Complex Analysis in Locally Convex Spaces

COMPLEX ANALYSIS IN LOCALLY CONVEX SPACES

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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

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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, ( 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')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


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;
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
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


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 ,
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


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.
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
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) ,
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 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) = ~

\


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

-
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


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 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 )
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 )


0 si i=l


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

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


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.


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');

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


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


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
(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


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 .


(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

,


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*


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 . 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 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

,
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


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
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
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
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


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


1

such t h a t

- log(k+l) s n(k) 9

6

q log(k+l)

for all

k.


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)
O a r e a r b i t r a r y 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 W of x i n X s u c h t h a t

If(x)-f(y)

Baire's X =

0 n=l

integer

[ 5

E

f o r aZZ y i n Kn W and aZZ f i n FI. I f X is a compZete m e t r i c s p a c e and

Theorem

Fn where e a c h F n i s c Z o s e d t h e n t h e r e e x i s t s an n o such t h a t F

has non-empty i n t e r i o r .

"0

Let

(Xa)aEr

topological

be a collection of topological spaces.

s p a c e X i s t h e topoZogica2 i n d u c t i v e l i m i t

the inductive l i m i t i n the category of topological and continuous mappings) e x i s t s f o r each a i n

r

a n d we w r i t e X =

a mapping i

Xa

:

a

Xa

(or

spaces

if there

X such t h a t X

h a s t h e f i n e s t topology r e n d e r i n g each ia c o n t i n u o u s 2.

The

I

A v e c t o r s p a c e E endowed w i t h a t o p o l o g y f o r w h i c h

v e c t o r a d d i t i o n and s c a l a r m u l t i p l i c a t i o n a r e c o n t i n u o u s is c a l l e d a topoZogicaZ v e c t o r s p a c e .

If the topological

vector

space E has a neighbourhood base a t t h e o r i g i n c o n s i s t i n g of closed,

convex, balanced absorbing sets then E i s c a l l e d

a ZocaZZy e o n v e x s p a c e . ( A C E i s convex i f x , y E A and o 5 A 5 1 =.j Ax + ( l - A ) y E A , A C E i s baZanced i f x E A a n d o 5 I A l 5 1 a Ax E A , A C E i s a b s o r b i n g i f f o r e a c h x i n E t h e r e e x i s t s 6 > 0 s u c h t h a t Ax E A f o r a l l I A l 5 6 ) . I f E i s a l o c a l l y convex space then t h e topology of E

i s g e n e r a t e d by a f a m i l y o f semi-norms

(Pa)aEr.

E is

400

Appendix II

Hausdorff i f and o n l y i f f o r each non-zero x i n E t h e r e

r

a in

e x i s t s an

such thatpa(x)

c o n v e x s p a c e i s norniabZe

Irl

can choose

=

#

0 .

A Hausdorff l o c a l l y

( r e s p . m e t r i z a b Z e ) i f a n d o n l y i f we

1 (resp.

~ ~ w ~A )c o .m p l e t e

rI

normable

( r e s p . 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 i s c a l l e d a Banach ( r e s p . Fre/chet) s p a c e . Let

(EaIaer

b e a c o l e c t i c n o f l o c a l l y convex s p a c e s .

The l o c a l l y c o n v e x s p a c e E i s t h e ZocalZy convex i n d u c t i v e

L i m i t (or the inductive l i m i t i n the category of locally convex spaces and continuous l i n e a r nappings) of(E ) ~1

t h e r e e x i s t s f o r each a i n

r

a l i n e a r m a p p i n g i,

a ~ irf

:Ea

+

E

such t h a t E h a 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 f o r which each i a i s continuous.

The t o p o l o g i c a l and l o c a l l y c o n v e x

inductive limits of a

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

may n o t c o i n c i d e . normed

A l o c a l l y convex

inductive l i m i t of

( r e s p . Banach) s p a c e s i s c a l l e d a bornoZogicaZ

( r e s p . ~ Z t r a b o r n o Z c g i c a Z )s p a c e .

A subset B of a locally

convex s p a c e E which i s a b s o r b e d by e v e r y neighbourhood o f z e r o i s c a l l e d bounded ( i . e .

i f V i s a neighbourhood of zero

then t h e r e e x i s t s a p o s i t i v e 6 such t h a t A B C V f o r a l l

1x1 2

6).

Proposition

The folZowing c o n d i t i o n s on t h e 1ocalZy eonvGx

space E are eqziivazent, ( a ) E is b o r n o Z o g i c a Z , ( b ) t h e c o n v e x baZanced s u b s e t s of E w h i c h a b s o r b aZZ

bounded s e t a r e n e i g h b o u r h o o d s of z e r o , ( c ) if F is a l o c a l l y c o n v e x s p a c e and T is a l i n e a r niapping from E Cnto F w h i c h maps bounded s e t s onto bounded

s e t s then T i s continuous. I f e v e r y c l o s e d convex balanced absorbing 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 i s a n e i g h b o u r h o o d o f z e r o t h e n we say E i s barrelzed.

T h e s u p r e m u m a n d t h e sum o f a n a r b i t r a r y

f a m i l y o f c o n t i n u o u s semi-norms on a b a r r e l l e d l o c a l l y convex

Definitions and results

40 1

space are continuous whenever t h e y are f i n i t e .

The

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 is barrelled.

By B a i r e ' s T h e o r e m F r G c h e t s p a c e s a r e

barrelled. A l o c a l l y convex s p a c e i s c a l l e d infrabariqelled i f e v e r y

c l o s e d convex b a l a n c e d s e t which a b s o r b s a l l bounded

sets is

a neighbourhood of zero. H a h n - B a n a c h TCeorem ( a ) l f $ i s c continucus linear f u n c t i o n a l f i . e . s c a l a r v a l u e d ) on t h e s u b s p a c e F of t h e l o c a l l y conz'ex s p a c e E t h e n t h e r e e x i s t s a c o n t i n u o u s l i n e a r

form

$

PI,

on E s u c h t h a t $1

8

;:

= $.

( b ) I f A and B a r e d i s j o i n t c o n v e x s u b s e t s of

the locally

c o n v e x s p a c e E and A h a s non-empty i n t e r i o r t h e n t h e r e e x i s t s a c o n t i n u o u s Z i n e a r f u n c t i o n $ on E s u c k t h a t

The Hahn-Banach t h e o r e m i m p l i e s t h a t t h e c o n t i n u o u s dual of E,

E',

separates the points of E .

The strong topology

on E 1 , 8 , 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 s u b s e t s o f E .

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

(resp. bornological) space is quasicomplete (resp. complete). The f o r m u l a

(Jx)($)

= $(x),

c o n t i n u o u s l i n e a r mapping J

=(Ei)'

xeE a n d

defines a

(El)' B 8' ( a s s e t s ) we s a y E i s s e m i - r e f l e x i v e .

J(E) i n f r a b a r r e l l e d i f and o n l y i f E

spaces).

$ € E l ,

from E i n t o

2

J(E)

If

E is

( a s l o c a l l y convex

An i n f r a b a r r e l l e d s e m i - r e f l e x i v e s p a c e i s

called reflexive.

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

r e f l e x i v e and t h e s t r o n g d u a l o f a s e m i - r e f l e x i v e space is barrelled.

I f F i s a c o l l e c t i o n o f l i n e a r f u n c t i o n a l s on t h e v e c t o r space E

we l e t o ( E , F )

denote t h e l o c a l l y convex topology

g e n e r a t e d by t h e semi-norms

(P )

$ JIEF'

~

~

( = x

I$(x) I

)

all

X E

E.

Appendix I1

402

I f E i s a l o c a l l y convex s p a c e t h e n t h e o ( E , E ' )

s u b s e t s o f E are bounded.

A Banach s p a c e E i s r e f l e x i v e i f

and o n l y i f t h e c l o s e d u n i t b a l l o f E i s a ( E , E ' ) The c l o s e d u n i t b a l l o f t h e d u a l E ' E i s always u(E',E)

compact.

o f a normed l i n e a r s p a c e

compact.

Mackey-Arens Theorem

dual E ' .

bounded

L e t E be a Zocally convex space w i t h

A l o c a l l y c o n v e x topoZ-ogy

T

on E i s c o m p a t i b l e

w i t h t h e o r i g i n a Z t o p o l o g y ( i . e . E ' = ( E , T ) ' ) i f and o n l y i f

2 o(E,E')

and T i s r,'eaker t h a n t k e t o p o l o g y of u n i f o r m c o n v e r g e n c e o n t h e c o n v e x b a l a n c e d o ( E ' , E ) compact s u b s e t s of E l . T

A l o c a l l y c o n v e x s p a c e endowed w i t h t h e f i n e s t

locally

c o n v e x t o p o l o g y c o m p a t b l e w t h i t s own d u a l i t y i s c a l l e d a Mackey s p a c e . I f e v e r y c l o s e d bounded s c b s e t o f a l o c a l l y convex s p a c e E i s c o m p a c t we s a y E i s s e m i - M o n t e l .

An i n f r a b a r r e l l e d

s e m i - M o n t e 1 s p a c e i s c a l l e d a MonteZ s p a c e .

The s t r o n g

d u a l o f a Montel s p a c e i s a Montel s p a c e . I f p i c s e m i - n o r m o n t h e v e c t o r s p a c e E we l e t E

P

=

(EG-l

(0)

,p)

(i.e. E

P

i s t h e Banach s p a c e o b t a i n e d by

f a c t o r i n g o u t t h e k e r n e l o f p and c o m p l e t i n g t h e normed E

linear space ( / - i

(ol, p)).

A l o c a l l y convex space E i s a

Schwartz s p a c e i f € o r e a c h c o n t i n u o u s semi-norm p on 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 q on E , q ) p ,

such t h a t

-

t h e c a n o n i c a l mapping ( i . e . t h e mapping i n d u c e d b y t h e i d e n t i t y o n E) f r o m E

E i s compact. A l i n e a r mapping 9 P T b e t w e e n t h e Banach s p a c e s E a n d F i s nuclear i f t h e r e e x i s t m a s e q u e n c e (A ) i n 11,a b o u n d e d s e q u e n c e ( x ~ ) i~n =F ~a n d n n=l W

W

a bounded s e q u e n c e ($n)n=l

f o r every x i n E.

in E'

s u c h t h a t Tx =

1"

n=l

An

$n(x)~n

Definitions and results

403

A l o c a l l y convex s p a c e E i s n u c l e a r i f and o n l y i f f o r

e a c h c o n t i n u o u s s e m i - n o r m p on E t h e r e e x i s t s a c o n t i n u o u s

-

s e m i - n o r m q on E , E

9

E

P

q

p,

such t h a t t h e c a n o n i c a l mapping

is nuclear.

The s t r o n g d u a l o f a c o m p l e t e S c h w a r t z s p a c e i s ultrabornological.

I f E i s a Frgchet space then

(E',T

0

) is

a Schwartz space, i n p a r t i c u l a r t h e s t r o n g d u a l o f a FrgchetMontel s p a c e i s a Schwartz s p a c e .

The f o l l o w i n g c h a r t s

i l l u s t r a t e v a r i o u s r e l a t i o n s h i p s between t h e d i f f e r e n t spaces defined above. E a l o c a l l y convex s p a c e

I

bornological

b a r r e l l e d --f

infrabarrelled

Reflexive

Semi -

Semi - r e f l e x i v e

Mackey

E 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 .

I1 (a)

u l t r ab o r n o 1o g i c a 1 t-7) b o r n o 1o g i c a 1< d b a r r e 11e d

infrabarrelled (b) I11

nuclear

__j

S c h w a r t z >--

semi-bIontel

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

n u c l e a r>- .

Schwartz

j

Montel

A l o c a l l y convex s p a c e E i s a D F space i f

Appendix I1

404

(i)

E a d m i t s a f u n d a m e n t a l s e q u e n c e o f bounded s e t s

(i.e.

(Bn)n=1

each B

n

i s bounded and e a c h bounded

s u b s e t o f E i s c o n t a i n e d i n some B ) n (ii)

I f (Un)n i s a s e q u e n c e o f c l o s e d convex b a l a n c e d s u b s e t s o f E and

a b s o r b s a l l bounded s e t s t h e n

Un

Un

is

a neighbourhood o f z e r o .

The s t r o n g d u a l o f a F r g c h e t s p a c e i s a DF s p a c e a n d t h e s t r o n g d u a l o f a DF s p a c e i s a F r z c h e t s p a c e .

The

c o l l e c t i o n o f b o r n o l o g i c a l DF spaces c o i n c i d e s with t h e c o l l e c t i o n o f c o u n t a b l e l o c a l l y convex i n d u c t i v e l i m i t s o f A q u a s i - c o m p l e t e DF s p a c e i s c o m p l e t e .

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

A p o i n t w i s e bounded f a m i l y o f s e p a r a t e l y c o n t i n u o u s

b i l i n e a r f o r m s on a p r o d u c t o f DF s p a c e s i s e q u i c o n t i n u o u s . to

A sequence o f vectors

( en ) n = l i n a l o c a l l y c o n v e x s p a c e E i s

c a l l e d a basis i f f o r each x i n E t h e r e e x i s t s a unique sequence of s c a l a r s x x = lim m+m

1"

n=l

n

xnen =

I f t h e m a p p i n g s Pm

such t h a t

lw x n e n

n =1

: -E

E,

1"

pm(C" x n e n 1 = xnen n=l n =1

a r e c o n t i n u o u s f o r a l l m t h e b a s i s i s c a l l e d a Schauder b a s i s and i f t h e f a m i l y (Pm)z=l o f l i n e a r mappings i s e q u i c o n t i n u o u s t h e b a s i s i s c a l l e d an equi-Sckauder

(or equicontinuous) b a s i s .

If l i m

J C N

1

ncJ

x e = x n n

J finite

f o r e v e r y x i n E t h e b a s i s (en):=1

i s c a l l e d uneonditiona2.

An e q u i - S c h a u d e r b a s i s i n a n u c l e a r s p a c e i s u n c o n d i t i o n a l . A l o c a l l y convex s p a c e E h a s t h e approximation property

i f f o r each compact s u b s e t K o f E ,

each neighbourhood V o f

z e r o i n E and e a c h p o s i t i v e 6 t h e r e e x i s t s a c o n t i n u o u s

405

Definitions and results l i n e a r o p e r a t o r T from E i n t o E s u c h t h a t dim T ( E ) c + = i f and o n l y i f t h e i d e n t i t y

a n d x - T x E ~ aV l l x i n K ( i . e .

m a p p i n g on E c a n b e u n i f o r m l y a p p r o x i m a t e d on c o m p a c t s e t s by f i n i t e rank o p e r a t o r s ) .

E h a s t h e kounded approximation

p r o p e r t y i f t h e i d e n t i t y m a p p i n g on E c a n b e a p p r o x i m a t e d u n i f o r m l y on c o m p a c t s e t s b y a s e q u e n c e o f f i n i t e r a n k operators.

A l o c a l l y convex space 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 .

Schauder b a s i s has

R

Nuclear space have t h e

m, a p p r o x i n a t i o n p r o p e r t y . T h e B a n a c h s p a c e d ( R 2 , e ) , d i m ([,)= 2 with t h e s t r o n g topology does not have t h e approximation p r p y

Let

(En)n b e an i n c r e a s i n g and e x h a u s t i v e s e q u e n c e

o f s u b s p a c e s o f t h e v e c t o r spclce E ( i . e .

and E =

UE ) n n

and suppose e a c h E

t o p o l o g y -rn s u c h t h a t

T

I

n+l E

=

n

T

n n

.

EnCEn+l

all n

has a l o c a l l y convex The v e c t o r s p a c e E

endowed w i t h t h e 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 t o p o l o g y , ~ , o f t h e sequence (En)n i s c a l l e d t h e s t r i c t Znductive l i m i t of t h e sequence

m

For each n

T

'

En

= T

n

and each

b o u n d e d s u b s e t o f E i s c o n t a i n e d a n d b o u n d e d i n som e E

.

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

Open Mappi ng Theo r em

A c o n t i n u o u s l i n e a r mapping from E

o n t o F is o p e n if any o f t h e f o Z Z o w i n g conditions h o l d ; (i)

E and F a r e P r g c h e t s p a c e s ,

(ii)

E and F a r e t h e s t o n g d u a l s of F r g c h e t - S c h w a r t z

(iii)

E and F a r e c o u n t a b Z e

spaces,

ZocaZZy c o n v e i i n d u c t i v e l i m i t s

o f Frzchet spaces. Let A b e a s u b s e t o f a v e c t o r s p a c e V .

A point x

i s a n intern.aZ p o i n t o f A i f t h e r e e x i s t s a v e c t o r y # o i n E such t h a t {xo + Ay,-1 5 h

5

+ 11

C

A.

A point

x i s an

extreme p o i n t o f A i f it i s n o t an i n t e r n a l p o i n t o f A .

406

Appendix I1

Krein-Milman Theorem

A compact c c n v e x s u b s e t of a ZocalZg

c o n v e x s p a c e is equaZ t o t h t c Z o s e d c o n v e x huZZ of i t s extreme p o i n t s . L e t E and F b e v e c t o r s p a c e s o v e r C and l e t Ba(EJF) d e n o t e t h e s p a c e o f a l l b i l i n e a r forms on E x F .

Each

in E x F defines a linear functional,

element (x,y)

x @ y , on B ( E ; F ) by t h e f o r m u l a a x @ y ( b ) = b ( x J y ) where b

E

Ba(E,F).

The l i n e a r s u b s p a c e o f B a ( E , F ) * s p a n n e d b y { x @ y ; ( x , y )

E

Ex FI

i s c a l l e d t h e t e n s o r p r o d u c t o f E and F and i s w r i t t e n

E

OF.

I f E and F a r e l o c a l l y convex space t h e n t h e f i n 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 f o r w h i c h t h e c a n o n i c a l mapping o f E x F i n t o E &)F

is continuous (resp. separately

continuous) i s called the projective (resp. inductive) tensor product topology.

The v e c t o r s p a c e E Q F endowed w i t h

t h e p r o j e c t i v e ( r e s p . i n d u c t i v e t o p o l o g y ) i s d e n o t e d by F(resp. F and E

( E @ F , T ~ ) )a n d t h e c o m p l e t i o n s a r e w r i t t e n a s

BF

respectively.

T h e p r o j e c t i v e t o p o l o g y i s g e n e r a t e d by t h e semi-norms

w h e r e p a n d q r a n g e o v e r t h e c o n t i n u o u s s e m i - n o r m s on E a n d F respectively. We h a v e

(E

0,

F)

=

B(E,F)

= the space of a l l

c o n t i n u o u s b i l i n e a r f o r m s on E x F and

a

(E F) ' = 8 ( E J F ) = t h e s p a c e o f a l l s e p a r a t e l y c o n t i n u o u s b i l i n e a r forms on E x F . T h e t o p o Z o g y of b i e q u i c o n t i n u o u s c o n v e r g e n c e on E @ F i s

401

Definitions and results

g e n e r a t e d by t h e semi-norms

where U and V r a n g e o v e r t h e e q u i c o n t i n u o u s s u b s e t s o f E ' and

F' r e s p e c t i v e l y .

The s p a c e E @ F

endowed w i t h t h i s t o p o l o g y

Q

i s w r i t t e n a s E x F a n d i t s c o m p l e t i o n i s d e n o t e d by E

.A

Oc

F.

For any l o c a l l y convex s p a c e s E and F t h e f o l l o w i n g canoncial inclusions a r e continuous

A l o c a l l y convex s p a c e E h a s t h e approximation p r o p e r t y

i f and o n l y i f

E l

i s dense i n g ( E ; E ) ,

@ 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 convergence on compact s e t s ( n o t e t h a t E may b e i d e n t i f i e d w i t h t h e f i n i t e r a n k l i n e a r m a p p i n g s

E'@

from E i n t o i t s e l f ) . E

Ot;F =

A

E

a

fi

I f E and F a r e n u c l e a r s p a c e s t h e n

F i s a nuclear space.

General r e f e r e n c e s f o r l o c a l l y convex s p a c e s a r e J . Horvath 13481,

and R .

H.d.

T z a f r i r i [447],

L.

and A .

Schaefer [625],

A.

f o r nuclear spaces A.

Pietsch [570]

Grothendieck [288]

f o r Banach s p a c e s J . L i n d e n s t r a u s s a n d

Edwards [ 2 2 9 ] ,

Grothendieck [287]

and f o r n u c l e a r F r g c h e t s p a c e s E.Dubinsky

[212].

Let f b e a complex v a l u e d f u n c t i o n d e f i n e d on an

3.

open s u b s e t U o f every point a

We s a y f i s hoZornorphic o n U i f t o

(Cn.

o f U t h e r e corresponds a neighbourhood V of

a and a power s e r i e s

c

a.

1

E

N all i

a = ( al,.

.

c

al

. .

.

c1

n

(z,

- al)

al

.

,

.(zn-an)

, a n ) , which c o n v e r g e s t o f ( z ) a l l

Z E

V.

a

n

,

Appendix I1

408

L e t H(U) d e n o t e t h e s p a c e o f a l l h o l o m o r p h i c 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 open t o p o l o g y .

H(U) i s a

F r g c h e t n u c l e a r s p a c e and i n p a r t i c u l a r a Montel s p a c e .

This

l a t t e r r e s u l t , w h i c h s a y s t h a t a n y s e q u e n c e i n H(U) w h i c h i s u n i f o r m l y b o u n d e d on c o m p a c t s e t s c o n t a i n s a c o n v e r g e n t s u b s e q u e n c e , m o t i v a t e d t h e t e r m i n o l o g y Montel s p a c e i n l o c a l l y convex a n a l y s i s . Cauchy I n t e g r a l Formula n f ( z i ) i = l ; }zi - s i J 5 pi)

If m l J

. .

.,mn

Let f

Cu

E

H(U)

and s u p p o s e

where pi

o

~ Z iZ.

are non-negative i n t e g e r s then

m +m2 . . + m , 1 n

=

1 n (m) ml!

. . .

f(zl,.

m !

(Z1-E1)

f o r any s e t

> o all

i

(ml,..,

*

-

zn)

d z l . .dzn

(zn-En)

m +l n

n I f f EH(U) and { ( z i ) i = l ; l z i - E i / z

The Cauchy I n e q u a l i t i e s C U where p

.

m 1+ 1

pi]

then

i

mn ) o f n o n - n e g a t i v e i n t e g e r s .

L i o u v i l l e ' s Theorem

A bounded h o Z o m o r p h i c f u n c t i o n o n

Fn

is a c o n s t a n t . Maximum M o d u l u s T h e o r e m s u p If(7-1 z EU

mapping.

I

=

If(zo)

I

I f

f

E

H ( U ) , U c o n n e c t e d , and

for some z

in

u

then f i s a constant

Definitions and results

409

Hartogs'

Theorem on S e p a r a t e A n a l y t i c i t y I f U and V n a r e o p e n s u b s e t s of C and C m r e s p e c t i v e Z y and f :U x V

Then f

E

-

H ( U x V)

v

fx : and

u

fY :

if t h e functions

-

c, fx(Y) = f ( x , y ) c, fY(X)

= f(x,y)

a r e hoZornorphic for e v e r y x i n U and y i n V r e s p e c t i u e Z y . Let U b e an open s u b s e t o f C n .

The hoZornorphic huZZ

( o r t o b e m o r e p r e c i s e t h e H(U) h o l o m o r p h i c h u l l ) o f a s u b s e t A of U i s defined as

Iz

E.

u;

jf(z)

1

5

sup l f ( c ) SEA

1

all f e H ( u ) ~

A d o m a i n U i s s a i d t o b e hoZornorphicaZZy

convex i f t h e

holomorphic h u l l o f each compact s u b s e t o f U i s a g a i n a compact s u b s e t o f U . General r e f e r e n c e s f o r s e v e r a l complex v a r i a b l e s t h e o r y a r e L . Harmander [ 3 4 7 ] and R .

Gunning and H .

Rossi

[294].

C.

This Page Intentionally Left Blank

Appendix 111

NOTES ON SOME EXERCISES

CHAPTER ONE

1.63

This exercise is related to the result of S. Kakutani and V. Klee

[369] which says that the finite open topology on a vector space E is Direct proofs are to be found locally convex if and only if dim(E) 6

No.

in S. Dineen [186] and J . A . Barroso, M.C. Matos and L. Nachbin [51].

In

dealing with the finite open topology, one shouldbe wary of the following curious fact:

if E

is an infinite dimensional vector space, there exists

a subset U of E such that U n F is a neighbourhood of zero for every finite dimensional subspace F of E but U is not a finite open neighbourhood of zero. A class of topologies which lie between the finite open topology and locally convex topologies and which arise in the theory of plurisubharmonic functions and holomorphic functions on locally convex spaces are the pseudo-convex topologies. These are studied in P. Lelong [431,435,436] and C . O . Kiselman [382,383,388]. __ 1.68

See also exercise 2.60.

This method of differences was used by M. Frgchet [240] to define

polynomials on an abstract space. 1.69 -

A function which i s continuous when restricted to the complement of

a set of first category is called a B-continuous function. These functions arise in measure theory and are useful since the pointwise limit of Bcontinuous functions on a Baire space is B-continuous. F o r general results concerning B-continuous functions we refer to H. Hahn [302] and J . C . Oxtoby [560]. Applications of B-continuous functions to polynomial and holomorphic functions on Banach spaces can be found in S. Mazur and W. Orlicz [481,482] and M.A. Zorn [724]. 1.70 -

This result can b e found in P.J. Boland and S. Dineen [91]. 41 1

The

412

Appendix III

proof is not difficult and should help motivate proofs of exercises 1.73 and 1.74. See example 5.46 f o r a more general result. 1.71 This result is due to P . J . Boland and S. Dineen 1911. The proof uses the concepts of surjective limit (section 6.2) and very strongly convergent sequence (definition 2.50).

See also example 5.46 and corollary

6.26. 1.72 __

This result is proved in P.J. Boland [84]. A more general result

is proved in chapter 5. 1.73 This result is due to L.A. de Moraes [498]. The proof is technical and involves concepts similar to those of 1.70. Recently, de Moraes has shown that the conditions of the exercise are equivalent to the condition that E

admits a continuous norm.

1.76

This is a polynomial version of the Banach-Dieudonng theorem and is

~

due to J. Mujica [504]. An alternative proof can be found in R.A. Ryan [620]. 1.82 __

This result is an infinite dimensional version of Hartogs’ theorem

on separate analyticity. See the notes and remarks on exercise 2.76.

1.83

The proof of this result is given in S. Dineen [189,190,191].

It

uses very technical surjective limits (see chapter 6) and i s a particular case of a more general result. We feel that a direct proof should exist € o r the space

1.84

&(X).

The space

See also proposition 6.29. co(r),

r

uncountable,is a useful counterexample

space in infinite dimensional holomorphy (see, for instance, B. Josefson [358,360], Ph. Noverraz [552], J. Globevnik [275] and S. Dineen [190,193]). The theory of surjective limits partially explains the behaviour o f

co(r) and the geometry of the unit ball also plays a role. The first part o f this exercise is quite easy.

The second part is due to R.M. Aron [21], and

we refer to B. Josefson [360] for applications.

See also A . Pelczynski and

Z. Semadeni 15661. 1.86

This result is due to S. Banach [45], and generalises to symmetric

Notes on some exercises

413

n - l i n e a r forms t h e well-known l i n e a r r e s u l t t h a t a s e l f - a d j o i n t compact o p e r a t o r from a H i l b e r t s p a c e i n t o i t s e l f h a s a n e i g e n v a l u e ( c h a r a c t e r i s t i c v a l u e ) whose a b s o l u t e v a l u e i s e q u a l t o t h e norm o f t h e o p e r a t o r . a c t e r i z a t i o n o f polynomials on

L:(M),

c a n b e r e p r e s e n t e d by means of

L2

FI

A char-

a l o c a l l y compact s p a c e , which

k e r n e l s i s g i v e n i n T.A.W.

Dwyer [214].

T h i s r e s u l t i s due t o A . P e l c z y n s k i [564] and i s r e l a t e d t o t h e

1.87

r e s u l t s of

e x e r c i s e s 1 . 8 8 and 2.67.

T h i s r e s u l t s a y s t h a t a Banach s p a c e h a s t h e polynomial Dunford-

1.88

P e t t i s p r o p e r t y i f and o n l y i f it h a s t h e ( l i n e a r ) D u n f o r d - P e t t i s p r o p e r t y . I t i s due t o R . Ryan [619] and answers a q u e s t i o n posed by A. P e l c z p s k i [565].

F u r t h e r i n f o r m a t i o n on t h e D u n f o r d - P e t t i s . p r o p e r t y may b e found i n

A . Grothendieck [288] and J . D i e s t e l and 3 . Uhl [172].

Use theorem 27 and t h e Hahn-Banach theorem.

1.89

See a l s o A . Grothen-

d i e c k [287; c h a p t e r 2 , p r o p o s i t i o n 101. __ 1.90

T h i s r e s u l t may b e found i n C . P . Gupta [295].

1.91

This r e s u l t is n o t d i f f i c u l t t o prove (se e P . J .

Dineen [ 8 8 ] ) .

R . Ryan h a s a n u n p u b l i s h e d p r o o f u s i n g t e n s o r p r o d u c t s .

1.92 -

See S . Dineen [177].

1.93

T h i s r e s u l t i s due t o K . F l o r e t [23

1.95 -

For

~

rary

n

Boland and S.

n=l

t h i s r e s u l t i s due t o R.S

i t i s due t o R. Aron [ 2 1 ] .

I. P h i l l i p s [568] and f o r a r b t -

The p r o o f u s e s i n d u c t i o n and a v a r i a n t

of p r o p o s i t i o n 1.1.

1.96

This r e s u l t , t o g e t h e r with o t h e r i n t e r e s t i n g p r o p e r t i e s o f poly-

n o m i a l s on c l a s s i c a l Banach s p a c e s , may b e found i n R . M . Aron [ 2 1 ] .

Appendix III

414

CHAPTER TWO 2.61 ___

See the notes on exercise 1.63.

2.64

This is a weak implies strong holomorphy result.

It is due to

N. Dunford [213, p.3541 who requires only weak holomorphicity with respect toadetermining manifold in

Ffi .

A weaker result of a similar kind on the

analytic dependence of an operator valued function on a parameter is due to A . E . Taylor [676].

A proof, using the Cauchy integral formula, is given by

A.E. Taylor in [679].

2.65

This result also follows from corollary 2.45.

This result (and exercise 2.66) is due to L. Nachbin 1516,5201. It

shows that conditions on the range of a Ggteaux holomorphic function can provide information about its continuity properties. A different type of examination of the range (how to densely approximate a predetermined range) was initiated by R. Aron [22] and developed in a series of papers by

J. Globevnik (see the remarks on exercise 4.78).

See also D. Pisanelli

[575] for exercise 2.66. 2.67 -

This result is due to R. Aron and M. Schottenloher [31].

that the range space plays a role in this result.

Notice

See R. Ryan [620] f o r

the analogous result for weakly compact holomorphic mappings. 2.68

8'

This result arose in studying holomorphic functions on (P.J. Boland and S . Dineen [92]).

3

and

See also exercises 1.70,1.71,1.73,

1.74,exarnple 5.46 and corollary 6.26. __ 2.72

This result is due to M. EstGves and C. HervGs [231,232].

show, in fact, that one only need assume that

f

They

is universally measurable.

See also Ph. Noverraz [554]. 2.73 -

This result can be found in R. Aron and J. Cima [27].

See A . E .

Taylor [679, theorem 31 for a related result. 2.74

On Frgchet o r

B3w

spaces pointwise boundedness of linear

functionals implies equicontinuity or local boundedness.

Equicontinuity

plus pointwise convergence implies uniform convergence on compact sets and shows that

I:=,

$:

is hypocontinuous and thus continuous since the

Notes on some exercises domain space is a

k-space.

415

Part (b) follows from the finite dimensional

nature of the weak topology. 2.75 __

The first result of this kind for Banach spaces was proved by M . A .

Zorn [724].

Generalizations to Frgchet spaces and

a38

spaces were given

by Ph. Noverraz [536] and A . Hirschowitz [341] respectively. Subsequently, it was found that all these results could be derived from Zorn's result for Banach spaces by noting that Fre'chet spaces and 8 3 8 spaces are superinductive limits of Banach spaces.

In this fashion, one obtains the result of

the present exercise, which may be found in D. Pisanelli [578], J . F . Colombeau [141] and D. Lazet [423].

Further generalizations are proved by

using surjective limits (S. Dineen [190,191]).

A . Hirschowitz [341] shows

that one cannot extend this result to arbitrary locally convex spaces (see also J . F . Colombeau [140]). __ 2.76

This is a generalization of Hartogs' theorem on separate analytic-

ity. For holomorphic functions on

CxE, E

a Banach space, it is due to

A . E . Taylor [678] and for holomorphic functions on a product o f Banach

spaces it is due to M . A . Zorn [724].

Zorn's proof uses a category argument.

The extension to Frgchet spaces (Ph. Noverraz [536,538]) and to

2 38

spaces (A. Hirschowitz [341]) can be obtained, as in the previous exercise, by noting that these spaces are superinductive limits of Banach spaces. Further infinite dimensional versions of Hartogs' theorem are to be found' in J. Sebastiz e Silva [649,653], D. Pisanelli [578], H. Alexander [5], J . Bochnak and J. Siciak [74], D. Lazet [423], J . F . Colombeau [141], S.

Dineen [190], M . C . Matos [454,465,466] and N. Thanh Van [684]. Separately holomorphic functions arise in examples 2.13 and 2.14, proposition 5.34, corollary 5 . 3 5 , examples 5.36 and 5.50 and exercise 3.80. __ 2.80

This result as well as those in 2.81 and 2 . 8 2 may be proved using

surjective limits (see chapter 6 and S. Dineen [189,190]).

They originally

appeared as corollaries of more technical results and it may be possible to find a direct proof. 2.83

See the comments on exercise 1.84.

Use the method of example 2.31.

Note that the result is not true f o r arbitrary range spaces. Can you find a non-separable Banach range space for which the conclusion is still valid? See also [358] and [193].

416

2.84

Appendix 111

The

T~

topology lies between the compact open topology and the

topology of pointwise convergence. It is always strictly finer than the pointwise topology but may coincide with the compact open topology in infinite dimensional spaces, e.g.

E = C").

One can easily generalise to infin-

ite dimensions the classical Vitali and Monte1 theorems using this topology (see f o r instance, M.C. Matos [462] and chapter 3).

The results of this

exercise are due to D. Pisanelli [578]. 2.85 -

Use exercise 2.79 to show that each bounded set of holomorphic

functions factors through a finite dimensional subspace.

2.87

A careful study of example 2.47 should help with this exercise.

The result may be found in S. Dineen [l85]. 2.88 -

This exercise and exercise 2.89 are due to R. Pomes [584].

See

also the footnote on p.42 of [185].

2.91 -

A proof of this exercise and of exercises 2.92 and 2.93 may be found in S. Dineen [190].

2.94

To generalise this result to arbitrary locally convex spaces, one must first define very strongly convergent nets. The result is then a rather easy consequence of any one of a number of factorization results. A

proof is given in [184] and a generalization appears in [190].

_-2.96

See A . Hirschowitz [339].

2.97

Use uniqueness of the Taylor series expansion about points o f

2.98 -

See J. Mujica [503].

2.99

This result says that condition (a) of proposition 2.56 is suffic-

Is

? continuous? K

ient to characterize bounded subsets of H(K) when K is a convex balanced compact subset of a metrizable locally convex space. This is because on balanced sets, the Taylor series expansion at the origin converges in any of the topologies we discuss. This is a useful property and most of chapter 3 is motivated by this observation. The case K = { O l is due to R.R. Baldino [43]. For further information on condition (a) of proposition 2 . 5 6

417

Notes on some exercises w e r e f e r t o 52.6 and § 6 . 1 2.100

By t h e Dixmier-Ng theorem ( s e e R . B . Holmes [345, p.2111 f o r

d e t a i l s ) a Banach s p a c e w i t h c l o s e d u n i t b a l l

i s a d u a l Banach s p a c e i f

B

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

(B,r)

i s compact.

theorem and t o n o t e t h a t t h e u n i t b a l l o f T~

T

on

such

E

To prove t h e e x e r c i s e , i t s u f f i c e s t o u s e t h i s i s , by A s c o l i ' s theorem,

Hm(U)

compact. T h i s e x e r c i s e h a s a n i n t e r e s t i n g s e q u e l which i s t y p i c a l o f t h e

a c c i d e n t s t h a t f r e q u e n t l y o c c u r on r o u t e t o a mathematical d i s c o v e r y .

J.

Mujica, on l o o k i n g o v e r t h e t e x t , n o t i c e d t h i s e x e r c i s e and a s k e d m e how t o prove i t .

I t o l d him, as I had t o t e l l a few o t h e r s , t h a t I had s e e n a

Mujica worked o u t t h e above

p r o o f o f e v e r y e x e r c i s e b u t e x e r c i s e 2.100.

s o l u t i o n and i n f i n d i n g i t , n o t i c e d t h a t t h e i n t r o d u c t i o n o f a second t o p o l ogy, which r e n d e r e d c e r t a i n sets compact, a l s o e n t e r e d i n t o t h e c o m p l e t e n e s s problem f o r

H(K)

(theorem 6 . 1 ) .

T h i s l e d him t o a g e n e r a l i z a t i o n o f t h e

Dixmier-Ng theorem and t o s h o r t e l e g a n t p r o o f s of c o r o l l a r y 3.42 and theorem 6.1.

Mujica proved t h e f o l l o w i n g : Let

E

b e a b o r n o l o g i c a l l o c a l l y convex s p a c e .

fundamental f a m i l y

(Ba)

a Hausdorff l o c a l l y convex t o p o l o g y compact.

Assume t h e r e e x i s t a

o f convex, b a l a n c e d , bounded s u b s e t s o f T

on

E

Then t h e r e e x i s t s a c l o s e d s u b s p a c e

such t h a t each of

F

E;i

B

E

is

and T-

such t h a t

E S' ( F ' , T ) . w As an immediate c o r o l l a r y , h e showed t h a t

( F ' , T ~ ) f o r a s u i t a b l e Frgchet space

F,

H(K)

whenever

i s isomorphic t o K

i s a compact s u b s e t

o f a Frgchet space. The above i n d i c a t e s a f u r t h e r r o l e f o r t h e

T~

topology, j u s t i f i e s t h e

i n c l u s i o n o f " d i f f i c u l t " e x e r c i s e s and s h o u l d a l s o encourage t h e r e a d e r t o look f o r new improved p r o o f s o f t h e main r e s u l t s we p r e s e n t .

Unfortunately

i t was t o o l a t e t o i n c l u d e h l u j i c a ' s p r o o f i n t h e main t e x t .

2.101

T h i s r e s u l t i s due t o J . Mujica [SO31 who a l s o shows t h a t

(H(U),T~) is a locally metrizable space.

m

convex a l g e b r a when

U

i s a n open s u b s e t o f a

The fundamental r e f e r e n c e f o r l o c a l l y

m

convex a l g e b r a s

418

Appendix III

i s E . A . M i c h a e l ' s memoir [ 4 9 4 ] .

See a l s o J . Muj c a [SO21

2.103

T h i s r e s u l t i s due t o J . A .

2.104

A l o c a l l y convex s p a c e i n which e v e r y compact set i s s t r i c t l y

compact i s s a i d t o have p r o p e r t y 1.54).

Barroso [46

(s).

( s e e 51.4 and i n p a r t i c u l a r , lemma

T h i s r e s u l t , t o g e t h e r w i t h o t h e r r e s u l t s on t h e t o p o l o g i c a l v e c t o r

space s t r u c t u r e of 2.105

Let

s u b s e t s of

E

HM(E), m

(Kn)n=l

b e a fundamental system o f convex b a l a n c e d compact

such t h a t

neighbourhood of z e r o

may b e found i n D . Lazet [ 4 2 3 ] .

Vn

nKnCKn+l

for all

such t h a t

]If

]IK

n

+v

n

n.