(U),~o) If

U

is a nuclear space. E

=

E

and

is a reflexive A-nuclear space with

as the sequence occurring in t h e definition o f A (6 n);=1 nuclearity, then 6 K is a compact subset of E whenever K E (lemma 5 . 1 9 ) .

is a compact subset o f T

0

continuous semi-norm o n

1m e N (N)

m

a z m

H(E),

E

is also

(H(E),.ro)

H(E)

-io

n u c l e a r space

continuous and hence

is an A-nuclear space.

(H(U),T~)

space,

Corollary 5.23 =

is any

then the semi-norm

whenever

U

a f u l l y n u c l e a r space w i t h a b a s i s .

(U)

p

The m o n o m i a l s form a n a b s o Z u t e b a s i s f o r t h e

Corollary 5.22

A-nuclear

Hence if

then Let

HHy(U)

(H(E),.rO) E

I f

is a n o p e n p o l y d i s c i n E i s a reflexive

i s a l s o an A - n u c l e a r

space.

be a f u z z y n u c l e a r s p a c e with a b a s i s

f o r any open s u b s e t

= (HHy(U),~o) and any open p o l y d i s c U i n E.

(H(U),T~);

U

of

E.

Moreover,

9 (HHy(U),.ro)~ f o r

F o r corollary 5.22, it suffices to notice that each monom-

ial is continuous, that a subspace o f a nuclear space is nuclear and a subspace o f an A-nuclear space which contains the absolute basis is also A-nuclear.

T h e first part o f corollary

5.23-follows from the fact that we only used the boundedness o f hypoanalytic functions on compact sets in proving theorem 5.21. The second equality is a consequence o f the continuity of !nonomials and the completeness o f the space (HHy(U),~o). Theorem 5.21 shows that any bounded subset of (HHy(U),.ro)

is

contained in the closure of a bounded subset o f (H(U),ro). An application o f lemma 5.6 now completes the proof o f corollary 5.23. If

U

is a connected Reinhardt domain containing the

241

Holornoiphic functions on nuclear spaces with a basis o r i g i n , t h e n , by u s i n g f i n i t e d i m e n s i o n a l r e s u l t s on a n a l y t i c c o n t i n u a t i o n , o n e c a n show t h a t e a c h

f

E

H H Y W )2

(resp. H(U)

1

h a s a u n i q u e e x t e n s i o n as a h y p o a n a l y t i c ( r e s p . holomorphic) f u n c t i o n t o t h e modular h u l l o f

The method o f theorem 5 . 2 1

U.

c a n a l s o b e e a s i l y e x t e n d e d t o show t h a t t h e m o n o m i a l s f o r m a n absolute basis for

( H H y ( U ) , ~ o ) whenever

is a modularly

U

d e c r e a s i n g open s u b s e t o f a f u l l y n u c l e a r s p a c e w i t h a b a s i s . We n o w p r o v e a n a n a l o g o u s r e s u l t f o r t h e

H(U).

To s h o w , h o w e v e r ,

t o p o l o g y on

T~

( H ( U ) , T ~ ) i s n u c l e a r we m u s t

is an A-nuclear space.

E

assume t h a t

that

Proposition 5.24

Let U be a n o p e n polydisc i n a fully nucZear space with a basis E. Then the m o n o m i a Z s f o r m a n absolute basis for ( H ( U ) , T ~ ) . If E is also a n A-nuclear space, t h e n ( H ( U ) , r W ) a n A-nuclear space. Proof

Let

p

be a

is a nuclear space and

c o n t i n u o u s semi-norm on

T~

p o r t e d by t h e compact p o l y d i s c

let

b e a n open p o l y d i s c ,

V

By lemma 5 . 1 8 ,

in

K

Let

U.

t h e r e e x i s t an open p o l y d i s c

W

In=, 6

lamzml

where

a

sion of

m

Let

J

1 m - Ilamz

s

grn

f

‘K+W

H(U) H(U)

E

IIflIV

E

in >

1

z

E

z

m

and <

m.

and a for all. K+W

and

i n t h e monomial e x p a n -

(H(U),T~).

E

is

“6

is the coeff cient of

CK+W

P(g) If

n

then

f

such t h a t

K C V C U ,

s e q u e n c e -o f r e1 a l n u m b e r s 6 = (6 6,, n n = l where n and < such t h a t 6(K+W) C V . If mEN

(H(E),ru)

b e a p o s i t i v e r e a l number s u c h t h a t IlglI K + W

for all

i s any f i n i t e s u b s e t o f

N“)

g

in then

H(U).

242

Chapter 5

1

Since

i s f i n i t e , i t f o l l o w s t h a t t h e monomials 6" m c N "1 form an u n c o n d i t i o n a l b a s i s f o r (H(U),-cU). Moreover, s i n c e

t h e semi-norm

is

c o n t i n u o u s and h e n c e t h e monomials form an a b s o l u t e

-cU

basis for s'pace.

N o w suppose

(H(U),rU).

Let

m

(6n)n=1

be t h e sequence occurring i n t h e d e f i n -

i t i o n of A-nuclearity. m

numbers,

In=, 6'< m

1

n

m,

and

i s a l s o an A - n u c l e a r

E

Choose a s e q u e n c e o f p o s i t i v e r e a l

such t h a t 6'6'K

1 < 6' 6 6

n

n

for all

n,

i s a r e l a t i v e l y compact s u b s e t of

T h i s i s p o s s i b l e b y lemma 5 . 1 8 .

U.

243

Holornophic functions on nuclear spaces with a basis

f = C mcN "1 U

a z m

m

which contains

neighbourhood

Hence

W

in

6'6'K.

C

mcN

and let

b e an open subset o f

V

By lemma 5.18, there exists a 6'6'K

o f zero such that

q ,is ported by

Since

H(U)

6'6'K

pO

,q ( zm) # O q ( zm)

and

6 ' 6 ' W C V.

(N)

<

and all

p

If

U=E

then letting

6n

in the above proof, we see that

A-nuclear space.

m

meN

the Grothendieck-Pietsch criterion implies that a nuclear space.

Hence

continuous.

T

1

+

= 6;

(H(U),.rw)

is

for all

(H(E),-cw)

n

is an

This completes the proof.

In proving proposition 5.24, we have also shown the follm owing: if f E H(u), f(z) amz , where U is an open mEN polydisc in a fully nuclear space with a basis, and K is a

=I

compact subset of zero such that

U,

then there exists a neighbourhood

V

We now investigate when the monomials form a basis for H(U)

endowed with the barrelled and bornological topologies

of

244

Chapter 5

associated with

and

T~

T

~

.

We f i r s t s h o w t h a t t h e y a l w a y s

form an u n c o n d i t i o n a l equicontinuous b a s i s f o r t h e s e t o p o l o g i e s b u t t o s h o w t h a t t h e b a s i s i s a b s o l u t e we n e e d e x t r a h y p o t h e s e s on

E.

and

U

Proposition 5.25

If

E

nuclear space

is a n o p e n p o l y d i s c in a fully

U

with a basis,

t h e n t h e monomials form a n

unconditional equicontinuous basis for T

=

T

o"u'

Proof

or

b ' T w ,b

' 0 ,

Y

(H(U),T),

6 .

We h a v e a l r e a d y p r o v e d t h i s r e s u l t f o r

s i n c e an a b s o l u t e b a s i s i s a n u n c o n d i t i o n a l , equicontinuous,

basis.

If

and

T

T

and h e n c e a n

is an A-nuclear space, then we

E

may a p p l y p r o p o s i t i o n 5 . 1 3 t o s h o w t h a t t h e m o n o m i a l s f o r m a n absolute basis for

(H(U),T),

ever, does not cover

or

T h i s , howw,b' t h e g e n e r a l c a s e w h i c h we now p r o v e . We T

=T

o,b

Y

f i r s t show t h a t t h e m o n o m i a l s f o r m a n u n c o n d i t i o n a l b a s i s f o r ( H ( U ) , T & ) . Let every Vn

z =

in

Iz

E

J

of

and l e t

f

E

H(U).

f(z) =

1

mEN

For each p o s i t i v e integer

U.

/ I m E aJm z m /

U;

Then

s n

(N)

n

a zm m

for

let

f o r every f i n i t e subset

N")} be the interior of

Wn

Vn.

By o u r o b s e r v a t i o n co

preceding t h i s proposition, it follows t h a t U. If p i n c r e a s i n g c o u n t a b l e open cover o f c o n t i n u o u s semi-norm on

(Wn)n=l is a

H(U), t h e n t h e r e e x i s t

is an

C > 0

and

a p o s i t i v e i n t e g e r such t h a t p(g)

s c

IlglI

wN

f o r every

g

in

H(U).

J finite

and

{

I ~ E a Jm z m

JCN"),

J finite

is

T&

bounded.

N

245

Holomorphic functions on nuclear spaces with a basis

By lemma

lmsJ,J finite am zm

3*28,

+

f

as

J + W

in

;;Tlfo

as J + m a zm + 'mEJ,J finite m n ! lml=n in ( ? ("E),T~) for each non-negative integer n . Since the monomials form an absolute basis for (H(U),TW) this shows (H(U),T~)

if and only if

that they also form an unconditional basis for

( H ( U ) ,T6).

Since T~ t T this also shows that the monomials form ~ b T,o,b~ an unconditional basis for (H(U) ,Tw,b) and (H(U) ,TO,b).

(H(U),-r6)

fa

then

+

f

in

(H(U),-ro)

as

and hence,

since the monomials form a Schauder basis for

(H(U),T~)

aa + am as a + m for each m in N") and the monomials m form a Schauder basis for (H(U),T,),(H(U),T~,~) and

(H(U) 9To,b) .

This completes the proof.

If U is a n open polydisc in a fully Corollary 5 . 2 6 is the barrelled topology nuclear shace with a basis, then T~ assoeiated with T ~ .

Proof T

0

If

Let

on

p

Since

H(U).

is a

T

p'(f)

=

be the barrelled topology associated with

T

(H(U) ,-r6)

is barrelled,

cont n u o u s semi-norm o n

6

H(U)

T I T S T 0 6' then

sup

JCN")

J fini e is also a T~ finite subset

is

continuous semi-norm and p ' of N") the semi-norm

continuous and hence the set

T~

5 p

F o r each

J

V = {fEH(U);p(f)

$

l}

is T O (and hence T ) closed, convex, balanced and absorbing. Thus V is a T-neighbourhood o f zero and p' is continuous. Hence

T

3

T

6

and this completes the proof.

246

Chapter 5

space T

=

E

U

If

Corollary 5.27

with a basis,

T o>TwJ T

o , bJ?w, b

O r

i s a n open p o l y d i s c i n a f u z z y nucZear t h e n t h e foZZowing a r e e q u i v a Z e n t for T6;

(a)

(H(U),T)

is c o m p l e t e ,

(b)

(H(U)

, T )

i s quasicompzete,

(el

(H(U)

,T)

i s s e q u e n t i a l l y complete,

if

id)

is a s e t of c o m p l e x numbers

{arn} mE N “1

and

m amz 1

{ImEJ

i s a r-Cauchy

If (el

T

=

T

if and

or

0

I

net,

mEN

is a s e t of c o m p l e x numbers

“1

(N)

then

t h e n t h e above a r e equivaZent t o

w

{am}

, J finite

JCN“)

p(amz

m

) <

m

f o r every

mEN

c o n t i n u o u s semi-norm on

H(U)

then

We omit the proof since it is similar to the analogous result proved f o r holomorphic functions on balanced open sets in chzipter 3 (proposition 3 . 3 6 ) .

E i s a r e f Z e x i v e A-nuclear space, t h e n i s a n A-nucZear s p a c e f o r T = T ~ , ~T ’ ~ o , b ’ ~ w , obr

Proposition 5.28 (H(E),r)

If

Tc6’

E

Proof

If

shown

(H(E),.ro)

is a reflexive A-nuclear space, then we have is A-nuclear (corollary 5.22) and

is A-nuclear (proposition 5.24).

By proposition 5 . 1 3 ,

(H(E),T~)

the

241

Holomorphic functions on nuclear spaces with a basis associated bornological topologies, A - n u c l e a r t o p o l o g i e s on

H(E).

and

o,b

‘ w,b’

are a l s o

T h e r e r e m a i n s o n l y t h e case = ‘I w h i c h we now d i s c u s s . 6 Our p r o o f f o r t h i s c a s e c a n a l s o b e e a s i l y a d a p t e d t o g i v e 02

d i r e c t proofs for the other topologies. Let 6 = (6n)n=1 t h e sequence occurring i n t h e d e f i n i t i o n of A-nuclearity. Since

E

is a complete A-nuclear space,

i s a l i n e a r i s o m o r p h i s m when

H(E)

t h e mapping:

i s endowed w i t h t h e

topology ( i t i s a l s o an isomorphism f o r any o f t h e o t h e r topologies). then H(E).

p

If

XEB.mEN

is a

T~

‘6

i s a T~ bounded s u b s e t o f H(E) is also a bounded s u b s e t of

{ f

If

{ (62)matzm}

be

“1

c o n t i n u o u s semi-norm on

H(E)

and

then

f o r all

1

m amz

E

H(E).

mEN

The a b o v e shows t h a t

p1

i s bounded on t h e

T~

bounded sub-

sets of H(E) and h e n c e i s T~ continuous since (H(E) , T ~ ) is a bornological space. As t h e monomials form a n u n c o n d i t i o n al basis for

(H(E),-c6),

p

6

pl.

2 48

Chapter 5

t h i s shows t h a t

( H ( E ) , T ~ ) i s an A-nuclear space and completes

the proof. U

If

is a n o p e n s u b s e t of a l o c a l l y c o n v e x s p a c e arid H(U)

is a Z o c a l l y c o n v e x topology o n (H(U),r)'

a r e c a l l e d a n a l y t i c f u n c t i o n a l s (or t o b e more

precise,

analytic functionals on

space form,

(H(U],T)?

U).

can be interpreted,

T

t h e n t h e e l e m e n t s of We s h o w t h a t t h e

v i a t h e "Borel"

as a s p a c e o f holomorphic germs on

trans-

U

whenever

UM

is

a n open p o l y d i s c i n a f u l l y n u c l e a r space with a b a s i s . C o n t i n u i t y p r o p e r t i e s o f t h e germs w i l l depend on o u r c h o i c e o f T

-

f o r example

t i n u o u s germs and a n a l y t i c germs.

T~

a n a l y t i c f u n c t i o n a l s g i v e r i s e t o con-

T

a n a l y t i c f u n c t i o n a l s g i v e r i s e t o hypo-

T h i s i n t e r p r e t a t i o n a l l o w s u s t o s e t up a

correspondence between p r o p e r t i e s o f

H(U)

t r a n s l a t e s e e m i n g l y d i f f i c u l t problems on t r a c t a b l e problems on find that

(H(U),ro)

and H(U)

and conversely.

H(UM)

H(UM)

and t o

i n t o more

For

instance, we

i s i n f r a b a r r e l l e d i f and o n l y i f

H(U')

is a regular inductive l i m i t . The key t o o u r i n v e s t i g a t i o n s i s t h e e x i s t e n c e o f an absolute basis i n

(H(U),T)

and,

indeed,

certain results

a b o u t a n a l y t i c f u n c t i o n a l s a l r e a d y f o l l o w f r o m t h e r e s u l t s we have j u s t proved. that

(H(E),r);

P r o p o s i t i o n s 5 . 1 2 a n d 5 . 2 8 t o g e t h e r show i s an A - n u c l e a r s p a c e (and h e n c e h a s an

a b s o l u t e b a s i s and i s n u c l e a r ) s p a c e and

T

E

i s a r e f l e x i v e A-nuclear

i s any one o f o u r u s u a l t o p o l o g i e s on

Propositions 5.9,5.21 ( H ( U ) 9 T o , b) '

i f

and 5 . 2 5 imply t h a t

H(E).

( H ( U ) , T ~ ) ~a n d

b o t h have a n a b s o l u t e b a s i s whenever

U

i s an

open p o l y d i s c i n a f u l l y n u c l e a r space with a b a s i s while propositions 5.9,5.24 (H(U),Tw,b)i

and 5.25 imply t h a t

( H ( U ) , T ~ ) ; and

b o t h h a v e a n a b s o l u t e b a s i s when

U

i s a n open

polydisc i n a r e f l e x i v e A-nuclear space. T h e B o r e 1 t r a n s f o r m d e f i n e d h e r e I S N O T T H E SAME a s t h a t g i v e n i n c h a p t e r s 1 a n d 3 , b u t i t i s u s e d f o r more o r l e s s t h e same p u r p o s e s .

F o r t h i s r e a s o n , we a d o p t t h e s a m e t e r m i n o l o g y

b u t u s e a d i f f e r e n t n o t a t i o n t o d i s t i n g u i s h between t h e

249

Holomorphic functions on nuclear spaces with a basis d i f f e r e n t Borel transforms Let basis

b e an open p o l y d i s c i n a f u l l y n u c l e a r space w i t h a

U

E Z A(P).

all

m

for

(H(U),r6),

If

N").

E

T

E

(H(U),T~)' we l e t

S i n c e t h e monomials form a n u n c o n d i t i o n a l b a s i s

T

i s f u l l y determined by T,%T,

The Bore2 t r a n s f o r m of E; S A(P')

N

B1'

w h e r e t h e r i g h t hand s i d e of

and i s G-holomorphic. mapping. E

N

"1.

is g i v e n b y t h e s e t of p o i n t s converges).

tf

neighbourhood o f

%

One e a s i l y s e e s t h a t

0

in

E'

is an injective

The r e l a t i o n s h i p b e t w e e n o u r two B o r e l t r a n s f o r m s , cc,

and

B T

mE

i s defined on a subset of

(*)

i s t h u s d e f i n e d on a

BT

{bm)

by t h e formula

( t h e d o m a i n of d e f i n i t i o n of

N

bm = T ( Z ~ ) f o r

i s contained i n t h e following formula.

B,

(H(U),T&)'

and

bm = T ( z m )

for all

mcN

If then

and

where (

' .1

m

1

Iml!

=

ml!

...

m

n

!

i f

m = (ml,.

..,mn,O,. ..).

We now l o o k m o r e c l o s e l y a t t h e B o r e l t r a n s f o r m o f analytic functionals. Theorem 5 . 2 9

Let

space w i t h a b a s i s .

U

T~

be an open p o l y d i s c i n a f u l l y n u c l e a r

The B o r e l t r a n s f o r m ,

t o p o Z o g i c a Z i s o m o r p h i s m from

(H(U)

isomorphism e s t a b l i s h e s a one-to-one

,T~);

N

B,

onto

i s a l i n e a r and H(UM).

This

correspondence between

Chapter 5

250

equicontinuous subsets of ( H ( U ) , T ~ ) I and sets o f germs which a r e defined and uniformZy bounded o n a neighbourhood of U M . Let

Proof in

T

K

polydisc

U

s m

There e x i s t

(T(f)/

such t h a t

6,>l

i s a neighbourhood of

s

all

f

n,

i s a r e l a t i v e l y compact s u b s e t of

SK

UM

N"),

in

and a compact

C>O

C / ( f ( ( K f o r every

we may c h o o s e a s e q u e n c e o f p o s i t i v e

m

6 = (6n)n=1

< m and In=1 n lemma 5.17,(6K)*

any

, T ~ I) .

such t h a t

By l e m m a 5 . 1 8 ,

H(U).

real numbers, m

(H(U)

E

in

in

E' B

By

U.

and,

for

-C gm

where

and (6K)

bm

N

M

T(zm).

d e f i n e s a holomorphic f u n c t i o n on t h e i n t e r i o r o f

BT

.

=

We h a v e t h u s s h o w n t h a t t h e B o r e 1 t r a n s f o r m m a p s

(H(U),T~)' into depends o n l y on

H(UM) C

and

and moreover, K

s i n c e o u r bound on

d

BT

w e h a v e a l s o shown t h a t t h e i m a g e

o f an equicontinuous subset o f

( H ( U ) , T ~ ) ' i s a s e t of germs

which i s d e f i n e d and u n i f o r m l y bounded on a neighbourhood o f M U . We now p r o c e e d i n t h e o t h e r d i r e c t i o n . Let

g

E

H(UM).

d i s c neighbourhood of

UM

is

g,

?(w)

By l e m m a 5 . 1 7 , UM,

V,

and

t h e r e e x i s t an open polyE

H(V),

such t h a t =

I

mEN

(N) bmwm

for all

w E V

and

whose germ on

25 1

Holomotphic functions on nuclear spaces with a basis

By lemma 5.18, V and C ' can be chosen uniformly for any family o f germs which are defined and uniformly bounded on a fixed neighbourhood o f

1

for each

If

mE N (N)

mEN")

lambml

and

a z m

m

IIwmllV

m C 1 / l a m z 11 M . V

$

UM.

=

Let

in

H(U).

+a,

then

bm = 0

and hence

Otherwise

and

Hence

and

Tg

E

(H(U),T~)'.

Since

d

BTg(w)

=

$(w)

for all

WCV

this implies that the Bore1 transform is a linear isomorphism from

(H(U),T~)'

ute basis for

is

T~

onto

(H(U)

H(UM).

, T ~ )

A s the monomials form a n absol-

the semi-norm

continuous and thus we have established the required

result about equicontinuous subsets o f Finally, we show that On

N

B

(H(U),T~)'.

is a topological isomorphism.

we have two locally convex topologies - the natural inductive limit topology, T ~ , and the topology transferred by. H(U')

252

%

Chapter 5 from

that

( H ( U ) , T ~ ) ~ T, =

T .

T

c o m p l e t e t h e p r o o f , we m u s t s h o w M We u s e lemma 5 . 2 . (H(U ) , T ~ ) i s a b a r r e l l e d

B.

.To

~

l o c a l l y c o n v e x s p a c e s i n c e it i s a n i n d u c t i v e l i m i t o f Banach spaces. By c o r o l l a r y 5 . 2 3 , ( H ( U ) , . r o ) i = (HHy ('1 9 T o ) and hence, s i n c e ( H H y ( U ) , ~ o ) i s a complete n u c l e a r and t h u s a semi-reflexive

space,

(H(UM) ,

To c o m p l e t e t h e p r o o f ,

i s also a barrelled space.

T ~ )

u s i n g lemma 5 . 2 ,

w e m u s t show t h a t t h e M monomials form an a b s o l u t e b a s i s f o r b o t h (H(U ) , T ~ ) a n d ( H ( U M ) p B ) . By p r o p o s i t i o n 5 . 9 , basis for (H(U') ,T B ) . If

g

H(UM)

E

containing g

on

g

=

1

g

=

1

N

mc N

mE

N

t h e n t h e r e e x i s t s an open p o l y d i s c

and

UM

H(UM).

7

E

(N) b m w m

(N)

1 mEN

Since

H(V)

such t h a t

V

d e f i n e s t h e germ

By p r o p o s i t i o n 5 . 2 4 ,

in

b wm m

and hence

M

(H(U ) , T ~ ] . I f

p

is a

T .

H ( u ~ ) , let

bmwm

"1

(H(V),T~)

in

c o n t i n u o u s semi-norm on

for e v e r y

t h e monomials form a n a b s o l u t e

in

H(UM)

I

i s c o n t i n u o u s f o r every open p o l y d i s c P I (H(V) Jw)

containing

UM

V

and t h e monomials form an a b s o l u t e b a s i s f o r

( H ( V ) , T ~ ) it f o l l o w s t h a t e v e r y open p o l y d i s c

'1

(H(V)

containing

V

,Tu)

UM

is

and h e n c e

continuous f o r 4

p

is

T.

continuous. Hence t h e monomials form a n a b s o l u t e b a s i s f o r M (H(U ) , T ~ ) a n d t h i s c o m p l e t e s t h e p r o o f . C o r o l l a r y 5.30

Let

U

b e an o p e n p o l y d i s c i n a fully

25 3

Holomorphic functions on nuclear spaces with a basis

with a basis.

E

nuclear space erties:

Consider t h e f o l l o w i n g prop-

(a)

(H(U),.ro)

i s a bornological space,

(b)

(H(U)

i s an infrabarreZled space,

(c)

H ( u ~ ) =

, T ~ )

11 \ l V )

l i m

(H~(v),

regular

'Ls a

----f

v 2 UM

V open i n E '

inductive l i m i t , id)

H(UM)

B

i s complete,

(fj

bounded l i n e a r f u n c t i o n a l s o n continuous, H ~ U M ) is q u a s i - c o m p l e t e ,

(g)

H ( u ~ ) i s sequentially complete,

(el

H(U)

T~

are

T

( a ) < = >( b ) < = > ( c ) = > ( d ) < =(>e ) < = > ( f ) < = > ( g ) .

then

E

Furthermore, i f

i s A - n u c l e a r a l l of t h e a b o v e p r o p e r t i e s U = E.

a r e e q u i v a l e n t when Proof

I n any l o c a l l y convex space

(d)=>(f)=>(g).

Since

e a s i l y show t h a t

H(UM)

(g)=>(d).

( a ) = > ( b ) , and

Now s u p p o s e

b e a semi-norm on

H(U)

subsets of

By p r o p o s i t i o n 5 . 2 5 ,

H(U).

(a)=>(e)=>

has an a b s o l u t e b a s i s , (b)

which i s bounded on

one can

holds.

Let

p

bounded

T~

w e may s u p p o s e

J finite

for every

Let

absorbs every t h i s shows t h a t continuous.

Then

V

bounded s u b s e t o f

T

e a s i l y seen t o b e T~

c 11.

V = {fEH(U);p(f) T

V

0

c l o s e d and

i s convex, balanced and H(U).

(H(U),.ro)

Since

V

is

is infrabarrelled

i s a neighbourhood of z e r o and h e n c e

Thus

(b)=>(a).

(b)

and

(c)

p

is

are equivalent

25 4

Chapter 5

by theorem 5 . 2 9 ,

s i n c e a l o c a l l y convex s p a c e

is infra-

F

b a r r e l l e d i f and o n l y i f t h e e q u i c o n t i n u o u s s u b s e t s and t h e

F'

s t r o n g l y bounded s u b s e t s o f NOW

H ( u ~ ) i s complete.

suppose

and 5.25,

coincide.

6

BY p r o p o s i t i o n s 5 . 9 , 5 . 2 1 ,

t h e monomials form an a b s o l u t e b a s i s f o r b o t h ( H ( U ) , T ~ , ~. ) ~I f

( H ( U ) , T ~ ) ~a n d

T

( H ( U ) , T ~ , ~ ) t' h e n t h e

E

p a r t i a l sums i n t h e monomial e x p a n s i o n o f in

(H(U),ro)l

and hence

t h i s completes t h e proof Now s u p p o s e (H(E),.ro) only i f

and

(H(U),ro)l.

E

i s an A-nuclear

E

p r o p o s i t i o n 5.28,

T

for arbitrary

form a Cauchy n e t

T

Thus

(d)=>(e)

and

U.

s p a c e and

U = E.

By

t h e monomials form an a b s o l u t e b a s i s f o r b o t h

( H ( E ) , T ~ , ~ ) .By lemma 5 . 1 ,

T~

( H ( E ) , T ~ ) ' = ( H ( E ) , T ~ , ~ ) 'a n d h e n c e

= T

o,b (e)=>(a).

i f and This

completes t h e proof. Corollary 5.31

basis. on

H(U)

Proof

Then

E

Let T~

= T

be a f u Z l y nucZear space w i t h a on

0,b

H(E)

i f and o n l y i f

f o r e v e r y open p o l y d i s c By c o r o l l a r y 5 . 3 0 ,

T

i n E.

U =

T

0,b is a regular inductive l i m i t . 0

on

H(E)

T~

= T

0,b

i f and o n l y

if H(OEl) Since t h e space of B germs a b o u t any compact p o l y d i s c i s r e g u l a r i f and o n l y i f t h e

s p a c e o f germs a t t h e o r i g i n i s a l s o r e g u l a r ,

a further applic-

a t i o n o f c o r o l l a r y 5.30 completes t h e p r o o f . Corollary 5.32

U

i s an o p e n p o Z y d i s c i n a F r g e h e t n u c l e a r s p a c e w i t h a b a s i s , t h e n T~ = T~ o n H(U) i f and onZy i f

H(UM)

Example 5 . 3 3

I f

i s a regular inductive l i m i t . (a)

If

admit a c o n t i n u o u s norm, 2.52).

then

i s a Frgchet s p a c e which does n o t T

Hence, by c o r o l l a r y 5.31,

has a basis then p a r t i c u l ar ,

EN

E

H(OEl)

H(Oc(N)) 'is

#

on

y6

if

E

H(E),

(example

i s a l s o n u c l e a r and

is not a regular inductive l i m i t . not a regular inductive l i m i t since

d o e s n o t a d m i t a c o n t i n u o u s norm.

We h a v e a l r e a d y p r o v e d

In

255

Holomotphic functions on nuclear spaces with a basis t h i s d i r e c t l y i n example 3 . 4 7 . that a

83n

s p a c e w i t h a b a s i s and

norm. (b) of

E

More g e n e r a l l y ,

t h e a b o v e shows

i s n o t a c o m p l e t e i n d u c t i v e l i m i t whenever

H(OE)

does not admit a continuous

Ei

i s a F r g c h e t s p a c e and

E

If

then H(K)

(H"(V),

l i m

=

---t

is

E

II

i s a compact s u b s e t

K

llv)

V 3K V open

is a regular inductive l i m i t c o r o l l a r y 5.30, whenever

U

i s a k-space,

E

since

( p r o p o s i t i o n 2 . 5 5 ) and h e n c e , by T

i s an open p o l y d i s c i n a & J k

0

on

= T~

H(U)

space with a b a s i s .

This i s a p a r t i c u l a r case of t h e r e s u l t proved d i r e c t l y i n example 2.47. We now c h a r a c t e r i z e t h e B o r e 1 t r a n s f o r m o f functionals.

T~

analytic

T h i s c h a r a c t e r i z a t i o n was o r i g i n a l l y u s e d t o

prove t h e t o p o l o g i c a l isomorphism o f theorem 5.29,

and l e a d s t o

a s i m p l e c r i t e r i o n f o r comparing

H(U),

U

T

and

0

on

T~

when

i s an open p o l y d i s c i n a f u l l y n u c l e a r s p a c e w i t h a b a s i s .

Proposition 5.34

U

Let

be a n o p e n p o 2 y d i s c i n a f u 2 2 y &

B, is a n u c l e a r space w i t h a b a s i s . The Bore2 t r a n s f o r m , M v e c t o r s p a c e i s o m o r p h i s m from (H(U) , T ~ ') o n t o HHy(U ) . V

Moreover, a s u b s e t

( H ( U ) , T ~ ) ' i s e q u i c o n t i n u o u s i f and

of &

o n 2 y if t h e germs i n B(V) a r e d e f i n e d and uniformly bounded M o n t h e compact s u b s e t s o f some n e i g h b o u r h o o d of U . Proof K

in

have

Let

T

E

( H ( U ) , T ~ ) ' . T h e r e e x i s t s a compact p o l y d i s c

s u c h t h a t f o r e v e r y open p o l y d i s c

U

IT(f)l

G

C(V)

d e p e n d s o n l y on

T

IlflIV and

for all V.

f

Moreover,

in

V,

K C V C U ,

H(U)

where

the set of all

we c(V)

T

which

s a t i s f i e s t h e above i n e q u a l i t i e s forms an equicontinuous subset of (H(U),T~)'. bourhood V of m

6 = (6n)n=13

By lemma 5 . 1 8 , K

6n >1

we c a n c h o o s e f o r e a c h n e i g h -

a s e q u e n c e o f p o s i t i v e r e a l numbers m 1 m, and a n f o r a l l n and In=, dn

256

Chapter 5

open p o l y d i s c

and

subset of

U

in

If

r

N").

= {mcN(N);

in

W

6( K + W ) C V.

11 zmllV

) ) z r n ) l V<

By lemma 5 . 1 7 ,

such t h a t

E

BT

E

then

=

.

m}

i s a r e l a t i v e l y compact

6K

Let

IIwmll

bm = T ( z m ) VM

m

f o r each Let

= 0.

Then

Y

HHY(U').

By t h e u n i f o r m i t y o f o u r b o u n d s

we h a v e a l s o s h o w n t h a t t h e B o r e 1 t r a n s f o r m m a p s e q u i c o n t i n u o u s subsets of

(H(U),rw)'

onto subsets of

H(UM)

which are

d e f i n e d and u n i f o r m l y bounded on t h e compact s u b s e t s o f neighbourhood of

some

U". rJ

i s a s u r j e c t i v e mapping. Let V b e an open p o l y d i s c neighbourhood of U M a n d l e t .& = ( C K ) K E ~ . B

We now s h o w t h a t

b e a s e t o f p o s i t i v e r e a l numbers indexed by t h e f a m i l y , of compact p o l y d i s c s i n

Ha

Let If

g

m IIbmw If

E

H a

I/K

<

f = X

then CK

"1 mEN

V .

{gEHHy(V); I l g ) l K < C K

=

g =

1

b wm m N (N) f o r each K in k mE

a z m

m

E

k ,

( H ( U ) , T ~ ) and

f o r each

in

in

(HHy(V),To)

and g

K

in

m E

HA

N").

we let

k and

1.

251

Holornorphic functions on nuclear spaces with a basis

For any open polydisc

W

in

containing

VM we can find a 6 n > 1 all n and sequence o f real numbers, 6 = (6n)n,l, 1 m, and W1 a polydisc neighbourhood o f zero in E n such that 6VM is a relatively compact subset o f U and (vM+wl) is a compact polydisc 6(v"+w1) c W. B Y lemma 5 . 1 7 , U

m

in

V.

C = C

Let

C m -b m z

lambml

6m

I1

(VM+W1 1

i

I/zrnll

is finite and

*

Vb'+W1

such

IIzml/

'I

M . C

6

IIw

.

[If

IIwmI/

M

=

v +wl

m

Otherwise

M

=

1.

then /IzmlI

v +wl

For all

( V +W1)

m

i

i

for every Since

f

in VM

H(U)

is a compact subset o f

U

and

W

was N

T g E ( H ( U ) , T ~ ) I . A s BTg = g this proves that B is surjective, and since our bounds a r e uniform over g in H A , we have a l s o shown that (Tg )g E H is a n equicontinuous subset o f (H(U),rU)l. This completes the proof. arbitrary, we have shown that

25 8

Chapter 5

Corollary 5.35

E

Let

be a f u l l y nuclear space w i t h a

The f o l l o w i n g a r e e q u i v a l e n t :

basis. (a)

T

0

H(U)

on

= T

for e v e r y o p e n p o l y d i s c

U

E;

in

( H ( U ) , T ~ ) ; = ( H ( U ) , T ~ ) ; I for e v e r y o p e n p o l y d i s c

ibi

(H(U),ro)'

(c)

(H(U),T~)

=

U

for e v e r y o p e n p o l y d i s c

E,

i n

for e v e r y o p e n s u b s e t V of El 8' i s c o m p l e t e for e v e r y o p e n s u b s e t V

id)

HHy(V) = H(V)

(el

(H(V),.ro)

ifi

o f Eb, t h e b o u n d e d s u b s e t s of ( H ( V ) , - r o ) a r e loea2Z.y b o u n d e d for e v e r y o p e n s u b s e t V o f E L . I t is clear that

Proof

U

E,

in

and

(d)=>(e).

Since

t h e monomials form a n a b s o l u t e b a s i s f o r b o t h

(a)=>(b)=>(c)

(H(U),-ro)

and

( H ( U ) , T ~ ) , lemma 5 . 1 s h o w s t h a t (d) are e q u i v a l e n t .

( c ) and

(c)=>(a).

Since

(

H

m

By p r o p o s i t i o n 5 . 3 4 ) = HHy(V)

open p o l y d i s c i n a f u l l y n u c l e a r space w i t h a ' b a s i s

(f)

If where

i s s a t i s f i e d and

ImEJ is a

T

g =

i s an open polydisc i n

V

-bounded

amzm

JCN"),J

subset of

sup (g(z)1

Hence

<

m

and

"1 mEN

a z m

m

E

HHY(V)

9

then

finite

H(V)

K C V C U ,

V,

1

EA,

and hence i s l o c a l l y bounded K

Hence f o r e a c h compact p o l y d i s c open p o l y d i s c

(corollary

(e) are equivalent.

(d) and

5.23),

f o r any

M <

and

g

in

E

H(V).

V

t h e r e e x i s t s an

such t h a t

T h i s shows t h a t

ZEV

( f ) = >( d )

.

Finally,

if

(a) i s s a t i s f i e d , then t h e equicontinuous

259

Holomorphic functions on nuclear spaces with a basis

subsets of ( H ( U ) , T ~ ) and ( H ( U ) , T ~ ') coincide. By theorem 5 . 2 9 , and proposition 5 . 3 4 , this means that (a)=>(f) and completes the proof. Example 5 . 3 6 (a) If U is an o p e n p o l y d i s c in a Frcchet nuclear space with a basis or in a 8 3 n space with a basis, then

=

T~

T

on

w

H(U).

This result follows from corollary

5 . 3 5 since condition (d) is easily seen to be satisfied.

We

have already proved this result for arbitrary open subsets o f

a3'Lz

spaces (example 2 . 4 7 )

and for entire functions o n

Frzchet nuclear spaces (corollary 3 . 5 4 ) . we showed that T~ # T~ on H ( C N x (E")).

I n example 1 . 3 9 , This is a particular

(b)

case of the following result which is an immediate consequence o f corollary 5 . 3 5 and example 1 . 2 3 .

If

is a n i n f i n i t e d i m e n s i o n a l f u l l y n u c l e a r s p a c e w i t h T~ # T~ on H(E x E l ) .

E

a basis,

then

B

We also obtain a topological characterization of (H(U),rw); in certain situations. This is illustrated by the following proposition. Proposition 5 . 3 7

Let

U

b e a n o p e n p o l y d i s c in a f u l l y E.

n u c l e a r space w i t h a b a s i s (a) (b)

(H(U),T~);

EHy(U

M

1

= lim +

(HHy(V),ro),

V3UM V open

(H(U),-cw);l

has t h e monomials a s an a b s o l u t e b a s i s (H(U) , T ~ ) i s s e m i - r e f l e x i v e ,

and (c)

=

The f o l l o w i n g a r e e q u i v a l e n t :

t h e T~ bounded s u b s e t s o f bounded.

Moreover,

if

E

i s an A-nuclear space,

equivalent t o the following: (d)

(H(U)

, T ~ )

(e)

(H(U)

, T ~ )

H(U)

is s e m i - r e f l e x i v e , is q u a s i - c o m p l e t e .

are l o c a l l y

t h e n t h e above a r e

Chapter 5

2 60

Proof The m o n o m i a l s f o r m a n a b s o l u t e b a s i s f o r H H Y by theorem 5.21. By t h e o r e m 5 . 2 9 , lemma 5 . 1 7 a n d c o r o l l a r y 5 . 2 3 , H H y ( U M ) ’ may b e i d e n t i f i e d , v i a t h e B o r e 1 t r a n s f o r m , w i t h H(U). (b)

An a p p l i c a t i o n o f lemma 5 . 1 now shows t h a t are equivalent.

be a K

T

in

Now s u p p o s e

bounded s u b s e t o f

w

H(U).

and Let

For each compact p o l y d i s c

t h e r e e x i s t s an open p o l y d i s c

U

(a)

is satisfied.

(c)

W

such t h a t

6,>1 6 = ( 6 n ) mn = 1 , an open p o l y d i s c i n E

Choose a s e q u e n c e o f p o s i t i v e r e a l numbers for all such t h a t

n

and

m

Cn=l

1

6(K+V) C K + W .

<

m,

and

V

Hence

and

i s a l o c a l l y bounded and h e n c e a Since

T

bounded s u b s e t o f

H

26 1

Holomoiphic Jirnctions on nuclear spaces with a basis t h i s proves t h a t N (N 1 (H(U),T~)~ h a s t h e monomials a s an a b s o l u t e b a s i s .

f o r any s e t o f s c a l a r s

{bm}

mE

then t h e r e e x i s t s a that

x

aml

{I

B =

where

1

lies in

i s s e m i - r e f l e x i v e and

Suppose be a

T

EN'^)

(b)

.

is satisfied.

bounded s u b s e t o f

w

am = suplakl

x

m

@ E

H(U).

X E r

( c ) = >( b )

B

m

f o r every

in

Let

H(U).

3A E r

amz

N (N)

suplamlz ,

a n d we h a v e shown

H(U)

bounded s u b s e t

w

/(wmII

A m

mE

I t now f o l l o w s t h a t

T

I am[ 6

( @ ( w m ) /=

If

1

and h e n c e

C =

mE

For each

N

m

m

mE N (H(U),T@)

Hence

{I

( N ) amz

(N)

in

A m

amz

1

A E r

let

N")

and l e t

A

1

f o r every

rnE

N

(N)

bmwm

in

(H(U)

( H ( U ) , T ~ ) ~ +,,

m i a l s form an a b s o l u t e b a s i s f o r c o n t i n u o u s form on

(H(U),T@)A.

i d e n t i f i e d with an element of

Hence,

H(U),

f o r e v e r y compact p o l y d i s c

By n u c l e a r i t y ,

i t now f o l l o w s t h a t

subset of

If

H(U)

E

and h e n c e

in

B

(b)=>(c)

and h e n c e

b a s i s by p r o p o s i t i o n 5 . 9 .

Hence

U,

such t h a t

.

i s an A-nuclear space then

by p r o p o s i t i o n 5 . 2 4 ,

T

+,

w

may b e

that is,

K

W

K

is a

By s e m i - r e f l e x i v i t y

open p o l y d i s c

containing

,T~)'.

t h e r e e x i s t s an A zm s y p llam < m.

I(w

i s a l o c a l l y bounded

(H(U),T~) is nuclear

( H ( U ) , T ~ ) h~a s a n a b s o l u t e (b)

and

(d)

a r e equivalent.

Chapter 5

262

In g e n e r a l , it i s e a s i l y seen t h a t satisfied then

(c)=>(e).

If

(e)

is

( H ( U ) , T ~ ) i s a q u a s i c o m p l e t e n u c l e a r s p a c e and

hence it i s semi-reflexive.

Thus

( e ) + (d)

and t h i s completes

the proof. The Bore1 t r a n s f o r m o f

T~

analytic functionals is

t r e a t e d i n e x e r c i s e 5.81,

H O L O M O R P H I C FUNCTIONS ON

55.3

DN

SPACES W I T H A BASIS

Using t h e r e s u l t s o f t h e preceding i o n s o f t h e t e c h n i q u e s u s e d t o show

T

s e c t i o n and m o d i f i c a t =

on

T~

i s a Banach s p a c e w i t h a n u n c o n d i t i o n a l b a s i s

show t h a t nuclear

=

T

DN

T~

on

H(U)

when

H(E)

when

E

(section 4.3),

we

i s an open p o l y d i s c i n a

U

space with a basis.

We b e g i n b y r e c a l l i n g s o m e f u n d a m e n t a l f a c t s a b o u t

DN

spaces. S,

t h e space of rapidly decreasing sequences,

is the

Frgchet nuclear space with a b a s i s consisting of a l l sequences, m

(Zn)n=l

o f complex numbers such t h a t

is finite for all positive integers m

g e n e r a t e d b y t h e norms

( P , ) ~ = ~ .s

m.

The t o p o l o g y o f

is a universal generator

f o r t h e c o l l e c t i o n o f n u c l e a r l o c a l l y convex s p a c e s , i . e . ,

l o c a l l y convex s p a c e

E

is

s a

i s n u c l e a r i f and o n l y i f i t i s

isomorphic t o a s u b s p a c e o f

sA

f o r some i n d e x i n g s e t

A.

iz

depends on t h e c a r d i n a l i t y o f a fundamental neighbourhood system a t t h e o r i g i n i n

E.

In p a r t i c u l a r ,

s p a c e i s i s o m o r p h i c t o a c l o s e d s u b s p a c e of D e f i n i t i o n 5.38

Let

E

n.

E

i s a

DN

s

N

.

b e a m e t r i z a b l e ZocaZZy c o n v e x

s p a c e w i t h g e n e r a t i n g f a m i l y o f semi-norms

for a 2 2

any F r g c h e t n u c l e a r

m

(pn)n=l,

pn 6 pn+l

(dominated norm) space i f t h e r e i s a

263

Holomorphic functions on nuclear spaces with a basis

P

c o n t i n u o u s norm k

there e x i s t

E

on

s u c h t h a t for a n y p o s i t i v e i n t e g e r

a positive integer

n

and

such t h a t

C>O

The f u n d a m e n t a l r e s u l t c o n c e r n i n g n u c l e a r

spaces i s

DN

the following proposition. Proposition 5.39 DN

i s a

A metr2izable n u c l e a r l o c a l l y convex space

s p a c e i f and o n l y i f i t i s i s o m o r p h i c t o a s u b s p a c e

s.

of

Now l e t

be a Frgchet nuclear space with a basis.

E

is isomorphic t o

w

m

=

and

m

m

m,

for all

(Wm,n ) n = 1

for all

w h e r e w e may s u p p o s e

A(P)

w

~

+ 2 ~w

E

m

P = ( w ~ ) ~ = ~

, ~ for all m,n

m

and

n

(by t h e Grothendieck-Pietsch c r i t e r i o n f o r n u c l e a r -

ity). The c o l l e c t i o n where t h e is the

row i s t h e

mth

n th

may b e d i s p l a y e d a s a n i n f i n i t e m a t r i x

P

For e a c h p o s i t i v e i n t e g e r

vm and l e t

07

= {(zn)n=l

m

(ni)i=l

column

=

07

{(zn)n=l

m

let

E

o f a1

continuous weights on

E.

i s a s t r i c t l y increasing sequence

of positive integers with (n,)

n th

s u p z nwm,n n

E;

E

denote t h e se

[PI

Now s u p p o s e

LJ

weight and t h e

mth

coordinate.

n E;

1

=

SUP

n

.

Let

Chapter 5

264

nm s n

for

m

It is immediate that V

if

U(ni)i,l

k

E,

The sequence

I/(

C =

1

n

E

then

1

-

m

) n = l ( (1 .

a

C

K

then there exists a

V 3 U(nl,

K

...,n k ) ,

K is is contained in a compact If

)n=l lies in E . Let n Now choose a strictly increasing sequence

a = (

II ( O , O , ... ,o>- 1 N

U(ni)i,l

and

say

m

o f positive integers,

Then

m

such that

a compact subset of polydisc in

E

is a compact subset of

is a neighbourhood of

positive integer

m=1,2, . . . I .

nm+l'

m

C CU(ni)i,l

"i

(ni)i=l, n l = l ,

such that

1 , ... )I/

c

for all

and consequently t h e s e t s

CU(ni)i,l

,

a

n .1+1

i

r a n g e s o v e r a l l p o s i t i v e r e a l numbers and

i.

m

m

(ni)i=l

as

ranges

o v e r a l l s t r i c t f y i n c r e a s i n g s e q u e n c e s of p o s i t i v e i n t e g e r s with

n l = 1,

f o r m a f u n d a m e n t a l s y s t e m of c o m p a c t s u b s e t s o f

E. We now give a characterization of nuclear

DN

spaces

with a basis. Proposition 5.40 L e t E be a Frgchet nuclear space w i t h a basis. The f o l l o w i n g a r e e q u i v a l e n t :

265

Holomorphic functions on nuclear spaces with a basis

(a)

E

i s i s o m o r p h i c t o a s u b s p a c e of

(b)

E

i s a

(c)

E

i s isomorphic t o

a

m+l,n

( W m + ,l n )

Id)

E

2

w

m,n

,

<

wm,n

W

m

( w m , n 1 2 2‘

and

cw l , n

11.

m

where P = ( w ~ ) ~ = ~ , m and f o r e a c h

such t h a t for all n,

k,n

i s isomorphic t o A ( P ) w h e r e P = (w m ) mm = l ’ m for a l l m and t h e f o l l o w i n g w m = (Wm, n ) n = l hold: E

( i ) wm,.,

-

Bm,n

then

m

for a l l

> 0

(ii) i f

B,,,

W

-

2

m+l ,n W

m,n 1 all m

(Wm,n(Bm,nP)n=l

for a l l m

m

and n

and

n

and

[PI f o r any m and p.

E

positive integers Proof

and

there e x i s t a positive

C>O W

m

A(P)

m

positive integer k

for a l l

for all

(Wm,n ) n = 1

integer

m

m+2,n

i s isomorphic t o

wm =

(e)

A(P)

where for a l l m, wm = (Wm,n ) n = 1 for a l l m and n, and

P = (w m ) mm = l ’ W

s,

space,

DN

( a ) and ( b ) a r e e q u i v a l e n t b y p r o p o s i t i o n 5 . 3 9 .

do n o t p r o v e t h e e q u i v a l e n c e o f

(b),

( c ) and

(d) h e r e .

We

See

t h e n o t e s a n d r e m a r k s a t t h e end o f t h i s c h a p t e r f o r a r e f e r ence.

and

(c)=>(e)

n

.

we h a v e W

m + l ,n W

W

Since

W

4

-

m,n

6

m+2,n W

m,n

m+l,n W

(Wm+1,nI2

W

m+2,n

for a l l

m

and h e n c e

m+l,n

m+j+l,n W

W

Wm,n

m+j , n

for all positive integers

m,

n

and

j .

266

Chapter 5

fir'

W

Hence W

wm , n ( R m , n ) P and

m+p,n

m + l ,n w m , n ( ____ W m,n

=

j=O

(WIJ

m

A m n W

m+j , n

)P)n=l

E

m,

assuming ( e ) , t h a t

p

for all positive integers

[PI

and

m.

1, n

The case

p

m=l,

arbitrary is t r i v i a l .

is true for the positive integer

m

induction hypothesis there exist

C1>O

j

w

--

(c)+(e).

( e ) = > ( d ) . We f i r s t p r o v e b y i n d u c t i o n o n W

Wm+j+l,n

Now s u p p o s e t h e a b o v e

and f o r a l l

p.

By o u r

and a p o s i t i v e i n t e g e r

such t h a t for all

By c o n d i t i o n

(e) there e x i s t

C2> 0

n.

and a p o s i t i v e i n t e g e r

such t h a t for all

n.

Hence

c1 . c*

<

C = JC,. C2

where Thus

w.j , n w k , n

W

W

1,n

If we let

(

~

and

m+l,n )P< 1, n

p=2

I,

c

w

' c2

wt,n

= j+k.

&,n

for all

w e o b t a i n ( d ) and hence

completes t h e proof.

n

and

(e)=>(d).

This

k

267

Holornotphic functions on nuclear spaces with a basis Condition

(e) of p r o p o s i t i o n 5.40 a r o s e i n our study of

h o l o m o r p h i c f u n c t i o n s on F r g c h e t n u c l e a r s p a c e s w i t h a b a s i s a n d i s t h e o n l y o n e o f t h e a b o v e e q u i v a l e n t c o n d i t i o n s t h a t we s h a l l u s e f r o m now o n .

I n t h e o r i g i n a l p a p e r s on h o l o m o r p h i c

f u n c t i o n s on n u c l e a r s p a c e s , a F r g c h e t n u c l e a r s p a c e w h i c h s a t i s f i e d c o n d i t i o n ( e ) was known a s a B - n u c l e a r r e l a t i o n s h i p between B-nuclear

s p a c e s and

DN

space.

The

spaces with a

b a s i s was n o t i c e d a f t e r w a r d s . W

be a s t r i c t l y i n c r e a s (an)nz 1 m qCLn< m i n g sequence of p o s i t i v e r e a l numbers s u c h t h a t

Example 5 . 4 1

(a)

Let

a =

In=,

q, O
f o r some and l e t

E = A(P)

Let

where

f o r every positive integer

w

m,n

P =

=

1 m c1 )

(

m't

for all

n

( w ~ ) ~a n = d ~ wm =

m.

Since

m

and

n

m

(Wm,n ) n = 1

wm+l,n

is finite for a l l basis.

m

for all

and

E

m,

Furthermore,

and

i s a Frgchet nuclear space with a

n,

is a nuclear

If w e l e t

E

since

an=n

DN

space with a b a s i s

for all

n

we o b t a i n

c o m p a c t o p e n t o p o l o g y a n d i f we l e t we o b t a i n

s.

H(C)

with the

an = log(n+l) for a l l

n

268

Chapter 5 The s p a c e

i s c a l l e d a power s e r i e s s p a c e of i n f i n -

h(P)

i t e type and w e u s e t h e n o t a t i o n m

Let

(b)

~

~

(

af o )r t h i s s p a c e .

be a s t r i c t l y increasing sequence. "n q is f i n i t e for

a = (a,In=1

of p o s i t i v e r e a l numbers and suppose q, O
all

and l e t

P

for all

p

'

an f o r a l l n = p P.n (Wp,n);=l'O
Let

=

and

w

p',

.

O
( a ) and c a l l Po A power s e r i e s s p a c e

i t a power s e r i e s s p a c e of f i n i t e t y p e .

of f i n i t e type is not a E,

and

space.

DN

t h e n we o b t a i n

po=l

O
i s a FrGchet n u c l e a r

A(€')

0'

p,

We d e n o t e t h i s s p a c e b y

space with a basis.

n

and

I f we l e t

H(D),

an=n

for all

t h e open u n i t d i s c i n

D

w i t h t h e compact open t o p o l o g y . I t i s p o s s i b l e t o c o n s t r u c t f u r t h e r examples of

(c)

DN

s p a c e s w i t h a b a s i s by c h o o s i n g w e i g h t s o f even more r a p i d

For example,

growth. ers

m

and

all

m.

Let

Since

for all Also,

n.

w

for all positive integ-

= 2~~

m, n P = (w ) " m m=l

wm =

where

( W m ,n ) n = l

for

is a Frcchet nuclear space with a basis.

A(P)

m,

let

since

w

W

m,n

(---

m+l ,n W

m,n

I

m

P

2n

=

(2"

f o r any p o s i t i v e i n t e g e r s large,

h(P)

is a

DN

m+l

IP

.(2n

m

space.

m

6

2"

m+2

IP

and

p

and a l l

n

sufficiently

I t c a n b e shown t h a t

n o t i s o m o r p h i c t o a power s e r i e s s p a c e .

A(P)

is

269

Holomorphic functions on nuclear spaces with a basis

I n p r o v i n g o u r m a i n r e s u l t , w e s h a l l make f r e q u e n t u s e o f t h e f o l l o w i n g e q u a l i t y c o n c e r n i n g t h e supremum o f a m o n o m i a l o n a polydisc:

co

V = { ( z , , ) ~ = ~ s; u p ( z n a n l

if

Theorem 5 . 4 2

Proof

By c o r o l l a r y 5 . 3 1 ,

H(E)

(nl,:..,nk)

i n t e g e r s with

is

continuous.

T

m

I1

in

a positive integer

H(U).

on

T~

T~

on

= T~

that

T

0

Let

integer

T~

N"),

= T

0'

bounded

T

be such a

k

there exist

6

and

c(k) > 0

c(k)

II zmll

such t h a t

6U(nl'

* .

T h e n we c l a i m ,

N").

j

and

c(k+l)>1

*

,nk) if

6' > 6,

there exist

such t h a t

I f n o t , w e can choose f o r each p o s i t i v e i n t e g e r

N"). E

"1

N

a s t r i c t l y i n c r e a s i n g sequence of p o s i t i v e

nl = 1

IIT(zm)

I

=

i t s u f f i c e s t o show

S u p p o s e f o r some n o n - n e g a t i v e

m.

T~

which i s bounded on t h e

H(E),

subsets of H(E), linear functional.

f o r every

m E

i t s u f f i c e s t o show t h a t a n y B a n a c h v a l u e d l i n e a r

f u n c t i o n a l on

in

then

E,

S i n c e we h a v e a l s o s h o w n , e x a m p l e 5 . 3 6 ( a ) ,

H(E).

6>0,

and

is an o p e n p o Z y d i s c i n a c o m p l e t e

U

s p a c e w i t h a basis,

DN

nuclear

on

If

11

1.

m = (ml,m 2 , . . . , m n , 0 , . . .

and

$

j,

such t h a t

We f i r s t s h o w t h a t

O t h e r w i s e we c a n c h o o s e s u p / m j j = m. m I and a p o s i t i v e i n t e g a s t r i c t l y i n c r e a s i n g sequence (jn)n=l, er

J

such t h a t

Imj

n

I

= J

for all

jn.

Since

270

Chapter 5

U (nl,.

.. ,nk,

j+n,) 3 V k c l

for all

j

t h e sequence

i s u n i f o r m l y bounded on a f i x e d n e i g h b o u r h o o d o f z e r o i n and hence i s a

bounded s u b s e t of

T

and

6"('E)

E

H(E).

This contradicts the fact that

By t a k i n g a s u b s e q u e n c e i f n e c e s s a r y we may t h u s

for all

n.

suppose

Imj/

-f

+m

as

j

-f

m.

For each j l e t m . = ( r . , s . ) where r . are t h e f i r s t 1 1 1 I j + n -1 c o o r d i n a t e s a n d s. are t h e remaining coordinates of k 3 m.. We i d e n t i f y r . a n d s . w i t h e l e m e n t s o f N") in the 3 1 I usual fashion. For a l l j and a l l z m. r. s . z J = z J z J .

C o n s i d e r t h e f o l l o w i n g two p o s s i b i l i t i e s :

5.

where point and

(**)

S .

i s t h e v a l u e o f t h e monomial

(@k,n)_ (@k,n)n=l.

Since

B

~ 2 , 1 ~f o r a l l

cover a l l p o s s i b i l i t i e s .

z

k

and

at the

n

(*)

Holomorphic functions on nuclear spaces with a basis We f i r s t s u p p o s e t h a t

Hence

(*)

is satisfied.

27 1

Chapter 5

212

m

for all

N"),

E

we m u s t h a v e

This is a contradiction,

since

and h e n c e

6' > 6 ,

(*)

cannot

hold. We now s u p p o s e

(**)

holds.

By t a k i n g a s u b s e q u e n c e i f

n e c e s s a r y we may s u p p o s e

and

i s a s t r i c t l y increasing sequence.

lmjl Let

m. f(z)

lj=1 m

=

m IIZ

S i n c e e a c h monomial

Z J

'11

U(nl,

i s c o n t i n u o u s and

imp,lies t h a t

theorem 2 . 2 8

. . . ,n k , j + n k )

f

n

let

Q

;;In

m

(an)n,l

= anun

E.

be an a r b i t r a r y element o f

R

where

E.

For each

m

( u ~ ) ~i s= t ~h e u n i t v e c t o r b a s i s o f

a p o s i t i v e i n t e g e r such t h a t 1 2 l e t m j , m j , . . . , m! all nbR. For each j J coordinate of m . E N ") and l e t 3 E.

Choose

=

i s a Frgchet space,

i s an e n t i r e function i f t h e

above s e r i e s converges a t a l l p o i n t s of Let

E

d

an

E

Vk+l

for

be t h e f i r s t

27 3

Holornorphic functions on nuclear spaces with a basis

We h a v e 1

m a j

6

m.

'(/u(nl,. ..

11'

2

m . m. clJ c 2 J .

. .

II m.

ceJ ~ ( a , m ~ )

, n k , j+nk)

where

for all

j

such t h a t

j>a

( t h e t e r m s between

II

and

j+nk

a r e a l s o l e s s t h a n o n e b u t we n e e d a s h a r p e r e s t i m a t e ) . NOW

given any p o s i t i v e i n t e g e r

and h e n c e

nk+j

have

>

II1;

and t h u s

l a w k , n ( ~ k , n ) P I6 1

p,

for all

in particular for all

j

E )[ P ( w ~ , ~ ( R ~ , ~ ) P ; I= ~

n

2

k1

> II.

Hence i f

s u f f i c i e n t l y l a r g e , we

Chapter 5

214 where

for Since all

As

1s

i h 1.

i m. 5 / m j l 1 i . Hence

for all

is greater than

w

zero and h e n c e

f

E

1

and

i

and

P

j

we h a v e

0 < r . <1

for

is arbitrary, the l i m i t is

H(E).

Hence N

f(z)

=

1"

m.

1

j = 1 (6') lmj

Z

I

i s a l s o an e n t i r e f u n c t i o n on

is a that

T

bounded s u b s e t o f

J

~~zmjl~U(n . .l ,,n. k , j + n k )

E.

H(E).

The s e q u e n c e

This contradicts the f a c t

275

Holornotphic functions on nuclear spaces with a basis

and e s t a b l i s h e s o u r c l a i m . m

c o n t i n u o u s and (nVl)n=l i s an i n c r e a s i n g T6 c o u n t a b l e open c o v e r o f E there exist c ( l ) > O and 61 a Since

is

T

p o s i t i v e i n t e g e r such t h a t

f

f o r every

in

H(E).

In p a r t i c u l a r

for all

m

E

m

("In=2

Let

nl

where

N

= 1

be a sequence o f p o s i t i v e r e a l numbers, m

6, = 6 i s f i n i t e . By t h e n=l a b o v e , we c a n c h o o s e i n d u c t i v e l y a s t r i c t l y i n c r e a s i n g s e q u e n c e all

6,>l

n,

such t h a t

m

(nk)k=l, n l = 1 ,

of p o s i t i v e integers, p o s i t i v e numbers

and a s e q u e n c e o f

( ~ ( k ) ) ; , ~ such t h a t

. . ., n k )

Let

m

K = 6U (ni)i=l.

If

V

lemma 5 . 1 8 , E > 1 all n

K

i s a compact p o l y d s c i n

i s any neighbourhood o f

K

a sequence o f r e a l numbers, 1 n and < m, and W

L=l

n

E.

t h e n we c a n c h o o s e , b y E

=

m

( E ~ ) ~w i=t h ~

a neighbourhood o f

276

Chapter 5 E(K+W)C

zero such that K

V.

Since

there exists a positive integer S

...

SkU(nl,.

Hence, €or any

€

E

. . ,nk)C

K+IV

k

is a neighbourhood of such that

6 U(nl,.

. . ,nk) C K + W

H(E),

(proposition 5.25)

V was arbitrary, this shows that compact subset K o f E. Hence T is this completes the proof. Since

T

is ported by the

T

continuous and

w

Theorem 5.24 immediately leads to a strengthening of some

of our earlier results. (a)

If

E

particular if

The following are now easily verified.

is a nuclear E

=

s or

H((C))

DN

space with a basis (in then

(H(E),ro)

is a reflex-

ive A-nuclear space. (b)

If

U

is an open polydisc in a nuclear

with a basis, then basis.

(H(U),.ro)

DN

space

is a fully nuclear space with a

(c) If E i s the strong dual of a nuclear D N space with a basis then H(OE) = lim (Hm(V), I/ is a complete +

V30,V open

]Iv)

271

Holomorphic functions on nuclear spaces with a basis regular inductive l i m i t . Thus w e h a v e examples o f non-rnetrizable

l o c a l l y convex

s p a c e s i n which t h e s p a c e o f germs a b o u t t h e o r i g i n i s c o m p l e t e and r e g u l a r . I n c h a p t e r 6 , w e prove, u s i n g t e n s o r p r o d u c t s and a r e s u l t of Grothendieck,

I f

on

55.4

E

H(E)

t h e following converse t o theorem 5.42.

i s a F r g c h e t n u c l e a r s p a c e w i t h a b a s i s and then

E

is a

DN

T

~

=

space.

TOPOLOGICAL PROPERTIES INHERITED B Y STRICT INDUCTIVE LIMITS A N D SUBSPACES The r e s u l t s a n d m e t h o d s d e v e l o p e d i n t h e f i r s t t h r e e

s e c t i o n s o f t h i s c h a p t e r h a v e a number o f i n t e r e s t i n g c o n s e quences such as a k e r n e l s theorem f o r a n a l y t i c f u n c t i o n a l s , regularity r e s u l t concerning

H(K),

where

compact s e t i n c e r t a i n n o n - m e t r i z a b l e and r e p r e s e n t a t i o n theorems f o r spaces of i n f i n i t e type.

K

a

is an a r b i t r a r y

l o c a l l y convex s p a c e s ,

H(Am(a)i)

as power s e r i e s

Some o f t h e s e w i l l b e d i s c u s s e d i n

t h e next chapter. We c o n f i n e o u r s e l v e s i n t h i s s e c t i o n t o a p p l i c a t i o n s w h i c h

y i e l d new e x a m p l e s c o n c e r n i n g t h e r e l a t i o n s h i p b e t w e e n t h e topologies

TO , ~ W

and

T&.

As w e have a l r e a d y had

s u c c e s s w i t h F r z c h e t n u c l e a r and

a4'n

some

spaces, it is n a t u r a l

t h a t w e i n v e s t i g a t e holomorphic f u n c t i o n s on s u b s p a c e s and i n d u c t i v e and p r o j e c t i v e l i m i t s of t h e s e s p a c e s . The p r o j e c t i v e l i m i t c a s e i s n o t v e r y s a t i s f a c t o r y ( e . g . consider

(EN x

C"))

and any p o s i t i v e r e s u l t s w e o b t a i n i n t h i s

d i r e c t i o n are g i v e n i n t h e s e c t i o n on s u r j e c t i v e l i m i t s i n chapter 6.

Since arbitrary inductive l i m i t s are too general

we c o n f i n e o u r s e l v e s t o

(countable)

s t r i c t i n d u c t i v e l i m i t s of-

T

~

Chapter 5

218

Frgchet n u c l e a r spaces ( t h e s t r i c t i n d u c t i v e l i m i t of & J 4 L spaces i s a

8JQ

This class,

a s we s h a l l s e e , i s r a t h e r r e s t r i c t i v e b u t d o e s

s p a c e a n d s o l e a d s t o n o new e x a m p l e s ) .

y i e l d new n o n - t r i v i a l e x a m p l e s .

The t e c h n i q u e s u s e d ,

from t h o s e d e v e l o p e d i n t h i s c h a p t e r , those used f o r

a

3 q

apart

a r e somewhat s i m i l a r t o

s p a c e s a n d d i r e c t sums o f Banach s p a c e s

Our i n v e s t i g a t i o n s i n t o h o l o m o r p h i c

( c h a p t e r s 1 , 2 a n d 4).

f u n c t i o n s on s u b s p a c e s l e a d t o a correspondence between h o l o m o r p h i c e x t e n s i o n t h e o r e m s ( s e e c h a p t e r 4 f o r t h e Banach s p a c e c a s e ) and t h e c o m p l e t e n e s s o f q u o t i e n t s p a c e s o f h o l o m o r p h i c functions. We b e g i n b y d i s c u s s i n g h o l o m o r p h i c f u n c t i o n s o n f u l l y n u c l e a r s p a c e s w i t h a b a s i s which can b e r e p r e s e n t e d a s a s t r i c t inductive l i m i t of Frgchet nuclear spaces.

A typical

e x a m p l e o f s u c h a s p a c e i s t h e c o u n t a b l e d i r e c t sum o f F r g c h e t nuclear spaces with a b a s i s .

Our n e x t r e s u l t shows t h a t t h i s

i s , i n f a c t , t h e o n l y p o s s i b l e example. Lemma 5 . 4 3

E = l i m En

Let

be a s t r i c t i n d u c t i v e l i m i t o f E

F r g c h e t n u c l e a r spaces a n 2 s u p p o s e Then

OD

E

t h e n each Proof

En

DN

is a

space

space w i t h a b a s i s .

m

Let

i s a Frgehet n u c l e a r space

M o r e o v e r , if e a c h

DN

is a

Fn

Fn

where each

l n = 1 Fn

w i t h a Schauder b a s i s .

has a Schauder b a s i s .

(en)n=l

be a Schauder b a s i s f o r

Since

E.

E

m

i s an a b s o l u t e b a s i s f o r i s a f u l l y nuclear space, (en)n=l E. For each p o s i t i v e i n t e g e r n let Fn be t h e c l o s e d sub-

space of Since

E

g e n e r a t e d by

FnCEn

follows that

all

a l s o i s each

e

m .d E j

j
for

is a s t r i c t inductive l i m i t ,

E

and i f each

n

Let

Fn.

F =

En

is a

En

m

Fn.

DN

space then so

Using t h e n a t u r a l

i n j e c t i o n s from

Fn

inductive l i m i t ,

we o b t a i n a c o n t i n u o u s i n j e c t i o n II

into

E.

If

it

i s a Frgchet n u c l e a r space with a Schauder

Fn

b a s i s f o r each

I e m ;e m € E n ,

and

n

x =

into m

i s a bounded s e q u e n c e i n l i m i t is regular,

and t h e c o n s t r u c t i o n of an

xnen€E E,

then t h e sequence

from

F

{xnen}t=l

and s i n c e e v e r y s t r i c t i n d u c t i v e

there exists a positive integer

m

such

279

Holomoiphic functions on nuclear spaces with a basis that

Em

xnen'

n.

for all

Hence

{en;xn

# 01 C E m

as

and,

the basis is absolute

m

i s a Cauchy s e q u e n c e i n s u r j e c t i v e mapping.

and

Fn

X E

Thus

F.

By t h e o p e n m a p p i n g t h e o r e m ( b e t w e e n

c o u n t a b l e i n d u c t i v e l i m i t s o f B a i r e s p a c e s ) TI phism and hence

W

E =

Proposition 5.44 DN

Proof

By lemma 5 . 4 3 ,

nucZear s p a c e s .

is a

En

b e a fuZly n u c l e a r s p a c e w i t h a

can be r e p r e s e n t e d a s a s t r i c t i n d u c t i v e

Z i r n i t of

each

E

Then

=

T~

we may s u p p o s e

-r6 E =

space with a basis.

DN

i s a homeomor-

This completes t h e proof.

Fn.

Let E

b a s i s and s u p p o s e

is a

IT

H(E).

on

ln=lE n W

Hence

where

is a

E

r e f l e x i v e A - n u c l e a r s p a c e a n d s o , b y lemma 5 . 1 a n d p r o p o s i t i o n 5.28,

Let

i t s u f f i c e s t o show

T

E

(H(E),T~)I.

For each

n

choose a neighbourhood o f zero i n

s u c h t h a t e a c h monomial on possible since each then

n

W

z = (Zn)n=l

j =

IW

(jn,m n,m=l h(j)

E

z n c En

where

Hence e a c h monomial on

This i s

Vn.

a d m i t s a c o n t i n u o u s norm.

En

sufficiently large.

i s bounded on

En

En,Vn,

For each

E

( ~ ~ 1 " ) = sup{ncN;3

all n,

n zn

i s indexed by

and

zn

If = 0

pl

= (Zn,m)m=l

z

E

E

for all

and so

(N2)").

If

we l e t

m

such t h a t

We c l a i m t h e r e e x i s t s a p o s i t i v e i n t e g e r

Z n

0).

such t h a t

280 if

Chapter 5 jE(N2)")

and

h(j) >no,

thenm

b y r e s t r i c t i n g o u r s e l v e s t o some

can f i n d a s e q u e n c e . i n h(jn) = n

T(zJn) # 0

and

s e q u e n c e of complex numbers. l

m

subset of Let

i f n e c e s s a r y , we

k (jn)n=l,

n.

for all

such t h a t m

Let

be any

We f i r s t s h o w t h a t

i s a l o c a l l y bounded,

{ a n z n}n,l

l k = lE n m

(N2)"),

I f not, then

T(zj) =O.

and hence a

bounded,

T &

H(E). be a r b i t r a r y ,

say

. . . ,L

I/anzJn((

where

L'>R

and

ZEE

M = 1 + sup n=l,

Now s u p p o s e

z

E

R

En.

Let

V = z + V1 x v 2

x...x

VQ

c ~ + ~ , . . . , c a~ r~e p o s i t i v e r e a l

numbers s u c h t h a t

If

c > 0

then

is positive since

where

e(L')

Hence,

by choosing

c

h[jQ,+l) = R'+l.

s u f f i c i e n t l y s m a l l a n d p o s i t i v e , we

have

Since

h(jn)

=

n

t h e same e s t i m a t e a l s o h o l d s f o r a l l

jn anz , n i Q 1 . T h u s we c a n c h o o s e a s e q u e n c e o f p o s i t i v e r e a l m numbers ( c ~ ) & =s ~ uch t h a t

Holomorphic functions on nuclear spaces with a basis T h i s shows t h a t t h e sequence i s a very strongly

Jn

)mnZ1

is locally

w e also see t h a t t h e sequence

in fact,

and,

bounded

(an z

28 1

convergent sequence i n

T&

{zjnlm

n=1

H(E).

n

contini?ous. Hence t h e r e e x i s t s a p o s i t i v e i n t e g e r Jn "0 such t h a t T(z ) = 0 if h(jn) > no. Let F = n = lE n ' is

F

T~

1

is a

space with a basis.

DN

0

If

t h e n , by t h e above,

Since

on

i s a complemented s u b s p a c e o f

F

H(F)

T(flF) = T(f)

As t h e b a s i s i n

F

1 a m T ( z m )I N

meN

for every

By t h e o r e m 5 . 4 2 , c>O

N

and

K

is a

T

N

T

T

is

is a

T

c /(f/IK f o r every

,<

H(E).

N (N)

E

E

a z m

m

we h a v e E

H(F).

c o n t i n u o u s and h e n c e t h e r e

f

F

in

such t h a t H(F).

and so f o r any

c o n t i n u o u s l i n e a r f u n c t i o n a l on

T

Since

c o n t i n u o u s semi-norm on

T~

a compact s u b s e t o f

i s a l s o a compact s u b s e t o f

T

1

mE

exist

Hence

in

f o r every

H(F).

l'?(f)l

f

extends t o a basis i n

<

(H(E),T~) is barrelled,

K

N

by t h e formula

Lu

1

w e may d e f i n e

E

f

in

H(E)

and t h i s c o m p l e t e s t h e p r o o f . Example 5 . 4 5

If

n

i s any open s u b s e t o f

Rn

then

H(E)

282

3

Chapter 5

,$ (Q)

=

(the space of

f u n c t i o n s on

& ,m

w i t h compact

s u p p o r t endowed w i t h t h e s t r i c t i n d u c t i v e l i m i t t o p o l o g y o f

&

the spaces Hence =

To

= T

T~ T

0

on

,b

[-n,+n], n

on

6

H(E).

H(U)

Example 5 . 4 6 each T

on

w

Then

=

T~

on

W

and hence

.

in 8

U

i s a .83Q En i f and o n l y i f

where each T

=

H(E;)

In p a r t i c u l a r

H(2').

each

If

m

E = n=l En

Let

s")

1",1S we a l s o h a v e

By c o r o l l a r y 5 . 3 1 ,

i s a f i n i t e dimensional space.

En

#

i s isomorphic t o

)

f o r every open p o l y d i s c

space with a basis. T~

Z)

E

is a f i n i t e dimensional space then

En =

T~

may s u p p o s e

T

on

W

H(E)

by example 5 . 3 6 .

s cN

O t h e r w i s e , we

i s an i n f i n i t e dimensional space.

El

EB

By

c o r o l l a r i e s 5 . 2 3 a n d 5 . 3 5 , w e may c o m p l e t e t h e p r o o f by s h o w i n g t h e e x i s t e n c e o f a non-convergent ,T~),

Let

(Vn);=l

the origin in I J J ~ E

f

=

Cauchy s e q u e n c e i n

.

(H(E;)

be a fundamental neighbourhood system a t

(El);.

En

such t h a t

m

n

For each

Ilonl\

=

'n

n

choose

x:=l(En)E;

f o r some i n t e g e r

I:=1

n I I $ n + n IIK

f o r e a c h compact s u b s e t

of

E

El

and

for all

n.

Let

Cn=l+nQn* S i n c e each compact s u b s e t o f

in

9,

$n # 0

and

m

f

i s c o n t a i n e d a n d comp-act

we h a v e

N(K) n Zn=1 ll$n$nllK

=

of

K

k,

E' B

E'.

form a Cauchy s e q u e n c e i n

B

m

(Wn)n=l,

zero i n

such t h a t

f o r each

n,

m

H e n c e t h e p a r t i a l summands If

HIE;).

t h e r e would e x i s t a s e q u e n c e (En);

<

Wn

t h i s i s i m p o s s i b l e and completes t h e p r o o f .

f E

H(Eb)

then

a neighbourhood of

28 3

Holornorphic functions on nuclear spaces w i t h a basis

Ne now c o n s i d e r h o l o m o r p h i c f u n c t i o n s o n c l o s e d s u b s p a c e s o f We f i r s t p r o v e t h e f o l l o w i n g lemma

f u l l y nuclear spaces.

c o n c e r n i n g l i n e a r f u n c t i o n a l s on a s u b s p a c e o f a f u l l y n u c l e a r space. Lemma 5 . 4 7

Let

n u c l e a r space

E

Proof

Since

space,

b e a c l o s e d s u b s p a c e of t h e f u l l y

F

t h e n , if

FL

= I $ E E ' ; $ I ~= 0 } ,

is a closed subspace of a f u l l y nuclear

F

it i s a complete n u c l e a r space and hence i s s e m i - r e f l e x -

Thus F' i s a b a r r e l l e d space and (F')'d F. Also E' B B B i s a c o m p l e t e r e f l e x i v e s p a c e and hence EA/FI is a barrelled

ive.

l o c a l l y convex s p a c e . spaces.

Thus

B

FA

and F' a r e b o t h Mackey B s u f f i c e s t o show

T O

n

E

E'

To c o m p l e t e t h e p r o o f , / i t )'

If

T

=" F .

( E i / F ~ ) '

E

then

c a n o n i c a l p r o j e c t i o n from reflexive,

E l

t h e r e e x i s t s an

B

x

in

$ ( x ) # 0.

Since

$

E

F

,

n($)

This contradicts the fact that

By t h e H a h n - B a n a c h

F.

such t h a t $(F) = 0 and t h u s

= 0

$(x) # 0

T O

and s o

This completes t h e proof.

If

E

?IF

H(E);flF = g}.

and I f we l e t

= 01

HE(F) r:

r e s t r i c t i o n m a p p i n g f r o m H(E) into E E Kernel ( r F ) and HE(F) = r a n g e ( r F ) Theorem 5 . 4 8

n u c l e a r space

Let E.

F

.

=

n($)

and = 0.

F ' = Ei/FI. B

i s a subspace o f a l o c a l l y convex s p a c e

F

H(F)I = If such t h a t

is the

Il

Since E is A/F I' such t h a t T o n ( $ ) = $(x)

E

L

where

B

onto

f o r every $ in E' Suppose x B' in E' theorem t h e r e e x i s t s a $ I

E'

E

we l e t

{ g E H ( F ) ; 3 Z E H(E) denote t h e natural

H(F)

then

H(Ff

b e a c l o s e d s u b s p a c e of a f u z z y

Then (HE(F),ro)

(by

=

284

Chapter 5

Proof

E rF : H(E)

The mapping

s u r j e c t i o n and h a s k e r n e l

-t

i s a continuous

HE(F)

H(F)L.

We c o m p l e t e t h e p r o o f b y

showing t h a t t h e i n v e r s e mapping from

HE(F)

i t s u f f i c e s t o show

By t h e o r e m 1 . 2 7 a n d p r o p o s i t i o n 1 . 4 1 , t h a t f o r e a c h compact s u b s e t subset

of

M

f o r every If that

and

F

f

in

c > 0

~ ~ ( $= 1 i n f { l ( $ I / K : T E E ' ,

subset

If

all in

of

L

ji

E

suppose i

F,

n

6

P

lT=l

and

=

$1,

( n ~ ) . If n

))IJJ~/\

liZl

of zero i n

W

\]Ti\\n

<

m

-

and

Y n (jii) = P

$EF',

is a positive integer,

<

m

E

yiiF

in

is a

then

W

pIF = li=l(jii)n

where

f o r some n e i g h b o u r h o o d

t h e n , b y t h e Hahn-Banach theorem,

bourhood

l i m m-

and

@lF

t h e n lemma 5 . 4 7 i m p l i e s

E

nl

and h e n c e t h e r e e x i s t s a compact

F' B such t h a t

F

E'

t h e r e e x i s t a compact

H(E).

i s a compact s u b s e t o f

K

E

of

such t h a t

c o n t i n u o u s semi-norm on

NOW

K

onto

and = jii

Ti

€

all

( H ( E ) , T ~ ) and

all

i.

Since P

V

E

of

F' 0

there e x i s t s a neigh-

E',

N

jii

lF

=

i,

PiF

such t h a t

w e have

285

Holomoiphic functions on nuclear spaces with a basis

Hence

= nL(p)

f o r every

8 ("E).

in

P

If f = C ___ n=O n!

then

9

Hence t h e r e e x i s t s in

H(E).

such t h a t

c>O

q ( f ) ,< c n L ( f )

for every

An a p p l i c a t i o n o f p r o p o s i t i o n 1 . 4 1 i m p l i e s t h e

e x i s t e n c e o f a compact s u b s e t q ( f ) 6 cllf/(M f o r every

f

M

in

of

such t h a t

F

H(E).

This completes t h e

proof.

We now a p p l y t h e o r e m 5 . 4 8 t o t h e f o l l o w i n g p r o b l e m s :

(a)

When i s

HE(F) = H(F)?

f u n c t i o n on

i.e.

when c a n e v e r y h o l o m o r p h i c

b e e x t e n d e d t o a holomorphic f u n c t i o n on

F

E? (b)

H(E)

What t o p o l o g i c a l p r o p e r t i e s o f

are inherited by

H(F)? S i n c e e v e r y c o n t i n u o u s l i n e a r form on Hahn-Banach

theorem,

extends,by the

t o a continuous l i n e a r form on

t h e polynomials o f f i n i t e t y p e on it f o l l o w s t h a t

F

HE(F)

F

are

T~

i s a dense subspace o f

E

and

dense i n H(F)

, T ~ ) .

H(F) W e

immediately o b t a i n t h e following c o r o l l a r i e s t o theorem 5.48: Corollary 5.49

Let

F

b e a c l o s e d s u b s p a c e of t h e f u l l y

f

286

Chapter 5 E.

n u c l e a r space

i s complete then every hoZomorphic f u n c t i o n on

(bl

F

( H ( E ) , T ~ ) is c o m p Z e t e ,

If

F

f u n c t i o n on

Corollary 5.50 E

space

t h e n e v e r y hoZornorphic

e x z e n d s t o a hoZornorphic f u n c t i o n o n

if and o n l y i f

E

( H ( E ) . T ~ ) / ~ ( ~is) ~c o m p Z e t e .

F

If

F

We h a v e a l r e a d y s e e n ( c o r o l l a r y 3 . 5 6 )

i s a Frgchet nuclear space. (H(E)

Since

H(F)l

it f o l l o w s t h a t

, T ~ )

extends t o a

E.

h o l o m o r p h i c function o n

space of

83g

is a c Z o s e d s u b s p a c e o f a

t h e n e v e r y h o Z o m o r p h i c function o n

Proof

E.

e x t e n d s t o a hoZornorphic function o n

a l s o a Frgchet nuclear space.

that

(H(E),T~)

i s a closed sub-

( H ( E ) , T ~ ) / . ~ ( ~ i) s ~

An a p p l i c a t i o n o f c o r o l l a r y

5 . 4 9 ( a ) now c o m p l e t e s t h e p r o o f . Corollary 5.51

If E

nucZear space

is a c Z o s e d s u b s p a c e of a F r g c h e t

F

t h e n e v e r y h o Z o m o r p h i c function o n

F

extend

E i f and onZy if is a c o m p Z e t e ZocaZZy c o n v e x s p a c e .

e x t e n d s t o a hoZomorphic f u n c t i o n on (H(E)

, T ~ )

/H(F)l We now t u r n t o problem ( b ) . If Proposition 5.52 T~

= T

6 subspace

on F

Proof p

If

: H(E)

then

p +

be a R

T~

on

H(F)

for a n y c Z o s e d

c o n t i n u o u s semi-norm on

T&

b e d e f i n e d by t h e formula

$(f)

H(F). = p(flF).

i s an i n c r e a s i n g c o u n t a b l e o p e n c o v e r o f

E

Wnzl

F.

C

C

\\f\\v,

consequently p

=

i s an i n c r e a s i n g countable open cover of

(Vn n

N

i s a fuZZy n u c Z e a r s p a c e and

(vn);=l

Hence t h e r e e x i s t s

T(f)

E T~

E.

of

Let d

Let

H(E)

c > 0

for all T~

f

and

N

E

H(E).

c o n t i n u o u s on

i n d u c e s a c o n t i n u o u s semi-norm

then

a p o s i t i v e i n t e g e r such t h a t Hence H(E). q

on

4. /

p

Since

is @ p

-r6

and

= o

JH(F)L (H(E),T~) / H ( F ) ~ .

By

287

Holornotphic functions on nuclear spaces with a basis t h e r e e x i s t a p o s i t i v e number

theorem 5.48,

K

subset

f o r every p(f)

F

of f

in

H(E).

y(f)

Hence,

In particular

6 clIflIK.

@(nE).

such t h a t

= p(fIF)

= q(f+H(F)

f o r every p(p)

f

c(~PI(~

6

and a compact

c in

L

) 6 c(lf(lK

HE(F),

f o r every

P

in

Hence

n

m

.

d n f ( o ) E H(E) By t h e o r e m 3 . 1 9 , t h e h o m o g e n In=, n! eous polynomials form an a b s o l u t e decomposition f o r (H(E),ro) f o r every and t h u s

is

p

T

continuous.

0

This completes t h e proof.

A similar r e s u l t holds f o r the

Proposition 5.53

n u c l e a r space

F

If

E

and

p

be a

is a c l o s e d s u b s p a c e o f a f u l l y = T

T~

topology.

T~

H(E)

on

then

= T

T~

on

H(F).

Let

Proof Suppose

p

i s p o r t e d by t h e compact s u b s e t

N

define

on

p

H(E)

&

p(f)

If

c o n t i n u o u s semi-norm on

T~

bourhood o f

= p(fIF)

in

K

We

by t h e formula f o r every

i s a neighbourhood of

W

F.

of

K

H(F).

F.

K

f

in

in E

H(E).

then

Hence t h e r e e x i s t s

Wn F

i s a neigh-

c(Wn F ) > O

such

that

Thus N

P

1

N

p

H(Ff

semi-norm

c o n t i n u o u s semi-norm on

is a = 0

q

and

T~

= T

H(E),

.

on

e x i s t a p o s i t i v e number

0

on

c

N

p

H(E).

Since

defines a continuous

By t h e o r e m 5 . 4 8 ,

and a compact s u b s e t

L

of

there

F

288

Chapter 5

f o r every

f

in

H(E).

i s now c o m p l e t e d a s i n t h e p r e c e d i n g p r o p o s i t i o n .

The p r o o f

A s c o r o l l a r i e s t o t h e above p r o p o s i t i o n s ,

we may d r o p t h e

b a s i s r e q u i r e m e n t i n some o f o u r p r e v i o u s r e s u l t s . Corollary 5.54

then

T~

=

E

I f

on

T~

is a c o m p l e t e n u c l e a r

E

i s a complete n u c l e a r

s p a c e i f and o n l y i f i t i s a c l o s e d s u b s p a c e o f 5.42,

T~

=

on

T &

space

H(E).

By p r o p o s i t i o n 5 . 3 9 ,

Proof

,DN

H(s)

s.

DN

By t h e o r e m

and an a p p l i c a t i o n o f p r o p o s i t i o n

5 . 5 2 now c o m p l e t e s t h e p r o o f . A similar proof using proposition 5.44, =

T

T &

on

H(F)

whenever

F

shows t h a t

i s a c l o s e d sub'space o f

2

.

We a l s o o b t a i n a new p r o o f o f c o r o l l a r y 3 . 5 4 Corollary 5.55 T

0

=

on

T

Proof

E

If

E

i s a Fre'chet n u c l e a r s p a c e t h e n

H(E). i s a closed subspace of

nuclear space with a basis, To

=

15.5

on

T

N

.

sN

i s a Frgchet

and h e n c e by example 5 . 3 6 ( a ) ,

~ ( s ~ )B Y. p r o p o s i t i o n 5 . 5 3 ,

To

=

T

0

on

H(E).

EXERCISES

5 . 5_ 6* _ A-nuclear 5.57* -

s

Show t h a t a F r g c h e t n u c l e a r s p a c e w i t h a b a s i s i s a n space. Let

(E,T)

be an A-nuclear space.

t h e t o p o l o g y o f u n i f o r m c o n v e r g e n c e on

the

convex modularly d e c r e a s i n g s u b s e t s o f

E'.

Show t h a t u(Ei,E)

T

com~act

is

289

Hobmorphic functions on nuclear spaces with a basis 5.58 -

Show t h a t a n y B a n a c h s p a c e w i t h a n a b s o l u t e b a s i s i s

isomorphic t o

R

If

5.59* -

~

.

i s a s t r i c t inductive l i m i t of Frgchet

E = l i m En --f

s p a c e s and E

n has an unconditional b a s i s ,

E

where each

Fn

show t h a t

i s a Frgchet space w i t h an uncon-

Fn

ditional basis. 5.60 -

If

A

~

i n f i n i t e type,

exists

h m ( ~ ) a r e p o w e r s e r i e s s p a c e of

show t h a t

A

If

c an

log(n+l) a n

5.62 -

<

m )

6

an > O

n

all

5.63 -

when

g

If

_ 5 . 6_ 4

B,

6 Can

for all

n.

(xn)”,=,

H(C), g (n)

g(z) =

+ 0

as

AR(a)

(resp.

0

=

R i m ) .

lies in

l:=Oan~ n +

n

s

i f and o n l y

,

where

m .

i s a n open p o l y d i s c i n a f u l l y n u c l e a r space

U

If

in

such t h a t

with a basis,

~ ~ ( 5i f) a n d o n l y i f t h e r e

log(n+l) an (resp.

l i m n+R <

Show t h a t a s e q u e n c e

if there exists a

c

i s a p o w e r s e r i e s . s p a c e , show t h a t

AR(a)

i s n u c l e a r i f and o n l y i f sup n

(a)

1-

such t h a t

C > 0

5.61

~ an ) d

(

-

show t h a t

U = U

MM .

i s a l o c a l l y convex s p a c e w i t h t h e approx-

E

imation property

(i.e.

t h e i d e n t i t y mapping on

E

can be

u n i f o r m l y a p p r o x i m a t e d on compact s e t s by f i n i t e r a n k o p e r a t o r s ) show t h a t t h e a l g e b r a g e n e r a t e d b y (H(U),To) _ 5 , 6_ 5*

If

n

, T ~ )=

5.66* basis,

and

E’

i s dense i n

U

of

E.

i s a quasi-complete d u a l n u c l e a r space and

E

i s an open s u b s e t o f (HtU)

1

f o r any b a l a n c e d open s u b s e t

E,

show t h a t

Hn(U)

= HHy(U)

U

and

HHy(u).

Let

be a r e f l e x i v e n u c l e a r space with a Schauder

E

b e a modularly d e c r e a s i n g open s u b s e t o f E l B’ Show t h a t t h e m o n o m i a l s f o r m a n a b s o l u t e b a s i s f o r (H(U),To) and l e t

U

290

Chapter 5

5.67" -

Let

b e a r e f l e x i v e n u c l e a r space with Schauder

E

b a s i s and l e t

Ei.

subset of

N

Show t h a t e a c h

and t h a t t h e mapping

f

E

from

(HHy(U) , T o )

Y

f

HHy(U)

f

-f

(resp.

H(U))

in

-4

N

f

can

HHy(U)

(resp.

i s a l i n e a r isomorphism

-4

onto

Show

a modularly d e c r e a s i n g open

U,U,is

b e e x t e n d e d i n a u n i q u e way t o a f u n c t i o n

H(G))

Ei.

b e a c o n n e c t e d R e i n h a r d t domain i n

U

t h a t t h e modular h u l l o f

(HHy(U) , T o )

and from

( H ( U ) ,To)

onto

( H ( G ) , T ~ ) . Hence deduce t h a t t h e monomials f 0 r . m a n a b s o l u t e basis for

5.68*

(H(U) Let

, T ~ )

and

(HHy(U) ,

T ~ ) .

be a n open p o l y d i s c i n a h e r e d i t a r y Lindelgf

U

f u l l y nuclear space with a basis.

If

(H(U),T&)

show t h a t t h e monomials f o r m a n a b s o l u t e b a s i s f o r

5.69*

Let

Show t h a t t h e s e t o f a l l

l i n e a r f u n c t i o n a l s on with

(H(U),T~).

be an open p o l y d i s c i n a f u l l y n u c l e a r space

U

with a basis.

i s complete

H(U),

spec(H(U)

T

6

,T&),

multiplicative . may b e i d e n t i f i e d

U.

5.70*

Let

that

H(K)

be a f u l l y nuclear space with a b a s i s .

E

l i m ( H ( V ) , ~ ~ )f o r a n y c o m p a c t p o l y d i s c

=

4

Show in

K

V 3 K

E

if and o n l y if

disc

in

V

__ 5.71*

Let

Let K

( H ( V ) , T ~ )= ( H ( V ) , r ( , ) )

f o r every open poly-

E.

b e a l o c a l l y convex space w i t h completion

E

b e a compact s u b s e t o f

l o o k e d upon as a s u b s e t o f

A

E.

E

and l e t

Show t h a t

A

K

denote A

H(K) = H(K)

h

E.

K

alge-

b r a i c a l l y and t o p o l o g i c a l l y .

5.72 which t h e

G i v e an e x a m p l e o f a r e f l e x i v e A - n u c l e a r bounded s u b s e t s o f

T

b u t i n which t h e 5.73* -

Let

U

T~

H(E)

space

E

in

a r e l o c a l l y bounded

bounded s u b s e t s a r e n o t .

b e an open p o l y d i s c i n a r e f l e x i v e A-nuclear

29 1

Holomoiphic functions on nuclear spaces with a basis space

and l e t

E

w = (Wn)n=l 00

be t h e mapping which t a k e s

f

M R ~ : H ~ ~ +( ~u

~

o n t o i t s germ a t t h e p o i n t

w

uM.

E

Let

~

and l e t tRw d e n o t e t h e t r a n s p o s e o f t h i s mapping. Show t h a t t R w ( H ( E ) ) C H(U) a n d t R w ( H H y ( E ) ) C H H y ( U ) . Hence d e d u c e t h a t

Rw : (H(uM)

,

o(H(UM),H(U)))

i s c o n t i n u o u s and t h a t Mackey s p a c e i f

(H(U),y0)

(H(E),.ro)

-

(H(OE,),o(H(OE,),H(E)))

(resp.

(resp.

4

6

(H(U),T~)) is a

( H ( E ) , T ~ ) ) i s a Mackey

space. If

5.74* -

E

and

limit,

i s a q u o t i e n t s p a c e o f t h e l o c a l l y convex s p a c e

F

H(OE) = l i m (Hm(V),II ---+ O E V , V open show t h a t

Ilv)

H(OF) = lirn

(Hm(V),I(

--t

OEV,V

regular inductive l i m i t .

is a regular inductive (Iv)

is also a

open

Use t h i s r e s u l t t o g i v e a n a l t e r -

n a t i v e proof of c o r o l l a r y 5.55.

Why c a n n o t t h i s r e s u l t b e u s e d

t o give a f u r t h e r proof of proposition 5.52? Let

5.75* -

b e a Banach s p a c e .

E

A sequence

( x 00~ ) ~i n= ~E

i s s a i d t o be rapidly decreasing i f f o r every i n t e g e r

n p I/xn

I\+ 0

as

n+-.

Let

ES

denote t h e vector space

p, E

endowed w i t h t h e t o p o l o g y o f u n i f o r m c o n v e r g e n c e o n t h e r a p i d l y

E’.

decreasing sequences i n

and

Show t h a t

ES

HM(Es) = H H y ( E s ) ’

5.76*

E = CN

If

x

C(N)

show t h a t

( p ( n E ) , ~ o ) i s a borno-

l o g i c a l s p a c e which i s n o t b a r r e l l e d f o r any 5.77*

If

5 . 7_ 8* _

and

If

limit,

.

)i

( ( H ( U ) , T ~ ) ~ f (H(U),.rg).

K

i s a compact s u b s e t o f a f u l l y n u c l e a r s p a c e

H(K) = l i m

4

V3K,V

space.

n z 2.

i s a b a l a n c e d open s u b s e t o f a F r e c h e t n u c l e a r

U

s p a c e show t h a t

E

i s a nuclear space

show t h a t

(Hm(V),II

11,)

is a regular inductive

open

H(K)

i s a quasi-complete

l o c a l l y convex

(

0

292

Chapter 5

5.79" -

Show t h a t a f u l l y n u c l e a r s p a c e w i t h a b a s i s i s

5.80 -

Let

ultrabornological. be an open p o l y d i s c i n a f u l l y n u c l e a r space

U

with a basis.

i s a s e t of complex numbers,

{bm} meN "1 show t h a t t h e r e e x i s t s a T * " such t h a t

1

mEN

If

= bm

T(zm)

( N ) la,b,I

analytic functional

m

for all

5.81* -

Let

numbers,

show t h a t t h e r e e x i s t s a

with

E

infinite subset

5.82*

Let

(m,n)'

and f o r

N")

(mama

j

1 b'

in

=

n

,

i s not

5.83"

T

i f and o n l y i f e a c h J'

. i s an i n t e g e r and

an

ma,,,

m

and

...

0,

t

n

a

n+l

Let

> a +n

n

let

)

an p o s i t i o n

m

let

N")

lo

T(zj) 0

a n a l y t i c f u n c t i o n a l on

contains an i n f i n i t e subset

. . .,

0,

otherwise.

Show t h a t t h e r e e x i s t s a such t h a t

space. and suppose

N

For any p o s i t i v e i n t e g e r s

=

H(u).

i s a s e t o f complex

T~

E

E

b e a n u c l e a r power s e r i e s o f t y p e 1.

A,(a)

Suppose each

(an)n=l* for all n.

m

"1

mEN

m

for all

of

J

{bm}

C m E J , b m w mE H ( O E , ) B

m

a =

If

= bm

T(zm)

such t h a t

a zm

be a r e f l e x i v e A-nuclear

E

(H(E),T~) i s complete.

U

i f and o n l y i f

N

c m c N "1

for every

<

E

on

T

= b.

1

T~

a n a l y t i c f u n c t i o n a l on

f o r every

in

j

N").

A,(a)

Show t h a t

T

continuous.

For each non-negative i n t e g e r

n

let

LA(nE)

denote the vector space of continuous alternating forms on t h e l o c a l l y convex space

E.

Let

n

linear

293

Holomorphic functions on nuclear spaces with a basis

HAW)

aA ( n E ) , C n , O / I P n l ( K"

a,

=

m

{(Pn)n=O;Pn~

m

= P ~ ( { P ~ } ~ < = m ~j )

a n d endow

HA(E)

w i t h t h e t o p o l o g y g e n e r a t e d b y t h e semi-

norms

as

r a n g e s o v e r t h e compact s u b s e t s o f

E

pK

K

is a f u l l y nuclear space with a basis,

a c l o s e d complemented s u b s p a c e o f Hence deduce t h a t E

is a nuclear

5.84* -

If

HA(E)

n=1,2,.

..

.

is

~ ) .

is a reflexive nuclear space i f

n

show

form a Schauder b a s i s f o r

If, i n addition,

E

has t h e

p r o p e r t y and i t s b a s i s i s s h r i n k i n g ,

( p("E)

t h e y a l s o form a b a s i s f o r

55.6

HA(E)

i s a Banach s p a c e w i t h a Schauder b a s i s ,

t h a t t h e monomials o f d e g r e e

Dunford-Pettis

( H ( E ) ,T

If

space with a basis.

DN

E

(6( n E ) , ~ o ) ,

show t h a t

E.

show t h a t

,B).

NOTES A N D REMARKS Although D .

Hilbert

[ 3 3 2 ] f i r s t s u g g e s t e d , i n 1909, a

monomial e x p a n s i o n a p p r o a c h t o h o l o m o r p h i c f u n c t i o n s i n i n f i n i t e l y many v a r i a b l e s ,

t h i s i d e a was n o t d e v e l o p e d u n t i l

r e c e n t l y and most o f t h e r e s u l t s p r e s e n t e d i n t h i s c h a p t e r were discovered w i t h i n t h e last f o u r y e a r s .

I n d e e d , many o f

the original research a r t i c l e s are still only available i n p r e p r i n t form. N u c l e a r mappings and n u c l e a r s p a c e s , a s well a s most o f t h e i r fundamental p r o p e r t i e s ,

are due t o A.

Grothendieck

[287].

Further accounts of the l i n e a r theory are given i n t h e very r e a d a b l e book o f A . i n the notes of Y.C.

Pietsch Wong

[570],

[717].

in P.

t h i s c h a p t e r i s given i n S. Dineen [198], from a s l i g h t l y d i f f e r e n t p o i n t o f view, S . Dineen

Kr6e [ 4 0 3 , 4 0 4 ] ,

and

A survey of t h e r e s u l t s of

and a f u l l a c c o u n t , can b e found i n

[197].

Lemma 5 . 1 i s due t o P . J . A p a r t from i t s u s e s i n [ 9 1 ] ,

Boland and S . Dineen

[91].

we h a v e u s e d i t i n t h i s c h a p t e r

t o s h o r t e n a number o f t h e o r i g i n a l p r o o f s .

Proposition 5.4

294

Chapter 5

i s t h e c l a s s i c a l characterization of nuclear sequence spaces due t o A.

Lemma 5 . 6 ,

[570].

i s given i n A.

G r o t h e n d i e c k [287] and a p r o o f

c o r o l l a r i e s 5 . 7 and 5 . 8 ,

Pietsch

p r o p o s i t i o n 5 . 9 and

c o r o l l a r y 5 . 1 0 a r e a l l p r o b a b l y known t o r e s e a r c h w o r k e r s i n n u c l e a r space t h e o r y , b u t w e have been unable t o l o c a t e an exact reference.

A-nuclear

s p a c e s were i n t r o d u c e d by S . Dineen

[2023 as a t o o l i n studying holomorphic f u n c t i o n s

P r o p o s i t i o n 5 . 1 2 i s d u e t o S . Dineen

of independent i n t e r e s t .

w h i l e p r o p o s t i o n 5 . 1 3 and c o r o l l a r y 5 . 1 4 a r e g i v e n i n

[202 P.J.

b u t may b e

Boland and S . Dineen 1913. F u l l y n u c l e a r s p a c e s and f u l l y n u c l e a r s p a c e s w i t h a b a s i s

were i n t r o d u c e d by P . J .

Boland and S . Dineen

(H(E),ro)

s o l v e t h e b a s i s problem f o r space with a basis.

This a r t i c l e ,

[go] i n o r d e r t o

when

[go],

is a nuclear

E

contained t h e motiva-

t i o n f o r many o f t h e r e s u l t s i n t h i s c h a p t e r ,

introduced t h e

concept of multiplicative polar

( d e f i n i t i o n 5.16)

t h e key r e s u l t s ,

Lemma 5 . 1 9

5 . 1 7 and 5 . 1 8 .

and c o n t a i n s

i s due t o S. Dineen

[202]. Theorem 5 . 2 1 ,

a r e due t o P . J .

corollaries 5.22,5.23

Boland and S . Dineen

and p r o p o s i t i o n 5.24

[go],

and S. Dineen

t h e l a t t e r c o n t a i n i n g t h e r e s u l t s on A - n u c l e a r i t y . using a proof

similar t o t h a t of theorem 5.21, A.

h a s shown t h a t

(H(E),-Io)

decomposition ( i . e . whenever

E

5.29 i s due t o P . J .

dim(En)

<

f o r each

n

i n d e f i n i t i o n 3.7)

Propositions 5.25,5.28

a r e g i v e n i n S . Dineen

Boland and S . Dineen

proof of t h i s proposition,

[go].

[202],

Boland and S. Dineen

and

Theorem

The o r i g i n a l

Boland and S . Dineen

w h i l e p r o p o s i t i o n 5.34 [go].

i s due t o

C o r o l l a r y 5 . 3 5 and example

b o t h o f which i n v o l v e a n a p p l i c a t i o n o f l e m m a 5.1,

due t o P . J .

Boland and S . Dineen

has

Corollaries 5.30,5.31,

5 . 3 2 a n d e x a m p l e 5 . 3 3 may b e f o u n d i n P . J . and S . Dineen

[202].

which used p r o p o s i t i o n 5 . 3 4 ,

b e e n s h o r t e n e d h e r e b y u s i n g lemma 5 . 1 .

5.36,

[56]

i s a F r g c h e t - S c h w a r t z s p a c e w i t h a f i n i t e dimen-

c o r o l l a r i e s 5.26,5.27

P.J.

Benndorf

admits a f i n i t e dimensional Schauder

s i o n a l Schauder decomposition.

[91],

[202],

Recently,

1911.

are

Proposition 5.37 i s a

s l i g h t l y improved v e r s i o n o f a r e s u l t proved i n P . J .

Boland and

295

Holomorphic functions on nuclear spaces with a basis S . Dineen

[91]

I n h i s i n v e s t i g a t i o n s on t h e mathematical f o u n d a t i o n s o f q u a n t u m f i e l d t h e o r y w i t h i n f i n i t e l y many d e g r e e s o f P.

freedom,

Kr6e [ 4 0 7 , 4 0 8 , 4 0 9 , 4 1 0 ] u s e d t h e n u c l e a r i t y r e s u l t o f P . J .

Boland

[86] and L .

Waelbroeck

[713!

l o c a l l y convex space.

This,

and s u b s e -

was a b o r n o l o g i c a l

together with the r e s u l t s of

f u r t h e r q u e s t i o n s o f Krge c o n c e r n i n g

s e c t i o n s 5 . 1 and 5 . 2 ,

H(E)

(theorem 3.64)

(H(s),.ro)

q u e n t l y h e w i s h e d t o know i f

and t h e p o s s i b i l i t y of a k e r n e l s theorem f o r a n a l y t i c

f u n c t i o n a l s i n i n f i n i t e l y many v a r i a b l e s ( s e e c h a p t e r 6 ) m o t i v a t e d much o f t h e r e s e a r c h d e s c r i b e d i n s e c t i o n s 5 . 3 a n d 5.4. To s o l v e t h e s e p r o b l e m s , t h e concept of B-nuclear Subsequently, B-nuclear

D.

Vogt,

Dineen [195,202],

S.

introduced

s p a c e and proved theorem 5 . 4 2 .

i n a p r i v a t e c o m m u n i c a t i o n , showed t h a t

s p a c e s and n u c l e a r DN-spaces w i t h a b a s i s c o i n c i d e d

DN

(proposition 5.40).

spaces a r e due t o D.

Vogt and h a v e

played an important r o l e i n t h e development of s t r u c t u r e We r e f e r t o D .

theorems f o r Frgchet nuclear spaces.

Vogt

[703]

f o r an e x c e l l e n t survey a r t i c l e on n u c l e a r DN s p a c e s and t o E.

Dubinsky

[212] f o r a comprehensive account o f r e c e n t d e v e l -

opments i n nuclear Frgchet space theory. ences f o r p r o p o s i t i o n 5.39 are D. [211].

Vogt

Proposition 5.40 i s due t o D .

Lemma 5 . 4 3 i s d u e t o S . D i n e e n case of proposition P.J.

[198,199].

is given i n P . J .

[704] and M.

Vogt

[199].

[702,703]. The p a r t i c u l a r

Valdivia

while the general r e s u l t An e a r l i e r p a r t i a l

Boland and S . Dineen [ 9 1 ] . i.e.

Vogt

B ( n ) & s “1 .

Example 5.46 f o l l o w s from r e s u l t s proved i n P . J . Dineen

result for D.

[690] have proved, i n d e p e n d e n t l y , t h e

deep r e s u l t quoted i n example 5.45, S.

Dubinsky

5.44 d e s c r i b e d i n example 5.45 i s due t o

Boland and S . Dineen [ 9 2 ] ,

a p p e a r s i n S . Dineen H ( a )

The o r i g i n a l r e f e r -

[702] and E .

Boland and

[91].

The r e m a i n i n g r e s u l t s i n s e c t i o n 5 . 4 a r e t o b e f o u n d i n

296

Chapter 5

S . Dineen

[199], where t h e connection between e x t e n s i o n

theorems f o r holomorphic f u n c t i o n s on s u b s p a c e s and t o p o l o g i c a l H(E)

properties of

is established.

Corollary 5.50 (the

h o l o m o r p h i c H a h n - B a n a c h t h e o r e m ) was f i r s t p r o v e d b y P . J . In [161],

J.F.

Boland

[83] using a d i r e c t approach.

Colombeau

and B .

P e r r o t g e n e r a l i s e t h i s c o r o l l a r y by showing t h a t a

F r g c h e t - S c h w a r t z v a l u e d e n t i r e f u n c t i o n on a c l o s e d n u c l e a r

a38

subspace of a to

E.

space,

E,

can be extended holomorphically

An e x t e n s i o n t h e o r e m f o r n u c l e a r h o l o m o r p h i c f u n c t i o n s

on Banach s p a c e s i s g i v e n i n R .

Aron and P .

Berner [26] and

u s i n g t h i s theorem and a u n i f o r m f a c t o r i z a t i o n theorem f o r holomorphic f u n c t i o n s on J.F.

of

23q

(see exercise 2.105),

spaces

Colombeau and J . M u j i c a [ 1 5 6 ] g i v e a n a l t e r n a t i v e p r o o f

corollary 5.50.

of theorem 5.48, 6.54),

A further proof

of t h i s c o r o l l a r y and a l s o

using t h e symmetric t e n s o r a l g e b r a ( d e f i n i t i o n

has r e c e n t l y been o b t a i n e d by R .

I n [488],

R.

Meise and D .

Meise and D .

Vogt

[487].

Vogt show t h a t t h e h o l o m o r p h i c Hahn-

Banach theorem i s n o t v a l i d f o r c e r t a i n n u c l e a r F r g c h e t s p a c e s . A different kind of

e x t e n s i o n r e s u l t f o r e n t i r e f u n c t i o n s on a

n u c l e a r subspace o f a l o c a l l y convex space i s g i v e n i n A . Martineau

[453].

In chapter 6,

s e c t i o n 4 , w e p r o v e a number o f s t r u c t u r e

theorems f o r holomorphic f u n c t i o n s on i n f i n i t e t y p e power series spaces. The r o l e o f n u c l e a r i t y i n i n f i n i t e d i m e n s i o n a l holomorphy i s much m o r e e x t e n s i v e t h a n t h a t o u t l i n e d i n t h i s b o o k . i s mainly due t o o u r choice of t o p i c s .

This

I n Appendix I , w e s e e

t h a t i t a p p e a r s i n t h e s t u d y o f c o n v o l u t i o n o p e r a t o r s on s p a c e s of holomorphic f u n c t i o n s ,

5

i n s o l v i n g t h e Levi problem and t h e

p r o b l e m a n d i n i n f i n i t e d i m e n s i o n a l h o l o m o r p h i c s h e a f theory.

A p a r t f r o m t h e s e t o p i c s we a l s o f i n d n u c l e a r i t y a p p e a r i n g i n A.

M a r t i n e a u ' s s t u d y [453] o f t h e s u p p o r t s o f a n a l y t i c functiorr

als i n several variables,

c o n t i n u a t i o n [712]

~

i n L.

i n B.

W a e l b r o e c k ' s r e s u l t on a n a l y t i c

Kramm's

[398,399,400]

/

interesting

c l a s s i f i c a t i o n theorems f o r Frechet n u c l e a r a l g e b r a s , a n a l y t i c s p a c e s and S t e i n a l g e b r a s and i n M .

a &?

S c h o t t e n l o h e r [&61

Chapter 6 GERMS, SURJECTIVE LIMITS, E -PRODUCTS AND POWER SERIES SPACES

The l a s t two c h a p t e r s d e a l t w i t h s c a l a r - v a l u e d holomorphic f u n c t i o n s d e f i n e d on s p e c i a l domains i n s p e c i a l s p a c e s . return i n t h i s chapter t o the general theory, f u r t h e r methods - t h e

We

and p r e s e n t t h r e e

t o p o l o g y , s u r j e c t i v e l i m i t s , and

T~

&-products - f o r studying t h e r e l a t i o n s h i p between t h e t o p o l ogies

T

0'

T

and

w

T

~

.

The

T~

t o p o l o g y aims a t r e m o v i n g

g e o m e t r i c r e s t r i c t i o n s on t h e domain,

surjective l i m i t s are

used t o g e n e r a t e spaces o f holomorphic i n t e r e s t and u s i n g

6 - p r o d u c t s we s t u d y v e c t o r - v a l u e d f u n c t i o n s . A p a r t f r o m t h e p r o b l e m o f t h e d i f f e r e n t t o p o l o g i e s we a l s o d i s c u s s i n t h i s chapter o t h e r problems of general i n t e r e s t such as t h e r e p r e s e n t a t i o n o f a n a l y t i c f u n c t i o n a l s and t h e comple'teness of

H(K).

The f i n a l

section of t h i s chapter i s devoted t o

holomorphic f u n c t i o n s on t h e s t r o n g d u a l s o f c e r t a i n power In t h i s case, t h e r e s u l t s o f c h a p t e r f i v e are

s e r i e s spaces.

c o m b i n e d w i t h some i n t e r e s t i n g e s t i m a t e s t o o b t a i n a number o f representation theorems.

§6.1

HOLOMORPHIC G E R M S O N C O M P A C T SETS I n c h a p t e r s 2 , 3 a n d 5 we o b t a i n e d v a r i o u s r e s u l t s

c o n c e r n i n g t h e r e g u l a r i t y and c o m p l e t e n e s s o f compact s u b s e t o f a l o c a l l y convex s p a c e .

H(K),

K

a

The p o s i t i v e a n d

n e g a t i v e r e s u l t s we o b t a i n e d s h o w t h a t t h e s e a r e i n d e e d c o m p l e x q u e s t i o n s and n o t u n r e l a t e d t o one a n o t h e r . assumed t h a t condition

-

K e.g.

I n m o s t c a s e s , we

s a t i s f i e d a rather r e s t r i c t i v e geometric that

K

consisted of a single point o r w a s a

298

Chapter 6

balanced set o r a polydisc. internal structure of

T h i s e f f e c t i v e l y meant t h a t t h e

p l a y e d no p a r t i n o u r i n v e s t i g a t i o n s

K

a n d t h a t e s s e n t i a l l y we w e r e s t u d y i n g l o c a l p h e n o m e n a .

H e r e we

l o o k a t t h e g l o b a l t h e o r y b y c o n s i d e r i n g a r b i t r a r y compact s e t s . We d o n o t c o n c e r n o u r s e l v e s w i t h t h e r e g u l a r i t y a n d c o m p l e t e ness of

-

H(0)

as w e have considered t h e s e q u e s t i o n s i n chap-

t e r s 3 and 5 and s h a l l g i v e a f u r t h e r example i n t h e n e x t

-

section

b u t look a t

H(K)

where

K

i s a compact s u b s e t of a

Frgchet space o r of a f u l l y nuclear space with a basis.

The

m e t h o d o f p r o o f u s e d f o r m e t r i z a b l e s p a c e s was m o t i v a t e d b y t h e proof o f p r o p o s i t i o n 3.40 and t h i s ,

i n turn, provided t h e

m o t i v a t i o n f o r the f u l l y n u c l e a r space case.

We h a v e a l r e a d y s e e n i n p r o p o s i t i o n 2 . 5 5 , r e g u l a r when

K

that

H(K)

is

i s a compact s u b s e t o f a F r g c h e t s p a c e and

K

t h a t i t i s c o m p l e t e when

is balanced.

We now e x t e n d t h e

l a t t e r r e s u l t t o a r b i t r a r y compact s e t s . Theorem 6 . 1

then

If K i s a compact s u b s e t o f a F r g c h e t s p a c e i s cornpZete.

H(K)

Proof

Since

is a

H(K)

t h a t it i s quasicomplete. t o p o l o g y on H(K)

H(K).

Let

Let T

space,

DF

-rl

E

i t s u f f i c e s t o show

denote the inductive l i m i t

b e t h e l o c a l l y convex topology on

g e n e r a t e d by a l l semi-norms

which have e i t h e r o f t h e

following forms:

is a

where

p

where

(xnInzl,

m

T~

c o n t i n u o u s semi-norm on

m

(XA)n=1

H(O),

a r e two s e q u e n c e s i n

K,

m

(yn)n=l,

are n u l l sequences i n E, x n + y n = x n’ + yn’ f o r a l l and (Y;l);=l n and (kn)m i s a s t r i c t l y i n c r e a s i n g sequence o f p o s i t i v e n=l

Germs, surjective limits,

E

299

-products and p o w e r series spaces

integers

By p r o p o s i t i o n 2 . 5 6 , subsets of

H(K)

and

We now s h o w t h a t (gB)BEB be a

n

f

T

T

-rl

and

T ~

d e f i n e t h e same b o u n d e d

.

(H(K),T)

is quasi-complete.

bounded Cauchy n e t i n

T

H(K).

If

Let X E

and

K

is a positive integer,

o f t h e form

( g(

net in

then using T c o n t i n u o u s semi-norms fin we s e e t h a t (d g B ( x ) / n ! ) B E B i s a Cauchy

(*),

n ~ ) , ~ u ) s. i n c e

(corollary 3.42),

(0'

( n ~ ) , ~ u )i s c o m p l e t e

t h e r e e x i s t s an element of

@("E),

Pn,x,

such t h a t

( g B ) B E B i s bounded t h e r e e x i s t a neighbourhood

As

zero i n €3

B E B,

for all

and

Y

and

y,y'

in

arbitrary.

4n

exists

qnl

E

X E

gB

E

H(K+4W)

and a l l non-negative

K

for all

w W

satisfy

and

6

In=, m

x+y = x ' + y '

x.

n.

Bo

E

T

B

For any

Pn , x ( y ) .

and l e t

c o n t i n u o u s semi-norm on such t h a t

in

This implies

Let 6'

0

x

in

x,x' be

C h o o s e nl a p o s i t i v e i n t e g e r such t h a t 6 E . F o r a n y f E H(K) let

is a

of

W

B

Hence

for all n 4" let f(x)(y) =

llPn,xll,v

Since

such t h a t

M,O

s u P l l g B l l K + 4 w= M < m . BEB

and

that

and

E

H(K)

there

K E

K

300

Chapter 6

Hence

and

Thus, there exists an

f

and

for all

f(x)(y)

f(x+y)

=

in

Hm(K+W) x

in

such that K and y

Due to the form o f the semi-norms immediate that (H(K),T)

gf3

+

f

as

is quasi-complete.

ogy associated with

T

barrelled space and

T

hence

+

(H(K),T~)

on 4

and

Let H(K).

in T~

(**)

it is

(H(K),T). Hence be the barrelled topol-

Since

(H(K),T~)

it follows that

T~

(H(K),T~)

This shows that (H(K),.r2) over, by proposition 3 . 6 , h e n c e complete.

m

and

(*)

/ I f l / K + w6 M in W .

is a and

T~

have the same bounded sets.

is a barrelled DF space. More(H(K),.r2) is quasi-complete and

We complete the proof by showing that

T

~ T~=

T h e situation now is rather similar to that o f proposition and an examination o f the proof o f that proposition

3.40,

shows that we only need find a fundamental system o f bounded

m k subsets o f H(K), (Bn)n=l, such that closed for any finite sequence o f numbers to complete the proof.

AnBn

is

T~

in order

(~~)i=~

Let be a decreasing fundamental neighbourhood system at the origin in E consisting o f convex balanced open sets and f o r each H" (K+v,) . Let

r

h

E

n

l:=lAnBn

Bn

let

+

h

E

be the closed unit ball o f

H(K)

By corollary 3.39, Bn (H(K+V,),T~). F o r each y in r

y

h

E

Y>n

+

E

m .

Bn.

in t h e

T

topology as

is a compact subset o f

let

h, =

l n = 1An h Y,n where By using subnets, if necessary, we see that there

301

Germs, surjective limits, e -products and power series spaces exists a in

hn

E

(H(K+V,),T~)

rv

hy + h

in

Bn, n = l , . . . , k , for all n.

This implies t h a t n.

all

as

(H(K+Vk),~o)

%(x) n!

Iy

Hence

h = h

closed subset of

y

and

is a

l!=l~nBn

B

B

of

H(K)

H(K) V

some n e i g h b o u r h o o d o f

for every

f

in

B

in

K

for

K

and -r2,

an a r b i t -

i s c o n t a i n e d and bounded i n of

K

(i.e.

Hm(V),

i f and only i f t h e elements

K,

Taylor series development o f elements of W of

+ w

and hence

T,

s a t i s f y u n i f o r m Cauchy e s t i m a t e s o v e r

neighbourhood

y

of a l o c a l l y convex s p a c e .

f o r some n e i g h b o u r h o o d of

x

for all

n!

We now l o o k a t t h e r e g u l a r i t y o f

A subset

as

hn

+

This completes t h e proof.

H(K).

r a r y compact s u b s e t

hy,n

m .

-f

'nh(x)

-

~

such t h a t Hence

B

and t h e l o c a l

K

is coherent i n

i f and o n l y i f t h e r e e x i s t s a

zero such t h a t

whenever

x,x'

E

y,y'

K,

E

and

W

x+y = x'+y'). We h a v e p r e v i o u s l y u s e d t h i s r e d u c t i o n i n o u r a n a l y s i s , a s f o r example i n p r o p o s i t i o n 2 . 5 6 ,

where t h e semi-norms

were u s e d t o o b t a i n C a u c h y e s t i m a t e s a n d t h e s e m i - n o r m s were u s e d t o p r o v e c o h e r e n c e .

If

H(0)

(*)

(**)

is regular, then we

h a v e Cauchy estimates and i t i s p o s s i b l e t h a t t h i s a l s o i m p l i e s coherence.

We a r e n o t ,

however,

able t o prove t h i s .

To p r o v e c o h e r e n c e , w e n e e d e x t r a h y p o t h e s e s a n d t h e s e

can t a k e various forms.

O n e may p l a c e c o n d i t i o n s o n

a s l o c a l c o n n e c t e d n e s s , or c o n d i t i o n s o n i l i t y o r a c o m b i n a t i o n o f c o n d i t i o n s on

E K

K,

such

such as metrizaband

E.

We s h a l l

302

Chapter 6

assume t h a t

i s m e t r i z a b l e and t h a t

K

E

satisfies a certain

t e c h n i c a l c o n d i t i o n which appears t o b e s a t i s f i e d by most, not all,

s p a c e s f o r which

examples o f non-metrizable H(K)

is regular.

H(0)

if

This gives us

l o c a l l y convex s p a c e s i n which Our m e t h o d s a r e

K.

i s r e g u l a r f o r e v e r y compact s e t

e a s i l y seen t o be i n f l u e n c e d by t h e p r o o f s o f p r o p o s i t i o n 2.56 and theorem 6 . 1 . Proposition 6.2

a)

K

b)

H(0)

c)

if

b e a c o m p a c t s u b s e t of a ZocaZZy

K

Let

E

convex space

and s u p p o s e

i s metrizable, is r e g u Z a r , i s a c o n v e x b a l a n c e d o p e n s u b s e t of

V

(fn)n c H ( V ) ,

and

fn f 0

fn(xn) # 0

such t h a t H(K)

then Proof

n,

for e a c h

t h e r e e x i s t s a bounded s e q u e n c e i n

E

then (xn)nmZl,

V,

n

for a22

i s a regular inductive l i m i t . Let

b e a bounded

B

subset of

H(K).

Since each

semi-norm o f t h e form

where on

p

H(K)

i s a c o n t i n u o u s semi-norm on

and

H(0)

a neighbourhood

I/ f o r every

in

Now s u p p o s e

K,

zero i n

E,

such t h a t

i s continuous

of

V

0

in

and

E

MzO

such t h a t

A

x

nets in

H(O),

is regular it follows that there exists

K,

and

f

in

is not coherent.

B

( x a )a E

nzO

r

(ya) acr y,,yA E V

and and all

(x:)

acr *

B.

Then t h e r e e x i s t s two two n e t s c o n v e r g i n g t o and

(fa)aEr a net i n B (Y;)aEr’ a, xa+ya = x ’ + y ’ f o r a l l a and a a

303

Germs, surjective limits,E -products and power series spaces

Since set

is metrizable

K

{Y,-Y;IaEr W

and

(Xn)n=l

i s a l s o m e t r i z a b l e and hence t h e

K-K

(X;)n=l

be t h e corresponding sequences i n

For each p o s i t i v e i n t e g e r

3 f ,

(x,)

n

(XI

j!

Each

hn

n

I;=,

-

and each

such t h a t

x

A .

in

# 0

let

V

(X+YA-Y,)

j!

and

V

for all

n.

# 0.

hn(yn)

m

t h e r e e x i s t s a bounded s e q u e n c e

hn(zn)

K.

d'fan(xr;)

i s a holomorphic f u n c t i o n on

By c o n d i t i o n ( c ) , V

m

( ~ ~ - y ; ) ~ =L e~t .

contains a n u l l sequence

( z ~ ) ~i n= ~

By t h e i d e n t i t y t h e o r e m

f o r h o l o m o r p h i c f u n c t i o n s o f o n e c o m p l e x v a r i a b l e we c a n c h o o s e

a n u l l sequence i n n.

for all

2~~ > 0

m

C,

loss of,generality,

Hence

such t h a t

Anzn+O

w e may s u p p o s e

Xnzn+yA-yn~V f o r a l l

n.

Xnzn

E

all

> n

n

V

and,

=

without

and

Now choose i n d u c t i v e l y a s t r i c t l y m

(kn)n=l

increasing sequence of p o s i t i v e integers, 2 k n B,

Ihn(Anzn)l

n-tm

as

such t h a t

and

Let q(f)

2.kn

=

f o r every If

f

f c H(K)

in

H(K).

then t h e r e e x i s t a neighbourhood

and a p o s i t i v e i n t e g e r

IIf(IK+4W

6

M,

Xnzn

E

no W

and

such t h a t

W

of

0,

M>O

f e Hm(K+4W),

Xnzn+yA-yn

E

W

all

n

2

n

.

304

Chapter 6

Hence

Since

H(K)

Since

q(fa?

is barrelled k n Bn > n L 2

q

is a continuous semi-norm. m and (fa)n=l is bounded n B is a coherent family

for all n this leads to a contradiction. Hence and

H(K)

is a regular inductive limit.

This completes the

proof. If

E

is a fully nuclear space with a basis, then every

E

compact subset o f

is metrizable.

Condition (c) o f prop-

osition 6 . 2 is satisfied by E if E;i admits a continuous norm. Hence we have the following corollary to proposition 6 . 2 and this applies, in particular, to strong duals o f n u c l e a r

DN

spaces.

Eb

then

H(K)

subset

If

K

E

E

If

Corollary 6 . 3 such t h a t

i s a fuzzy n u c Z e a r s p a c e w i t h a b a s i s

a d m i t s a c o n t i n u o u s norm and

H(OE) i s regular i s a r e g u l a r i n d u c t i v e Z i m i t f o r e v e r y compact of E . is a nuclear locally convex space, E

compact subset o f

and

V

K

is a

is a neighbourhood o f zero in

E

then, using Cauchy estimates, one can show there exists a neighbourhood W o f zero such that H ( K ) and Hm(K+W) induce the same uniform structure and hence the same topology o n t h e unit ball o f H m ( K + W ) . Since H m ( K + W ) is a Banach space, corollary 6 . 3 yields the following result. Corollary 6.4

If

E

i s a fully nuclear s p a c e w i t h a b a s i s ,

305

Germs, surjective limits, E -products and power series spaces Eb

a d m i t s a c o n t i n u o u s norm a n d

H(OE)

i s regular,

n chapter 2, on

H(U)

we d e f i n e d ( d e f i n i t i o n 2 . 5 9 )

for

then

of

K

is q u a s i c o m p l e t e f o r a n y c o m p a c t s u b s e t

E.

the

T

71

top-

an open s u b s e t o f a l o c a l l y convex

U

T h i s t o p o l o g y h a s good l o c a l p r o p e r t i e s and c o i n c i d e s i n certain cases,

W

and

71

i n d e e d i t has been c o n j e c t u r e d

always c o i n c i d e .

w

We now e x a m i n e t h i s t o p -

and b e g i n by showing t h a t i t i s i n d e e d well d e f i n e d . Lemma 6.5

space

E.

=

lirn

+ KE

&(U)

Proof

t h e l o c a l l y convex

Then a l g e b r a i c a l l y

H(U)

where

be a n o p e n s u b s e t of

U

Let

H(K)

k(U,

i s t h e s e t of a l l c o m p a c t s u b s e t s of

U.

Under t h e n a t u r a l r e s t r i c t i o n mappings

i s c l e a r l y a projective system.

{ H ( K )' K E A ( U ) mapping A :

H(U)-

lirn

c-

The c a n o n i c a l

H(K),

K E .k(U 1

where

and

[fIK

i s t h e h o l o m o r p h i c germ on

K

induced by

l i n e a r and i n j e c t i v e .

I t r e m a i n s t o show t h a t

ive.

E

Let

(EK)KEt(u)

l i m

H(K)

f,

is

is surject-

A

We d e f i n e a

be given.

t-

function

f

on

U

by

f E H(U) and compact s u b s e t o f U,

claim

KEP<(U) f(x) = fIxl(x)

for all

A(f) = (fK)KEk(U). If then since

K

x

in

i s any

U.

We

306

Chapter 6

n!

n!

f o r any compact s u b s e t negative integer Hence i f

x E U

of zero such t h a t in

V

of

K

containing

U

and

i s a convex balanced neighbourhood

V

x+VCU

and

fix)

H"(x+V)

E

f{x+y}(X+Y)

=

f [X,X+Yl

=

n!

t h e n f o r any

y

U,

x

E

f K

n!

E

H(U).

Moreover,

and

n

is arbitrary,

n!

if

K

[fIK

=

n! Hence

fK.

i s a compact sub-

then

n!

and c o n s e q u e n t l y

(X+Y)

{x+Ay; 0 i X i 11

[x,x+y] =

T h i s shows t h a t

set of

and any non-

we have f(x+y)

where

x

n.

A(f)

=

( f K l K E& ( u ) .

This

completes t h e proof. Remark 6.6 U an a r b i t r a r y - M!e h a v e T ~ S 7TS1T w o n I l ( U ) , Our n e x t p r o p o s i t i o n open s u b s e t of a l o c a l l y convex s p a c e . d e s c r i b e s a s i t u a t i o n i n which

T~

and

T~

coincide.

Proposition 6.7

If U is a b a Z a n c e d o p e n s u b s e t of a ZocaZZy c o n v e x s p a c e , t h e n T~ = T on H(U). Proof Suppose

Let p

p

be a

T

c o n t i n u o u s semi-norm on

i s p o r t e d by t h e compact b a l a n c e d s u b s e t

H(U). K

of

U.

307

Germs, surjective limits,E -products and power series spaces By t h e o r e m 3 . 2 2 , we may s u y p o s e , w i t h o u t l o s s o f g e n e r a l i t y , that

Let

If

V

i s a neighbourhood of

such t h a t

then there ex i s t s

K

p ( f ) 6 ~ ( V ) l ( f / (f o~ r e v e r y

f

in

c(V)> 0

H(U).

Hence f o r

every

N

is and t h u s p i s a c o n t i n u o u s semi-norm on H(K). If 'U,K t h e c a n o n i c a l mapping o f H(U) i n t o H(K), then p = OpU,K

and h e n c e than

T~

5

is

T~

continuous.

Since

T~

i s always f i n e r

t h i s completes t h e proof.

We now show t h a t Frschet-Schwarz

T~

spaces.

i s a s h e a f o r l o c a l t o p o l o g y on

To p r o v e t h i s , we n e e d some p r e l i m -

The f i r s t , a n open mapping theorem,

inary results.

we g i v e

without proof. Proposition 6.8

t h e semi-Monte2 s p a c e i s bounded i n o p e n mapping.

T

Let E

E

whenever

b e a c o n t i n u o u s l i n e a r mapping from

into the B

DF

F.

space

i s bounded i n

F

If then

T-I(B)

T

i s an

Chapter 6

308 Proposition 6.9

If

Schwartz space E, Proof

Since

is a c o m p a c t s u b s e t of a F r g c h e t -

K

is a ,X13d'space.

H(K)

then

i s a complete

H(K)

show t h a t f o r e v e r y n e i g h b o u r h o o d e x i s t s a neighbourhood

B

If

=

E

Hm(K+W).

Let

p

it suffices to

space,

of zero i n

V

c 11

there

E

of zero contained i n

W

Ilf 11 K + V

H"(K+V) ;

DF

such t h a t

V

i s a precompact s u b s e t o f

b e t h e Minkowski f u n c t i o n a l o f

E

Since

V.

i s a S c h w a r t z s p a c e , t h e r e e x i s t s a c o n t i n u o u s semi-norm on

q,q 32p,

s u c h t h a t t h e c a n o n i c a l mapping from

E

into E = (E/ ,p) i s precompact. By q-l(O) P P-l(O) L i o u v i l l e ' s t h e o r e m we may s u p p o s e , w i t h o u t l o s s o f g e n e r a l i t y ,

E

(E/

=

9

that

p

K+W

,q)

q

and

a r e norms.

i s a compact s u b s e t o f

t h e c l o s u r e oE

K+V

compact s u b s e t o f

in

A

E

Let

.

W

P (H(K+V),r,)

{ x ~ E ; q ( x ) <} .

=

= K + Interior

V1

(v)

Hence

is

is a

B

By p r o p o s i t i o n 3 . 3 7 ,

and hence e v e r y n e t i n

c o n t a i n s a s u b n e t which i s c o n v e r g e n t i n

v

where B

This

H"(K+W).

completes t h e proof. Theorem 6.10

The

lT topology

is a sheaf o r local topology on Fre'chet-

Schwartz spaces. Proof

Let

let

be an open subset of the Frgchet-Schwartz

U

be an open covering of

(Ui)iEI

denote t h e r e s t r i c t i o n mapping from

H(U)

theorem we must show t h a t t h e mapping l i m (H(Ui),rT) c

F o r each

U.

into

J:f

E

iE1,

H(Ui).

space

let

E

and

P

U,Ui To prove our

( H ( U ) , T ~ )+ ( f l u . ).

E

1 1 E I

i s a l i n e a r isomorphism.

i

i s e a s i l y seen t o be a l i n e a r b i j e c t i o n .

J

mappings from

(H(U),rv)

compact subset hence

J

Let

of q

T~

on

K

of

U

into

H(K),

p

i s continuous. p

U,Ui

be a continuous semi-norm on

t h e r e e x i s t a compact subset H(K)

a r e continuous f o r every

IIK,

t h e mappings

such t h a t

p s q o IIK.

K As

Since t h e canonical

a r e a l l continuous and

( H ( U ) , T ~ ) . By t h e d e f i n i t i o n of K

U

and a continuous semi-norm

i s compact and

E

i s metrizable

Germs, surjective limits, we may w r i t e

OK.

as

K

f o r some j i n m H(K) + II H ( K ; ) j=1 t h e definition of

E

309

-products and power series spaces

i s a compact s u b s e t o f U . J m I . The n a t u r a l i n j e c t i v e l i n e a r mapping A = TI A;; j=1 i s c o n t i n u o u s . By r e g u l a r i t y o f t h e i n d u c t i v e l i m i t s i n J=1

where each

K. J

J

J

.

H(K.), j = 1 , . . , m , one s e e s t h a t A-l(B) i s bounded J m f o r e a c h bounded s u b s e t B o f n H(Kj). F u r t h e r m o r e , j =1 i s a DF s p a c e and H(K) i s a Monte1 s p a c e by p r o p o s i t i o n

i n H(K) m n H(K.) j=1 3 6.9. An a p p l i c a t i o n o f p r o p o s i t i o n 6.8 now i m p l i e s t h a t Hence t h e r e e x i s t s c o n t i n u o u s semi-norms

on

qj

H(K.) J

such t h a t q(g) d sup j=1,. Since

A . o II J

q . (A.(g)) 1 3

f o r all

. .,m

g

in

i s open.

A

for

j=l,

..., m

H(K)

-

K -*Kj

p ( f ) 6 q(nK(f)) 6 sup qj(nK j = 1 , . ,m j

~ ~ , ~ . ( ff o) r) e v e r y

..

f

in

H(u).

1

q . o I I K , i s a c o n t i n u o u s semi-norm on ( H ( U j ) , ~ . r r ) f o r each ' 3 tliis i m p l i e s t h a t J-' i.s c o n t i n u o u s and c o m p l e t e s t h e p r o o f .

Since

j

A s a c o r o l l a r y we improve a r e s u l t o f c h a p t e r f i v e . C o r o l l a r y 6.11 E

space

By c o r o l l a r y 5.23,

T~

Since

0

and

i s a baZanced open subset of a Frgehet nucZear T~ = T

c

-c0

whenever U T

U

with a b a s i s , then

Proof E.

If

T~

T

7

=

on H(U). whenever

T

c1 T 0 t h i s i m p l i e s t h a t

i s a n open p o l y d i s c i n

E.

T~

U

i s a n open p o l y d i s c i n

and

c o i n c i d e on

T~

By lemma 2 . 3 9 , and theorem 6 . 1 0 ,

a r e l o c a l t o p o l o g i e s and hence, s i n c e open p o l y d i s c s form a

fundamental neighbourhood b a s i s f o r t h e neighbourhoods o f z e r o i n

E

since

=

and

T

T

are both t r a n s l a t i o n i n v a r i a n t topologies,

H(U) f o r any open s u b s e t whenever

U

H(U)

U

of

E.

By p r o p o s i t i o n 6 . 7 ,

i s b a l a n c e d and open and hence

T~

= T

.

T

=

T

T

0

7

on

and T

1

on

H(U)

T h i s completes t h e

proof.

We now g i v e a few o t h e r r e s u l t s which show f u r t h e r p o s s i b i l i t i e s o f

Chapter 6

3 10

applying germs to obtain global results about the

topology

T~

Let U be an open subset of a locally convex space E

and let K be

a compact subset of U. We let

If U is an open subset o f a ZocaZly convex space,

Proposition 6.12 then (H(U),rw)

lim

=

kcU

HU(K)

K compact

Proof

=

lim

kcU

-

HU(K)

.

K compact

The algebraic identification follows at once, as in lemma 6.5,

from the fact that holomorphic functions are locally defined.

If K

is a

compact subset of U we let IIK denote the canonical mapping from H(U) into HU(K). If pK is a continuous semi-norm on HU(K) then we claim that p K o n K is a continuous semi-norm on (H(U),T~). If V is open and K C V C U then there exists c(V) > 0 such that pK(f) 6 c(V) I/f/lV for every f in HU(K). Hence

and this proves our claim. Thus the canonical linear bijection from (H(U),.cU) onto lim HU(K) is continuous. KCU K compact

Conversely, let p

be a

T

0

continuous semi-norm on H(U)

which is

ported by the compact subset K of U. If f,g E HU(K) and nK(f) = n,(g) then f and g coincide on a neighbourhood V

of

K.

31 1

Germs, surjective limits, E -products and power series spaces

Hence p(f) = p(g) since p(f-g) 5 c(V)(If-g/IV = 0. Thus, we may write p = pK o n K where pK is a well defined semi-norm on HU(K). Since p is ported by

K the restriction of pK to each Hm(V)nH(U)

and hence pK

is a continuous semi-norm on HU(K). (H(U),T~) = lim HU(K).

is continuous

This shows that

f-

KCU,K compgct The canonical injection from HU(K) into H (K) U

is clearly continuous

and hence it suffices to show that any continuous semi-norm p extends to a continuous semi-norm on Ff,(K). of p

to H"(V)n

H"(V)

n H(U)

p(f)

6

H(U)

Let pv be the restriction

for any open set V,

is dense in the Banach space

-

KCVCU.

(H"(V)

c(V) ((fl/Vfor every f in Hm(V)f7 H ( U ) ,

n

-

and

TIHu(K) = p.

If H"(V)n H(U) = H"(V) for all V neighbourhood system of K then H(K) =

Since

H(U),

pv

sion to a continuous semi-norm pv on (H"(V) n H(Uj, d rr/ z = pv for every V. p p on H"(K) by 'ir continuous semi-norm on HU(K)

on HU(K)

11

Ilv)

and

has a unique exten-

11 ( I v ) .

We define is a well defined

This completes the proof.

belonging to some fundamental = HU(K) for

gu(K) and if H(K)

a Eundamental system of compact subsets of U

then proposition 6.12 implies that T = T on H(U). This is the case if U is balanced and also 0 7 1 occurs in the next example. Example 6.13

Let

R

be a holomorphically convex open subset of (En.

. Any compact subset of nxcCN is contained in a compact set of the form KxL where K is a holomorphically convex compact subset of n and L is

a balanced convex compact subset of EN. If f E H(KxL) then f depends only on a finite number of coordinates and hence, by a reduction to finite dimensions and an application of the finite dimensional Oka-Weil approximation theorem (see Appendix I), f can be uniformly approximated on some N neighbourhood of KxL by holomorphic functions on G x C . Hence (H(RX~$,T,? T

0

= T

0

=

(H(QxE N 1

on H(U)

, ~ ~ )We. shall use this result in 56.3 to show

for any open subset

3

of

CN.

We now use proposition 6.12 to show that (H(U),T~) is complete whenever U is an open subset of a quasi-normable metrizable locally convex space.

Chapter 6

312 Definition 6.14

The i n d u c t i v e limit

( E , T ) = l i m ( E L YT ,L Y )

7

i s b o u n d e d l y r e t r a c t i v e i f f o r e a c h bounded s u b s e t t h e r e e x i s t s an

B C ( E a , ~ L Y ) and

such t h a t

a

T

B ~

E

of = B T

~

~

B

Boundedly r e t r a c t i v e i n d u c t i v e l i m i t s a r e r e g u l a r and i f each

i s quasi-complete then

ELY

Proposition 6.15

(E,T)

Let

is a l s o quasi-complete.

E

( E n , ~ n ) be a c o u n t a b Z e

= l i m

* n

boundedZy r e t r a c t i v e i n d u c t i v e Z i m i t o f Banach s p a c e s , and l e t F

b e a s u b s p a c e of

E.

If

-+ n nl

(F,T')

then

is b o u n d e d l y r e t r a c t i v e and c o m p l e t e .

For e a c h p o s i t i v e i n t e g e r

Proof

c l o s e d u n i t b a l l of

( 7 , ~ ' ) .S i n c e

integer

n

Now

Bnfl F

and

LY

xB

xB

E

x

+

x

c

is a

DF

B

n

let

Bn

be the

-cn

b e a bounded s u b s e t o f

space there e x i s t

a positive

such t h a t

h > O

-T

and t h e r e e x i s t

BnnF

a positive integer

m

such t h a t

> O

If then

(p,~')

and

T I

and l e t

En

E

h(BnO F) in

Em

X(BnCI F )

+

as

x

E

B+m

A(Bnn F ) and

TmC A(Bmfi F ) T m .

T'

as

B+-

in

(F,T)

.

Germs, sutjective limits, Hence

B C A (Bm r \ F )

shows t h a t

E

313

-products and p o w e r series spaces

Tm

and T and T a g r e e on B. This m i s a boundedly r e t r a c t i v e i n d u c t i v e l i m i t

(F,T')

and completes t h e p r o o f .

We a l s o n e e d t h e f o l l o w i n g r e s u l t . Proposition 6.16

( E , T ) = l i m ( E n , ~ n ) be a c o u n t a b l e

Let

-2

i n d u c t i v e l i m i t o f Banach s p a c e s .

l i m ( E n , ~ n ) i s bound-

+ n

Then

co

e d l y r e t r a c t i v e if and onZy i f f o r e a c h nu22 s e q u e n c e in

E

there e x i s t s a positive integer m

co

i s a (x,,,)~=~

( x , ) , = , C ( E ~ ~ T ~ )and Definition 6.17

T

n

such t h a t

n

n u l l sequence. E

A l o c a l l y convex space W

e x i s t s a z e r o neighbourhood B

i s quasi-

of z e r o t h e r e

V

n o r m a b l e i f f o r any g i v e n n e i g h b o u r h o o d c a n f i n d a bounded s u b s e t

such t h a t f o r every E

of

( x ~ ) ~ = ~

we

a>O

WCB+aV.

with

Every normed l i n e a r s p a c e i s q u a s i - n o r m a b l e and a l o c a l l y convex s p a c e i s a Schwartz s p a c e i f and o n l y i f it i s q u a s i normable and i t s bounded sets a r e precompact.

Thus a F r g c h e t -

Monte1 s p a c e i s q u a s i - n o r m a b l e i f and o n l y i f it i s a F r c c h e t Schwartz space, Proposition 6.18

I f

is a c o m p a c t s u b s e t of a q u a s i -

K

normabZe r n e t r i z a b Z e s p a c e

E

then

H(K)

(Hm(V),

= l i m

+

VDK,

v

I(

I(v)

open

i s a boundedZy r e t r a c t i v e i n d u c t i v e l i m i t . We a p p l y p r o p o s i t i o n 6 . 1 6 .

Proof

sequence i n

H(K).

Since

H(K)

Let

m

be a null

(fn)n=l

is a regular inductive l i m i t

( p r o p o s i t i o n 2 . 5 5 ) , t h e r e e x i s t s a convex balanced neighbourhood

V

of zero such t h a t

IlfnlI K + V = M <

m.

Since

(fn)E=,CHm(K+V) E

n a convex balanced neighbourhood t h a t f o r every

a>O

and

is quasi-normable t h e r e e x i s t s W

of

zero,

ZWCV,

we c a n f i n d a b o u n d e d s u b s e t

B

such of

E

3 14

Chapter 6

with

We c o m p l e t e t h e p r o o f b y s h o w i n g t h a t

WCB+clV.

i s a null

sequence i n

Since

fn(x+Y)

=

(H~(K+w),

/I /I K+W).

fim d fJX)

Cm=o

( f n ) nm = l

(y)

for every

x

in

K

m! and

y

in

for all

x

in

K

1 O < 6 c T

Given

For a n y

where with

and

W

n,

,

xeK,

dmfn(x)

/\m

m! fn(X)

.

and a l l

choose ylcB,

n

B y2

i t s u f f i c e s t o show

bounded i n E

V

and

is t h e symmetric

Since

E

y1+6y2

n

E

w

y1 = y 1 + 6 y z - 6 y Z €W + 6 V C T1V + S V C V

m!

m!

WCB+6V.

l i n e a r form a s s o c i a t e d

that

+

such t h a t

V

we s e e

315

Germs, surjective limits, E -products and power series spaces A

Since

p(f)

semi-norm on

for all

m

dnf (x)

sup

=

f

XEK

is a continuous

and

H(K)

and

H(K),

this implies

n

and this completes the proof

is a n o p e n s u b s e t of a quasi-normabZe

If U

Corollary 6.19

E

metrizabZe ZocaZZy c o n v e x s p a c e compZete. By proposition 6 . 1 8 ,

Proof

for any compact subset

K

of

then

H(K)

U.

(H(u),T~)

is

is boundedly retractive

By proposition 6.15,

UNJK V open is also boundedly retractive and hence complete. Since = lim (lim H"(V)nH(U)) (proposition 6.12) (H(U),.rU) c-

--f

KCU U 3V3K K compact and a projective limit o f complete spaces is complete, this shows that

(H(U),.rw)

is complete.

This completes the proof.

A weak converse to proposition 6 . 1 8 is also true as one

can easily prove the following: if E H(K)

=

is a distinguished Frgchet space and lim (Hm(V), I / I l v ) is boundedly d

V3K V open retractive for some non-empty compact subset

316

Chapter 6 K

of

E

then

E

In particular, E

is quasi-normable.

H(OE)

i s n o t b o u n d e d l y r e t r a c t i v e when

i s a F r g c h e t Monte1 s p a c e which i s n o t a F r g c h e t Schwartz

space.

16.2

SURJECTIVE LIMITS O F L O C A L L Y C O N V E X SPACES We now d e s c r i b e a m e t h o d o f d e c o m p o s i n g s p a c e s o f h o l o m o r -

p h i c f u n c t i o n s i n t o a u n i o n o f more a d a p t a b l e s u b s p a c e s . Alternatively,

t h i s m e t h o d may b e d e s c r i b e d a s a way o f g e n e r -

a t i n g l o c a l l y convex s p a c e s w i t h u s e f u l holomorphic p r o p e r t i e s . Our m e t h o d , t h e u s e o f s u r j e c t i v e l i m i t s a n d L i o u v i l l e ’ s i s b a s e d o n t h e f a c t o r i z a t i o n r e s u l t s o f c h a p t e r two

theorem,

a n d a r i s e s n a t u r a l l y i n many p r o b l e m s o f i n f i n i t e d i m e n s i o n a l

I t s range o f u s e f u l n e s s f o r problems of topologies

holomorphy. on

i s n o t a s g r e a t a s i n some o t h e r a r e a s a s f o r

H(U)

i n s t a n c e i n s o l v i n g t h e Levi problem. Definition 6.20

A c o l Z e c t i o n of l o c a l l y c o n v e x s p a c e s and

(Ei,ni)iEA

l i n e a r mappings

i s called a s u r j e c t i v e represen-

t a t i o n of t h e l o c a l l y e o n v e x s p a c e t i n u o u s l i n e a r mapping f r o m

E

E

ni

i f each Ei

onto

and

(ni

-1

i s a con(Vi))iEA

forms a b a s e ( a n d n o t a s u b b a s e ) f o r t h e f i l t e r of n e i g h b o u r 0 i n E as Vi ranges o v e r t h e neighbourhoods o f h o o d s of 0 i n Ei and i r a n g e s o v e r A . E i s called the surjective l i m i t of ( E i , ~ i ) i c A and we w r i t e E = l i m (Ei,ni). f-

iEA

I f each

ni

i s a n open mapping,

we call

l i m (Ei,iIi)

ci EA

an open s u r j e c t i v e l i m i t and i f f o r each subset

K

such t h a t

of

Ei

ni(Ki)

i E A

and e a c h compact

t h e r e e x i s t s a compact s u b s e t = K

then we say

l i m (Ei,ni)

f-

Ki

of

E

i s a compact

i EA

surjective l i m i t . E v e r y l o c a l l y c o n v e x s p a c e i s a s u r j e c t i v e l i m i t o f normed

Germs, surjective limits,

E

317

-products and power series spaces

nuclear spaces are s u r j e c t i v e limits of separ-

linear spaces,

s p a c e s and a l o c a l l y convex s p a c e which h a s

able inner product

t h e weak t o p o l o g y i s a s u r j e c t i v e l i m i t o f f i n i t e d i m e n s i o n a l spaces.

i s a s u r j e c t i v e l i m i t o f TI. E. 'iEA E i iaAl 1 ranges over all the f i n i t e subsets of A. This

Example 6 . 2 1 where

A1

s u r j e c t i v e l i m i t i s e a s i l y seen t o b e open and compact. Example 6 . 2 2 and

& ,

If

is a completely regular Hausdorff space

X

i s t h e s p a c e of a l l c o n t i n u o u s complex v a l u e d

(X)

f u n c t i o n s on

endowed w i t h t h e t o p o l o g y o f u n i f o r m c o n v e r -

X

g e n c e on t h e compact s u b s e t s o f .Q,cx)

=

l i m ( &(K), f-

11

then

X,

llK)

KCX where

K

r a n g e s o v e r t h e compact s u b s e t s of

and

X

&(K)

i s t h e Banach s p a c e o f complex v a l u e d c o n t i n u o u s f u n c t i o n s on K

endowed w i t h t h e s u p norm t o p o l o g y .

X

Since

i s a complet-

e l y r e g u l a r space, t h e T i e t z e extension theorem implies t h a t

I(

l i m (,&(K),

t-

I/K)

i s a compact s u r j e c t i v e l i m i t and t h e open

KCX

mapping theorem f o r Banach s p a c e s i m p l i e s t h a t i t i s a n open surjective l i m i t . The s t r o n g d u a l o f a s t r i c t i n d u c t i v e l i m i t

Example 6 . 2 3

o f F r c c h e t Monte1 s p a c e s i s a n open and compact s u r j e c t i v e l i m i t of

33W

Proof

Let

.spaces. E

=

l i m ( E n , ~ n ) be a strict inductive l i m i t ----f

n

o f Frgchet-Monte1 spaces.

Since

E

induces on

En

its

o r i g i n a l t o p o l o g y , w e s e e , by t h e Hahn-Banach theorem,

that

t h e transpose of the canonical injection of

E

s u r j e c t i v e mapping from

on

E'

Eb

onto

(En);.

En

into

is a

The s t r o n g t o p o l o g y

i s t h e t o p o l o g y o f u n i f o r m c o n v e r g e n c e on t h e bounded

subsets of

E

and,

s i n c e e a c h bounded s u b s e t o f

E

318

Chapter 6

i s c o n t a i n e d a n d c o m p a c t i n some

t h e topology on

E' is B t h e weakest t o p o l o g y f o r which a l l t h e t r a n s p o s e mappings a r e En,

E' is a surjective l i m i t of B An a p p l i c a t i o n o f t h e o p e n m a p p i n g t h e o r e m s h o w s t h a t i t i s

continuous.

Hence

an open s u r j e c t i v e l i m i t . (En);.

Now l e t

b e a compact s u b s e t o f

Kn

There e x i s t s a convex balanced neighbourhood

zero in

whose p o l a r i n

En

(En);,

contains

Vo,

of

V

Since

Kn.

E

is a strict inductive l i m i t ,

t h e r e e x i s t s a neighbourhood

W

of

Wn

in

0

s

I$(WnEn)I a

$EE'

such t h a t

E

Monte1 s p a c e

Hence i f

En.

lv(W)

I

4

and

1

$IEn = $.

i s a compact s u b s e t

Wo

of

In particular, we note t h a t

CNx

C")

=

there exists

E'.

B

is a

E

As

completes t h e proof.

then

+EVO

a n d , by t h e Hahn-Banach theorem,

1

such t h a t

N

Vy)

This

is a

l i m C c-

n

compact and open s u r j e c t i v e l i m i t o f a 3 R s p a c e s w i t h a b a s i s . The u s e f u l n e s s o f s u r j e c t i v e l i m i t s stems from t h e f o l l o w i n g r e s u l t which i s e a s i l y proved u s i n g t h e method o f proposition 2.24.

lemma 6 . 2 4

E = l i m

Let

(E.,n.) 1

f-

u

iEA

H(U) i

if

=

in

iEA

H(ni(U))

and

A

iEA

c H(U)

f

1

f-

X

N

H(ni(U))

if

f

E

H(U)

such t h a t

such t h a t

bounded s u b s e t of

E.

Then

then there e x i s t s an N

f = foni).

Moreover,

is a l s o a c o m p a c t s u r j e c t i v e l i m i t and

1

f a e t o r s uniforrnzy through

X CH(ni(U))

subset o f

E

(i.e.

(E.,n.)

E = l i m

d

be a c o n v e x b a l a n c e d o p e n s u b s e t of

U

l i m i t and Zet

b e an o p e n s u r j e c t i v e

1

H(U)

ieA

fi.e.

X = nifi))

then

i f and o n l y

if

X

, 4

X

there e x i s t s

is a

is

G

T~ T~

bounded

H[ni(u)).

To o b t a i n p o s i t i v e r e s u l t s i t i s t h u s n e c e s s a r y t o f i n d c o n d i t i o n s u n d e r which a f a m i l y o f f u n c t i o n s on a s u r j e c t i v e

319

Germs, surjective limits, E -products and power series spaces

limit factors uniformly through some Ei. Frequently such conditions involve the structure of the indexing set A . For example, an analysis of

(X)

of

If T

in the surjective representation

leads to the following result. i s a c o m p l e t e l y r e g u l a r Hausdorff space t h e n t h e

X

H(&,(X))

bounded s u b s e t s o f

0

A

consequently

= -c6

T

on

a r e 1ocaZZ.y b o u n d e d and

H(& (X)))

if and o n l y i f ,&(X)

o,b i s an i n f r a b a r r e l l e d l o c a l l y convex space.

This result can be proved using methods similar to those employed in the proof of proposition 6.29.

We will not under-

take here a detailed study of the inde.xing set but instead confine ourselves to a few representative examples. Let

Proposition 6.25

l i m i t o f Frgchet-Montel U

t i n u o u s norm.

I f

HHy(U)

and t h e

= H(U)

l o c a l l y bounded.

E = lim E be a s t r i c t i n d u c t i v e - - b n n s p a c e s and s u p p o s e E a d m i t s a c o n -

i s a n o p e n s u b s e t of EL T~ bounded s u b s e t s o f

Hence

(H(U),-co)

then H(U)

i s c o m p l e t e and

H(U).

on

Proof

are T

0,b

=

T

We may suppose without loss of generality, that U is an E i = lim ((En)b,nn) cn

is convex and balanced and since

open surjective limit, u = ~,'(]L~(U)) for some positive integer m. Let V be the unit ball of a continuous norm on E . Since E and E t ; are complete Monte1 spaces, Vo is a compact subset of We first claim that V o is a deter-

Ei.

mining set for hypoanalytic functions o n f

E

HHy(U)

and

fIVon

=

0,

then

f

U;

i.e. if

0.

By using a Taylor series expansion, we see that if

6

320

Chapter 6

An d f(O) n! that

L o n

VO

u

for all

0

=

n

and hence it suffices to show

is a determining set for hypoanalytic homogeneous

polynomials.

n=l

If

this is clear since hypocontinuous

homogeneous polynomials o f degree continuous linear forms o n

E'

1

are nothing more than

greater than

1

E,

i.e. elements o f

B'

is the unit ball o f a continuous norm o n

N o w let

E.

and suppose we have shown that

VO

and V n be is deter-

mining for all hypocontinuous homogeneous polynomials o f degree strictly less than n . Let P be a hypocontinuous n-homogeneous polynomial which vanishes o n symmetric

n

Vo

and let

P.

linear form corresponding to

ization formula (theorem 1.5) we see that and vanishes on

Vo

x

Vo.

..

L

L

be the

By the polaris hypocontinuous

since

xV0

i=l,. . , n Fix

x

in

V O .

Then

Lx:E'

B

+ C

defined by

L(x,z, . . . , z) is an n - 1 homogeneous hypocontinuous polynomial which vanishes o n Vo. By our induction hypothesis Lx(z)

=

Lx : 0 . Now let y be arbitrary and let Ly:'L' + C be B B defined by Ly(z) = L(z,y, ...,y). Then Ly is a hypocontinuous linear form o n E' which vanishes o n V o , and hence by B

I n particular, we have Ly(y) = P(y) = 0 induction, o n E ' 6' for any y in E A . Hence V o is a determining set for hypoanalytic functions on U. Now let f E Hrjy(U). Since

E' B

=

lim ((En);,nn) n

is an open and compact surjective limit,

4-

it suffices to show that

f

factors through some (E;lB.If

were not true, then for each integer zn and zn+yn E U such that

F o r each

n

the function

z

-+

n,

f(z+yn)-f(z)

msn,

this

there exist

defines a non-zero

hypoanalytic function o n some convex balanced neighbourhood o f

32 1

Germs, surjective limits, E -products and power series spaces EB

zero i n

1

and hence t h e r e e x i s t s

x n ~ V o n 7 U such t h a t

f(xn+yn) # f ( x n ) . For a l l

n t m

Hence

gn w h i c h m a p s

i s a non-constant

f(xn+Xyn)-f(xn) gn(0) # gn(l),

the function

I

> n

it f o l l o w s t h a t

U.

Ixn+hnyn};=m

xn

to

C since

such t h a t

E (c

m

Since

v e r y s t r o n g l y convergent sequence and of

An

n 5 m.

for all

E

entire function,

a n d h e n c e we c a n c h o o s e

If(xn+hnyn)

h

(yn),=,

1

is a

for all

E V O ~ Y U

n

i s a r e l a t i v e l y compact s u b s e t

This contradicts the fact that

i s unbounded on

f

and hence f f a c t o r s t h r o u g h s o m e (E ) ' a n d {Xn+hnYn};=m nt3 f E H ( U ) . Now s u p p o s e (fa)aEr is a T bounded s u b s e t o f

We c l a i m t h a t

H(U).

(fa)aEr

f a c t o r s u n i f o r m l y t h r o u g h some

(E ) ' . S i n c e E; i s a compact s u r j e c t i v e l i m i t and (En)A is a n B space f o r each n t h i s would c o m p l e t e t h e p r o o f ( s e e

Jjgq

r e l a t i v e l y compact i n (fn):=1

t h e n w e c o u l d f i n d , as i n

I f t h i s were n o t s o ,

example 2 . 4 7 ) .

the f i r s t part of the proof,

m

a sequence

( X ~ + X ~ Y , ) , =w ~hich i s

and a sequence o f f u n c t i o n s

U

~ ( f a ) a E r such t h a t

Ifn (xn+anyn)

This contradicts the fact t h a t

I

is

(fa)aEr

n

>

for all

n.

bounded and

T

completes t h e proof. Corollary 6.26 H(U)'

' 0

'U

a

Proof

s p a c e s and

T

#

o,b

=

p(f)

open subset

= IIfllR

of

U T~

;I

T~

3'.

w,b

=

T

t h e n on

6'

i s a c o n t i n u o u s norm on T

0,b Since

; T ~ , ~ .

The r e q u i r e m e n t

6.25

T

is a s t r i c t inductive l i m i t of Frgchet nuclear

proposition 6.25 implies 5 . 4 6 show

i s an open s u b s e t o f

U

If <

of

=

T

w,b

on

T~

$nn=l

S'

a.

H(U)

Hence f o r any

examples 2.52

and

This completes t h e proof.

a c o n t i n u o u s norm on

i s n o t s u p e r f l u o u s as t h e example

corollary 5.35

=

and example 5 . 3 6 ( b ) ) .

i n proposition

E

shows ( s e e

C"XC(~)

N e v e r t h e l e s s i f w e do

n o t h a v e a c o n t i n u o u s n o r m w e may r e p l a c e

T~

by

T~

in

322

Chapter 6

proposition 6.25 and we obtain a similar result. Let

Proposition 6.27

E = lim (En,nn)

b e a n o p e n and

c-

n

c o m p a c t s u r j e c t i v e Zimit o f $ F W 2 o p e n s u b s e t of

The

E.

locaZZy b o u n d e d and

T

w,b

T~

= T

s p a c e s and l e t

bounded s u b s e t s of

on

6

U

be an are

H(U)

H(U).

As in the previous proposition we may suppose where U is a convex balanced open subset o f E. Let B be a T bounded subset o f H(U). .It suffices to show that B factors uniformly through some En. If not, Proof

U

=

n m- 1 (nm(U))

then w e can find a sequence in

u,

and

(Xn);=l

=

0

(Xn+Yn);=l’ for all

n.

B,

(fn)mnYl,

such that

two sequences in

fn(xn+yn)

# fn(xn)

By considering, for positive

and nn(yn) integers k and r , the T~ continuous seminorm Pk,r(f) = sup Id k f(0)(xn)r(yn)n-rl we s e e that Cd k f(0)JfEB factors n for each k . Hence we can find a uniformly through some E nk m sequence o f scalars, two sequences o f increasing positive integers,

for all

(R~):=~

and

(kn);=l

sucii t h at

n.

T h e semi-norm

m

on H(O), O E E, is continuous since the sequence (yL )n=l n is very strongly convergent and H(0) is a barrelled locally convex space. Since the canonical mapping (H(U),r,J H(0) is continuous it follows that p continuous semi-norm on H ( U ) . Hence sup p(f) w fE B +

T

is also a <

m.

323

Germs, surjective limits,E -products and power series spaces This contradicts the f a c t t h a t B

f a c t o r s t h r o u g h some

En

T

bounded s u b s e t s of

H(V),

s p a c e , a r e l o c a l l y bounded

p(f,

n

) > n

and i s a l s o

n.

Hence

bounded.

Since

for all T

an open s u b s e t o f a

V

(example 2.47),

23q

t h i s completes t h e

proof. I t i s worth n o t i n g t h a t even though p r o p o s i t i o n s 6.25 and 6 . 2 7 a r e s i m i l a r i n s t a t e m e n t ( b o t h h y p o t h e s i s and c o n c l u s i o n ) , q u i t e d i f f e r e n t methods are used t o g e t uniform f a c t o r izations. C o m b i n i n g p r o p o s i t i o n s 6 . 2 5 a n d 6 . 2 7 we o b t a i n t h e following r e s u l t . Proposition 6.28

-

E = l i m E

Let

n

n

be a s t r i c t i n d u c t i v e

l i m i t of F r z c h e t Montel ( r e s p . Frgchet Schwartz, Frgchet nuc l e a r ) s p a c e s . Then (H(E;I,T~)

-

(H(E;),T~,~= ) lim(H(En);) 9 ~ o ) n

=

i s a s t r i c t i n d u c t i v e Z i m i t of F r g c h e t - M o n t e 2

( r e s p . Frgchet

Schwartz,

F r g c h e t n u c l e a r ) s p a c e s and t h e

T

of

a r e ZocaZZy b o u n d e d .

(H(E;),.r6)

H(E)

E

c o m p l e t e and i f on

Moreover,

w

bounded s u b s e t s

a d m i t s a c o n t i n u o u s norm, t h e n

T~

H(EA).

Proof

I n example 2.47,

If

Frgchet space.

En

we s h o w e d t h a t

i s Montel

(H(En)b) , T ~ ) i s Montel

then

c o r o l l a r y 3.38 ion 6.9,

=

T

0,b

is a

Schwartz, nuclear),

Schwartz, nuclear), by

(resp. a modification o f the proof of proposit-

corollary 3.65).

Since limit,

(resp.

(resp.

(H(En);),~o)

is

E l

B

=

l i m (E f-

n

it f o l l o w s t h a t

) I

i s a n open and compact s u r j e c t i v e

l i m

(H(En)A),~o) i s a s t r i c t inductive

n B --f

n

324

Chapter 6 /

I i - m i t o f F r e c h e t s p a c e s and c o n s e q u e n t l y it i s c o m p l e t e .

As

T and t h e i n d u c t i v e l i m i t t o p o l o g y on H ( E i ) are both 6 b o r n o l o g i c a l a n d h a v e t h e same b o u n d e d s e t s , b y p r o p o s i t i o n

they are equal.

6.27,

This completes t h e proof.

Our f i n a l a p p l i c a t i o n o f s u r j e c t i v e l i m i t s i s t o h o l o m o r p h i c germs and i n t h i s example, w e do n o t assume t h a t t h e indexing set is countable. P r o p o s i t i o n 6.29

space a n d suppose

uous f u n c t i o n s on

d e n o t e t h e s e t o f a l l bounded c o n t i n By t h e T i e t z e e x t e n s i o n t h e o r e m

X.

i s a dense subspace

balanced open s u b s e t o f

w e can choose n.

be a completely regular Hausdorff is an infrabarrelzed 'ZocaZly convex

is a regular inductive limit of Banach is a compact metrizable subset of ,&(X).

,eb(X)

Let

,gb(X)

(X)

L.&

space. Then H ( K ) spaces whenever K Proof

X

Let

xn

E

of a ( X ) . and

&(X)

VnAb(X)

If

V

(fn);=l

i s a convex

CH(

(X))

fn(xn) # 0

such t h a t

then

for all

By t h e i d e n t i t y t h e o r e m f o r h o l o m o r p h i c f u n c t i o n s o f o n e m

complex v a r i a b l e , choose a sequence o f s c a l a r s ( A ~ ) ~s u = ch ~ xnIIX 6 1 a n d fn(h,xn) # 0 f o r a l l n . Hence that IX,I.ll h,x,

as

+ 0

n+m

in

&(X)

and

o s i t i o n 6 . 2 i t s u f f i c e s t o show

fn(Xnxn)

#

0.

By p r o p -

i s r e g u l a r t o complete

H(0)

the proof. Let

B

b e a bounded s u b s e t o f

H(0)

and l e t

I t o b v i o u s l y s u f f i c e s t o show t h e r e e x i s t s a n e i g h b o u r h o o d o f z e r o i n &(X) let

Wp

=

for all

{xEX;

f,g

such t h a t

E

Vx

supIIFII,, F Eg open i n

h(X1 S u p p o r t ( g ) C V x

P(f+g) # P(f)}.

<

X,

and

a.

xcVx

For each

P

there exists

in

V N

B

325

Germs, surjective limits, e -products and power series spaces

We c l a i m

W

-

o f open s u b s e t s o f n

AnOK

and i f

#

Wpnn A n # + .

(gn)n=19

such t h a t

Pn€ B

&(X),

Pn(fn+gn) # Pn(fn)

h(X)

is dense i n

n.

for all

and

we may s u p p o s e

By t h e i d e n t i t y t h e o r e m f o r h o l o m o r -

p h i c f u n c t i o n s and s i n c e e a c h s e q u e n c e i n ,fib(X) w e a k l y c o n v e r g e n t , we may a l s o s u p p o s e sequence i n

Since support ( g n ) C A

is very is a null

such t h a t

-

n

n

all X

Ann K

and

and a l l

n

=

+

f o r any

sufficiently large

is very strongly convergent t o zero.

(gnInzl

L i o u v i l l e ' s theorem,

of

K

m

t h e sequence 1

(fn)mnZ1

(XI.

g i v e n compact s u b s e t

( Bn =:

(fn)iXl,

n.

for all

S i n c e fS.,(X) f n ~.fi,(X)

&

there exists

such t h a t

(gn)CAn

for

then

X

Hence t h e r e e x i s t s e q u e n c e s i n

a,

support

AnnW # $

such t h a t

sufficiently large.

AnnW # $

Since

and

n

for all

$

(An):=l,

X,

i s a n y compact s u b s e t of

K

If n o t ,

X.

t h e r e e x i s t s a sequence

is infrabarrelled,

t h e n s i n c e ,&(X) all

.v

{ U W p ; P E B } i s a compact s u b s e t o f

=

By

we may c h o o s e a s e q u e n c e o f s c a l a r s I P , ( f , + ~ ~ g ~ )> l n

for all

b e t h e d e g r e e o f t h e homogeneous p o l y n o m i a l

n.

Pn.

Let

kn

If

sup kn < t h e n t h e r e e x i s t a p o s i t i v e i n t e g e r N and a n sequence of integers n such t h a t kn. = N f o r a l l j . j 1 Hence (pn. i s a bounded s u b s e t o f H(0) and consequently I a bounded s u b s e t of L

=

i s a c o m p a c t s u b s e t o f $(X)

sUpllPn.IIL < 1

( (PcNE) , T ~ ) .

J

m.

The set

(fn+Bngn);=l

and hence

This contradicts the fact t h a t

for all

j.

(nj)y=l

such t h a t

On t h e o t h e r h a n d ,

u { O }

j

\IL>

nj

s u p kn = then w e can n choose a s t r i c t l y i n c r e a s i n g sequence of p o s i t i v e i n t e g e r s kn.> I

kn

j -1

if

IIPn

for all

j .

The semi-norm

326

Chapter 6

H(0).

is a continuous semi-norm o n

Since

all j we again get a contradiction. subset o f X . We now claim that each rv

F E B subset L If

Let

X

of

f,g

E

LJ

lim & ( K )

W

>

n . for

I is a compact

factors through &(W).

B

factors through h(L) since

for some compact

is an open surjective limit.

t-

KCX

a(X)

W.

hood o f

F

then

in

F

Hence

p(P,,) I

and s u p p o s e

g

vanishes on some neighbour-

h l € &(X)

Choose

such that h l is equal t o 1 W and support ( h l ) C { x ; g ( x ) = 0 1 . If X E L\Vl then there exists a neighbourhood Vx o f x such that F(fx+gx) = T ( f x ) for any fx,gx E ,@,(X), support ( g X ) c V x . Choose h x E & ( X ) such that h x ( x ) = 1

o n a neighbourhood

and support

V1

(hx)CVx.

u XE

contains

of

WuL

L\V

Let

-

TXhr

V]

Vx

N

k(x)

such that

(y€X;hx(y)>

xl,

and hence there exists

=

2 -

x d V1

all

1

lu

...

N

W c r L C V l u V X 1 d Vx

such that

=

.

uVx

L,

Vx

1

identically z e r o o n some neighbourhood of n = k ( x ) + h l ( x ) + 1. h (x) for every 1 = 1 xi 1 for every x in X . and I k l ( x ) / 2 -

kl(x)

. . . ,xn rv

uVx

n KuL.

X E

2

Y

k = k/

Let i=l,

. . . ,n . N

hl

= hl/kl

Now A

k + hl +

and

kl

A

,

N

I:= hi,E

1

on X

and

d

hi

E

T h e set

L\V1 k

Now choose

...

r4

1 T}.

= hX

X.

and

k

(Xj

E

is

Let kl

E

(X)

321

Germs, surjective limits, €-products and power series spaces

(since

, d

is identically zero o n a

k

K u L)

neighbourhood of

(since

A

support (hl)

CIx,g(x)

=

support (h 1 )

= 0))

F(f)

=

(since

PJ

support ( h i g ) C V x

i = l , . . . ,n) . Now suppose

vanishes o n

g

W

choose

1

o n some neighbourhood of

of

V.

hV

Since

converges to

E

&(X)

W.

V

For each neighbourhood

of

Hence

I(hVII'<1, hV is identically W and h V : 0 o n the complement the net hVg converges to zero as V F(f+g)

vanishes on a neighbourhood of through

for

such that

gIK : 0 K.

W.

i

=

K.

lim F(f+hVg)

V-t K

Thus each

=

F

F(f) E

%

as

hVg factors

Since

lima(X) is a compact surjective limit and KCX each To-bounded subset ofH(.&(W)) is locally bounded, this completes the proof. t-

§6.3

-PRODUCTS

In this section, we use &-products to study holomorphic functions defined on a product o f open sets and to extend results concerning scalar valued holomorphic functions to vector valued holomorphic functions.

When t h e d o m a i n s p a c e s

328

Chapter 6

have t h e a p p r o x i m a t i o n p r o p e r t y ,

the

E.-product

i s a tensor

product. Let

and

E

t h e dual of

E i

be l o c a l l y convex spaces.

F

w i l l denote

endowed w i t h t h e t o p o l o g y o f u n i f o r m c o n v e r -

E

gence on t h e convex b a l a n c e d compact s u b s e t s o f

E.

c l o s e d convex h u l l o f each compact s u b s e t of

i s compact

( f o r example,

if

i s quasicomplete)

E

E

If the

t h i s topology coincides

w i t h t h e compact open t o p o l o g y . D e f i n i t i o n 6.30

Let

E

F

and

be l o c a l l y convex spaces.

E E. F

i s t h e s p a c e o f a l l c o n t i n u o u s l i n e a r mappings f r o m

into

F

endowed w i t h t h e t o p o l o g y o f u n i f o r m c o n v e r g e n c e o n

t h e e q u i c o n t i n u o u s s u b s e t s of E ' . by

E:

We a l s o d e n o t e t h i s s p a c e

&(E;,F). I t i s n o t d i f f i c u l t t o show

QE;;F)

Y J!JF;;E). A %E(Ei;F) i s g i v e n by

f u n d a m e n t a l s y s t e m o f semi-norms on

where

ranges over

a

If

U

cs(E)

and

i s a n open s u b s e t of

over

B

(In a n d

cs(F). F

i s a quasicom-

p l e t e l o c a l l y convex s p a c e , t h e n t h e mapping from into

H(U)E F

f*(v)

= vof,

which t a k e s

f

to

H(U;F)

f*:F;--9H(U),

i s a l i n e a r isomorphism and w e have

H(U;F)

=

H(U) E F . Our aim i s t o e x t e n d t h i s f o r m u l a t o i n f i n i t e d i m e n s i o n s u s i n g t h e same m a p p i n g .

Our f i r s t e x a m p l e shows, h o w e v e r ,

that

t h i s may n o t a l w a y s b e p o s s i b l e . Example 6 . 3 1 s p a c e and l e t

Let f

E

be an i n f i n i t e dimensional Hilbert

b e t h e i d e n t i t y mapping from

S i n c e t h e compact open t o p o l o g y on t h a n t h e weak t o p o l o g y

f p! H ( E ; ; E i ) .

E'

E;

onto

E;.

is strictly finer

We s h o w t h a t

f*

is in

329

Germs, surjective limits, E -products and power series spaces

E EA

H(E:)

ed.

when

i s g i v e n t h e compact open t o p o l o g y .

H(E,')

Since

f*(v) = vof = v a ( E ; ) ' C H ( E ' ) ,

Since

(Ed):

from

into

(E;);

Lemma 6 . 3 2

space

= (Ed):,

(H(E;)

U

Let

be a n o p e n s u b s e t of a ZocaZly c o n v e x -+

(HHy(U),~o):

6(x)(f) = f(x).

mapping g i v e n by

i s well d e f i n -

,r0).

6 : U

and l e t

E

f*

i s a l s o a c o n t i n u o u s mapping

f*

denote the evaZuatim

Then

A)'

HHy(U;(HHy(U)>ro) Proof and

The l o c a l l y c o n v e x s p a c e ((HHy(U) ,ro)A) 6(X+AY) ( f )

for fixed function and

6

x,y

and

A

n

and a l l

f

(y)

E

s u f f i c i e n t l y small, t h e

i s holomorphic a t t h e o r i g i n i n

(c

i s a G-holomorphic mapping. b e a compact s u b s e t o f

K

C {f

HHy(U); /lfIIK 6

1 } O

equicontinuous subset of on

is complete

Since

2 f ( x ) n!

&(x+Ay)(f)

A +

Let

6(K)

I",=,

=

(HHy(U),ro)

HHy(U).

=

induced by

6(K)

I

(HHy(U),ro):

Clearly

ogy on

a n d t h e weak t o p o l o g y o n

K

Since

it follows t h a t

(HHy(U),r0)'

topology.

6

U.

6(K)

is an

and hence t h e topology

i s e q u a l t o t h e weak

is continuous f o r the i n i t i a l topol-

(HHy(U),ro)'.

Thus

6IK

i s continuous and t h i s completes t h e p r o o f . Proposition 6.33

convex space

E

convex space.

g i v e n by

6*($)

Let and Z e t

u

be an o p e n s u b s e t of a ZocaZZy F

be a q u a s i - c o m p l e t e

ZocaZZy

The mapping

= $06

i s a c a n o n i c a Z i s o m o r p h i s m of ZocaZZy c o n v e x s p a c e s and h e n c e

Chapter 6

3 30

Proof

By lemma 6 . 3 2 ,

i s well d e f i n e d and i t i s

6"

o b v i o u s l y l i n e a r and i n j e c t i v e .

We now s h o w t h a t

surjective.

We d e f i n e

f

Let

by t h e formula

E

HHY(U;F).

v(f*(w))

(HHy(U),~o)'.

If

w(vof)

=

i s a compact s u b s e t o f

K

f(K),L,

Hence

f o r every

( ( v o f / I K6 \ ( v ( \ v

E

F'

Thus

B E cs(F)

then

i s well defined.

f*

If

i s a r e l a t i v e l y compact s u b s e t o f

B(f*(w)) = 1/w/Iv Moreover,

v(B*f*(x)) and s o

and

f o r any

x

f* in

w

F'

f*(w)

is

(F;)'

E

=

F

( H H y ( U ) , ~ o ) . Hence

i s a c o n t i n u o u s l i n e a r mapping. U

= v(f*(d(x)))

6*f* = f .

F.

and f o r f i x e d

F'

in

v

endowed w i t h t h e c o m p a c t o p e n t o p o l o g y . and

then the

U

i s c o n t i n u o u s when

w(vof)

-f

and w i n

VEF'

i s a compact s u b s e t o f

c l o s e d convex h u l l o f t h e mapping

for every

is

6"

and =

v

in

F'

B(x)(vof)

= vof(x)

I t r e m a i n s t o show t h a t

is a topol-

6*

o g i c a l isomorphism. Let all

in B

K

b e a compact s u b s e t o f

hEHHY(U),

and l e t If

(HHy(U),~o)'. unit ball in

f o r any

f

and

F',

f*

6

V E

U,

let

a(h) =

be t h e polar of the

cs(F)

and

W

a

/Ih

/IK

unit ball

is the polar of the

then

as d e f i n e d above.

This completes theproof.

33 1

Germs, surjective limits, e -products and power series spaces Proposition 6.34

u

Let E

ZocatZy c o n v e x s p a c e s

V

and F

and

be o p e n s u b s e t s of t h e respectively.

HHy(UxV) = HHy(U) & HHy(V) = H H y ( U ; H H y ( V ) )

topologically

Then

~ Z g e b r a i c a l Z y and

l e a c h f u n c t i o n s p a c e is g i v e n t h e c o m p a c t o p e n

top0 logy I . Proof

Since

(HHy(V),~o) i s quasi-complete,

6.33 implies that f

E

H H Y ( U xV ) .

proposition

HHy(U) g HHy(V) = H H y ( U ; H H y ( V ) ) . IJ

We d e f i n e

N

f(x)(y) = f(x,y).

f : U + HHy(V)

Now l e t

by t h e formula

By u s i n g t h e C a u c h y i n t e g r a l f o r m u l a o n e

s e e s t h a t t h e mapping

n! i s hypoanalytic f o r any f i x e d

negative integer

n.

xo

in

U,

and any non-

XEE,

Hence t h e f u n c t i o n

n! belongs t o

Pa(”~;~,,(v)).

f o r any fixed

xo

in

Since

x

U,

in

E,

s u f f i c i e n t l y small it f o l l o w s t h a t Now l e t

K

respectively. continuous. a.

for all

a I a

0

E

IK

x

I c

and a l l

h

HG (U;HHy(V)).

U

and

V

i s c o n t i n u o u s i t is u n i f o r m l y

~ L K + x

E

f o r a given

- f(X3Y)

f

and

b e compact s u b s e t s o f

L

Hence i f

such t h a t If(Xa’Y)

and Since

ycV N

r1>0

as

a-m

we h a v e

Tl

y

in

L.

Thus

then there exists

Chapter 6

332

and

r*

f

E

HHy(U;HHy(V)1

Now s u p p o s e t h e formula

K

Let ively.

Let

and u

a

b e compact s u b s e t s of -+

u

and

v BE L +v

i s continuous and

g(u)\

HHy(UxV)

HHy(UxV) with

as

g(u,)

u n i f o r m l y on t h e compact s u b s e t s o f Hence

on

UxV

V

respect-

by

= g(u)(v).

L

E K

We d e f i n e

HHy(U;HHy(V)).

z(u,v)

Then

since

g

.

+

U

and

respectively.

a , B+m

g(u)

as

V.

a n d we may a l g e b r a i c a l l y i d e n t i f y

HHy(U;HHy(V)).

t h i s i s a l s o a t o p o l o g i c a l isomorphism and c o m p l e t e s t h e p r o o f . Corollary 6.35

c o n v e x s p a c e s and let space.

If

U,V

U

Let

and

F UxV

and

V

be o p e n s u b s e t s of localZy

be a q u a s i - c o m p l e t e a r e k-spaces,

then

locally c o n v e x

333

Germs, surjective limits, E -products and power series spaces

Corollary 6 . 3 5 applies if

U

and

V

are both open subsets of

FrGchet spaces or both are open subsets of our next proof, we use the fact that E C F

U83mA = E

In

spaces.

0, F

if

E

is

a locally convex space with the approximation property (see Appendix 11). As our first application of the 6 product, we prove a converse to theorem 5 . 4 2 .

-r0 =

T

6

E

If

Theorem 6 . 3 6 and

on

i s a F r z c h e t n u c l e a r s p a c e w i t h a basis

H(E)

E

then

is a

DN

space.

m Proof Let (enIn=1 be an absolute basis for F be the closed subspace of E spanned by

E m

and let Since

F and E , C and F are Frgchet nuclear spaces, an application of Corollary 6 . 3 5 shows that E =

If

(c x

on

= -r0

eLF1

T&

H(C)

H(E)

then the closed complemented subspace

of H ( E ) is a bornological space. By proposition 15, chapter 2 of A. Grothendieck's thesis, this implies that

F

contains an increasing fundamental system o f weights m

(Wm)m=l'

w m = (wm,n)

m

SUP(lXn(W

n

such that )€

m,n

W m,n

W

l'n

is finite for every positive integer m , all E>O and all 1 (X&l in F . Letting p = and taking pth roots we see that

Chapter 6

334 for a l l positive integers

m

and

F

and p r o p o s i t i o n 5.40 i m p l i e s t h a t

and a l l

p

(x~);=~ in

F.

Hence

i s a c o n t i n u o u s weight on

F

is a

that

space.

DN

is also a

E

Since

s

= C x F C s x s

E

t h i s means

s p a c e and completes t h e p r o o f .

DN

Theorems 5.42 and 6 . 3 6 t o g e t h e r g i v e t h e f o l l o w i n g : E

I f =

T o

is a F r g c h e t n u c l e a r s p a c e w i t h a b a s i s t h e n on

T6

if and o n l y i f

H(E)

E

is a

DN

space.

F o r o u r n e x t a p p l i c a t i o n , a k e r n e l s theorem f o r a n a l y t i c

functionals on c e r t a i n f u l l y n u c l e a r spaces, w e need a f u r t h e r type of tensor product. E G F

s p a c e s we l e t E@F

E

If

F

and

are l o c a l l y convex

d e n o t e t h e c o m p l e t i o n of t h e v e c t o r s p a c e

endowed w i t h t h e t o p o z o g y o f uniform c o n v e r g e n c e o n t h e

s e p a r a t e l y e q u i c o n t i n u o u s s u b s e t s of E x F.

c o n t i n u o u s b i z i n e a r forms o n Proposition 6.37

complete nuclear

Proof

Since

nuclear

DN

(H(V),T~)

If DN

U,V

spaces, and

U

t h e s e t of a l l s e p a r a t e l y

and

V

a r e o p e n p o l y d i s c s in

spaces w i t h a b a s i s ,

and

U x V

then

a r e open polydiscs i n complete

theorem 5.42 implies t h a t

(H(UXV),T~)

(H(U)

, T ~ ) ,

are f u l l y nuclear spaces.

By

corollary 6.35,

Since 2,

(H(UXV),T~)A

of A .

i s complete,

the corollary p.91,

Grothendieck's t h e s i s implies

chapter

335

Germs, surjective limits, e-products and power series spaces

This completes the proof. For

a3rL

spaces, the situation is much simpler and the

following result is easily proved:

Using techniques similar to the above, a more detailed analysis o f the E - p r o d u c t o f 8 2 2 s p a c e s and propositions 6.9 and 6 . 1 8 one may prove the following results: Proposition 6.38 If K 1 and Frgchet Schwartz spaces, t h e n

If U

Corollary 6 . 3 9 Schwartz spaces,

As

V

a r e c o m p a c t s u b s e t s of

a r e o p e n s u b s e t s of F r z c h e t

then

a further corollary, w e improve corollary 6.11 in the

special case o f

cN.

Corollary 6.40 T

and

K2

=

T

on

If

u

is a n o p e n s u b s e t of

cN

then

H(U).

Proof -

By analytic continuation, it is possible to find a pseudo-convex domain i2 spread over (c", n a positive

integer, and a n embedding o f U in 0 X Q N such that each holomorphic function on U has a unique extension to a holomorphic function o n

Q x

CN

and moreover,

336

Chapter 6

Hence N

( H ( U ) , T u ) '=" (H(R

cf ( H ( R

x

2 (H(R)

1,

Tu)

( e x a m p l e 6.13)

Q: N ) , T ~ ) (H(C

, T ~ )

N),T~) N

( H ( R ) , T ~& ) ( H ( Q :1 2'

(H(R

2'

(H(U)

x

aN) , y o ) ,T

~

(corollary

, ~ ~( e)x a m p l e

5

(corollary 6.35)

.)

This completes t h e proof

POWER SERIES SPACES O F INFINITE TYPE

86.4

I n c h a p t e r f i v e , we showed t h a t s p a c e s o f h o l o m o r p h i c f u n c t i o n s on c e r t a i n n u c l e a r s e q u e n c e s p a c e s c o u l d t h e m s e l v e s

In t h i s s e c t i o n we

be represented a s nuclear sequence spaces.

s t u d y t h e s e q u e n c e s p a c e s t h a t a r i s e when we c o n s i d e r h o l o m o r p h i c f u n c t i o n s on t h e s t r o n g d u a l s o f power s e r i e s s p a c e s o f i n f i n i t e type.

We b e g i n b y r e c a l l i n g t h e d e f i n i t i o n o f power

s e r i e s s p a c e o f i n f i n i t e t y p e and i n t r o d u c i n g some n o t a t i o n .

An i n c r e a s i n g s e q u e n c e numbers w i t h

~f

1 7 <

W

q n

l i m an = n-tm

-

then

a

'n

integer

positive real

q,

0
or e q u i v a l e n t ~ y

log(n+l) < a

n

wiZ.1 b e c a l l e d a n u c l e a r e x p o n e n t s e q u e n c e .

sup a2n < * n

of

w i l l b e c a l l e d an exp o n en t s eq u en ce.

f o r some SUP n

W

(an)n=l

a =

k

o r equiuatentty

sup n

kn

n

t h e n t h e exponent sequence

a s t a b l e exponent sequence.

<

m

I f

f o r any p o s i t i v e

a =

The s e q u e n c e s p a c e

03

i s called

337

Germs, surjective limits, €-products and power series spaces m

co

Am(a) = { ( x ~ ) ~ = ~ ; l x n l P a

n

<

O < p < m ) endowed

m,

w i t h t h e topoZogy generated by t h e norms

i s c a l l e d a power s e r i e s s p a c e o f i n f i n i t e t y p e .

It i s a

F r z c h e t s p a c e w i t h a b a s i s and i s n u c Z e a r i f and onZy i f

a

i s

a nuclear exponent sequence. i s a s t a b l e exponent sequence i f and o n l y i f Am(.) x Am(a). If a and a are exponent sequences

a

Am(a)

=

then

A a ( a ) 'g Am(;) i f and o n l y i f t h e r e e x i s t s D b 1 such 1 for all n. I n t h i s s i t u a t i o n we s a y a n I a n s Dan

that a

and

are equivalent.

CL

Let

M = N").

m = (m.)w 1 J=1

(aim)

=

and d e n o t e by

a

1j E N a j m j an i n c easing r e a

6 = &(a)

((a J m ) ImEE.,. L e t

family

6, = ( a l d ( n ) )

i s an exponent sequence and

we l e t

M

E

If

n

f o r each

The s e q u e n c e

d:N+M

&(a)

in

ingement o f t h e

be a b i j e c t i o n such t h a t N.

i s caZled t h e a s s o c i a t e d sequence o f

a. A rephrasing of

Proposition 6.41

a basis.

where

theorem 5.21 y i e l d s t h e following r e s u l t .

Let

A(P)

b e a fuZly n u c l e a r s p a c e w i t h

The mapping

M = N"),

am(f)

monomial e x p a n s i o n o f

i s t h e c o e f f i c i e n t of f

t o p o l o g i c a l isomorphism.

and

PM

=

zm

i n the

{ ( pm ) m E M ; p ~ P Ii ,s a l i n e a r

338

Chapter 6

Corollary 6.42

Let

a s s o c i a t e d sequence

Proof

Since

HHy(Am(a)A).

Hence

&(a) = 6 .

Am(a)'

a

Then

is a

3312.

space

H(Am(a)i) =

By p r o p o s i t i o n 6 . 4 1

(H(Am(a)A),rO)

theorem 5.11,

be a n u c l e a r exponent sequence w i t h

a

A,(6)

a n d we h a v e a l r e a d y n o t e d ,

that t h i s i s a nuclear space.

in

This completes

t h e proof. A s a c o r o l l a r y , we o b t a i n a Cauchy-Hadamard

formula f o r

e n t i r e f u n c t i o n s o n power s e r i e s s p a c e s o f i n f i n i t e t y p e . Corollary 6.43

Let

of i n f i n i t e type.

Then

Our a i m i s t o d e s c r i b e a.

A,(a)

b e a n u c l e a r power s e r i e s s p a c e

&(a),

up t o equivalence, f o r v a r i o u s

T h i s t o p i c h a s been t h e o b j e c t o f r e c e n t r e s e a r c h and a

good d e a l o f i n f o r m a t i o n i s known. e s t i n g and t e c h n i c a l .

The p r o o f s a r e b o t h i n t e r -

We c o n f i n e o u r s e l v e s t o a d e t a i l e d d e s -

c r i p t i o n o f a p a r t i c u l a r c a s e which i s t y p i c a l o f t h e g e n e r a l s i t u a t i o n a n d c o v e r s many o f t h e c l a s s i c a l s p a c e s . look a t a particular situation.

We f i r s t

339

Germs, surjective limits, E -products and power series spaces J

Proposition 6.44 Proof

(H(si),.ro) = s.

Since

where

= hm(u)

s

6.41 implies t h a t

= log(n+l), proposition n is isomorphic t o a nuclear

(H(s;),.ro)

power s e r i e s s p a c e

Am(6).

By n u c l e a r i t y

sup n

<

m .

Since

c

m

~

~

I

~

=

~

c

6n

t h e r e e x i s t s a s t r i c t l y i n c r e a s i n g sequence of p o s i t i v e integers k(n) 3 n

{k(n)}",l for all

such t h a t n

= 6k(n)

'k(n) log (n+l)

,<

log(n+l)

T h i s means t h a t

and

=

-" n

a

i

(log(n+l));=l

i s a s t a b l e exponent

following r e s u l t

(see also corollary 6.26).

We d e n o t e b y

and,

w e o b t a i n as a c o r o l l a r y t h e

Y s

t h e cardinality of the set

IAl

D e f i n i t i o n 6.46

N(t)

1.

This completes t h e proof.

~8

[O,m)

Hence

a r e e q u i v a l e n t sequences and

6

Using t h e f a c t t h a t

E

n.

n

sequence and t h a t

t

for all

and

'n

hW(a) = s .

U,

Let n

A.

b e an e x p o n e n t s e q u e n c e .

CY.

If

i s a positive integer, w e l e t

=

sup Nn(t) n

N-(t)

=

ICmcN(N); (ulm) < t } \ .

Since

un+

+m

as

=

n + m

I I r n c N ( N ) ; ( a ) m ) 6 t}/ a n d

it follows t h a t

N(t)

is

{

G

~

~

340

Chapter 6

finite for all

Ni)

and it i s a l s o immediate t h a t

t

i s c o n t i n u o u s from t h e r i g h t

Lemma 6 . 4 7

sequence

Let

be a n e x p o n e n t s e q u e n c e w i t h a s s o c i a t e d

a

i f

n ,< N ( t )

fc)

i f

N-(t)

N-.

6n

then

< n

,< t ,

t c 6n

then

.

(a) f o l l o w s i m m e d i a t e l y f r o m t h e d e f i n i t i o n o f N(t) 3 n

(b) If

Since

6,,

t h e increasing rearrangement of t h e s e t that

N-(t) < n

(el If

6 n 6 t .

elements i n

with

M

definition of

We s h a l l c a l l

N

Proposition 6.48 H:R+

R+

+

n

A s i n fb), i t f o l l o w s , b y t h e

n 2 t. a

b e an exponent sequence w i t h

a

Let

be a c o n t i n u o u s s t r i c t l y i n c r e a s i n g f u n c t i o n such

that

H(t)

f o r some p o s i t i v e

Q N(t) B

,< H ( B t )

and a l l

t

sufficiently large.

Then

are equivalent sequences.

("):=1

Proof some

term i n

((alm))mEM it follows

and n u m b e r i n g f u n c t i o n N .

6

and

elements i n

n th

t h e numbering f u n c t i o n of

Let

a s s o c i a t e d sequence

6

is the

N

then t h e r e a r e l e s s than

(alm) < t .

that

6,

n

then there are a t least

(alm) 6 t .

with

M = N

n.

left) for all

The f o l l o w i n g a r e t r u e :

6.

fbi

Proof

(resp.

(resp.

Nn

-

l i m 6n = w e have f o r a l l k larger than n+m 26k-1 3 6k-1+a1 = 6 2 dk. For each p o s i t i v e

Since ko E N ,

integer

n

k(no) > ko.

let If

j k(n) = i n f { j ; 6 . = 6 1 . n 2 n

then

J

n

Choose

no

such t h a t

341

Germs, surjective limits, €-products and power series spaces

( b y lemma 6 . 4 7 ( a ) ) 6

H - ~ is increasing)

(since

2H-'(n)

On t h e o t h e r h a n d H

-1

(n) 6 H

-1

( b y lemma 6 . 4 7 ( a ) )

(N(6,))

This completes t h e proof. We now c o n s t r u c t a f u n c t i o n w h i c h s a t i s f i e s t h e c o n d i t i o n s

of p r o p o s i t i o n 6 . 4 8 and u s e it t o e v a l u a t e & ( a ) Lemma 6.49

function

Let

=

for all

t

be a n e x p o n e n t s e q u e n c e w i t h numbering

and l e t

N

F(t)

a

for certain u .

sup n al E

R'.

tn

...

arm! F ( t ) 6 N(t)

Then

t

for a l l

sufficiently

large. Proof

Let

n

lj,lajxj

M n ( t ) = {xeR;;

t}.

Now

Nn(t)

is the

number o f p o i n t s w i t h i n t e g e r c o o r d i n a t e s c o n t a i n e d i n Since any c u b e . w i t h edge l e n g t h one i n follows t h a t

Volume(Mn(t))

6

Nn(t).

that V o l u m e (Mn ( t ) )

N(t) = sup Nn(t) n

Lemma 6.50

increasing.

Let

Mn(t). h a s volume one it

An e a s y c a l c u l a t i o n s h o w s

tn

=

a1

Since

Rn

...

u n!

n

t h i s completes t h e proof. co

a = (YnnP~n,l

Then t h e r e e x i s t s

where

D>O

p>O

such t h a t

and

y

i s

342

Chapter 6

N ( t ) 5 F ( D t ) l'p

for a l l

t

sufficiently l a r g e .

A = {jl, . . . , j n}

is a f i n i t e s e t of positive

Proof

If

integers,

t h e n t h e m e t h o d o f t h e p r e v i o u s lemma s h o w s t h a t

I{mcM;

and

support(m) = A

(alm) 5 t

1I

tn

I

a. J1

...

a. n !

'n

By d e f i n i t i o n

1{ j , ,..., j n 3

1 n!

n Ik=pj 6t k (the extra

n!

1 a. '1

...

a.

I

Jn

a r i s e s from t h e permutations of

For each integer

n

let

An(t)

=

{ j l,...,jn})

a. 5 t } Jk

{ j E N n ;

and l e t

With t h i s n o t a t i o n

We may a s s u m e , i f n e c e s s a r y b y r e p l a c i n g

yn

by

y;

= ynn9,

Germs, surjective limits, e -products and power series spaces 1

2 :: P

that

Cn(t)

=

E

N.

1

n!

343

Hence

cj€An(t)

1 a.

J1

...

( s i n c e t h e sequence

a. 'n

( Y ~ )i ~ s increasing)

where B n ( t ) = {xER";

since (**)

Bn(t)

(*)

x. 3 0 1

for

lsjsn,

l nj Z ly 3. x p3

6 t},

may b e i n t e r p r e t e d a s t h e l o w e r R i e m a n n sum o f

corresponding t o t h e p a r t i t i o n given by t h e p o i n t s i n with integer coefficients. In o r d e r t o e s t i m a t e t h i s i n t e g r a l , w e first introduce t h e

coordinates

Sk = t -1 ykx;,

16kSn,

We now p r o v e b y i n d u c t i o n t h a t

and w e g e t

344

Chapter 6

n=l.

Obviously this formula holds for

J

A(1-S)

'0

=

--

Since

Jol

Bn

1 [( --2) 1

[n(-

P

--2 Sp

*

1

( I - S)

!In

(

n(- - 1 )

dS

1 - 2)! [n( 1 - l)] !

P

1 [(n+l)(--

- l)]!

P

I)]!

(by the induction hypothesis)

1

-

" - P-

2)!1

n+1

1 [(n+l)( - -

P

I)]

!

It f o l l o w s t h a t t h e f o r m u l a h o l d s f o r a l l

n.

Hence

Germs, surjective limits, €-products and power series spaces

w h e r e we l e t

We now h a v e

b

l

+

1 y n = l 2n

(since

"1 1 F (2Pct)P 1 -

=

1 + F(2Pct)P

To complete t h e p r o o f , F(ZP+'ct) for a l l Let

t

2

1 $

(1 + F(2'ct))'.

we show t h a t

1 + F(2'ct)

sufficiently large. t

be any r e a l number.

Choose

no

such t h a t

345

Chapter 6

346

F(2P+1ct)

-

=

n

(2%)

O

...

a1

n !

a "0

O

Then F(2P+1ct)

sup

=

2n ( 2 P c t y

n

...

al

n

2 O(2Pct)

::

...

al

n

0

n

o

!

2 F ( 2 P c t ) :: 1 + F ( Z P c t )

2

since F(t)

c1

n

n !

c1

for

t 5

t

2

c1

1

-

a1

This completes the proof P r o p o s i t i o n 6.51

m

a = ("n)n=l

Let

and Z e t F(t)

t

for a l l

E

sup

=

R'.

b e an e x p o n e n t s e q u e n c e

tn

al

...

a n!

n

T h e e x p o n e n t s e q u e n c e associated w i t h

i s equivaZent t o t h e sequence

( F - l (n)):=,

i f

a

a,&,

satisfies

e i t h e r of t h e f o l l o w i n g e q u i v a l e n t conditions:

a)

c(

is s t a b l e a n d t h e r e e x i s t s

(ann-P)EcN sequence;

bi

there exists a positive integer 1

Proof

( a n "-p);=l

inf n

"kn __ u

p>O

s u c h that

is e q u i v a l e n t t o a n i n c r e a s i n g

6

n

F i r s t suppose

%n sup n a n

k

sueh t h a t

m .

(a/ i s s a t i s f i e d .

By h y p o t h e s i s m

i s equivalent t o an i n c r e a s i n g sequence (yn)n=l 1 a n d we may s u p p o s e 2 $ - E h i . Hence t h e r e e x i s t s B > 0 such P

341

Germs, surjective limits,E -products and power series spaces 1 a ,< an f B a n where a = ynnP for all B n n N b e t h e numbering f u n c t i o n s o f a and a

that

and

n for a l l

t c [O,

a1

...

co

(Bn)n=l

N

a n!

n

there exists 1 -

If

Let

respectively

m).

By lemma 6 . 4 9 , N(t)

n.

5 N ( B t ) i F(BDt)'

D> 0

such t h a t

1 6 F(B2Dt)'.

i s a s t a b l e e x p o n e n t s e q u e n c e , t h e n f o r some

we have B2n = sup n Bn

c

<

C>O

00

and h e n c e

By i n d u c t i o n o n n2

k

w e have

k

2k

k+l j =1

for all p o s i t i v e i n t e g e r s

n

and

k.

s t a b l e exponent sequence, so a l s o i s

Since m

m

( u ~ ) ~i s= a~ and hence,

a p p l y i n g t h e a b o v e i n e q u a l i t y t o t h i s s e q u e n c e , we s e e t h a t there exists

C1>

0

such t h a t

3 48

Chapter 6

for all

t

R‘,

E

al

.. .

n

and

a

n2

k ( n 2k ) !

k

positive integers

Hence k

tn n

for all

t

k,

P ’

2

>

al

R+

E

...

n

n!

cil

. ..

and e v e r y p o s i t i v e i n t e g e r

and a l l

By p r o p o s i t i o n

ct

6.48,

t

c1

k.

n2

k k(n2 ) !

Thus f o r a l l

sufficiently large

m

(an)n=l

and

(F

-1

m

(n)ln=l

are equivalent

sequences. We now s h o w t h a t Suppose (a)

( a ) and

is satisfied.

(b) a r e equivalent c o n d i t i o n s .

For any p o s i t i v e i n t e g e r s

w e have

Now c h o o s e

q

such t h a t

> 1

B

and t h u s

n

and

q

3 49

Germs, surjective limits, E -products and power series spaces

and t h i s shows t h a t If

(a)

-

is satisfied, then t r i v i a l l y

(b)

sequence.

By h y p o t h e s i s ,

and

such t h a t

A > 1

Hence

a

kj

> hj.al

k j , < n , < kj",

where

p =

Let show t h a t

(b).

is a stable

there exist a positive integer

a k n ? Aa

for all

and a l l

m

n

k

for every positive integer Consequently,

j .

for all

n.

n,

we h a v e

j

a n d h = -a l. log k yn

A

= a n-'

for all

n

n.

T o c o m p l e t e t h e p r o o f , we

m

( Y ~ ) ~i s= e ~q u i v a l e n t t o a n i n c r e a s i n g s e q u e n c e .

We f i r s t s h o w t h a t i t i s a n i n c r e a s i n g s e q u e n c e a l o n g t h e arithmetic progression

co

(kn)n,l

and t h e n modify c e r t a i n i n t e r -

mediate values t o obtain an equivalent increasing sequence. Now

and hence t h e sequence

c3

=

sup akn n n

.

For

(

Y

m

~

~

nd j snk

i) s ~i n = c r e~a s i n g . w e have

Let

Chapter 6

350

For any p o s i t i v e i n t e g e r s let

and

j

n

with

we

kn 6 j dkn"

(T) j -kn

yj

.

kn

=

m

Since t h e sequence

For any -

(Ykn)n=l

n

and

-

kn - < j s k n + '

Yj

d Ykn+l

6

Ykn+l S c3ykn

Yj

-< c 3 y

=

C3Ykn

6

I f

a

-

,

and

C3yj

-

Hence

C3Yj.

y

1

inf n

<

y

are equivalent

a

kn "n

< sup n

f o r some p o s i t i v e i n t e g e r

for a l l

and

i s a n u c Z e a r e x p o n e n t s e q u e n c e and

a

kn an

= y

w e have

kn s e q u e n c e s and t h i s c o m p l e t e s t h e p r o o f .

Theorem 6.52

-

y

kn kn i s an i n c r e a s i n g sequence.

(yj)j=l

j,

-kn

i s i n c r e a s i n g and

m

-

we s e e t h a t

n

all

kn + 1

k

<

m

then

n.

Proof

By c o r o l l a r y 6 . 4 2 , p r o p o s i t i o n s 6 . 4 8 a n d 6 . 5 1

( H ( A m ( a ) ~ ) , ~ o )= n _ ( F - l ) F(t)

=

where

sup n

F-l

tn al

...

a n! n

=

(F-l(n));=l

and

35 1

Germs, surjective limits, e -products and power series spaces for all Fn

t

For e a c h p o s i t i v e i n t e g e r

[O,m).

E.

: R+ + R+

n

define

by

We h a v e

F ( t ) = sup Fn(t) for all t E R+. Since Fn(t) + 0 n a s n -t m f o r e a c h t sup F n ( t ) = F (t) f o r some n(t) n integer n(t). Since

F n + p ) - Fn(t) =

the equation

Fn(t) = Fn+l(t)

t n = ( n + l ) an + l . t 3 tn

.

since

Moreover

tn a1

. . . arm!

(

t (n+l)an+l

-1)

has e x a c t l y one r o o t

Fn+l(t)

2 Fn(t)

if and o n l y i f

is an increasing sequence of real

(tn):=l

n u m b e r s we h a v e F(t) = Fn(t) = for

nan

t L

,i

k

for all

n nn

...

a1

(n+l)an+l.

a n!

=

a,n

a1

... a n

Fn(tn-l)

=

and

By p r o p o s i t i o n 6 . 5 1 , such t h a t

Consequently

h k 5 Fn+l(tn)

a =

n

For each p o s i t i v e i n t e g e r

A(n)

. . . arm!

such t h a t

an a1

tn

sup n

n

a1

...

a

n +1

(n+l)!

let a n

nn

A*(n) = - A(n) = a1 n! a

n +1 ( n + l )n+l n+l

n

...

. n

n

an + l n !

-

Fn(tn-l).

i s s t a b l e and hence t h e r e e x i s t s

a2n - an

c.

C>O

Chapter 6

352

Hence f o r any p o s i t i v e i n t e g e r

.

Since

where

to

c

A(2m) L

Since A*(n)

and,

j=1 A(2j-l)

(C2)2

m L

(C2)2n = C

A(l) =

n n n!

A(n)

=

h

2 2m (C )

4n

2

2"-l

6

A*(n)

n.

sup{n;A*(n)

greater than o r equal

6 k}.

k

=

n

n

-

n!

A(n)

let

By d e f i n i t i o n ,

A*(n(k)) L k L A*(n(k)+l) a n d c o n s e q u e n t 1y 2" (k) - 1 L A*(n(k)) ,< k c A * ( n ( k ) + l ) 5

(eC4 ) n ( k ) + l

This implies (n(k)-l)log2 Hence,

.

2 2"-'

For each p o s i t i v e i n t e g e r n(k)

j=l

C2j

w e a l s o have

1

=

by S t i r l i n g ' s f o r m u l a ,

for a l l

fi

.

6 A(l)

i s t h e s m a l l e s t power o f

2m

n.

m

we h a v e

i s a n i n c r e a s i n g f u n c t i o n , we h a v e

A

A(n)

m,

f o r each p o s i t i v e i n t e g e r

and by i n d u c t i o n , A(2m) = A ( l )

n

s logk

there exists 1

s ( n ( k ) + l ) logeC4

q>1

such t h a t

- log(k+l) s n(k) 9

6

q log(k+l)

for all

k.

<

(eC4)"

353

Germs, surjective limits, e -products and power series spaces for all F

and

k. -1

(k)

F-l(k)

Thus f o r a l l =

a

,.. ...

a nn !

k )n

*

1 -

=

nan = n ( k ) a n ( k )

c

(a1

(a1

is stable,

...

we h a v e

1 -

(a1

...

n = n(k)

with

2

c

Since

(a1

k

a n!A*(n))

n

2

n

1 -(log (k+l))a 1 9 [ q log(k+l)l

a n ! A*(n+l))n

1 -

n

a n!

n

(n+l)

n+l

n+l

(n+l)!

1 n

"n+l a1

...

it follows t h a t t h e r e e x i s t s

a

1 n+1

C1 > 0

such

that

This completes t h e proof. 00

Example 6 . 5 3

(a)

Let

a = (nP)n,l

Since w e have

where

p

is positive.

Chapter 6

35 4 "2n i n f __ n an

a

2n s u p __ n an

=

2P*

=

By t h e o r e m 6 . 5 2 ,

In p a r t i c u l a r ,

m

f o r any p o s i t i v e i n t e g e r (b)

If

a =

m

(Pnln=l

where

pn

denotes the

nth

prime

t h e n t h e f u n d a m e n t a l t h e o r e m o n t h e d e n s i t y o f p r i m e s shows

(H(A,(~)A),T,)

for all

2

Am(&)

where

n.

We c o n c l u d e t h i s s e c t i o n w i t h a q u i t e d i f f e r e n t

d e s c r i p t i o n o f t h e s p a c e o f holomorphic f u n c t i o n s on a f u l l y nuclear space with a basis. Definition 6.54

Let

E

symmetric t e n s o r a l g e b r a o f

be a l o c a l l y convex space. E

A

i s a complete commutative

35 5

Germs, surjective limits, E -products and power series spaces S(E)

l o c a l l y m u l t i p l i c a t i v e l y convex algebra i:E

together with a continuous i n j e c t i o n

with unit

S(E)

-f

which has t h e

following universal property: f o r a n y c o n t i n u o u s l i n e a r mapping

0

E

of

into

a complete l o c a l l y m u l t i p l i c a t i v e l y convex algebra A

( o r e q u i v a l e n t l y i n t o a Banach a l g e b r a )

with

u n i t s a t i s f y i n g $ ( x ) $ ( y ) = $ ( y ) + ( x ) for a l l x,y E E t h e r e e x i s t s a unique c o n t i n u o u s aZgebra @:S(E)

homomorphism

-f

By s t a n d a r d a r g u m e n t s ,

A

wi-kh

0

= Qoi.

one e a s i l y shows t h a t a l l symmetric

t e n s o r a l g e b r a s o f a l o c a l l y convex space ( i f t h e y e x i s t ) are i s o m o r p h i c as a l g e b r a s . Theorem 6 . 5 5 (H

HY

A(P)

Let

be a f l ~ l 7 y ynuclear space.

i:*(P)

+

(HHy(JW);)Jo)

i s t h e symmetric t e n s o r algebra o f Proof

By o u r p r e c e d i n g r e m a r k ,

(HHy(A(P);),~o)

x,y that

E

into A(P).

II

s P,

A(P).

The mapping

be a

A

all

n

where

for all

(pn)Zzl m

E

P

such

is the unit vector

If

converges absolutely i n j .

Let

$(x)$(y) = $(y)$(x)

By c o n t i n u i t y , t h e r e e x i s t s

Il+(en)

basis for

i t s u f f i c e s t o show t h a t

b e a c o n t i n u o u s l i n e a r mapping from

$

such t h a t

A

A(P).

has the required properties.

Banach a l g e b r a and l e t A (P)

Then

together with the canonical i n j e c t i o n

(A(P);),y0)

@

w h e r e we l e t

A

: H H y ( A ( P ) i ) , ~ o )+ A

is e a s i l y seen t o extend

+

b. = $(e.) 1 J given by

for all

and t h i s completes t h e p r o o f .

356

Chapter 6 T h e o r e m 6 . 5 5 i n d i c a t e s how t o d e f i n e a f u n c t i o n a l c a l c u l u s m

f o r a sequence algebra,

o r i n a c o m p l e t e m u l t i p l i c a t i v e l y convex a l g e b r a .

A,

m

I t s u f f i c e s , given

space

o f commuting e l e m e n t s i n a Banach

(bn)n=l

b = (bn)n=l,

such t h a t

A(P).

A

If

mapping

t o choose a f u l l y nuclear

(I\bn\lln=l

i s commutative,

then the continuous linear

: ( H H y ( A ( P ) A ) , ~ o )+ A

@

is the desired functional

c a l c u l u s and t h e j o i n t spectrum of

i s a c o n t i n u o u s weight on

00

i s t h e n a compact s u b s e t

o(b)

A(P)b.

The e a r l i e r r e s u l t s o f t h i s s e c t i o n i d e n t i f y t h e symmetric t e n s o r a l g e b r a o f v a r i o u s power s e r i e s s p a c e s o f i n f i n i t e t y p e . In particular,

p r o p o s i t i o n 6.44

Proposition 6.56

S(S)

2 s

implies the following r e s u l t .

where

r a p i d l y d e c r e a s i n g s e q u e n c e s and

is t h e s p a c e of i s t h e symmetric t e n s o r

S(s)

s.

a l g e b r a of

56.5

EXERCISES

6.57* ___

Let

space.

s

E =

n

CXEA

where each

Ea

For each f i n i t e s u b s e t

J

of

i s a l o c a l l y convex

E A

'

let

E.

=

TI

CXEJ H(OE) is

If

H(OE ) i s regular f o r every J show t h a t J regular. Show t h a t H(Op,) is a regular inductive l i m i t .

6.58* ___

Let

b e a l o c a l l y convex s p a c e .

E

If

H(OE)

is

r e g u l a r and d o e s n o t c o n t a i n a n o n t r i v i a l v e r y s t r o n g l y conv e r g e n t s e q u e n c e , show t h a t metrizable subset

__ 6.59*

If

show t h a t ___ 6.60*

H(K) If

contains

K

of

i s r e g u l a r f o r e v e r y compact

E.

i s a compact s u b s e t o f a Fre'chet n u c l e a r s p a c e

K

convex space

K

H(K)

i s a ,333-2 s p a c e . i s a compact s u b s e t of a m e t r i z a b l e l o c a l l y

K E

and

show t h a t

U

i s an open s u b s e t of

E

which

351

Germs, surjective limits, E -products and power series spaces

is a locally

m

convex a l g e b r a .

is a locally

m

convex a l g e b r a f o r any open s u b s e t

6.61 __

( 8 (nE) ,.rW) n

/

E

If

is a distinguished Frechet space, E

E.

show t h a t

i s quasi-normable.

A l o c a l l y convex s p a c e

convergence c r i t e r i o n i f g i v e n

s a t i s f i e s t h e s t r i c t Mackey

E

B C E

c l o s e d convex b a l a n c e d bounded s u b s e t such t h a t

of

U

i s boundedly r e t r a c t i v e f o r every p o s i t i v e i n t e g e r

i f and o n l y i f

6.62 -

(H(U),.rw)

H e n c e show t h a t

E

and

bounded t h e r e e x i s t s a A

of

E

containing

i n d u c e t h e same t o p o l o g y on

EA

E

B.

Show t h a t a n i n j e c t i v e i n d u c t i v e l i m i t o f B a n a c h s p a c e s i s boundedly r e t r a c t i v e i f and o n l y i f it i s r e g u l a r and

s a t i s f i e s t h e s t r i c t Mackey c o n v e r g e n c e c r i t e r i o n . Show t h a t t h e f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t o n a

6.63* -

l o c a l l y convex space E . (a)

Given a z e r o neighbourhood

W

in

E

there exists

a c l o s e d convex balanced zero neighbourhood V C W,

and a bounded s e t

for all (b)

(c)

E

such t h a t

V C B + 6V

6>0.

c o n t a i n s a f u n d a m e n t a l s y s t e m of semi-norms

such t h a t

Ei

f o r every

a

E

B

V,

i n d u c e s t h e norm t o p o l o g y o n

in

r,

c o n t a i n s a fundamental

E$

s y s t e m o f semi-norms

s u c h t h a t t h e normed t o p o l o g y o n P(mE,) c o i n c i d e s w i t h t h e induced topology of ( Q ( m ~ ) , ~ f) o r e v e r y (d)

E

a

in

r, 6

in

r

.

i s a n o p e n s u r j e c t i v e l i m i t o f normed l i n e a r

spaces.

r

r

358

Chapter 6

6 . 6_ 4 _

Show t h a t a n y l o c a l l y c o n v e x s p a c e w h i c h s a t i s f i e s t h e

equivalent conditions of t h e preceding exercise i s quasi-

If

normable.

i s a F r g c h e t - S c h w a r t z s p a c e which s a t i s f i e s

E

show t h a t

the condition of exercise 6.63, 6.65* ___

N

.

i s a s u r j e c t i v e l i m i t of

E = lirn (Ea,na)

If

E ZC

c-

a EA

normed l i n e a r s p a c e s , show t h a t t h e r e i s a n a t u r a l o r d e r on (i.e.

if

al,a2

surjections such t h a t

II

then there exist

A

E

“l’a3

Ii

0

Ea

:

na

“lYa3

+

3

- na

3

Ea

1

and

a 3 e A

and continuous

n

: E a

“2’“3

a n d iIa

1

o

n

LY03

+ E

3 = 11

“2

a2

A

“ 2

Hence

).

d e d u c e t h a t a s u r j e c t i v e l i m i t o f Banach s p a c e s i s an o p e n surjective l i m i t .

6.66

If

E = l i m

i s a s u r j e c t i v e l i m i t and e a c h

(Ea,na)

t-

Ea

a has t h e approximation property,

show t h a t

has the

E

approximation property. __ 6.67

If

E = l i m

i s a s u r j e c t i v e l i m i t ofcomplete

(Ea,IIa)

t-

a

l o c a l l y convex s p a c e s ,

show t h a t

n

E = l i m

A

where

(Ea,Iia)

f-

A

f

i

na:E

+

and

Ea

Ii

for all a. __ 6.68

A surjective l i m i t

E = l i m

(Ea,Xa)

is directed i f

t-

aEr

it enjoys t h e p r o p e r t y d e s c r i b e d i n e x e r c i s e 6.65.

i s a n open s u r j e c t i v e l i m i t i f and o n l y i f i s a n o p e n s u r j e c t i o n f o r e a c h a,B E r , asB.

E

II

a,

Show t h a t

B:E

a

+

Eg

Show t h a t a

d i r e c t e d s u r j e c t i v e l i m i t of Frgchet spaces i s an open s u r j ective l i m i t .

6.69

If

uous b a s i s ,

E

i s a l o c a l l y convex space w i t h an e q u i c o n t i n -

show t h a t

E

i s a s u r j e c t i v e l i m i t o f normed

359

Germs, surjective limits, E -products and p o w e r series spaces l i n e a r s p a c e s , each o f which h a s an e q u i c o n t i n u o u s b a s i s .

Show

i s an open s u r j e c t i v e l i m i t o f l o c a l l y convex

E

also that

s p a c e s , e a c h o f which h a s an e q u i c o n t i n u o u s b a s i s and a d m i t s a c o n t i n u o u s norm. __ 6.70*

By c o n s i d e r i n g t h e s p a c e

show t h a t i n g e n e r a l

r

co(r),

uncountable,

bounded s u b s e t s o f

T~

H(E) = l i m E i , t 1

d o n o t u n i f o r m l y f a c t o r t h r o u g h some

e v e n when we a r e d e a l -

Ei

i n g w i t h an open and compact s u r j e c t i v e l i m i t .

6.71

-

Let

V

b e a R e i n h a r d t domain,

containing the origin,

i n a Banach s p a c e w i t h a n u n c o n d i t i o n a l b a s i s .

6.72* -

Let

E

/

be a Frechet-Schwartz

space.

Show t h a t

Let

K =

6

K. i s a compact s u b s e t of E. If T E H(K)' I that there exists T . E H(K.)' f o r each j such t h a t n J J T = T. where each

1.J = 1

_ 6 . 7_ 3*

K. J

j -1

show

J '

If

show t h a t

i s a compact s u b s e t o f a l o c a l l y convex s p a c e

K

H(K) = l i m

(H(V),T~).

&

VDK,V o p e n

___ 6.74

Show t h a t t h e t - p r o d u c t

a

space.

6.75"

If

E,

is a

d3F6

o f two

$38

spaces i s again

s p a c e a n d a n i n d u c t i v e l i m i t of

Banach s p a c e s w i t h t h e a p p r o x i m a t i o n p r o p e r t y v i a c o m p a c t mappings,

show t h a t

e v e r y compact s u b s e t 6.76*

Let

K

H(K) K

E.

b e a compact s u b s e t o f

i s p o l y n o m i a l l y convex i n

p o l y n o m i a l s on

has t h e approximation property f o r of

(CN

Cn

f o r each

and suppose

CN n

E

N.

a r e sequentially dense i n

6.77* Let A(P) b e a s t a b l e n u c l e a r Fre'chet a d m i t s a c o n t i n u o u s norm. Show t h a t

nn(K)

Show t h a t t h e H(K). s p a c e which

Chapter 6

3 60

_ 6 . 7_ 8*

q

m

an = n

a = ("n)n=l'

Let

are p o s i t i v e r e a l numbers.

Am(&),

where

6n

Let

A(P)

6.79* -

of weights P.

i f f o r each

P

(log(n+l))q

Show t h a t

where

(log(n+l))P+l(log log(n+l))q

=

03

(an)n,l

i s a Schwartz s p a c e i f and o n l y

A(P)

in

there exist

P

E

P

E C + such t h a t a (Un)n=l 0 n 6 u n a nt f o r a l l n . i s a Montel space i f and o n l y i f f o r each

and

6.80*

P

such t h a t

Let

HM(U) = H(U)

l i m inf = 0. a A j # O , j + m a' nj b e a F r e c h e t Montel space.

A(P)

f o r any open s u b s e t

U

of

Show t h a t a l o c a l l y c o n v e x s p a c e

6.82 -

topology

o(E,E')

Show t h a t m

( a n ) n = l ~A ( P )

(n.)? there exists J J=1 an.

and each subsequence of i n t e g e r s E

n.

be a sequence space with s a t u r a t e d system

Show t h a t

A(P)

m

and =

for all

m

(aA)n=1

p

(H(Am(a)b),~O)

Show t h a t

n(P)A.

E

w i t h t h e weak

i s an open s u r j e c t i v e l i m i t o f f i n i t e

dimensional spaces.

56.6

NOTES A N D REMARKS The c o m p l e t e n e s s o f

l o c a l l y convex space [503].

H(K),

H e showed t h a t

K

a compact s u b s e t o f a

was f i r s t i n v e s t i g a t e d b y J . M u j i c a

E,

H(K)

i s c o m p l e t e whenever

m e t r i z a b l e l o c a l l y convex s p a c e w i t h p r o p e r t y cises 6 . 6 2 and 6 . 6 3 ) .

K.D.

B i e r s t e d t and R.

(B), Meise

E

is a

(see exer[69]

proved

t h e same r e s u l t f o r c o m p a c t s u b s e t s o f a F r g c h e t S c h w a r t z s p a c e and s u b s e q u e n t l y P.

A v i l e s and J . Mujica

[41]

extended t h i s

r e s u l t t o quasi-normable m e t r i z a b l e l o c a l l y convex s p a c e s . general r e s u l t that

H(K)

i s complete f o r any compact

of a m e t r i z a b l e l o c a l l y convex space, theorem 6 . 1 , S . D i n e e n [ZOO].

The

subset

i s due t o

361

Germs, surjective limits, E -products and power series spaces

Proposition 6 . 2 is due to R. Soraggi [669] while corollFurther examples aries 6.3 and 6.4 are d u e to S. Dineen [200]. including proposition 6 . 2 9 , concerning t h e regularity of when

K

H(K)

is a compact subset o f certain non-metrizable

locally convex spaces are given in R . Soraggi [667,668,669].. From the viewpoint o f holomorphic germs and analytic functiona l s , the following result o f J . Mujica [ S o l ] is also o f interest:

if

K

is a compact locally connected subset o f the met-

rizable Schwartz space

E = lim En, fn

where each

En

is a

normed linear space and the linking maps are precompact, then for each continuous linear functional T o n H(K) there exists a sequence o f vector measures such that 00

(ii)

f (iii)

if

1 m!

-

=

in

1

Smf(x)pm(dx)

P(mE,,))' l/m

norm o f

satisfying (i) and (iii),

a s an element

pm

then for each

Conversely, given a sequence H(K)

for every

H(K);

1 1 ~ ~ 1 is1 the ~

of &(K;

K

m

(um)m=l

then (ii)

n,

o f vector measures defines a n element o f

'.

Proposition 6 . 7 is due to S . B . Chae [ 1 2 0 ] . Proposition Baernstein [42], in his work o n the representation o f holomorphic functions by boundary integrals. 6.8 was discovered by A .

Proposition 6 . 9 , theorem 6 . 1 0 and corollary 6.11 are due to. K-D. Bierstedt and R . Meise [ 7 0 ] . See also E. Nelimarkka [525] for a further proof o f proposition 6.9. R . Meise has recently shown that T~ = T on any open subset o f a Frgchet nuclear space and thus the basis assumption in corollary 6.11 is not necessary. Example 6.13 is d u e to M. Schottenloher [644] who used it to prove corollary 6.40. Corollary 6.40 is also d u e

Chapter 6

362 independently, L.

and by a d i f f e r e n t method,

[53].

Nachbin

to J.A.

B a r r o s o and

The p r o o f g i v e n h e r e i s s l i g h t l y d i f f e r e n t

from e i t h e r of t h e above. The r e g u l a r i t y a n d c o m p l e t e n e s s o f i n d u c t i v e l i m i t s i s

see f o r i n s t a n c e , t h e

extensively discussed i n the literature, recent survey of K.

of K-D.

Floret

B i e r s t e d t and R .

and t h e f i r s t f e w s e c t i o n s

[238],

Meise

[70],

and h a s l e d t o t h e d e f i n -

i t i o n o f many s p e c i a l k i n d s o f i n d u c t i v e l i m i t s . research

[SO31 h a s l e d h i m t o d e f i n e " C a u c h y r e g u l a r "

l i m i t s and t h i s c o n c e p t , R.

Meise

J. M u j i c a ' s

[70],

as p o i n t e d o u t by K - D .

inductive

B i e r s t e d t and

c o i n c i d e s with t h e concept o f boundedly r e t r a c t -

i v e i n d u c t i v e l i m i t s i n t h e case of an i n j e c t i v e i n d u c t i v e

l i m i t o f Banach s p a c e s .

H.

Neus

s h o w e d t h a t many o f

[527],

these concepts coincide for countable inductive limits of Banach s p a c e s , and p r o v e d p r o p o s i t i o n 6 . 1 6 ,

i s an a b s t r a c t v e r s i o n ,

-

due t o K-D.

[69], o f a r e s u l t of J . Mujica inductive l i m i t

B i e r s t e d t and R.

[503].

( z ( V ) n H ( U ) ,1 1

l i m

Proposition 6.15 Meise

The i d e a o f u s i n g t h e

11")

i s due t o J . Mujica

KCVC U

[SO31 who p r o v e d p r o p o s i t i o n 6 . 1 2

and used i t t o p r o v e propo-

s i t i o n 6 . 1 8 and c o r o l l a r y 6 . 1 9 . S u r j e c t i v e limits are due independently t o S. 1901 and E . L i g o c k a

examples and a p p l i c a t i o n s t o

i n f i n i t e d i m e n s i o n a l holomorphy a r e g i v e n in[190] Further references are P. [ 2 0 7 ] , Ph. [463,467]

and R.

Berner

S. Dineen,

Noverraz

[552], M.

Soraggi

due t o L.A.

d e Moraes

P r o p o s i t i o n 6.27 R.

[498],

i s due t o P .

and

[443].

S.

Dineen

Schottenloher

Schottenloher [640], M.C.

[669].

independently,

[58,59,60,61,62],

Ph. Noverraz and M.

lemma 6 . 2 4 a r e g i v e n i n S . D i n e e n r e s u l t i s due,

[189,

[ 4 4 3 ] , (who u s e s t h e t e r m i n o l o g y b a s i c

system). Their basic properties,

[186,189,191,193],

Dineen

Matos

Examples 6 . 2 1 , 6 . 2 2 , 6 . 2 3 [190].

and

Proposition 6.25 i s

while a p a r t i c u l a r case of t h i s to P.J.

Boland and S. Dineen[gi].

B e r n e r [ 6 1 ] a n d S . D i n e e n [194].

Soraggi proves proposition 6.29 i n

[669].

In studying vector valued distributions,

L.

Schwartz

[648]

363

Germs, surjective limits,E -products and power series spaces compensated f o r t h e absence of t h e approximation p r o p e r t y by defining M.

6 -

products (definition 6.30).

Schottenloher

[631] i n t r o d u c e d

as a tool

e-products

i n i n f i n i t e dimensional holomorphy. In

[639] h e

p r o v e d lemma 6 . 3 2 ,

propositions 6.33,6.34,

a r y 6.35 and gave example 6.31. 6.34 is due t o A .

Hirschowitz

coroll-

A weak f o r m o f p r o p o s i t i o n

p43

,

p r o p o s i t i o n 3.41

and

w e i g h t e d v e r s i o n s o f t h e same p r o p o s i t i o n a r e g i v e n i n K.

Bierstedt

[66,p.44

Theorem 6 . 3 6 i s new.

and 551.

The i d e a

o f u s i n g t e n s o r p r o d u c t s and t h e c o n n e c t i o n between t h i s

o f A . Grothendieck

theorem and p r o p o s i t i o n 1 5 , c h a p t e r 2 ,

was p o i n t e d o u t t o t h e a u t h o r b y D . counterexample,

[287]

Earlier a d i r e c t

Vogt.

which a p p l i e d t o t h e n u c l e a r power s e r i e s s p a c e

c a s e , was g i v e n b y S . D i n e e n

[202],

(see exercise 5.82).

It

would b e o f i n t e r e s t t o e x t e n d t h i s counterexample t o t h e t h a t t h i s i s p o s s i b l e ) and t h u s

g e n e r a l case ( i t i s our b e l i e f

give a completely self-contained

proof.

We d o n o t know i f t h e

b a s i s hypothesis i n theorem 6.31 i s necessary. 6 . 3 7 i s due t o S . Dineen 6.39 are due t o K-D. applications of

6

[202].

P r o p o s i t i o n 6.38 and c o r o l l a r y

B i e r s t e d t and R .

-products

Proposition

Meise

[69,70].

Further

i n i n f i n i t e d i m e n s i o n a l holomorphy

a n d k e r n e l t h e o r e m s f o r a n a l y t i c f u n c t i o n a l s may b e f o u n d i n K-D. B.

B i e r s t e d t and R. Perrot

Meise [69,70]

and i n J . F .

Colombeau and

[157,158,159,161,162].

A l l t h e r e s u l t s o f s e c t i o n 6.4 a r e due t o M.

Meise a n d D .

Bb'rgens,

Vogt and most o f them a r e c o n t a i n e d i n

comprehensive p a p e r , p a r t i a l l y summarised i n [95],

[96].

R.

This

contains

many f u r t h e r i n t e r e s t i n g e x a m p l e s o f s t r u c t u r e t h e o r e m s f o r H(Am(a)i).

T h e same a u t h o r s h a v e w r i t t e n a f u r t h e r a r t i c l e

[97] on t h e A - n u c l e a r i t y o f s p a c e s o f h o l o m o r p h i c f u n c t i o n s using refinements of t h e techniques developed i n The s y m m e t r i c t e n s o r a l g e b r a duced by A.

Colojoar?i

theorem 6.55 f o r

DF

[139].

[96].

( d e f i n i t i o n 6.54)

was i n t r o -

She proved an a b s t r a c t form of

nuclear spaces but did not establish a

c o n n e c t i o n between h e r r e s u l t s and holomorphic f u n c t i o n s .

was d o n e i n

[96] and d e t a i l e d i n [487].

This

3 64

Chapter 6 The r e s u l t s and methods o f s e c t i o n 6 . 4 a r e s t i l l i n t h e

p r o c e s s o f f i n d i n g t h e i r f i n a l form and v e r y r e c e n t d e v e l o p -

ments s u g g e s t t h a t t h e y w i l l p l a y a v e r y i m p o r t a n t r o l e i n t h e future of the subject. D.

Vogt

We s h a l l o n l y m e n t i o n t h a t R .

Meise and

[485,486] have r e c e n t l y obtained a holomorphic

c r i t e r i o n f o r d i s t i n g u i s h i n g open p o l y d i s c s i n c e r t a i n n u c l e a r power s e r i e s s p a c e s a n d h a v e shown i n [489] t h a t t h e t h r e e topologies

To

, ~ u and

spcace with a basis,

T &

on

H(A(P)),

A(P)

a fully nuclear

can a l l b e i n t e r p r e t e d as normal t o p o l o g -

ies i n t h e sense of G.

KGthe [ 3 9 7 ] .

Appendix I

FURTHER DEVELOPMENTS IN INFINITE DIMENSIONAL HOLOMORPHY

In t h i s appendix, we provide a b r i e f survey of some r e s e a r c h c u r r e n t l y being developed within i n f i n i t e dimensional holomorphy.

The t o p i c s we d i s -

cuss emphasise t h e a l g e b r a i c , geometric and d i f f e r e n t i a l , r a t h e r than t h e topological a s p e c t s o f t h e theory.

We hope t h i s i n t r o d u c t i o n w i l l i n s p i r e

t h e reader t o f u r t h e r readings and t o an o v e r a l l a p p r e c i a t i o n of t h e u n i t y of t h e s u b j e c t . THE LEI7 PROBLEM

We begin by looking a t a s e t of conditions on a domain convex space

U

i n a locally

E.

(a)

U

i s a pseudo-convex domain;

(b)

U

i s holomorphically convex.

(c)

U

i s a domain of holomorphy.

(d)

U

i s t h e domain of e x i s t e n c e of a holomorphic

function; (e)

The

(f)

If

a

3

problem i s solvable i n

U;

i s a coherent a n a l y t i c sheaf, then

H1(U;S) = 0. A l l these conditions a r e equivalent when

E

i s a f i n i t e dimensional

space (see L . Hormander [347] and R. Gunning and H. Rossi [294]) and t h i s equivalence may be regarded a s one of t h e h i g h l i g h t s o f s e v e r a l complex vari a b l e theory.

Note t h a t condition (a) i s m e t r i c , (b) geometric, (c) and (d)

a n a l y t i c , (e) d i f f e r e n t i a l and ( f ) a l g e b r a i c . c l a s s i c a l Cartan-Thullen theorem [118], and (c) a r e equivalent. equivalent f o r domains i n

In 1911, E . E . C2.

In t h e case of

E = Cn,

the

published i n 1932, a s s e r t s t h a t (b) Levi [441] asked i f (a) and (d) were

This became known a s t h e Levi problem and

was solved by K . Oka [558] i n 1942 and extended t o domains i n

3 65

Cn

by K. O k a

Appendix I

366 [559]

in 1953 and by F. Norguet [530] and H.Bremermann

The implication (f)

=>

(e)

[loll in 1954.

is due to P. Dolbeault [208], (b) => (f) is

due to H. Cartan [115] and (a)

=>

(e) is proved by L. Hormander [346].

Attempts to extend these conditions and to show their equivalence on arbitrary locally convex spaces have never been routine and have led to many interesting developments and results. We now describe the evolution of this line of research in infinite dimensions together with some related topics such as plurisubharmonic functions, envelopes of holomorphy, etc. H.J.Bremermann

[lo31 in 1957 was the first to consider pseudo-convex

domains, domains of holomorphy and plurisubharmonic functions (see proposition 4.12) in infinite dimensions. space to be pseudo-convex if the boundary o f

IJ)

He defined a domain U

-log dU

(dU(x)

in a Banach

is the distance from x

to

is plurisubharmonic and showed that this was equival-

ent to the finite dimensional sections of U

being pseudo-convex. In 1960

he showed that the envelope of holomorphy o f a tubular domain in a Banach space was equal to its convex hull [lo41 and afterwards [lo51 extended a number of his results to linear topological vector spaces. C.O. Kiselman [381] proved that the upper regularization of a locally bounded countable family of plurisubharmonic functions on a Frgchet space was plurisubharmonic (this is known as the convergence theorem) and this was extended to arbitrary families on complete topological vector spaces by P. Lelong [425], G. Coeurg [126,129] and Ph. Noverraz [536]. In [425] P. Lelong began a systematic study of the basic properties of plurisubharmonic functions and polar sets in topological vector spaces. This direction of research is developed in P. Lelong [426,428,429,430,434],

G. Coeur; [127,128,129], M. Herve/ [326] and Ph. Noverraz [536,537,538,545]. By using multiplicative linear functionals, H. Alexander [5] showed, in his thesis, that a domain U rJ

extension U,

spread over E,

in a Banach space E

erty that the canonical mapping of

(H(U),.ro)

ological isomorphism. He also noted that if and only if

fl

E

admits a holomorphic

which is maximal with respect to the propinto

(H(U),ro)

(H(??),T~)

is a top-

is a barrelled space

is finite dimensional and thus could not conclude that

was the natural envelope of holomorphy of

U.

J.M. Exbrayat [233] is

the only accessible reference for Alexander's unpublished thesis.

367

Further developments

The next contributions are due to G. Coeur6 [128,129].

He defined

pseudo-convex domains spread over Banach spaces by using the distance function and showed that a domain spread X is pseudo-convex if and only if the plurisubharmonic hull of each compact subset of X is also compact. Ihis result was extended to locally convex spaces by Ph. Noverraz [544]. To overcome the inadequacies of the compact open topology encountered by Alexander, Coeur; defined the T~ topology on domains spread over separable Banach spaces and showed that any holomorphic extension of a domain leads to a

T~

topological isomorphism of the corresponding space of holo-

morphic functions. This result was later extended to domains spread over arbitrary Banach spaces by A. Hirschowitz 1338,3431. G. Coeur; also proved in [129] that a suitable subset

5(X)

of the

could be endowed with

the structure of a holomorphic manifold spread over E a holomorphic extension of X holomorphy and

H(X)

spectrum of H(X),

T~

X a domain spread over a separable Banach space E,

and identified with

and that, furthermore, if X

separates the points of X

then X

is a domain of 5 (X).

In 1969, two important contributions were made by A. Hirschowitz [ 3 3 5 , 3361. In [335], he showed that the Levi problem had a positive solution and this result was subsequently extended to Riemann domains over CN by M.C. Matos [456] and to domains spread over A C , A arbitrary, by V. Aurich [33]. In his analysis, A. Hirschowitz for open subsets of CN

showed that any pseudo-convex open subset U

of CN had the form

-'(nn(U)) for some positive integer n where nn is the natural U = IIn projection from CN onto Cn. This result, together with factorization properties of holomorphic functions on CN given by A. Hirschowitz in [335] and also obtained independently by C.E. Rickart [605], led eventually to the concept of surjective limit (see 56.1) and to a technique for overcoming the lack of a continuous norm in certain delicate situations. V. Aurich used the bornological topology associated with the compact open topology in A his investigation of the spectrum of H(U), U a domain spread over C [331. In [336], see also [337] for details, A. Hirschowitz showed that the unit ball of &([O,R]),

fi

the first uncountable ordinal, is not the domain

of existance of a holomorphic function, i.e. (c) #> (d).

This counter-

example to the Levi problem and B. Josefson's [358] example of a domain in co(r),

r uncountable, which is holomorphically convex but not a domain

of

Appendix I

368

holomorphy, i.e. (b) #> (c), rely heavily on the non-separability of,&[O,n] and r respectively and, indeed, it appears that countability assumptions have always, and probably always will, enter into solutions of the Levi problem. We note in passing that A. Hirschowitz introduces bounding sets in [336] and that this concept had also arisen in H. Alexander’s work on normal extensions

,

in S. Dineen’s investigation of locally convex top-

ologies on spaces of holomorphic functions, [177,178], and in M. Schottenloher’s study [631] of holomorphic convexity. In three further papers 1338,340,3431, A . Hirschowitz looked at various other aspects of analytic continuation over Banach spaces. He showed, using germs of holomorphic functions, that every domain spread over a Banach space has an envelope of holomorphy. His investigation of vectorvalued holomorphic functions showed that whenever C

valued holomorphic

functions can be extended then so also can Banach valued holomorphic mappings and that conditions (b) and (c) , (resp (d)), remain unchanged when holomorphic functions are replaced by Banach (resp. separable Banach) space valued mappings. In 1970, S. Dineen 11761 replaced H(U)

by Hb(U),

the set of holo-

morphic functions on U which are bounded on the bounded open subsets of

E

contained in U and at a positive distance from the boundary of U.

Since Hb(U)

has a natural Frgchet space structure he was able, by suit-

ably modifying conditions (b) and (c), to obtain a Banach space version o f the Cartan-Thullen theorem. This approach was developed by M.C. Matos [457,459,460] who proved similar Cartan-Thullen theorems for various subalgebras of H(U). Independently of S . Dineen 11761 and A . Hirschowitz 13381, M. Schottenloher [631,632] was considering a much more general situation by defining regular classes (see also G. Coeur6 [129]) and admissible coverings for domains spread over a Banach space. F o r each regular class he proved a Cartan-Thullen theorem. By looking at all regular classes and by generalizing the classical intersection theorem for Riemann domains to infinite dimensions, he showed that the envelope of holomorphy could be identified with a connected component of the T~ spectrum. In [640], he extended this result to domains spread over a collection o f locally convex spaces which included all metrizable spaces and alld43w spaces, (see also K. Rusek and J. Siciak [618]). In later papers, [633,635,638,639] he

Further developments

3 69

considered various other topics on analytic continuation in infinite dimensions such as vector-valued extensions and the extension problem for Mackey holomorphic functions. We now digress a little to describe more recent results on the spectrum of H(U). In [352], M. Isidro showed that Spec(H(U),To) 2 U when U is a convex balanced open subset of a complete locally convex space with the approximation property and this result was extended to polynomially convex domains in quasi-complete spaces with the approximation property by J. Mujica, [502,505]. In [504], J. Mujica proved that Spec(H(U),T6) Q U when U is a polynomially convex domain in a Frgchet space with the bounded approximation property and the counterexample of B. Josefson [358] shows that this result is not true for every polynomially convex domain in Banach spaces with the approximation property. M. Schottenloher proved Spec(H(U),ro) = U for U pseudo-convex in a Frgchet space with basis. The study of the spectrum is the study of the closed maximal ideals and a few authors have also studied finitely generated ideals i n

H(U). J. Mujica shows in [505] that H(U), U a polynomially convex domain in a Frzchet space with the approximation property, is the T~ closure of the ideal generated by anyfinite family of functions in H(U) without common zero. In [277], B. Gramsch and W. Kaballo prove the following result: if

A is a Banach algebra with identity e, U is a polynomially convex domain in a a3-R space with Schauder basis and (fj)i=lCH(U) have the property that for every x 1;=1

(x) j aj,xf.

=

in

U

there exists (a,,X)j=ltA such that n e then there exists (aj) j=lC A such that

n

ajfj(x) = e for every x in U. In particular, this shows that the ideal generated by any finite family of holomorphic functions without common zero in U . is equal to H(U) (see also M. Schottenloher [646]). Further results and examples on analytic continuation, the spectrum of Cartan-Thullen theorems and the envelope of holomorphy are given in H(X), the book of G. Coeur6 [131]. We now return to our main theme. The following fundamental property of pseudo-convex domains in a locally convex space is due to S. Dineen, [183,186] and Ph. Noverraz [540, 5441; if U is a pseudo-convex (resp. finitely polynomially convex) open subset of a locally convex space E , p E cs(E), Il is the natural surjection from E onto E,ker(p), and n(U)

Appendix I

370 h a s non-empty i n t e r i o r t h e n sections of

R(U)

U = Il

-1

(n(U))

and t h e f i n i t e dimensional

are pseudo-convex ( r e s p . p o l y n o m i a l l y c o n v e x ) .

Various o t h e r forms and r e f i n e m e n t s o f t h e above a r e known and t h e y a l l o w one t o t r a n s f e r problems, such as t h e Levi problem, from

U

to

n(U)

and t o g e n e r a t e l o c a l l y convex s p a c e s w i t h p r e a s s i g n e d p r o p e r t i e s . I n [ 1 7 5 ] , S . Dineen r e p l a c e d t h e H(U)

and on showing

T~

=

T&

t o p o l o g y by t h e

T

t o p o l o g y on

T~

(theorem 4.38) o b t a i n e d a C a r t a n - T h u l l e n

theorem, i . e . ( b ) < = >( c ) , f o r b a l a n c e d open s u b s e t s o f a Banach s p a c e w i t h an uncoriditional b a s i s .

The f o l l o w i n g y e a r , S . Dineen and A . Hirschowitz

[203] improved t h i s r e s u l t by showing t h a t a domain

U

i n a Banach s p a c e

w i t h a Schauder b a s i s i s a domain of holomorphy i f i t s f i n i t e d i m e n s i o n a l s e c t i o n s are p o l y n o m i a l l y convex.

T h i s r e s u l t was extended t o s e p a r a b l e

Banach s p a c e s w i t h t h e p r o j e c t i v e approximation p r o p e r t y by Ph. Noverraz [540,543,546] t o m e t r i z a b l e and h e r e d i t a r y Lindel'df s p a c e s w i t h a n e q u i Schauder b a s i s by S . Dineen [186], t o

838

s p a c e s w i t h a b a s i s by N . Popa

[586] and t o v a r i o u s o t h e r s p a c e s by R. Pomes [583,584].

S. Dineen a l s o

showed i n [186] t h a t t h e c o l l e c t i o n o f s p a c e s f o r which t h i s r e s u l t was v a l i d was c l o s e d under t h e o p e r a t i o n o f open s u r j e c t i v e l i m i t . In [179], S . Dineen showed t h a t an open s u b s e t o f a Banach s p a c e w i t h

a Schauder b a s i s i s p o l y n o m i a l l y convex i f and o n l y i f i t s f i n i t e dimens i o n a l s e c t i o n s have t h e same p r o p e r t y .

T h i s r e s u l t was extended t o Banach

s p a c e s w i t h t h e s t r o n g a p p r o x i m a t i o n p r o p e r t y by Ph. Noverraz [540,544] and t o v a r i o u s o t h e r s p a c e s , i n c l u d i n g n u c l e a r s p a c e s , by u s i n g s u r j e c t i v e l i m i t s i n S . Dineen [183,186] and Ph. Noverraz [540,544].

A l l these

r e s u l t s a r e c o n t a i n e d i n t h e v e r y g e n e r a l r e s u l t o f M. S c h o t t e n l o h e r [643] who proved t h a t t h e same e q u i v a l e n c e was v a l i d i n any l o c a l l y convex s p a c e with t h e approximation p r o p er t y . We now look a t two c l o s e l y r e l a t e d q u e s t i o n s c o n c e r n i n g p o l y n o m i a l s , Runge's theorem and t h e Oka-Weil theorem. polynomials a r e dense i n s ubs et of

Cn,

(H(U),T~),

i f and o n l y i f

U

U

Runge's theorem s t a t e s t h a t t h e a h o l o m o r p h i c a l l y convex open

i s p o l y n o m i a l l y convex w h i l e t h e Oka-

Weil theorem s t a t e s t h a t a holomorphic germ on a p o l y n o m i a l l y convex

compact s u b s e t nomials.

K

of

Cn

can b e u n i f o r m l y approximated on

K

by poly-

37 1

Further developments I n [605], C . E . R i c k a r t proved an Oka-Weil theorem f o r

C*.

S. Dineen

[179] e x t e n d e d Runge's theorem t o Banach s p a c e s w i t h a Schauder b a s i s and i n c o l l a b o r a t i o n w i t h Ph. Noverraz [539,541] proved an Oka-Weil theorem f o r t h e same c l a s s o f s p a c e s .

C . Matyszczyk [469] showed t h a t t h e p o l y n o m i a l s

a r e s e q u e n t i a l l y dense i n

( H ( U ; F ) , T ~ ) when

open s u b s e t o f

E

and

E

approximation proper i y .

and

U

i s a p o l y n o m i a l l y convex

a r e Banach s p a c e s w i t h t h e bounded

F

The n e x t s e t o f c o n t r i b u t i o n s were made indepenS. Dineen [183,186],

d e n t l y by Ph. Noverraz [540,543,546],

S c h o t t e n l o h e r [31] and E . Ligocka [443].

R . Aron and M.

Noverraz proved Runge's theorem

and t h e Oka-Weil theorem f o r l o c a l l y convex s p a c e s w i t h t h e s t r o n g approxi m a t i o n p r o p e r t y , w h i l e R . Aron and M . S c h o t t e n l o h e r [31] proved a v e c t o r v a l u e d Runge theorem f o r domains i n Banach s p a c e s w i t h t h e a p p r o x i m a t i o n property.

Ligocka proved an Oka-Weil theorem f o r l o c a l l y convex s p a c e s

which c o u l d be r e p r e s e n t e d as a p r o j e c t i v e l i m i t o f normed l i n e a r s p a c e s w i t h a Schauder b a s i s and t h i s r e s u l t i n c l u d e d t h o s e o f Dineen.

E . Ligocka

a l s o showed t h a t any p o l y n o m i a l l y convex compact s u b s e t of a complete l o c a l l y convex s p a c e had a fundamental neighbourhood system o f p o l y n o m i a l l y convex open s e t s .

J . Mujica [502] p o i n t e d o u t t h a t L i g o c k a ' s p r o o f e x t e n d s

t o q u a s i c o m p l e t e s p a c e s and hence f o r t h i s c o l l e c t i o n o f s p a c e s t h e OkaWeil and Runge theorems a r e e q u i v a l e n t ( s e e a l s o Y . Fujimoto [ 2 4 9 ] ) .

In

/

[470], C . Matyszczyk proved a n Oka-Weil theorem f o r F r e c h e t s p a c e s w i t h t h e a p p r o x i m a t i o n p r o p e r t y and t h i s was extended t o h o l o m o r p h i c a l l y complete m e t r i z a b l e l o c a l l y convex s p a c e s by M . S c h o t t e n l o h e r [643].

In [502],

J . Mujica o b t a i n e d a v e r y g e n e r a l r e s u l t by p r o v i n g t h e Oka-Weil theorem

f o r q u a s i - c o m p l e t e l o c a l l y convex s p a c e s w i t h t h e approximation p r o p e r t y and a p p l i e d t h i s r e s u l t t o c h a r a c t e r i s e t h e p o l y n o m i a l l y convex.

E . Ligocka [443] i s s t i l l open;

H(U),

U

The f o l l o w i n g s u b t l e problem posed by if

s u b s e t o f t h e l o c a l l y convex s p a c e subset o f

spectrum o f

F u r t h e r a p p r o x i m a t i o n theorems a r e g i v e n i n C . Maty-

szczyk [470] and J . Mujica [504].

A

T~

E ( t h e completion o f

K

E

i s a p o l y n o m i a l l y convex compact

is

K

a p o l y n o m i a l l y convex compact

E)?

The s t u d y o f t h e Levi problem l e d d u r i n g t h i s p e r i o d t o t h e i n v e s t i g a t i o n o f c o n c e p t s such as holomorphic c o m p l e t i o n ( s e e s e c t i o n 2 . 4 ) , pseudo-completion,

spaces, e t c .

We refer t o Ph. Noverraz [540,543,544,

546,5471, M. S c h o t t e n l o h e r [633,637,645], [135] f o r d e t a i l s .

S . Dineen [184,186] and G . Coeurg

These topics and fundamental p r o p e r t i e s o f pseudo-

convex domains and p l u r i s u b h a r m o n i c f u n c t i o n s are s t u d i e d i n t h e t e x t o f

Appendix I

312 Ph. Noverraz [ 5 4 5 ] .

More r e c e n t a r t i c l e s on p l u r i s u b h a r m o n i c f u n c t i o n s

and p o l a r s e t s are S. Dineen [193,196], E . Ligocka [444], M . E s t g v e s and C . Her&

[231,232], S. Dineen and Ph. Noverraz [205,206], P . Lelong [438,

439,4401, B. A u p e t i t [32],Ph. Noverraz [554,557] and C . O .

Kiselman [388].

The n e x t r e s u l t on t h e e q u i v a l e n c e o f t h e v a r i o u s c o n d i t i o n s i s due t o Ph. Noverraz [543,546]. subsets of

&3/3

S. Dineen [190].

He proved t h e C a r t a n - T h u l l e n theorem f o r open

s p a c e s and t h i s was extended t o

$!y???-

s p a c e s by

L . Gruman [289,290] was t h e f i r s t t o g i v e a complete

H e used

s o l u t i o n t o t h e Levi problem i n an i n f i n i t e d i m e n s i o n a l s p a c e .

2

the solution t o the

problem i n f i n i t e dimensions and an i n d u c t i v e

c o n s t r u c t i o n t o show t h a t pseudo-convex domains i n s e p a r a b l e H i l b e r t s p a c e s a r e domains o f e x i s t e n c e o f holomorphic f u n c t i o n s .

The t e c h n i q u e and

r e s u l t o f L . Gruman have i n f l u e n c e d a l m o s t a l l l a t e r s o l u t i o n s t o t h e Levi problem.

He a l s o showed t h a t a f i n i t e l y open pseudo-convex s u b s e t o f a

vector space over

i s t h e domain o f e x i s t e n c e o f a G-holomorphic

C

f u n c t i o n ( s e e a l s o S. Dineen [186,187],

J . Kajiwara [365,366,367,368],

Y . Fujimoto [ 2 4 9 ] ) .

Kiselman [291] t h e n s o l v e d t h e

L . Gruman and C.O.

and

Levi problem on Banach s p a c e s w i t h a Schauder b a s i s and Y . H e r v i e r [329] e x t e n d e d t h i s r e s u l t t o domains s p r e a d .

I n [546] and [548] Ph. Noverraz

e x t e n d e d t h e s o l u t i o n o f t h e Levi problem t o Banach s p a c e s w i t h t h e bounded a p p r o x i m a t i o n p r o p e r t y and proved, f o r t h e s e s p a c e s , t h e f o l l o w i n g Oka-Weil theorems:

(i)

UCU'

is

then

H(U')

if T~

U

and

h u l l o f each compact s u b s e t of

U

a r e pseudo-convex domains w i t h

U'

dense i n

i f and o n l y i f t h e

H(U)

i s contained i n

pseudo-convex open s e t and t h e compact s u b s e t H(U)

h u l l t h e n e v e r y holomorphic germ on

by holomorphic f u n c t i o n s on

U.

K

K

(ii)

U;

of

U

H(U') if

U

is a

i s equal t o i t s

can be approximated on

K

Both ( i ) and ( i i ) were g e n e r a l i z e d t o

domains s p r e a d o v e r F r g c h e t s p a c e s and

33&'s p a c e s

w i t h f i n i t e dimension-

a l Schauder d e c o m p o s i t i o n s by M. S c h o t t e n l o h e r [ 6 4 0 ] .

Ph. Noverraz [548]

and R . Pomes [583,584] t h e n s o l v e d t h e Levi problem f o r 3 3 J s p a c e s w i t h a Schauder b a s i s .

The n e x t i m p o r t a n t development i s due t o M. S c h o t t e n l o h e r [ 6 3 6 , 6 4 0 ] . He combined r e g u l a r c l a s s e s , a d m i s s i b l e c o v e r i n g s , s u r j e c t i v e l i m i t s and

a s u b t l e b u t v e r y c r u c i a l m o d i f i c a t i o n o f L . Gruman's c o n s t r u c t i o n t o s o l v e t h e Levi problem f o r domains s p r e a d o v e r h e r e d i t a r y Lindelb'f l o c a l l y convex s p a c e s w i t h a f i n i t e d i m e n s i o n a l Schauder d e c o m p o s i t i o n .

This

Further developments

373

collection of spaces contains all Frgchet spaces and all & $ ? Y l a Schauder basis.

spaces with

Particular cases of Schottenloher's result are given in

S. Dineen, Ph. Noverraz and M. Schottenloher [ 2 0 7 ] .

M. Schottenloher [636,

6401 and P . Berner [59,60] obtained, independently, the following result:

is an open surjective limit and every pseudo-convex domain

if E = lim E - a

CXEA

spread over E , ~ E A , is a domain of holomorphy (resp. domain of existence) a

then every pseudo-convex domain spread over E

is a domain of holomorphy

(resp. domain of existence). In [ 3 6 ] , V. Aurich showed that domains of existence of meromorphic functions in Banach spaces with Schauder bases are domains of existence of holomorphic functions. In [154], J.F. Colombeau and J. Mujica solved the Levi problem for open subsets of

60 3"rz. spaces.

They reduced the Levi problem on 3 3 k

spaces to the Levi problem on a Hilbert space, where L. Gruman's result applied, by using surjective limits and by combining the fact, noticed by space is also open with previous authors, that any open subset of a

&3n

respect to a weaker semi-metrizable locally convex topology with Grothendieck's result [ 2 8 6 , 2 8 8 ] that for any sequence of neighbourhoods of O , ( U . ) . in a

0 X . IU 1. J

DF

space there exists a sequence of scalars

(Xj)j

1 1

such that

is also a neighbourhood of zero (see also corollary 2.30).

This

approach has been developed by J.F. Colombeau and J. Mujica 11561 in the r

study of Hahn-Banach extension theorems and convolution equations. In [ 5 0 6 ] , J. Mujica solves the Levi problem for domains in

(E',ro)

E a separable Frgchet space with the approximation property by using topological methods. Mujica also proves in [SO61 that a holomorphically convex domain in ( E ' , T ~ ) , E a separable Frgchet space, is the domain of existence of a holomorphic function and this result was extended, using

quite different methods, by M. Valdivia [691] to the case where E

is an

arbitrary Frgchet space. M. Valdivia obtains a number of interpolation theorems for vector valued holomorphic functions in [691]. See also M. Schottenloher [636] . This completes our survey of the Levi problem and the Cartan-Thullen theorem in infinite dimensional spaces. Our analysis has hopefully shown their central role in infinite dimensional holomorphy and their importance

Appendix I

374

in motivating new ideas and concepts. This direction of research still contains many open problems, e.g. the Levi problem has not been solved and no Cartan-Thullen theorem exists for arbitrary domains in separable Banach

spaces. Indeed the reader will no doubt have observed that all known positive results on the Levi problem involve an approximation property assumption and this excludes certain separable Banach spaces. Further references for the above topics are J. Horvath [349], W. Bogdanowicz [77] D. Burghela and A . Duma [110], E. Ligocka and J. Siciak [446], E. Ligocka [444,445], M. Her&

[325,326], J. Bochnak and J. Siciak [75], C.E. Rickart

[606], S. Baryton [54], I.G. Craw [170], S . Dineen [193], G. Coeur6 [132, 133,1341, G. Katz [372], J. Kajiwara [365], V. Aurich [34,37], Y . Hervier [330], L.A. de Moraes [495,496,497], A . Bayoumi [SS], M.G. Zaidenberg [719], S.J. Greenfield [279] and Y. Fujimoto [249].

a

operator can be In finite dimensions fundamental solutions of the obtained from the potential kernel, i.e. from a fundamental solution of the Laplacian. L. Gross [284] (see also P . Lgvy [442]) has studied infinite dimensional generalizations of the potential kernel and found, because of the absence of a translation invariant (i.e. Lebesgue) measure on an infinite dimensional locally convex space, that the natural setting for finding fundamental solutions of the Laplacian was an abstract Wiener space with its associated Gaussian measure. A triple (j,H,B) is called an abstract Wiener space if H

is a separable Hilbert space, B is a Banach space, j

is a continuous injection of H onto a dense subspace of B

and the norm

of B is, via j, a "measurable" norm on H (if for instance, H=B and j is a Hilbert-Schmidt operator with non-zero eigenvalues, then (j,H,H) is an abstract Wiener space). The canonical Gaussian "measure" on H leads to a true measure on C.J. Henrich

B

for any abstract Wiener space

[322] was the first to investigate the

5

(j,H,B). equation in

an infinite dimensional setting. His approach was influenced by the work of L. Gross [284] on the infinite dimensiondl Laplacian, by H. Skoda's research [662] on the finite dimensional

5 equation and by the work of

L. Hormander [346] on 'L estimates for partial differential operators. C.J. Henrich's work is very fundamental, quite delicate (even the statement of the equation and the interpretation of the solution necessitate a careful examination) and his ideas have influenced later developments. His main result is the following:

375

Further developments

if H

is a separable Hilbert space and

w

is an

(0,l)

form on H which factors through an abstract Wiener space as a closed form of polynomial growth, then there exists a

,Ad -

an

(*I

function of polynomial growth on H,a, such that

= w.

The condition on w abstract Wiener space ial growth on B

in

means the following: there exists an

(*)

(j,H,B),

a

3 closed

(0,l)

form

-

w

o f polynom-

such that the following diagram commutes j _ _ _ _ f

A0”(B)

B

Equivalently we may say that

(*)

is a solution to the

3

equation on a

dense subspace o f H. In [421], B. Lascar shows that Henrich’s solution can be extended to the whole space (i.e. to H) as a distributional solution to the

a equation.

A summary of the work of C . J . Henrich is given in [364] by J. Kajiwara. The formula for Henrich’s solution is very technical, mainly because Gaussian measures are not invariant under translation and this leads to complicated terms when differentiating under the integral s i p . In [187], S. Dineen used transfinite induction and sheaf cohomology to show that each infinitely dteaux differentiable closed (0,l) form on a finitely open pseudo-convex subset Q of a complex vector space is the image by 3 of an infinitely G&eaux differentiable function on 0. I n his study of the representation o f distributions by boundary values

of holomorphic functions, D. Vogt [701] encountered the vector valued 5 spaces (definition 5 . 3 8 ) . He proved the foll-

problem and discovered DN owing result [701]. are equivalent: 1)

If E is a 3 3 h s p a c e , then the following conditions

each E-valued distribution of compact support in R may be

Appendix I

376

r e p r e s e n t e d as t h e boundary v a l u e o f an element o f H(C\R;E),

a

(2) t h e mapping

is a

(3) E b

:.hm,(R2 ; E )

- - + g ( R 2 ; E ) is surjective,

space.

DN

$ a = w on a convex open s u b s e t

A.Rapp [601,602] s o l v e d t h e e q u a t i o n

o f a Banach s p a c e w i t h r e g u l a r boundary when t h e c l o s e d form

i s of

w

s u f f i c i e n t l y slow growth n e a r t h e boundary and E . Ligocka [444,445] o b t a i n -

&'

ed a solution f o r

f u n c t i o n s o f bounded s u p p o r t on .a Banach s p a c e .

Both used s t r a i g h t f o r w a r d g e n e r a l i z a t i o n s o f t h e f i n i t e dimensional method. Next, P . Raboin made a number o f i m p o r t a n t c o n t r i b u t i o n s by r e t u r n i n g t o t h e approach of C . J . he d e f i n e d t h e space

Henrich L'

of

and u s i n g Gaussian measures. (0,q)

9 i n t e g r a b l e w i t h r e s p e c t t o t h e Gaussian measure

H i l b e r t space

H

i~ on

t a t i o n f o r t h e adjoint of

T

9

the separable

5 t o L 2 was 9 9 After obtaining an i n t e g r a l represen-

and showed t h a t t h e r e s t r i c t i o n

a c l o s e d o p e r a t o r with dense range.

I n [587,589]

d i f f e r e n t i a l forms which a r e s q u a r e

of

T

and e s t a b l i s h i n g a p r i o r i e s t i m a t e s ( i n t h e

manner o f L . Hormander [346] f o r t h e f i n i t e dimensional c a s e ) t h a t each c l o s e d form i n

H e proved t h a t e a c h of

L2

3

was t h e form i n

a

L21

image o f a member o f

I n [589], Raboin showed t h e e x i s t e n c e o f a

problem f o r

&"

closed

(0,l)

L2q.

was t h e image o f an element

whose r e s t r i c t i o n t o a c e r t a i n d e n s e subspace o f

function. the

L2q+l

.g" c l o s e d

he proved

H

"&""

was a

l'al''

solution t o

forms, bounded on bounded s e t s , and

e x t e n d e d t h i s r e s u l t i n [593], ( s e e a l s o

[ 5 9 0 , 5 9 1 , 5 9 2 ] ) , t o pseudo-convex

domains i n a H i l b e r t s p a c e by u s i n g a g e n e r a l i z e d Cauchy i n t e g r a l formula for

&-

functions.

I n [137], G . Coeurg g i v e s a n example o f a the unit ball

B

5 1 f u n c t i o n on

R1 c l o s e d

(0,l)

o f a H i l b e r t s p a c e which i s n o t t h e image by

5

form on

o f any

B.

The n a t u r a l s t e p from H i l b e r t s p a c e s t o n u c l e a r s p a c e s , s u g g e s t e d by C . J . Henrich

[322], was t a k e n by P . Raboin i n [588,590,591,592,593].

[593], h e proved t h a t any

,&"

closed

(0,l)

form, s a t i s f y i n g a modest

t e c h n i c a l c o n d i t i o n on a pseudo-convex open s u b s e t

R o f a 83-Qs p a c e

In

377

Further developments

with a basis was the image by

5 of a

,(.' function on R .

In [164],

J.F. Colombeau and B. Perrot prove that every ,&"

closed ( 0 , l ) form on a E is the image by 5 of a 4" function on E (see also the remark by P. Kre'k in 56.0 of [418]) and in [166] they extend this result

83n space

to pseudo-convex domains in E by D. Nosske 15311).

(this result was also found, independently,

The initial version of J.F. Colombeau's and

B. Perrot's solution to the 2 problem [166] was considerably simplified by a result on hypoellipticity (due to P. Mazet [480]) which yields, as a particular case, the following: Any Ggteaux

A"

is a (Frgchet)

solution to the

a" solution.

2

problem which is locally bounded

Recently, R. Meise and D. Vogt [488], have shown that the solvability of the

a

problem on a nuclear Frgchet space E

property DN

implies that E has

(definition 5.38).

Applications of the infinite dimensional

a operator to natural

/

Frechet algebras are given in B. Kramm [398] and to convolution operators by J.F. Colombeau, R. Gay and B. Perrot in [148]. Application of the 5 operator to the Cousin I problem are discussed below. SHEAF THEORY Sheaf theory and sheaf cohomology play an important role in several complex variable theory and it is probable, see for instance B. Kramm [398], that the same remark will eventually apply to infinite dimensional holomorphy. Of key importance for finite dimensional holomorphy are theorems A and B of H. Cartan [115]. Theorem B states that HP ( X , 3 ) = 0 f o r any and any coherent analytic sheaf 3 on the Stein manifold X. Theorem B can be used to solve the a problem and to resolve the Cousin I p3 1

problem (also called the additive Cousin problem) on holomorphically convex domains in En.

Classically the Cousin I problem was to find a several

complex variable version of the Mittag-Lefflertheorem - which showed the existence o f a meromorphic function in any domain of Q: with preassigned poles. The several complex variables version sought to characterise within the collection of principal parts on a domain X in Cn those which gave rise to a meromorphic function on in Ells].

X. This problem was solved by H. Cartan

318

Appendix 1

In recent years, various authors (e.g. L. Hormander [ 3 4 7 ] , C.E. Rickart [605] and P. Raboin [588]) have assigned the terminology "Cousin I problem" to a more general collection of problems of which the following is typical : given a covering

(Ui)iEI

locally convex space E, such that

of a domain X, and hij

E

spread over a

H(Ui"U.)

for all

3

i,j € 1

for all i,j and k in 1,does there exist a family (hi)iEI, hi E H(Ui) such that hi-h. = h.. on U i n U . for all i and j in

1

I?

11

3

Using Cech cohomology we see that (**) has a solution for any set of data 1 { U . ,h. .I if H (X,@) = 0 where denotes the sheaf of holomorphic germs 1 1J on X. It is easy to show that a generalised Mittag-Leffler theorem is valid on X whenever H 1(X,t3) = 0 . Banach algebra considerations motivated the first examples of sheaf cohomology with values in a sheaf of holomorphic germs in infinitely many variables. In [12], R. Arens proved that H p ( g , @ I ) = 0 for any pzl A where A is a Banach algebra with continuous dual A' and spectrum ;6 and where 8,, is the sheaf of weak* holomorphic germs on A'. this result to show H1($,Z)

{xEA, x invertible} /;exp(x)

He applied

; XEA}

(see also R. Arens [13] and H. Royden [610]). In [605], C.E. Rickart proved that Hp(K,o) = 0 for any p>l and any polynomially convex compact subset K of CA, A arbitrary. C.E. Rickart [605] also states and solves a Cousin I problem on the set K and applies it to prove the Rossi local maximum modulus principle for Banach algebras.

P. Silici showsin

[659]

that theorems A

and

B

are valid f o r

379

Further developments

IC . H1(U,CG) = 0 A

compact p o l y d i s c s i n proved t h a t

By u s i n g t r a n s f i n i t e i n d u c t i o n , S . Dineen [187] f o r any f i n i t e l y open pseudo-convex domain

U

i n a complex v e c t o r s p a c e , where P G i s t h e s h e a f o f GGteaux holomorphic

8

germs, and used t h i s r e s u l t t o s o l v e t h e Levi problem and t h e f o r Gzteaux holomorphic and d t e a u x

8" f u n c t i o n s .

problem

J . Kajiwara [368]

extended t h i s r e s u l t t o the h i g h e r cohomology groups on f i n i t e l y open pseudo-convex domains i n p r o j e c t i v e s p a c e ( s e e a l s o Y . Fujimoto [ 2 4 9 ] ) .

In

[192], S. Dineen showed t h a t Cousin I i s n o t s o l v a b l e , and h e n c e H1(U,U) # 0

2

and t h e

problem i s n o t s o l v a b l e , f o r any domain

U

in a

l o c a l l y convex s p a c e which does n o t admit a c o n t i n u o u s norm and i n [35] V . Aurich proved t h a t a g i v e n f a m i l y o f p r i n c i p a l p a r t s on a S t e i n m a n i f o l d

spread over

CA,

a r b i t r a r y , g i v e s r i s e t o a meromorphic f u n c t i o n if and

A

o n l y i f t h e p r i n c i p a l p a r t s a l l f a c t o r t h r o u g h some

Cn.

The n e x t development i s due t o P . Raboin [ 5 8 8 ] who proved, u s i n g h i s s o l u t i o n t o the

2 problem, t h e f o l l o w i n g Cousin I r e s u l t ;

pseudo convex domain i n a F r g c h e t n u c l e a r s p a c e

E

n is a

if

w i t h a b a s i s and

t h e n f o r e a c h convex compact i s a s e t o f Cousin I d a t a on {$li,gij}i,j b a l a n c e d s u b s e t K o f E t h e r e e x i s t s a f a m i l y I f i E H ( Q . n EK)}i such that

g..

11

=

fi-fj

on

Q i o Q . n E Kf o r

space w i t h cl osed u n i t b a l l t o t h e topology o f

1

In

f o r any pseudo-convex domain i n v o l v e d a s o l u t i o n of t h e that

B 3-n

of unity.

and each

K

EK.)

i

fi

and

j.

1

(EK

in

a

&33n

i s t h e Banach

i s holomorphic with r e s p e c t

[593], P . Raboin proved t h a t

U

5

all

1

H ( Up) = 0

space with a b a s i s .

H i s proof

problem, t h e Oka-Weil theorem and t h e f a c t

A" p a r t i t i o n s a3-l s p a c e s by J . F .

s p a c e s a r e h e r e d i t a r y L i n d e l a f s p a c e s and admit T h i s r e s u l t was e x t e n d e d t o a r b i t r a r y

Colombeau and B. P e r r o t [164,166].

Theorems A and B o f H. C a r t a n have been extended t o v e c t o r v a l u e d holomorphic f u n c t i o n s on a f i n i t e d i m e n s i o n a l s p a c e by L . Bungart [ l o g ] . T h i s completes o u r d i s c u s s i o n o f c o n d i t i o n s ( a ) , ( b ) , ...,(f ) f o r i n f i n i t e dimensional spaces. DIFFERENTIAL EQUATIONS We now d i s c u s s c o n v o l u t i o n o p e r a t o r s and p a r t i a l d i f f e r e n t i a l o p e r a t o r s

on s p a c e s of holomorphic f u n c t i o n s o v e r l o c a l l y convex s p a c e s . A s t h i s s u b j e c t forms p a r t o f a book i n p r e p a r a t i o n by J . F . Colombeau, o u r

Appendix Z

380

presentation will be brief and concentrate mainly on the role played by this topic in the general development of infinite dimensional holomorphy. C.P. Gupta [295] was the first to consider convolution operators on spaces of holomorphic functions over locally convex spaces and his approach influenced many later workers in this area. The main finite dimensional considerations of C.P. Gupta were the results and techniques o f B. Malgrange [448] and A. Martineau [452 A simplified description of the basic approach used by C.P. Gupta goes A a locally convex translation invariant space of

as follows. Given

holomorphic functions on the locally convex space E , a convolution operator on A is defined as a continuous linear operator from A into itself which commutes with all translations. For operator has the form where

C"

an

&

=

H(C)

each convolution

dn

1 -

The Bore1 transform establishes a one-to-one correspondence between convolution operators on

&

, the elements of

and a space of holomorphic

functions of exponential type on E ' . The existence and approximation problem for convolution operators is then transposed and solved as a division problem for holomorphic functions on El. C.P. Gupta's [295] investigation of convolution operators on Banach spaces led him to HNb(E), the space of holomorphic functions of nuclear bounded type on E, and to the correspondence HNb(E);I = Exp(E;). He showed that every convolution operator on H (E) was surjective and that Nb solutions of the associated homogeneous equation could be approximated by exponential polynomial solutions. Extensions of this method to more general classes of locally convex spaces and to other collections of holomorphic functions are given in C.P. Gupta [296,297], L . Nachbin [511], P.J. Boland [79,80,81,82], M.C. Matos [458,463,464,467], S . Dineen [177], P.J. Boland and S. Dineen [ 8 8 ] , T.A.W. Dwyer [218,221,222,223,225], P . Berner [62], D. Pisanelli [580,58l], J.F. Colombeau and M.C. Matos [150,151], J.F. Colombeau and B. Perrot [163,167], F. Colombeau, T.A.W. Dwyer and B. Perrot [147], and J.F. Colombeau and J. Mujica [156].

38 1

Further developments A d i f f e r e n t approach i s t a k e n by T.A.

Dwyer [214,215,216,219]

(see

a l s o 0 . Bonnin [94]) i n s t u d y i n g p a r t i a l d i f f e r e n t i a l o p e r a t o r s on holomorHe d e f i n e s t h e Fock s p a c e s

p h i c Fock s p a c e s o f H i l b e r t - S c h m i d t t y p e . Yp(E)

on a H i l b e r t s p a c e

(and a f t e r w a r d s on c o u n t a b l y H i l b e r t s p a c e s

E

and o t h e r c l a s s e s o f l o c a l l y convex s p a c e s , see a l s o J . Rzewuski [621,622]) and shows t h a t

/I PflI P

2

11 Pm(I

partial differential operator

. /I f Ilp

P(D)

7

f o r any f i n (E) and any P m Pn(D). Using t h i s i n e q u a l i t y

=

Dwyer showed t h a t a l l such p a r t i a l d i f f e r e n t i a l o p e r a t o r s map

3P (E)

3 P (E)

onto

and g e n e r a l i s e d a number o f f i n i t e d i m e n s i o n a l r e s u l t s ( s e e F . N o t a b l e a s p e c t s of Dwyer's work, s e e t h e refer-

T r e v e s [686], c h a p t e r 9 ) .

are h i s c o n c r e t e r e p r e s e n t a t i o n o f

e n c e s c i t e d above and [224,226,227], c o n v o l u t i o n o p e r a t o r s by means o f

( V o l t e r r a ) k e r n e l s , e t c . and h i s

'L

r e c o g n i t i o n of a r e l a t i o n s h i p between c e r t a i n a b s t r a c t d i f f e r e n t i a l e q u a t i o n s i n l o c a l l y convex s p a c e s and problems i n c o n t r o l t h e o r y , a n a l y t i c b i l i n e a r r e a l i z a t i o n s , quantum f i e l d t h e o r y , e t c . ( s e e a l s o J . F . Colombeau and B . P e r r o t [158,162], J . F . Colombeau [145], P . Kr6e [401,410,417] and P . Kr6e and R. Raczka [ 4 1 9 ] ) . The long term r e l e v e n c e o f c o n v o l u t i o n

o p e r a t o r s i n i n f i n i t e l y many v a r i a b l e s may w e l l depend on t h i s k i n d o f r e c o g n i t i o n and i n s i g h t . The most r e c e n t developments i n t h i s g e n e r a l d i r e c t i o n a r e due t o J . F . Colombeau, R . Gay and B . P e r r o t [148].

They p r o v e , u s i n g a p r e p a r -

a t i o n theorem f o r holomorphic f u n c t i o n s on a Banach s p a c e due t o J . P . f&' ( Q ) =

holomorphic f u n c t i o n

on a c o n n e c t e d domain

nuc le a r space

E

s o l u t i o n s of t h e T

f

(Q)

Q

i n a quasi-complete dual

and a p p l y t h i s r e s u l t t o g e t h e r w i t h t h e e x i s t e n c e o f

5

problem on

i s a c o n v o l u t i o n o p e r a t o r on

p € H M ( E ) t h e n any s o l u t i o n t r a n s f o r m o f a n element

U

f

ajQ

A"

spaces t o prove t h e following:

Exp(E')

with c h a r a c t e r i s t i c function

of t h e equation

o f %'(E)

Ramis

f o r any non-zero Mackey (or S i l v a )

[598] ( s e e b e l o w ) , t h a t

f o r which

Tf = 0

pU = 0.

i s t h e Bore1 The f i n i t e

d i m e n s i o n a l a n a l o g u e s o f t h e s e r e s u l t s are due t o L . Schwartz [647] and R . Gay [254] r e s p e c t i v e l y .

The t h e o r y o f c o n v o l u t i o n o p e r a t o r s drew a t t e n t i o n t o t h e r o l e o f n u c l e a r p o l y n o m i a l s i n t h e g e n e r a l t h e o r y of holomorphic f u n c t i o n s i n i n f i n i t e l y many v a r i a b l e s and p r o v i d e d t h e f i r s t examples o f a f u n c t i o n space r e p r e s e n t a t i o n o f i n f i n i t e dimensional a n a l y t i c f u n c t i o n a l s .

The

if

382

Appendix 1

appearance of nuclearity motivated L. Nachbin [508,509] to introduce the concept of holomorphy type as a means of investigating holomorphic functions whose derivatives pertained to a certain class of polynomials (e.g. compact, Hilbert-Schmidt, nuclear, etc.) and whose Taylor series expansion satisfied growth conditions relative to the canonical semi-norms on the underlying spaces of homogeneous polynomials. The theory of holomorphy types has been developed in various directions by L. Nachbin [508, 509,511,512], S. Dineen [177], R.M. Aron [15,16,17,19], T.A.W. Dwyer L214, 216,221,222,2231, P.J. Boland [78,80,81], S.B. Chae [119,120] and L.A. de Moraes [495,496,497], and led, eventually, through the work of Boland, to the theory of holomorphic functions on nuclear spaces as outlined i n chapters 1 , 3 , 5 and 6. The Bore1 transform and the correspondence between analytic functionals on H ( E ) and holomorphic functions of exponential type on E' were almost totally developed within the framework o f convolution operators as outlined above (see the references listed previously and also T.A.W. Dwyer [217,220]), and although in this text we have more o r less exclusively preferred the representation of analytic functionals by holomorphic germs the motivations and guidelines arising from the exponential representation were always suggestive and significant. ANALYTIC GEOMETRY The first comprehensive treatments of analytic sets in infinite dimemsions are due to P.J. Ramis [594] and G . Ruget [613], both of whom worked only in Banach spaces but were aware that many of their results and techniques extended to locally convex spaces. Most of the other important developments in this area are due to P . Mazet [479]. As in the finite dimensional case (see for instance M. He&

[324]) the local theory is

first developed by studying the ideal structure of the commutative ring (the space of holomorphic germs at the origin in the locally convex space E), and then applied to obtain global results. The ring @(E) is

Q(E)

an integral domain and a local ring but is Noetherian if and only if E finite dimensional. Since the Noetherian property of W(cCn) plays a

is

crucial role in the finite dimensional theory, new methods in commutative algebra had to be developed and these appear to be o f independent interest.

383

Further developments

Two of the key results in both the finite and infinite dimensional theories are the Weierstrass Factorization and Preparation Theorems. Weierstrass Factorization Theorem If g # 0 E *(E)

then there exists a decomposition of E, El @Ce,

that the restriction of g

to Ce has order p # 0 and for any

there exists a unique polynomial U(E)

(J.P. Ramis [594] and P. Mazet [ 4 7 9 ] ) .

r of degree < p and a unique q

such f

E

@(E)

in

such that f = g.q+r.

A distinguished polynomial relative to the decomposition El @Ce a mapping of the form (zl,z)-zP

P-1

1

+

is

ai(zt)z i

i=O where ai(zf)

E

@(El)

and ai(0) = 0 for i

=

0,...,p-1.

The Weierstrass Preparation Theorem If 0 # g E O(E) has order pbl then there exists a decomposition of E, El @Ce, such that g can be written in a unique fashion, g = h.P, where h of @ ( E )

is an invertible element

and P is a distinguished polynomial of degree p

relative to

the decomposition El 8 Ce. Using the above theorems one shows [594,479], that U(E) is a unique factorization domain. In [ 2 3 6 ] , M.J. Field uses the Factorization Theorem to prove that a germ in @ ( E )

is irreducible if and only if its

restriction to some sufficiently large finite dimensional subspace is irreducible.

A subset X of a complex manifold modelled on a locally convex space is called analytic if for each x in X there exists a triple (Vx,fx,Fx) where Vx is an open neighbourhood of x, Fx is a locally convex space, fx E H(Vx;Fx) and X n V x = {YE Vx;fx(y) = 0 1 . If Fx can be chosen to be finite dimensional (respectively one dimensional) for each x in X then we say X is finitely defined (respectively a principal analytic set or a hypersurface). Thus, a finitely defined analytic subset is one which is locally defined by a finite number of scalar equations. An example of an analytic subset, not generally finitely defined, is the spectrum of a

384

Appendix I

commutative Banach algebra polynomial on If

A.

This s e t i s t h e zero s e t of a 2 homogeneous

(see J . P . Ramis [598,p.32] and B. Kramm [398]).

A'

i s an a n a l y t i c subset of a complex manifold

X

l o c a l l y convex space

E

e x i s t s a decomposition

then

of

in

U

and a biholomorphic mapping

in

E

such t h a t

p o i n t s of x.

each a n a l y t i c germ

The i d e a l

U

We l e t

of

a

onto a neighbourhood of

0

X*

denote t h e r e g u l a r

i s c a l l e d t h e (geometric) codimension of

a,

V(f)a

I(Xa)

X

at

i n Ua(E),

X

at

a.

To

t h e space

by l e t t i n g

i s an i r r e d u c i b l e germ.

Xa

V

I(Xa) = { f ~ @ ~ ( E ) ; =f \0~) . a i s equal t o i t s r a d i c a l and i s a prime i d e a l i f and only

I(Xa)

associate the

where

of

$

we a s s o c i a t e an i d e a l

Xa

i f there

X

w i l l denote t h e a n a l y t i c germ of t h e a n a l y t i c subset

Xa

of holomorphic germs a t

if

modelled on a

an open neighbourhood

E,

$(XnV) = $(V)ftE1.

and dim(E2)

X

i s a regular point of

a E X

El 8 E 2

U

To each i d e a l

9

i n Ua(E)

we may

"object"

i s t h e a n a l y t i c germ a t

r e p r e s e n t a t i v e of t h e germ ( f o r example, i f

dim(E) <

m)

a

9

If

f.

then

defined by t h e zero s e t of some

&

i s generated by

V(3) =

ifl,.

V(fi)a

. .,fn}

i s an a n a l y t i c

germ of a f i n i t e l y defined a n a l y t i c subset b u t , unfortunately, f o r a r b i t -

9

trary

the object

V(3)

i s n o t i n general an a n a l y t i c germ.

The main problem f o r t h e l o c a l theory i s t h e following: ideals

9

i n W(E),

given t h a t

V(3)

i s an a n a l y t i c germ a t

f o r what 0,

do we

have Rad(3)

=

I(V(g))?

The f i r s t r e s u l t on t h i s problem, a N u l l s t e l l e n s a t z f o r p r i n c i p a l i d e a l s ,

i s due t o J . P . Ramis [598,p.29] (see a l s o M . J . F i e l d "2361 and t h e d i v i s i o n theorem of C . P . Gupta [295,proposition 131) and t h i s may be s t a t e d as follows : if

E

i s a Banach space,

glv(f) = 0 such t h a t

f

and

g

E

W(E)

and

then t h e r e e x i s t s a p o s i t i v e i n t e g e r gm

generated by

m

belongs t o t h e p r i n c i p a l i d e a l i n @(E) f.

385

Further developments In o r d e r t o i d e n t i f y c e r t a i n i d e a l s with germs o f a n a l y t i c s u b s e t s ,

J . P . Ramis introduced t h e concept of geometric i d e a l and proved t h e followIf

ing result:

i s a Banach space and

E

then t h e following a r e e q u i v a l e n t :

9

i s a prime i d e a l i n B(E)

i s a geometric i d e a l ( i . e . t h e r e e x i s t s a f i n i t e l y

(a)

1C I ( V ( 9

generated i d e a l

9

in &(E)

where

X

i s an i r r e d u c i b l e germ of a f i n i t e l y

(b)

= I(X)

such t h a t %

))

defined a n a l y t i c s u b s e t . (c)

ht(

is f i n i t e (the height of

)

, ht(]

is the

)

length of a maximal proper chain of prime i d e a l s j o i n i n g and

(0)

(d)

8

)

1.

has a normal decomposition ( i . e . t h e r e e x i s t s a

decomposition o f such t h a t @ ( E l ) n

1

with dimension

E1&E2,

E,

and

= (0)

V() )

n E2

=

(E2)<"

(0)).

The equivalence of (b) and (c) i s a N u l l s t e l l e n s a t z f o r prime i d e a l s of f i n i t e height.

9

at

The dimension o f

( i n f a c t it i s equal t o 0).

i n condition (d) i s an i n v a r i a n t of

E2

ht(Q ) )

and i s ca(1ed t h e codimension of

X

I t i s n o t d i f f i c u l t t o s e e t h a t t h i s concept of codimension

coincides with t h a t given e a r l i e r f o r r e g u l a r p o i n t s of an a n a l y t i c s u b s e t . I t i s n o t known i f every prime i d e a l of f i n i t e h e i g h t i n f i n i t e l y generated.

is

@(E)

An e s s e n t i a l p a r t o f t h e proof of t h e above r e s u l t

uses ramified coverings and t h i s concept i s a l s o important i n t h e l a t e r works o f P . Mazet [475,477,479]

and V. Aurich [ 3 8 ] .

The next developments a r e due t o P . Mazet [471,472,473] J . P . R a r n i s [598] and G . Ruget [613,614]).

(see a l s o

Mazet proved t h e following

result: if

3

i s an i d e a l i n O(E),

by

n

elements

have height

E

a Banach space,which i s generated

then t h e minimal prime i d e a l s containing

3

all

,
I n t h i s way, s i n c e

Rad(3)

i s then t h e i n t e r s e c t i o n o f a f i n i t e

number o f prime i d e a l s o f f i n i t e type, Mazet obtained a N u l l s t e l l e n s a t z f o r f i n i t e l y generated i d e a l s .

Afterwards, he introduced [474,476,479]

the

Appendix I

386

concept of a C.M,

ring which generalized the notion of a Cohen-Macauley

ring (see 0 . Zariski and P. Samuel [ 7 2 2 , Appendix 61).

He proved, using

ramified coverings and techniques from homological algebra, that @ ( E ) a

C.M,

ring for any locally convex space E

is

and subsequently obtained

the Nullstellensatz for finitely generated ideals in @ ( E ) .

8

If 9

9c 1

i s an ideal in LJ (E)

finitely generated ideal E l @ E2

we let ht(

9)

=

inflht(3 ) ;

and for any analytic subset X we let codim Xa [473] and

9

in 13 (E)

Codim V ( 9 )

=

3 prime and

ht(I(Xa)).

Any

admits a normal decomposition

i s , in this case, also equal to the dimen-

sion of E 2 . Finally we list some properties and examples of analytic subsets which follow (some not exactly immediately) from the above results. Many of the

results we quote are to be found in J . P . Ramis [598] and apply to analytic subsets of a Banach space. For results on locally convex spaces we refer to P . Mazet [479] where the reader will find applications of infinite dimendimensional analytic geometry to the theory of finite dimensional analytic

If X

k

is a finitely defined analytic germ then

X = U X . where each Xi is a finitely defined 1=1 1 irreducible analytic germ. This decomposition of X is unique modulo a permutation of the X.'s given that no proper inclusions are allowed (see P. Mazet [ 4 7 3 ] ) . The set of regular points X* of a finitely defined analytic subset X is dense in X. If X

is an analytic subset of an infinite dimensional

Banach manifold U

and Xx

does not contain a principal

analytic germ at any point x in X (or equivalently if inf Codim Xx 3 2 ) then every C-valued holomorphic XE

x

function on (see also M . J .

U\X

extends to a holomorphic function on U

Field [ 2 3 6 ] ) .

A locally compact analytic subset of an infinite dimensional

locally convex space has infinite codimension at each point. If m

is a meromorphic function on a complex Banach

387

Further developments manifold

m

then t h e s e t o f p o l e s of

U

a n a l y t i c subset o f

m

minancy of

U

is a principal

and t h e s e t o f p o i n t s o f i n d e t e r -

i s an a n a l y t i c s u b s e t of codimension 2 a t

each p o i n t . (vi)

i s an a n a l y t i c s u b s e t of a Banach manifold and codim X, < then x E X * . I f X E X* then If

X

codim Xx = l i m i n f (Codim X ) . YEX* Y

(vii)

A f i n i t e l y defined a n a l y t i c s u b s e t o f a Banach manifold i s

i r r e d u c i b l e i f and o n l y i f (viii)

i s a f i n i t e l y defined a n a l y t i c s u b s e t o f a complex

X

If

Banach manifold of

i s connected.

X*

U

X\ X* = S(X) , t h e s i n g u l a r s e t

then

i s an a n a l y t i c s u b s e t of

X,

and

U

1 + Codim Xx ,< Codim S(X)x

x

f o r every (ix)

If

in

S(X).

pX

Codim XX = p } .

{xEX;

=

a l l points.

-X =

Let

Then

U

Each p o i n t o f

p=l i n f i n i t e codimension ( a s a p o i n t o f i s an a n a l y t i c s u b s e t .

-X

X)

U

is a finitely

p,
X\upX '.

defined a n a l y t i c subset o f

if

upX

i s an a n a l y t i c s u b s e t o f a complex Banach manifold

X

we l e t

k

at has

"X

but it i s n o t known

The decomposition

X = opXymX p=l

i s known a s t h e canonical decomposition o f t h e a n a l y t i c s u b s e t X .

(x)

Let on

fEH(U;V), and

E

F

U

and then If

into

F

U.

If

f o r every

dim Coker ( d f ( x ) ) <-)

(i.e.

(xi)

E

V

f-'(K) f(X)

X

complex Banach manifolds modelled

X

r e s p e c t i v e l y , and l e t

analytic subset of from

and

df(x) x and

be a f i n i t e l y defined

i s a Fredholm o p e r a t o r

in

flX

U

(i.e.

dim k e r ( d f ( x ) ) <

i s a proper mapping

i s compact f o r every compact s u b s e t

K

i s a f i n i t e l y defined a n a l y t i c s u b s e t o f

i s a f i n i t e dimensional a n a l y t i c space,

E

of

X)

V.

is a locally

Appendix I

388 convex s p a c e ,

i s an open s u b s e t o f

Q

p r o p e r holomorphic mapping o f

X

into

If

and

U

and

R

then

are complex Banach manifolds and

V

is a

$I

@(X)

is a

a.

f i n i t e dimensional a n a l y t i c s u b s e t o f (xii)

E

H(U,V)

f E

i s a p r o p e r Fredholm mapping, t h e n t h e s e t o f c r i t i c a l v a l u e s of

f

is a f i n i t e l y defined a n a l y t i c subset o f

V.

The r e s u l t s o f (x) and ( x i ) are d i r e c t image theorems and g e n e r a l i s e a well known f i n i t e dimensional r e s u l t o f R . Remmert [ 6 0 3 ] .

(x)

i s due

t o J . P . R a m i s and G . Ruget [600] ( s e e a l s o J . P . R a m i s [ 5 9 8 ] and G . Ruget [613]).

( x i ) i s due t o D . Barlet and P. Mazet [475] ( s e e a l s o P . Mazet

[477,479])

and g e n e r a l i s e s a H i l b e r t space result of G . Ruget [613] and

a Frgchet s p a c e r e s u l t o f B. Saint-Loup [623].

( x i i ) i s due t o V. Aurich

[38] and u s e s a d i r e c t image theorem i n i t s p r o o f . Aurich [38] i s v e r y r e c e n t and v e r y i n t e r e s t i n g .

The a r t i c l e o f V . I t contains an i n f i n i t e

dimensional v e r s i o n o f t h e Remmert Graph Theorem, a l o c a l d e s c r i p t i o n o f holomorphic Fredholm mappings and shows how i n f i n i t e dimensional holomorphy may b e used t o u n i f y and extend r e s u l t s a r i s i n g i n o t h e r c o n t e x t s ( s e e a l s o D . Abts [ l ] , H . Arker [ 1 4 ] ) .

F u r t h e r r e f e r e n c e s t o i n f i n i t e dimensional a n a l y t i c geometry and a n a l y t i c sets a r e H. C a r t a n [116,117],

I . F . Donin

[209], A. Douady [210],

Zaidenberg [719] and J . P . R a m i s [594,595,596,597,599].

M.G.

J.P.

Ramis

[599] i s a s u r v e y a r t i c l e and c o n t a i n s a l i s t of open problems and conjectures. HOMOGENEOUS SPACES Domains

and

U

V

i n l o c a l l y convex s p a c e s

p h i c a l l y e q u i v a l e n t , we w ri t e mapping f-'

E

f

from

H(V;U))

.

U

onto

V

U

E

and

F

are holomor-

V , i f t h e r e e x i s t s a biholomorphic

(i.e.

f EH(U;V),

f

i s b i j e c t i v e and

The problem o f c l a s s i f y i n g h o l o m o r p h i c a l l y e q u i v a l e n t

domains i n l o c a l l y convex s p a c e s h a s l e d , mainly due t o t h e e f f o r t s o f L . A . Harris, W .

elegant r e s u l t s .

Kaup and J . P . Vigue', i n r e c e n t y e a r s t o many deep and T h i s branch o f i n f i n i t e dimensional holomorphy u s e s t h e

methods o f d i f f e r e n t i a l geometry,

C*

a l g e b r a s and Lie groups.

Before

d e s c r i b i n g b r i e f l y t h e i n f i n i t e dimensional t h e o r y , w e s k e t c h t h e r e l e v a n t p o r t i o n s o f t h e f i n i t e dimensional t h e o r y and refer t o S. VLgi [689],

389

Further developments

M. Koecher [392], B.A. Fuks [250,chapter 51 and S. Kobayashi [389] for further details. The Riemann mapping theorem states that any two proper simply connected domains in C

are equivalent. By proving that the domains

are not equivalent, H. Poincar6 [585] showed that the Riemann mapping theorem does not extend to En

and opened the way for a classification

theory in several variables. A short elementary proof of Poincare's result is given in H. Alexander [6]. The problem of holomorphically classifying all simply connected domains in Cn proved unwieldy and so attention focused on more "manageable'? classes of domains which still contained the more interesting examples - the homogeneous, symmetric and Siegel\domains. Let

A (U)

denote the set of all biholomorphic mappings from U

itself. The elements of

4 (U)

into

are called (holomorphic) automorphisms of

U. A domain U in a locally convex space E is called homogeneous if for each a,bE U there exists a $ in A (U) such that $(a) = b, i.e. 4 (U) acts transitively on U. A domain U is said to be symmetric if for each a in U there exists a $ in 4 (U) such that $ 2 = @ . $ = Id and a is the unique fixed point o f @. (U) In 1935, H. Cartan [114] prove that is a Lie group when U is a bounded domain in an. Using this result and structure theorems for Lie groups, E. Cartan [112] classified all irreducible bounded symmetric domains in Cn (a domain is irreducible if it is not equivalent to a non-trivial product of domains, any symmetric domain is equivalent to a finite product of irreducible symmetric domains). He showed that there are four classical (i.e. corresponding to "classical" Lie groups of matrices)classes of domains, now called Cartan domains of type I,II,III and IV, each of which can be represented by matrix inequalities (see below) and also exceptional domains in C16 and C2'. An

Appendix I

390

important r o l e , i n E . Cartan's and o t h e r r e s e a r c h e r s ' work i n t h i s a r e a , i s played by pseudo-metrics on

4 (U)

which a r e i n v a r i a n t under

U

d(U)

e q u i v a l e n t l y , f o r which each element of

or,

i s an isometry (see below

for details). A completely new approach was i n i t i a t e d by M. Koecher [390,391,392],

who discovered t h e r e l a t i o n s h i p between Cartan domains and Jordan algebras. This approach was adopted by L.A. H a r r i s [309] who introduced t h e concept of

J*

Let

algebra.

H

and

K

Banach space of a l l bounded l i n e a r operators from a l g e b r a i s a closed subspace &L A

E

&(A*

of

A(H;K)

i s t h e a d j o i n t of

d(K;H)

E

L(H;K)

be complex H i l b e r t spaces and l e t

A).

H

to

such t h a t By t a k i n g

K.

A

AA*A

E

H

and

denote t h e J*

& whenever K

finite

dimensional, one o b t a i n s t h e c l a s s i c a l Cartan domains a s t h e open u n i t b a l l b a l l s of t h e following

a l g e b r a s ( t h e Cartan F a c t o r s ) ,

J*

Type 1

CL =

d

Type I1

=

{A

€ X ( H ; H ) ,A*

Type 1 1 1

GL &

=

{A

E

Type I V

& i s a s e l f - a d j o i n t subalgebra of J ( H ; H ) AL

in Every

,&*

(H;K)

&H;H),

A* = -A}

.&*

such t h a t

i s a s c a l a r m u l t i p l e of t h e i d e n t i t y f o r each

a.

algebra i s a

J*

A

algebra and r e c e n t l y , L . A . H a r r i s [315]

has shown t h a t an a l g e b r a i c theory f o r for

= A}

J*

algebras, s i m i l a r t o t h a t known

a l g e b r a s , e x i s t s and includes a s p e c t r a l decomposition theorem f o r

s e l f - a d j o i n t o p e r a t o r s and a f u n c t i o n a l c a l c u l u s . We now t u r n t o t h e i n f i n i t e dimensional theory.

The f i r s t r e s u l t

obtained i n t h i s a r e a ( S . J . Greenfield and N.Wallach [ 2 8 1 ] ) i s s i m i l a r i n s p i r i t t o t h e f i r s t r e s u l t o f H. Poincare/ i n t h e f i n i t e dimensional theory. I t s t a t e s t h a t t h e open u n i t b a l l

B

of an i n f i n i t e dimensional H i l b e r t

space i s n o t holomorphically equivalent t o

BxB.

I n a f u r t h e r paper [282]

Dn i s t h e open u n i t b a l l of t h e i n f i n i t e dimensional Cartan I Factor $(Cn,a2) ( t h e case n = l was t h e same authors c h a r a c t e r i z e A ( D n )

where

a l s o found, independently, by A. Renaud [604]) and they show t h a t

,D

a r e holomorphically equivalent i f and only i f

n=m.

Dn

and

391

Further developments Harris showed t h a t t h e open u n i t b a l l o f a

I n [309], L . A .

J*

algebra

i s a homogeneous symmetric domain and any biholomorphic mapping between t h e

CL,

open u n i t b a l l s LoTg

where

TB

and

and

&ao,

B

L

a0

J* a l g e b r a s

of the

i s a s u r j e c t i v e l i n e a r isometry o f Go i n t o i t s e l f ,

i s a Mobius t r a n s f o r m a t i o n o f T* (A)

=

& and $ h a s t h e form

CL o n t o 63

i.e.

1 1 2 2 (I-BB*) (A+B)( I + B * A ) -(I-B*B) ~

.

I n t h e same p a p e r , h e showed t h a t no i n f i n i t e dimensional C a r t a n domain o f type I - I V i s holomorphically equivalent t o a n o n - t r i v i a l product o f b a l l s . I n a more r e c e n t p a p e r [314], L . A . Harris shows t h a t t h e i n f i n i t e dimens i o n a l analogues o f t h e classical C a r t a n domains of d i f f e r e n t t y p e s a r e n o t holomorphically e q uivalent. Schwarz's lemma.

A l l o f t h e above a u t h o r s make e x t e n s i v e u s e o f

I n [377], W . Kaup and H . Upmeier show t h a t Banach s p a c e s

w i t h b i h o l o m o r p h i c a l l y e q u i v a l e n t u n i t b a l l s a r e i s o m e t r i c a l l y isomorphic ( s e e a l s o L . A . Harris [ 3 1 3 , p . 3 8 8 ] ) . C . E a r l e and R. Hamilton [228]

i n t r o d u c e d t h e concept o f i n v a r i a n t

m e t r i c f o r holomorphic f u n c t i o n s i n i n f i n i t e dimensions i n o r d e r t o prove a f i x e d p o i n t theorem ( s e e e x e r c i s e 4 . 5 4 ) and i n r e c e n t y e a r s , i n v a r i a n t

metrics have been s t u d i e d and a p p l i e d i n i n f i n i t e dimensional holomorphy by L . A . Harris [313,314], J . P . Viguk [696,697,698,700], W. Kaup [375], M.

H e r d [327,328], S . J . G r e e n f i e l d and N.Wallach [281] and T . Hayden and T. S u f f r i d g e [319].

A d e t a i l e d and v e r y r e a d a b l e account o f Schwarz-Pick

systems o f pseudometrics can b e found i n L . A . Harris [313] ( t h i s p a p e r a l s o c o n t a i n s a set o f e x e r c i s e s and a l i s t o f open problems) and fundamental p r o p e r t i e s o f t h e Caratheodory metric are g i v e n i n J . P . Vigug [696, Appendix].

The c o r r e s p o n d i n g t h e o r y f o r

a l g e b r a s i s developed by L.A.

J*

Harris i n [314].

The f u n c t i o n PD(Z1,z2)

where

z1,z2

=

tanh-'

z1- 2 2

lie in the unit disc /

metric, t h e P o i n c a r e metric, on

D

=

D

log

z

1z1-z21+ 11-z 1 2 I

o f t h e complex p l a n e d e f i n e s a

with t h e u s e fu l p ro p e rt y t h a t

Appendix 1

392

Consequently, any holomorphic automorphism of D is a p D isometry. p D is not a Euclidean metric but is equivalent to the Euclidean metric on D inherited from C.

A system ( L . A . Harris [313,p.356]), which assigns a pseudo-metric t o each domain in every normed linear space is called a Schwarz-Pick system if the following conditions hold: (i) (ii)

the pseudo-metric assigned to D if

and

p1

and V

p2

are the pseudo-metrics assigned resp. to U

then p2(f(z),f(y))

f e H(U;V)

is the Poincare metric,

and x,y

E

< pl(x,y) for all

U.

The Caratheodory pseudo-metrics, Pc,

defined on U

by

form a Schwarz-Pick system and are the smallest of all pseudo-metrics

p

which satisfy (the Schwarz-Pick inequality) for all

fEH(U;D)

and all x,y

in U

On the other hand, the Kobayashi pseudometric PK, domain U by

pK(x,y)

where p;(x,y)

=

=

n inf{Ci=l~i( X ~ - ~ , X ~x)i;e u

inf {PD(z,w);z fEH(D;U)

= f(x),

all

defined on a

i, xo=x, xn=y}

w=f(y)},

also form a Schwarz-Pick system and are the largest pseudometrics which satisfy

393

Further developments p ( f ( z ) , f ( w ) ) 8 pD(z,w) If

Z,WE

D

and

i s a pseudo-metric on a bounded domain

p

f E H(D;U) U

i n t h e normed l i n e a r

a r i s i n g from a Schwarz-Pick system, then t h e r e e x i s t p o s i t i v e

E

space

for a l l

numbers

A

and

tanh-'

A

and consequently norm of

such t h a t

B

p

i s t o p o l o g i c a l l y equivalent t o t h e metric given by t h e

E.

The above pseudo-metrics, a s well a s t h e i n f i n i t e s i m a l F i n s l e r pseudometrics and t h e i r i n t e g r a t e d forms s t u d i e d i n [228] and [313], a r e an Their

important t o o l i n t h e works of L . A . Harris W . Kaup and J . P . Vigu6.

r o l e , however, i s not apparent from our b r i e f o u t l i n e h e r e s i n c e i n v a r i a n t m e t r i c s u s u a l l y appear i n t h e proof r a t h e r than i n t h e statement of r e s u l t s e . g . J . P . Vigu6 [696] shows t h a t U

i s a complete metric space when

(U,pc)

i s a bounded homogeneous domain i n a Banach space and then uses t h i s

r e s u l t t o show t h a t bounded homogeneous domains a r e domains o f holomorphy f o r bounded holomorphic functions. In [317] L . A . H a r r i s and W . Kaup prove t h a t t h e group of a l l l i n e a r i s o m e t r i e s of a homogeneous u n i t b a l l i n a Banach space i s a (Banach) Lie group (see P . de l a Harpe [303]), and using t h i s r e s u l t , J . P . Vigu'e [696] showed t h a t A(U) Banach space.

i s a Lie group f o r

with open u n i t b a l l J . P . Vigu;

U

a bounded symmetric domain i n a

J . P . Viguk [696,698] gives an example of a Banach space

Eo

such t h a t

A (Eo)

i s n o t a Lie group.

E

In [696],

proved t h a t bounded symmetric domains i n Banach spaces a r e

homogeneous - t h e converse i s n o t t r u e even i n f i n i t e dimensions - and holomorphically equivalent t o balanced domains. [696] include t h e endowing of

A

(U),

The methods of J . P

.

Vigu;

bounded, with t h e s t r u c t u r e of a

U

uniform and topological space ( c a l l e d t h e l o c a l uniform topology, a f i l t e r

3

-+

Vx

of

f E ~ ( U ) i f and only i f f o r each x

such t h a t

31vcLlvx

x

in

U

uniformly on

3

a neighbourhood Vx)

and showing t h a t

X

A(U)

i s a complete topological group (see a l s o W . Kau? [375] and J . M .

Isidro [554]).

He a l s o s t u d i e s t h e s e t

a t i o n s of a bounded domain

U

g(U)

o f i n f i n i t e s i m a l transform-

( i . e . t h e s e t of holomorphic v e c t o r f i e l d s

Appendix I

394

a r i s i n g f r o m the group homomorphisms o f +$(t)x

E.

a n a l y t i c ) and shows t h a t

U

a Lie a l g e b r a and a Banach s p a c e .

R

into

g(U)

fc(U)

with

(t,x)

E

RxU

has the natural structure of

The Lie a l g e b r a r e s u l t i s a l s o due

i n d e p e n d e n t l y t o H . Upmeier [688].

E

t r i p l e or a h e r m i t i a n J o r d a n t r i p l e ,

J*

A

and a mapping (a)

(5 , x , y ) + Z( ~ , x , y ) i s symmetric and CC

t h e mapping

for all

5, n, 5,

for all

5

x

x

and

-t

A morphism o f

and

antilinear

C

E

E

XEE

and a l l

e x p ( i t Z ( 5, 5,x))

t E

t h e mapping

R

i s an isometry o f

t r i p l e s i s a mapping

J*

f:(E,Z)

for a l l

f(Z(x,y,z)) = Zl(f(x),f(y),f(z)) J*

y

5,

in

[c)

i s a Banach s p a c e

such t h a t

Z;E3 + E

linear i n the variables

(b)

(E,Z),

E.

(E1,Z1)

+

x,y,z

in

such t h a t E.

t r i p l e s were i n t r o d u c e d by W . Kaup [375] who proved t h e deep r e s u l t

t h a t t h e c a t e g o r y o f s i m p l y c o n n e c t e d , symmetric, complex Banach m a n i f o l d s with base point i s equivalent t o t h e category of a l g e b r a may be endowed w i t h t h e s t r u c t u r e o f a of a

J* If

(E,Z)

is a E

I n [697], J . P . Vigu; J*

Every

J*

The d e s c r i p t i o n

triple

J*

t r i p l e and

5

E

E

let

denote t h e closed

E5

g e n e r a t e d by

shows t h a t t h e s i m p l y connected domain a s s o c i a t e d w i t h

(E,Z)

i s h o l o m o r p h i c a l l y e q u i v a l e n t t o a bounded domain

i f and o n l y i f t h e r e e x i s t s a p o s i t i v e r e a l number

5EE

triples.

triple.

t r i p l e g i v e n h e r e i s t a k e n from J . P . Vigu/e [679].

r e a l s u b s p a c e of

the

J*

J*

the restriction t o

eigenvalue i n t h e i n t e r v a l

E5

o f t h e mapping

-k

(-a,

11 5 11

)

.

x

+

k

such t h a t f o r a l l

Z(5,S,x) h a s a r e a l

395

Further developments R e c e n t l y , J . P . Vigug [700] h a s o b t a i n e d a s u f f i c i e n t c o n d i t i o n f o r t h e c o n v e x i t y o f c e r t a i n domains. homogeneous domain positive integer each of d e g r e e

U

No

and a f a m i l y

E

i s convex i f t h e r e e x i s t s a

(Pi)iEI

o f homogeneous polynomials,

for all

i EI}.

such t h a t

No,

( X E E ; Ipi(x)I

=

H e h a s shown t h a t a b a l a n c e d bounded

i n a Banach s p a c e

U

I 1

F u r t h e r r e s u l t s on homogeneous and symmetric domains i n normed l i n e a r s p a c e s may b e found i n W. Kaup [373,374 3761, L . L .

Stacho [670], A. Douady

[210], R . Braun, W . Kaup and H . Upmeier [ 9 9 , 1 0 0 ] , W. Kaup and H. Upmeier [378], S.B. Chae [121], J . P . Vigug [699

and L . A . Harris

3121.

A p p l i c a t i o n s o f i n f i n i t e dimensional bounded symmetric domains t o t h e o r e t i c a l p h y s i c s are g i v e n i n I.A. S e r e s e v s k i i [655]. The o n l y r e s u l t s w e know on t h e holomorphic c l a s s i f i c a t i o n o f domains i n non-normed l o c a l l y convex s p a c e s a r e v e r y r e c e n t and due t o R. Meise and D . Vogt [485,486].

These r e s u l t s a r o s e from t h e i r i n v e s t i g a t i o n o f holo-

morphic f u n c t i o n s on p o l y d i s c s i n n u c l e a r power s e r i e s s p a c e s ( s e e 5 6 . 4 ) . I n [ 4 8 5 ] , t h e y prove t h e f o l l o w i n g r e s u l t .

series s p a c e .

a = (an)n

If

and

be a n u c l e a r power Let A,(a) b = ( b n ) n are sequences o f p o s i t i v e ~

r e a l numbers such t h a t

open p o l y d i s c s i n Al(a)b t h e n Da and D,, 1 1 E A1 ( a ) ;i and b A1 ( a ) &. I n a l l y equivalent i f

are

show t h a t t h e p o l y d i