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
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Chapter 1
POLYNOMIALS ON LOCALLY CONVEX TOPOLOGICAL VECTOR SPACES
T h e r e a r e two s t a n d a r d methods o f i n t r o d u c i n g p o l y n o m i a l s , by u s i n g t e n s o r p r o d u c t s o r by u s i n g
n
We have found it more
l i n e a r mappings.
c o n v e n i e n t t o a d o p t t h e l a t t e r approach.
T h e . r e a d e r familiar w i t h t h e
t e n s o r p r o d u c t approach w i l l have l i t t l e d i f f i c u l t y i n c o r r e l a t i n g h i s e x p e r i e n c e w i t h o u r approach b u t w e b e l i e v e t h a t t r a n s l a t i n g r e s u l t s i n t h e o t h e r d i r e c t i o n i s more d i f f i c u l t . between v e c t o r s p a c e s o v e r
C.
I n 5 1 . 1 we d i s c u s s polynomial mappings
In 11.2 w e discuss various kinds of contin-
uous polynomial mappings between l o c a l l y convex s p a c e s .
Our main i n t e r e s t
i s i n polynomials c o n t i n u o u s w i t h r e s p e c t t o t h e g i v e n t o p o l o g y on a
l o c a l l y convex s p a c e b u t we f i n d t h a t hypocontinuous and Mackey ( o r S i l v a ) c o n t i n u o u s polynomials a l s o p l a y a n i m p o r t a n t r o l e i n o u r s t u d y .
51.3 i s
d e v o t e d t o endowing s p a c e s o f p o l y n o m i a l s w i t h l o c a l l y convex t o p o l o g i e s For t h e n o n - s p e c i a l i s t we p r o v -
and i n 11.4 a d u a l i t y t h e o r y i s d e v e l o p e d .
i d e an i n t r o d u c t i o n t o f u n c t i o n a l a n a l y s i s i n Appendix 11. 51.1
ALGEBRAIC C,R,N
PROPERTIES
and
POLYNOMIALS
OF
d e n o t e r e s p e c t i v e l y t h e complex numbers, t h e r e a l num-
Z
b e r s , t h e n a t u r a l numbers and t h e i n t e g e r s . and
msN
and
m
then
copies of
In t h i s section,
and
B
E
If
A
and
and
xnyn F
w i l l d e n o t e t h e element
n
are sets,
B
w i l l denote t h e C a r t e s i a n product o f
AnBm
n
copies of
A
"FX' ' C Y I.
n times m times
w i l l d e n o t e v e c t o r s p a c e s o v e r t h e complex
numbers.
For each ings from
nsN
into
E
we l e t b a ( n E ; F ) F.
denote t h e space of n
The s u b s c r i p t
a
n o t assume any c o n t i n u i t y p r o p e r t i e s . d e f i n e d on
En
with values i n
o t h e r s remain f i x e d .
J!a(nE;F)
F
linear mapp-
refers t o algebraic s i n c e we do
Hence i f
L E ~ ~ ( ~ E ;t hFe n)
L
is
and i s l i n e a r i n each v a r i a b l e when t h e i s a v e c t o r s p a c e o v e r t h e f i e l d of complex 1
2
Chapter 1 1 l i n e a r mappings a r e j u s t l i n e a r mappings and i n t h i s c a s e w e
numbers.
use t h e n o t a t i o n da(E;F). 2 l i n e a r mappings are a l s o c a l l e d b i l i n e a r 2 i s sometimes used i n p l a c e o f "ea( E ; F ) . mappings and t h e n o t a t i o n @,(E;F) n When F=C w e w r i t e dea( E) i n p l a c e o f &a(nE;C) and E* i n p l a c e o f La(E;C).
E*
dLa('E;F)
as t h e s e t o f a l l c o n s t a n t mappings from
E.
i s called the algebraic dual o f
space can be i d e n t i f i e d with
When
n=O w e d e f i n e
F
into
E
and t h i s
i n a natural fashion.
F
The f o l l o w i n g a l g e b r a i c i d e n t i t y i s e a s i l y e s t a b l i s h e d and i s u s e f u l i n p r o v i n g r e s u l t s by i n d u c t i o n . P r o p o s i t i o n 1.1
then
E
If
and
F
are vector spaces over
and
C
nEN
z ~ ( ~ E ; F )I- L ~ ( E ; x ~ ( ~ - ' E ; F ) : ) B,(~-'E;~~(E;F)).
Proof
The mapping
T
T(xl'.-.Xn)
=
-f
T1
+
T2,
T
Xa(nE;F),
E
[Tl(xl)3(x2,...Jn)
=
g i v e n by
~ ~ , ~ ~ l ~ . . . , ~ n ~
e s t a b l i s h e s t h e r e q u i r e d correspondence. Definition 1 . 2
xl,
f o r any numbers.
L
An n-lineor mapping
symmetric i f
W l ' . . * ,xn)
..., xn E
E
=
.
L(xql)>
* *
and any permutation
We l e t ZS("E;F) a mappings from E i n t o
E
from
F
into
is said t o be
'Xo(n))
u
of t h e f i r s t
n
denote t h e vector space o f a l l symmetric F.
natural n
Zinear
By averaging over a l l permutations of t h e f i r s t
natural numbers we can associate, i n a canonical fashion, a symmetric n-linear mapping w i t h each n l i n e a r mapping and i n t h i s way we obtain a
n
st$(nE;F)
projection from f o l l o w i n g way.
where
L
Formally we a c h i e v e t h i s i n t h e
we s e t
E
s n denotes t h e s e t of permutations of the f i r s t We c a l l
s
IS
onto z:(nE;F).
s
t h e symmetrization o p e r a t o r .
if
L
E
%a(nE;F)
then
(b)
s(L) = L
i f and o n l y i f
(c)
s ( s ( ~ ) )=
S(L)
for all
s(L) L
L
Zz(nE;F)
E
E
bz(nE;F)
in
natural numbers.
The f o l l o w i n g p r o p e r t i e s of
are easily verified: (a)
n
x~(~E;F)
3
Polynomials on locally convex topological vector spaces (d) If
s
is a linear operator. i s a v e c t o r space we l e t
E
ing from
into
E
En
into
x
An)
denote t h e diagonal mapp-
onto t h e p o i n t
x".
A mapping from a ZocaZZy convex space
Definition 1 . 3
convex space
which maps
En
(or
A
E
t o a locally
which is the composition of the diagonal mapping from E
F
n
and an
E
Zinear mapping from
F
into
n
is called an
hom-
ogeneous poZynomia2. denote t h e vector space of a l l
We Zet QaCnE;F) nomials from
E
homogeneous poly-
F.
into
Thus we s e e t h a t i f t h e r e e x i s t s an
n
i s an
P:E+F
n
n
l i n e a r mapping
homogeneous polynomial i f and only from
L
F
into
E
such t h a t t h e
following diagram commutes E
~ ( x )= x
where
n
f o r every
A polynomial from
iaZs from
E
E
into
E
L
If
En
x
in
into
F
We l e t
F.
into
polynomials from Example 1 . 4
A ___f
E.
is a f i n i t e s m of homogeneous poZynom-
@,(E;F)
is a
l i n e a r (C-valued mapping on
2
it i s well known t h a t t h e r e e x i s t s an zAwt
for a l l If
z = (zl
A = (a. .) ij
,..., zn)
16idn 16 j6n
denote t h e v e c t o r space of a l l
F.
Cn
6
then
nxn
L(z,w)
=
P(Z) =
L(Z,Z) =
1_s i s_ n 1s j s n
1
1s i s n
L B,
B = - A + tA 2
zAzt = zBzt
f o r every
tA
z
L(z,w) =
C".
aijziwj. on
C"
'J
has t h e f a m i l i a r
J
s(L),
we o b t a i n a 2 - l i n e a r form
i s t h e symmetric matrix a s s o c i a t e d with where
E
then
a..z.z..
by i t s symmetrization,
I f we r e p l a c e whose matrix,
Since
1
such t h a t
w = (wl ,..., wn)
1s j s n Hence any C-valued 2-homogeneous polynomial, P , form
A
matrix
and a l l
ncN,
(c",
A,
i.e.
i s t h e transpose o f t h e matrix
in
Cn
d e f i n e t h e same 2-homogeneous polynomial.
it follows t h a t
L
and
A.
s(L)
Chapter I
4
More generally if L in E
dQa(nE;F)
E
then L(xn)
=
s(L)(xn)
for every x
and hence we do not, in general, have a one to one correspondence
between n-homogeneous polynomials and n-linear mappings. However, if we restrict ourselves to symmetric n-linear mappings we do obtain a unique correspondence. By the definition o f n-homogeneous polynomials and the symmetrization operator, the following diagram commutes
where the non-horizontal mappings are just restrictions to the diagonal. T+T
We denote the vertical mapping
by
is easily seen to be lin-
ear and we have already noted that it is surjective. As a consequence of the polarization formula we show that is injective and thus w e have a canonical bijective linear mapping between the space of symmetric n-linear A
mappings from E
E
into F
and the space of n-homogeneous polynomials from
into F. ( P o l a r i z a t i o n Formula)
Theorem 1.5 over
Proof
(c,
L
If E
and xl, ..., xn
Ex~(~E;F)
E
E
and F are vector spaces then
By linearity and symmetry
Zm.=n -
Hence 1 n 2 .n!
1
c=il
l,
El
. . . En
A
L(
n
1 EjXj)
j=l
=
Polynomials on locally convex topological vector spaces
- - 2"
1
1
L(Xl)
O,<m.sn m 1 ! . .. mn !
m
1
...
(x,)
m
n
.I
of
L(xl,. If
c l2
... e2n
and h e n c e t h e c o e f f i c i e n t
2"
=
i n t h e above e x p a n s i o n i s
f o r some
then
i
m.=O
m +1
ml+l
c
1 E . =+1
then
l,
. . ,xn)
m.>l
n
61
e. =?I 1
i sj s n
... = m n = l
ml=m2
m +1
... enn
1
Zm.=n If
ml+l El 6.=+1
5
E.
J1
l,
1.
f o r some
1
j
and hence mj+l+l E. J+1
1-1
=+1
..,'.
mn+l
Isisn,i#j
0.
=
Hence a l l t h e o t h e r terms v a n i s h and we have proved t h e p o l a r i z a t i o n formula. Corollary 1.6 Proof
The mapping
-:%:("E;F)
By t h e p o l a r i z a t i o n formula
i
and o n l y i f
i s i d e n t i c a l l y zero.
+
L
E
Qa("E;F)
i s a linear bijection. i s i d e n t i c a l l y zero i f
%-:(nE;F)
i s a l i n e a r mapping w i t h
Hence
z e r o k e r n e l and t h u s it i s i n j e c t i v e . T h i s completes t h e p r o o f . The p o l a r i z a t i o n f o r m u l a h a s many o t h e r a p p l i c a t i o n s , some o f . w h i c h we now d e s c r i b e .
These f o l l o w from t h e s p e c i f i c n a t u r e o f t h e formula which
a l l o w s u s t o compare t h e moduli o f symmetric n - l i n e a r forms and p o l y n o m i a l s S i n c e i t i s u s u a l l y e a s i e r t o compute t h e modulus o f an n
on c e r t a i n sets.
l i n e a r form, t h e n t h e modulus o f a p o l y n o m i a l , t h i s p r o v e s a e f u l . If
and
B
f
i s a f u n c t i o n d e f i n e d on a s u b s e t A o f
i s a semi-norm on
we let
F
llfll
u n d e r s t o o d from t h e c o n t e x t , we j u s t w r i t e Theorem 1 . 7
x&.A
and
F
are vector spaces over
baZanced subset of E
and
B
i s a semi-norm on
f o r any
Remark
L
in X;("E;F)
with values i n
= supB(f(x)).
E
I f
E
F
11
If
B
is f u l l y
/]A' CC,
A
is a convex
then
.
T h i s e q u a l i t y may t a k e t h e form
c a s e we o b t a i n i n f o r m a t i o n .
+ws+ms+-
F
b u t even i n t h i s
3
6
Chapter I
Proof
The l e f t - h a n d s i d e o f t h i s i n e q u a l i t y i s t r i v i a l s i n c e By t h e p o l a r i z a t i o n formula
i ( A ) C L(An).
lii,
x.
i = l , .. . , n
A,
E
and
C. = 2 1
then
,I &ixi
1 "
E
1=1
convex and b a l a n c e d , and hence
A,
since
A
is
Hence
T h i s completes t h e p r o o f . nn - i s t h e b e s t p o s s i b l e i n g e n e r a l (example 1 . 8 ) . In n! c e r t a i n c a s e s , however, i t i s p o s s i b l e t o improve t h i s e s t i m a t e - t h i s w i l l The c o n s t a n t
depend on
n
and on t h e geometry o f t h e s e t
A.
We g i v e a few examples t o
i l l u s t r a t e t h i s p o i n t (some w i t h o u t p r o o f ) and n o t e i n p a s s i n g t h a t t h e r e are s t i l l i n t e r e s t i n g open problems i n t h i s a r e a . Example 1 . 8 w i t h norm
where
L
Let
( t h e space o f a b s o l u t e l y convergent s e r i e s
E = R1
11 ( x ~ ) ~ I= / c n l x n l ) .
i
m
(Xm)m=l
xi =
Let
for
b e d e f i n e d by
L:En+C
.
i = l, . . , n .
i s n - l i n e a r and symmetric and one e a s i l y shows t h a t
l l L l l A :=
n
where
and t h e c o n s t a n t
A
is t h e u n i t b a l l of nn n!
R1.
llLlkn =
and
Hence
.
nn i n theorem 1 . 7 i s t h e b e s t p o s s i b l e . n!
Our n e x t example c o n c e r n s r e a l polynomials o v e r an i n n e r p r o d u c t s p a c e . The analogous problem f o r complex p o l y n o m i a l s i s s t i l l open a l t h o u g h some p a r t i a l r e s u l t s have been o b t a i n e d .
I
Polynomials on locally convex topological vector spaces Example 1 . 9 let
F
d e n o t e a real inner product space with u n i t b a l l d e n o t e a r e a l Banach s p a c e and suppose L E Y:(nH;F). Then
\]LIBn = llLlb
Let
H
where
B
i s t h e u n i t b a l l of
forms and r e a l p o l y n o m i a l s , w e always have <
l l ~ l b<
Case 1 Ow'
Dim(H)
and
< m
n=2.
Dim(H) < (xl,
...,x,)
aaB
n
m,
E
Bn
it f o l l -
arbitrary. such t h a t
# 0
such t h a t
J
(yl*, . . . , y n * ) E A
W e claim
y . * = 'yy
for all
such t h a t
y * + y:
# 0 and
= YD*
+
Yq*
IIYp*+ Y,*ll
=
j = l , ..., n .
n {(yl, ...,yn) E B ; b a ,
=
and Now choose
for
1
such t h a t t h e s e t A
and
11L(x,y) = ~ ( x + y -$(x-y) )
Since
L
Now choose
= z
and w e may suppose
IlLlbn
11
Case 2
z
6
1 IIL(x>Y) b T\lillB( IIx+Y112+(1x-YI(2)
that
Choose
IILIL
m.
and hence
a>O
H.
S i n c e t h e p o l a r i z a t i o n formula h o l d s f o r r e a l symmetric n - l i n e a r
Proof
o
B,
.
Hence t h e r e e x i s t s
j = l , ...,n.
( I L ( Y 1 , e . . 9 ~ n ) I I= I I L I I p
i s non-empty.
such t h a t
i
and
y* P
-
y;
j.
I f n o t t h e r e ex s t s
# 0.
Let
The f i r s t c a s e i m p l i e s t h a t
M + ( + < a , y
L
zi
= y r
if
(z l , . . . , z n )
- 1)
p
and
q
i # p , q and E
A.
Since
8
Clzapter 1
we have a r r i v e d a t a c o n t r a d i c t i o n and completed t h e p r o o f i n t h i s c a s e . Case 3
H
and
llLl/
arbitrary.
t h e r e e x i s t s a f i n i t e d i m e n s i o n a l subspace
Given €>O
that
n
' IILIIgn- t'
(Br\E)n
lfliB
c i s a r b i t r a r y we have shown t h a t
Since
= llLll
Bn
...
such
H
and completed t h e p r o o f .
The f o l l o w i n g r e s u l t r e d u c e s t o theorem 1 . 7 on t a k i n g nl=n2
of
E
By c a s e 2 it f o l l o w s t h a t
p=l,k=m,
n =l. k
P r o p o s i t i o n 1.10
Let
denote a Banach space and l e t
E
nl,
. . . ,nk
be
n +n +...+ n = m . Suppose x l , . . . ,xk are u n i t 1 2 k k \I z i x i / \ 6 ( \ z i I p ) l / P for a l l ( z , , . . . , z k ) E Ck i=l i=l
positive integers with vectors i n
E
and
l
where
"1
/Wl
"2
'X2
k
1
if L n
,... ' X k 1 I k
1
E
C(mE) nl!
6
...
nl/P
n1
A
and
nk! mm/P nk/P
.....nk
The n e x t p r o p o s i t i o n g i v e s , on t a k i n g
P r o p o s i t i o n 1.11
E
Let 2.
denote a power of
m!
p=2,
t o complex v a l u e d polynomials o f d e g r e e
m
i s t h e u n i t ball of E
Ilill,
a n e x t e n s i o n o f example 1.9
2".
denote a compZex
AP
space,
ldp<-,
and Let
Then
m! f o r any
L
where t h e norms are taken over t h e u n i t baZ1 o f
i n if,:(mE)
E.
We conclude t h i s s e c t i o n w i t h two r a t h e r s i m p l e b u t u s e f u l r e s u l t s . The f i r s t f o l l o w s r e a d i l y from t h e b i n o m i a l theorem and t h e second f o l l o w s from a n a p p l i c a t i o n o f t h e maximum modulus theorem f o r f u n c t i o n s o f one complex v a r i a b l e . Lemma 1 . 1 2
in
E
and
If L AEC
E
L;("E;F)
we have
and
P =
i
E
Oa("E;F)
then f o r any
x,y
9
Polynomials on locally convex topological vector spaces
P(x+y)
=
P(X)
If
E
and
i s a semi-norm on
F,
A
Lemma 1 . 1 3
Moreover, if
+
A#O,AXEA
P(Y)
c
n- 1 +
F
r =1
(;)L(x)n-r(Y)r
are vector spaces over
i s a balanced subset of
and
=
=
A
E
P
(E,
and
E
@a(nE;F), 6
XEE
then
is convex then
sup B(P(x+eiey)) YEA,BER sup B(P(eiex+y))
(since
i s balanced)
A
(by homogeneity)
YEA
0 ER
If
§1.2
AXEA
then
2
sup B(P(y)) YSA
=
1$\lb,A*
1 x+A C - A + A A
CONTINUOUS
=
( 1
+
(by t h e maximum modulus p r i n c i p l e )
x1 )A
and hence
POLYNOMIALS
Our main i n t e r e s t i n t h i s book i s i n s t u d y i n g c o n t i n u o u s polynomial and holomorphic mappings between l o c a l l y convex s p a c e s .
However, w e f i n d
i t u s e f u l (and n e c e s s a r y ) t o d e f i n e more g e n e r a l c l a s s e s o f mappings -
t h i s i s analogous t o t h e s i t u a t i o n i n f u n c t i o n a l a n a l y s i s where t h e weak t o p o l o g y i s used t o o b t a i n p r o p e r t i e s o f t h e norm topology.
10
Chapter 1 Let
and
E
b e t o p o l o g i c a l s p a c e s which are a l s o v e c t o r s p a c e s o v e r
F
U s u a l l y t h e r e w i l l b e some r e l a t i o n s h i p between t h e t o p o l o g i c a l and t h e
(E.
v e c t o r s p a c e s t r u c t u r e s b u t f o r t h e moment w e w i l l n o t make any such assumption.
We l e t
/ ; ( n E ; F ) , X(nE;F)
J ? ( ~ E ; F ) denote r e s p e c t i v e l y t h e
and
E
spaces of continuous n-homogeneous polynomials from E
tinuous n-linear mappings from E
n-linear mappings from
F
to
and the continuous symmetric
t h e above s p a c e s a r e v e c t o r s p a c e s and t h a t - ( d S ( n E ; F ) ) = S’(nE;F). spaces. a
cs(E)
E,
i s a semi-norm on
Note t h a t
Ea.
Ea
is
-
Ra
Ea
a(x) = 0
{ x e E ; a ( x ) < r ) and
B,(r)
i s the c l o s u r e of f
If
Ba(r)
Ba(r)
a r e l o c a l l y convex
F
in
II (x) = 0
E.
If
{xcE;a(x)
=
Since
E.
F
E
onto
and t h e open u n i t b a l l
i s a semi-norm on
a
from a t o p o l o g i c a l s p a c e i n t o
i s continuous for every
of
and
E
and
w i l l d e n o t e t h e normed l i n e a r s p a c e
i f and o n l y if
R a { x ~ E ; a ( x )< 1 ) .
=
nB
W e now suppose t h a t
J,’(~E;F)
=
w i l l d e n o t e t h e c a n o n i c a l s u r j e c t i o n from
Ba(r)
mapping
s(Y.(~E;F))
w i l l d e n o t e t h e s e t o f a l l c o n t i n u o u s semi-norms on
( E , ~ i ) / ~ - l ( ~ )and
of
the con-
I n a l l cases we c o n s i d e r we f i n d t h a t
F.
into
F,
into
=
we l e t
E
If
accs(E)
lim
ES (F)
then
(FB,IIB), a
i s c o n t i n u o u s i f and o n l y i f
F
B E C S ( F ) . We u s e t h i s f a c t i n e x t e n d i n g
r e s u l t s c o n c e r n i n g normed l i n e a r s p a c e v a l u e d polynomials and holomorphic f u n c t i o n s t o f u n c t i o n s w i t h v a l u e s i n a n a r b i t r a r y l o c a l l y convex s p a c e . Proposition 1.14
Let
E
be a l o c a l l y convex space over
C,
F
il
P E P ~ ( ( ” E ; F ) . The following are equivalent
normed l i n e a r space and suppose
(a)
P
i s everywhere continuous;
(bi
P
is continuous a t t h e o r i g i n ;
(el
P
i s bounded on some neighbourhood of the origin;
(dl
P
i s a l o c a l l y bounded f u n c t i o n i i . e . bounded on a neighbourhood of each p o i n t ) .
Proof 1.13
The i m p l i c a t i o n s ( c ) <=> ( d ) .
suppose
A=P.
( a ) => (b) => ( c )
W e now show
( c ) => ( a ) .
Let
By t h e p o l a r i z a t i o n formula and
b a l a n c e d neighbourhood o f z e r o be a r b i t r a r y .
are t r i v i a l and by lemma
Choose
a >O
V
such t h a t
such t h a t
A
(c)
XS(nE;F)
= M < m . Let Vn By Lemma 1 . 1 2
IlAIl
axo E V .
E
and
t h e r e e x i s t s a convex x0€ E
11
Polynomials on locally convex topological vector spaces
Hence
i s continuous a t
P
xo
and
( c ) => ( a ) .
This completes t h e proof.
Let E and F be ZocnZZy convex spaces over E and Corollary 1 . 1 5 n if and only if P i s continuous a t Zet P E Pa( E;F). Then P E @("E;F) one point. I t s u f f i c e s t o use p r o p o s i t i o n 1 . 1 4 and t h e p r o j e c t i v e l i m i t represent a t i o n of
by normed l i n e a r spaces.
F
We now look a t a very u s e f u l f a c t o r i z a t i o n lemma. l o c a l l y convex spaces, Hence
?(("E,;F)
a
E
cs(E)
and
P
!?(nE,;F)
E
may be i d e n t i f i e d with a subspace of
If
E
then
P
F
are
When
F
and 0
nLYE $fnE;F)
6'("E;F).
i s a normed l i n e a r space t h e f a c t o r i z a t i o n lemma says t h a t t h e union of a l l This i s not s u r p r i s i n g i n view of lemma such subspaces covers p("E;F). 1.13.
Lemma 1.16 F
and
(Factorization Lemma).
i s a ZocaZZy convex space
i s a nomed Zinear space then
f o r every p o s i t i v e i n t e g e r Proof
Let
P
E
B(nE;F)
n. and suppose
symmetric n - l i n e a r mapping.
Since
exists a
llpll
a ( x ) < 1, n $[
E
If
E
cs(E)
such t h a t
a(y) = 0
0EFI.
n A ( x ) ~ ( X ~ ) ~= - 1 ~]
1
R=O
polynomial from follows t h a t
g
Ax'
where
to
E
M<m.
The function
Now suppose
x , y ~E ,
g(A) = $oP(x+Ay) =
@ ( A ( X ) ~ ( ~ ) A~n - -R ~ ] . i s a
):(
of degree
i s the associated
i s a normed l i n e a r space t h e r e =
R=O
Ks(nE;F)
n.
sup a(x+Ay) = a ( x ) < 1 it AEC i s a bounded polynomial and hence has degree 0 . By t h e
C
Hahn-Banach theorem t h e form
and
F
A
C
R
Since
ObRsn-1. Since any z E E has R it follows t h a t A(z) (y)n-R = 0 i f Z E E
A(x) (y)n-R = 0 a(x')
Q
for
12
Chapter 1
and Y
a(y) = 0 , R#n.
E,
b e d e f i n e d on
P
n,(z1)
=
'TI
a
(?
) = z
Thus
2" P
u n i t b a l l of
E,
P(i,).
Hence
P(z+y) = P(z)
by
$(z)
then
71
=
P(3
if
(z2-z1) = 0 I
if
a(y) = 0 . Y
~ E Es a t i s f i e s
' T I J Z )= z.
Let If
and hence
_
i s well d e f i n e d and
z
f o r any
TO'TI=
P.
~ ( 2 =~ P) ( t l + ( ? 2 - i 1 ) ) = S i n c e n,(B,(l)) is the
w e have
T h i s completes t h e p r o o f . The f o l l o w i n g example shows t h a t t h i s r e s u l t d o e s n o t e x t e n d t o arbitrary
F.
Example 1 . 1 7
Let
d e n o t e t h e s e t o f a l l sequences o f complexnumbers
CN
i s g i v e n t h e p r o d u c t t o p o l o g y o r t h e t o p o l o g y o f c o o r d i n a t e convergence
CN
and i s a F r & c h e t s p a c e ( i . e . a c o m p l e t e m e t r i z a b l e l o c a l l y convex s p a c e ) . If
a
N
cs(C )
6
then
hence i s isomorphic t o
The i d e n t i t y L
of
E
'J
Cn
f o r some
belongs t o
I
&((CN),;CN)
a E C S (CN) C N . Hence
i s a f i n i t e dimensional normed l i n e a r s p a c e and
E
I
&![C
then
u
na.
N ;C N )
L((GN),)
and
We c l a i m
I(C N )
= C
N.
If
i s a f i n i t e dimensional subspace
a E cs (CN)
Our n e x t r e s u l t i s a f a c t o r i z a t i o n lemma f o r a r b i t r a r y p o l y n o m i a l s . The p r o o f f o l l o w s d i r e c t l y from l e m m a 1.16.
Lemma 1.18
If
G-J("E;F) =
and
E
F
are Locally convex spaces over
C
then
LJ P ( ~ E , ; F ~ ) BECS(F) a ~ c s ( E )
Before d e f i n i n g hypocontinuous and S i l v a c o n t i n u o u s polynomials we f i r s t p r o v e a r e s u l t a b o u t p o l y n o m i a l s on a Banach s p a c e .
This r e s u l t is t r u e
f o r any Baire s p a c e and i s used i n p r o v i n g Zorn's theorem i n t h e n e x t chapter. Lemma 1.19
Let
E
l o c a l l y eonvex space.
be a eompZex Banach space and
F
an arbitrary
If {Pm}i=l i s a sequence i n @("E;F)
and
13
Polynomials on locally convex topological vector spaces
We may suppose without l o s s of generality that F
Proof
linear space. Let Am E&'(~E;F)
where
im =
is a normed
Pm for every integer m.
By the polarization formula Am converges pointwise on E
to a symmetric
n-linear form A and hence P = E 2a(nE;F). We complete the proof by induction. Let B denote the unit ball of E. If n=l then Pm €;t(E;F) and by the uniform boundedness principle Hence IIPIIBs M and P is continuous by proposition 1.14. result holds for n=k. I f
Pm
E
6(k+1E;F)
for all m
= M <-.
Now suppose the
then, by induction,
Zsisk+l
By the uniform boundedness principle the collection of linear mappings
1 is bounded on the unit ball of E, (A (Y2,.*'3Yk+l) Yi'B 2fiSk+l i.e. . sup " (Y, .. .,Yn)II< YiEB ltick+l Thus P and A are continuous. This completes the proof. 9
The hypocontinuous and the Mackey continuous polynomial mappings between two locally convex spaces E and F can be defined in two different ways, either by defining new topologies on E o r by using properties of the compact and bounded subsets of E. We describe both methods.
A n element
p
of
@a(nE;F)
i s said t o be hypocontinuous i f
14
Chapter I
it i s continuous on t h e compact subsets o f
E.
pHy(nE;F)
We l e t
denote
the vector space o f a l l hypocontinuous n-homogeneous poZynomia2s from E
to
F.
Da(E;F)
An element of
i s said t o be Mackey ( o r S i l v a ) continuous F.
if it maps bounded subsets of E onto bounded subsets o f
We l e t
@M(nE;F) denote t h e vector space of a22 Mackey continuous n-homogeneous E
polynomials from
F.
into
Hypocontinuous and Mackey c o n t i n u o u s n - l i n e a r forms a r e d e f i n e d i n a n S i n c e t h e v e c t o r sum of compact ( r e s p . bounded) s u b s e t s o f
a n a l o g o u s way.
a l o c a l l y convex s p a c e i s compact ( r e s p . bounded) it f o l l o w s from t h e s y m m e t r i z a t i o n formula and t h e p o l a r i z a t i o n formula t h a t f o r any l o c a l l y convex s p a c e s
and
E
s(&,fE;F))
If
x
we have
F
giy(nE;F)
=
XHy(nE;F) n f;("E;F)
=
is a topological space ue l e t
Xk
denote the'space
with the
X
f i n e s t topology which coincides with the original topology on the compact subsets o f
If
X.
x
xk
=
we c a l l
X
a k-space.
If
is a locally
E
convex s p a c e t h e n it i s n o t t r u e i n g e n e r a l t h a t Ek i s a l s o a l o c a l l y convex space.
Any m e t r i z a b l e s p a c e i s a k - s p a c e .
A l o c a l l y convex s p a c e i s a
semi-Monte1 s p a c e if i t s c l o s e d bounded s e t s a r e compact. b a r r e l l e d semi-blontel s p a c e i s c a l l e d a Montel s p a c e .
An i n f r a -
The s t r o n g d u a l o f
a n i n f i n i t e d i m e n s i o n a l F r g c h e t Montel s p a c e ( a &I>mspace) o f a k-space which i s n o t m e t r i z a b l e .
i s a n example
The f o l l o w i n g p r o p o s i t i o n f o l l o w s
e a s i l y from t h e d e f i n i t i o n above.
Proposition 1.20
Hence i f convex s p a c e
E = Ek
If
E
=
B C ~ ( E;~ F ))
and
then
F
are ZocaZZy convex spaces then
Q H Y ( " ~ ; ~ )=
Q c ~ E ; F ) f o r any l o c a l l y
F.
W e now look a t Mackey c o n t i n u o u s p o l y n o m i a l s .
convex s p a c e w i t h t o p o l o g y
Mackey convergent t o scalars,
x,
T.
A sequence M
we w r i t e
(A~)E=~ \ i, n \ -f
01
as
xn-+ x,
n+-
Let
E
be a locally
( x ~ )i n~ E
is said t o be if there e x i s t s a sequence o f
, such t h a t
1 n(xn-x)
-f
0
in
(E,T)
15
Polynomials on locally convex topological vector spaces
-.
as
n
xn
y x as
-+
n-,
E
of
A
A subset
i s said t o be
( x ~ E) A,~
M-closed i f
The M-closed s u b s e t s o f
XEA.
implies
satisfy the
E
c l o s u r e axioms f o r a t o p o l o g y which w e c a l l t h e topology of the M-closure T ~ .W e
and d e n o t e by If
a l s o write
then t h e space
T = T~
EM
i n place of
(E,rM).
i s c a l l e d a superinductive space.
(E,T)
F r g c h e t s p a c e s and t h e s t r o n g d u a l s o f Frcchet-Schwartz s p a c e s a r e superinductive spaces. convex.
i s t h e s t r o n g d u a l o f a Frgchet-Monte1
(E,T)
For i n s t a n c e i f
(d.42 s p a c e s )
i s not necessarily l o c a l l y
The t o p o l o g y
space then t h e following a r e e q u i v a l e n t ;
is a 2 3 - 3 s p a c e ,
(a)
(E,T)
(b)
T = T~
(c)
( E , T ~ ) i s a l o c a l l y convex s p a c e .
on
E,
S i n c e c l o s e d sets are s e q u e n t i a l l y c l o s e d and Mackey c o n v e r g e n t sequences a r e convergent T~
in
T~
3
T~
on any l o c a l l y convex s p a c e
>,T
i s t h e k-topology a s s o c i a t e d w i t h M (E,rM) i f and o n l y i f xn x as
Proposition 1.21
any integer
n
F i r s t suppose
Proof
B(P(
F
and
E.
in
xn
+
x
as
be locally convex spaces.
Then f o r
P
E
@(n(EM);F) b u t
P
i s n o t bounded on t h e
B(P(xm)) > m
m 1 B(P(xm)) mn)) 1/2 X
=
>
m.
for all
1 m2 f o r a l l
m.
E,
(x,),,
This implies t h a t This contradicts t h e
n
fact t h a t
Hence
P
i s bounded on bounded s e t s and
8 Now suppose
P
n+-
(E,T).
Then we c a n f i n d a bounded sequence i n
B E C S ( F ) such t h a t
m
+
c ~ ~ ( ~ E ; =F )Q(~(E,);F).
bounded s u b s e t s of and
E
Let
n
-
Note a l s o t h a t
T.
-f
where
(E,T),
E
c~(E,);F)
c
(pM(nE;F).
IP~(~E;F). To show t h a t
P
E
@(n(EM);F)
it s u f f i c e s
16
Chapter 1 x as m - t m i m p l i e s P(xm) + P ( x ) as m b e a sequence o f scalars such t h a t 1x1, -t +
t o prove t h a t
Let hm(xm-x)
+
s u b s e t of
m+
x
as
0 E.
If
m
+.
A
ki(nE;F)
E
m
.
and
B = { A , ( X ~ - X ) }u~{XI i s a bounded
The s e t
m.
m
and
x=P
then
A
i s bounded on
Bn.
Hence
m=1,2,..
m = 1,2,.
i s a bounded s u b s e t o f
F.
I t now f o l l o w s t h a t P(xm)
- P(x)
shown t h a t
R n-R A(xm-x) (x)
P(x+xm-x)-P(x)
=
P(xm) + P ( x )
Corollary 1 . 2 2
as
=
m-+m
If E and
n
-t
1 R=l
as m + m if k & n . Since n R n-R (R) A(xm-x) (x) we have
0
and t h i s completes t h e p r o o f .
F
are ZocaZZy convex spaces then
~ , , ( " E ; F ) 3 P M ( " ~ ; ~3 )
for every
.
Q,,("E;F)
3 P ( n ~ ; ~ )
N.
Pa(nE;F)
may a l s o b e r e g a r d e d as a s p a c e o f c o n t i n u o u s polynomials i f we
p l a c e on
E
space, i s
the f i n i t e Z y open topoZogy tf
open i f V n F
tf.
A subset
V
of E
,
a vector
i s open for every f i n i t e dirnensionaZ subspace
F
of E where each f i n i t e dirnensionaZ space i s given i t s unique norm top0 zogy . When do we have e q u a l i t y i n c o r o l l a r y 1.22?
T h i s i s an i n t e r e s t i n g
q u e s t i o n and t h e answer p l a y s an i m p o r t a n t r o l e i n Chapter 5. a l r e a d y n o t e d cases i n which we do have e q u a l i t y .
We have
These r e s u l t e d from
c o n s i d e r i n g l o c a l l y convex s p a c e s as o b j e c t s i n t h e c a t e g o r y of t o p o l o g i c a l s p a c e s and c o n t i n u o u s mappings.
T h i s c a t e g o r y i s much t o o l a r g e f o r o u r
p u r p o s e s and r a r e l y p r o v i d e s n e c e s s a r y and s u f f i c i e n t c r i t e r i a .
In t h e
p r e s e n t case i t p r o v i d e s s u f f i c i e n t b u t n o t n e c e s s a r y c o n d i t i o n s f o r c e r tain equalities.
On t h e o t h e r hand, w e can a l s o look a t l o c a l l y convex
s p a c e s as o b j e c t s i n t h e c a t e g o r y of t o p o l o g i c a l v e c t o r s p a c e s .
T h i s cat-
egory i s t o o small and g e n e r a l l y o n l y g i v e s c o n d i t i o n s of t h e o p p o s i t e k i n d
-
i n t h e p r e s e n t s i t u a t i o n w e g e t c o n d i t i o n s which are n e c e s s a r y .
For example, i n o u r t e r m i n o l o g y a l o c a l l y convex s p a c e s p a c e if and o n l y if
6fM(E;F)
=
d(E;F)
E
is a bornological
f o r e v e r y l o c a l l y convex s p a c e
17
Polynomials on locally convex topological vector spaces I t i s , however, u s e f u l t o keep t h e s e t y p e
F.
o f r e s u l t s i n mind a s t h e y
p r o v i d e u s e f u l estimates and g i v e u s o u r p r e l i m i n a r y examples. The c o r r e c t s e t t i n g f o r t h e p r e s e n t c h a p t e r i s t h e "category" o f l o c a l l y convex s p a c e s and co n t i n u o u s m u l t i l i n e a r mappings and w e now g i v e some e x a m p l e s w h i c h r e f l e c t t h e
E = F x F' where F i s a l o c a l l y convex spa c e B i s i t s s t r o n g d u al ( i . e . F ' i s t h e spa c e o f a l l c ontinuous
Example 1 . 2 3 and
F' B
linear
Let
mappings on
F
bounded s u b s e t s o f Let
A
E
n a t u r e of t h i s "category".
A : E x E
and
f;(*E)
F). -+
Q: be d e f i n e d by
i ( x , x t ) = XI(%).
a normed l i n e a r s p ace. w e always have
w i t h t h e t o p o l o g y o f uniform convergence on t h e
iE
i s c o n t i n u o u s i f and o n l y i f
By t h e d e f i n i t i o n o f t h e s t r o n g topology on
PM(2E) and f r e q u e n t l y
form.
CN (C")
F'
i s hypocontinuous.
Our most u s e f u l counterexample i n t h i s book, CN x C(N), and
is
E
h a s t h e above
i s t h e s p a c e o f a l l complex s eq u en ce s w i t h t h e produc t topology i s t h e s p ace of a l l f i n i t e complex se que nc e s with t h e d i r e c t
sum to p o l o g y .
I n l a t e r c h a p t e r s we s h a l l u s e t h e f o l l o w i n g p r o p e r t i e s o f
it i s a r e f l e x i v e n u c l e a r s p ace w i t h a n a b s o l u t e b a s i s , it i s
CN x C " ) ;
a s t r i c t i n d u c t i v e l i m i t o f FrLchet n u c l e a r s p a c e s and a n open compact s u r j e c t i v e l i m i t of
( s t r o n g d u a l of F r 6 c h e t n u c l e a r ) s p a c e s , each
compact s e t i s c o n t a i n e d i n a s e t o f t h e form subset of
CN
and
KxL
where
K
i s a compact
i s a c l o s e d bounded f i n i t e dim e nsiona l s u b s e t o f
L
e v e r y neighbourhood o f t h e o r i g i n c o n t a i n s a neighbourhood o f t h e o r i g i n which h a s t h e . form U x CN-' x V where LEN, U i s a neighbourhood
(C(N),
i s a neighbourhood o f z e r o i n
CL and V
o f zero i n
_-Example 1.24
Let
E
(C").
b e a c o u n t a b l e i n d u c t i v e l i m i t o f normed l i n e a r
s p a c e s i n t h e c a t e g o r y o f l o c a l l y convex s p a c e s and c o n t i n u o u s l i n e a r
l i m En and E i s a b o r n o l o g i c a l spa c e which con-+ n m t a i n s a fundamental sequence o f bounded s e t s , (Bn)n,l. W e may suppose t h a t
mappings. each
Bn
Thus
E
=
i s convex and b al an ced .
S i n c e a l o c a l l y convex s p a c e i s b o r n o l -
o g i c a l if and o n l y i f ev er y convex b a l a n c e d s e t which a b s o r b s evEry bounded set i s a neighbourhood o f z e r o , w e f i n d t h a t s e t s o f t h e form
~n,lXnBn
Chapter 1
18
form a b a s i s of neighbourhoods of zero i n a l l sequences o f p o s i t i v e r e a l numbers
(ln=lAnBn m
m
ranges over m = Iln=lhnbn; bnEEn, m
i s convex and balanced and
IZ=lAnBn
a r b i t r a r y } ) . This follows s i n c e
as
E
absorbs every bounded s e t and hence i s a neighbourhood of zero.
Conversely
i s a convex balanced neighbourhood of zero then f o r every ntN m an t h e r e e x i s t s a n > 0 such t h a t anBn C V and hence V 3 Bn. Zn if
V
In=1
The countable d i r e c t sum of normed l i n e a r spaces and t h e strong dual I t can a l s o be
of a Frgchet Monte1 space a r e examples of such spaces.
e a s i l y shown t h a t t h i s c l a s s of spaces coincides with t h e c l a s s of bornological
DF
spaces.
E
We now show t h a t f o r such convex space
F.
$JM(nE;F) = @("E;F)
f o r every l o c a l l y
We may assume without l o s s o f g e n e r a l i t y t h a t
normed l i n e a r space.
Let
P
E
pM(nE;F)
and suppose
is a
F
A E%("E;F)
where
A = P.
B1 i s bounded I I P / I < M < B1 have been chosen so t h a t
Since
A2,
..., Am
If
x
m
E
I
AiBi, i=l
p(X+Xy)
1
i=l
=
and
YEB,,,+~
P(x)
XiBi
n
A >O
-.
Let
A1
= 1
and suppose
then
1
( " , A ( x ) " - ~ ( ~ ) X~R . and hence
IIAlL
i s f i n i t e and we can choose
+
R=l
+
where
ll A l l m Since
A
E
-
A;(~E;F),
s u f f i c i e n t l y small s o t h a t
X
=
A m+ 1
19
Polynomials on locally convex topological vector spaces m+l
s i=l
I
1 1
By induction that
?4(1 -i=l 2 i - 1
m
we can choose a sequence of p o s i t i v e numbers
llpll
such
(Xm)m=l
s 2M. m =1
Hence
P
i s bounded on a neighbourhood of zero and i s continuous by
p r o p o s i t i o n 1.14.
We a l s o n o t e t h a t t h e above proof shows t h a t i f
i s a subset of :J(nE;F), F i s a normed l i n e a r space, and ('cr)aE~ f o r every m then the collection is a locally s;p IIPall B, < (Pa)aEA bounded family of functions. Let
Example 1 . 2 5 convex space.
and only i f each I f each
E
=
Ci=lEm
PHy(nE;F')
where each
=
?("E;F)
Em
i s a normed l i n e a r space then
Em
i s a metrizable l o c a l l y
f o r every i n t e g e r
nz 2
if
i s a normed l i n e a r space.
Em
by example 1.24. Since
E
Then
Q("E;F)
=
pHy(nE;F)
i s not a normed l i n e a r space.
El
Conversely, suppose
i s a countable i n d u c t i v e l i m i t , it has a fundamental neighbourhood
system a t t h e o r i g i n c o n s i s t i n g of s e t s of t h e form a neighbourhood of zero i n
Em
i s contained and compact i n
f o r each
lm=iE
m
($m)m=2 denote a sequence i n (Elf and l e t Using t h e n a t u r a l embedding of each Em i n P
i t follows t h a t
P
Now suppose
=
E
P
lz=2$m$:-1
where
Pa("E).
Vm
and each compact s u b s e t of k.
is E
Let
Qm # 0 E EA f o r every m32. E and t h e p o l a r i z a t i o n formula
Since
@HY(nE) E
p("E).
neighbourhood of zero i n IIPII
Vm
Then t h e r e e x i s t s a sequence
Em, such t h a t
6 1.
Cm=1
E
Lo
f o r some p o s i t i v e i n t e g e r
m
i t follows t h a t
lm=lVm
m
(Vm)m=l,
Vm
a
20
Chapter I
For each
q,m(ym) # 0 .
choose
ym Vm such t h a t n-1 P(x+ym) = @,(XI ( ( ~ ~ ( y ~ ) ) and hence m22
Hence t h e r e e x i s t s a neighbourhood o f z e r o
II@ml)
v1 <
w
f o r every
ma2.
in
V1
a sequence i n
Hence we c a n c o n s t r u c t
Ei
11 @mil!,,m
such t h a t such t h a t
P
i n any l o c a l l y convex s p a c e
P
(Wm)i,2,
0,
and
m.
f o r every i n t e g e r
=
Since
CC
c a n b e embedded
as a c l o s e d complemented s u b s p a c e t h i s
F
q(nE;F) #
example can b e m o d i f i e d t o show t h a t
i s m e t r i z a b l e and n o t normable and
pH,(nE;F)
whenever
El
1122.
We now i n t r o d u c e a f u r t h e r s p a c e o f polynomials nomials.
then
V1
Q
such t h a t
El
@(nE).
g!
x
i s m e t r i z a b l e and n o t norm-
El
c o n t a i n s a neighbourhood system a t
a b l e t h e n it (@m):=z
However i f
If
-
t h e nuclear p o l y -
In c o n t r a s t t o t h e o t h e r s p a c e s o f polynomials we have i n t r o d u c e d ,
t h e s e form a s u b s p a c e of t h e s p a c e o f c o n t i n u o u s p o l y n o m i a l s .
The c o n c e p t
as a d u a l s p a c e and t o
o f n u c l e a r polynomial a l l o w s u s t o r e g a r d 6'("E)
The c o n c e p t i s a l s o u s e f u l i n d e v e l o p i n g t h e
develop a d u a l i t y theory.
t h e o r y of holomorphic f u n c t i o n s on n u c l e a r s p a c e s . D e f i n i t i o n 1.26
(a)
E
Let
F
and
be l o c a l l y convex spaces.
n
L E $ ( ~ E ; F )i s called a nuclear F
into
U
bourhood (A k ) i = l
k
E
and
R
yk
E,
in
(Yk>i,l
and
E
a bounded subset
from E (b)
E
and sequences where + i , k E B
for a l l
k
B
of
F,
m
( @ i , k ) k = l , i = l , .* q > n f o r aZZ i and
8
such t h a t
L ( X 1 > . . . > X n ) = l i = l A k @ l , k ( X 1 )' . * (X1, ...,X,) E E n .
We Let j t ' , f ~ ; ~ )
l i n e a r mapping from
i f there e x i s t a convex balanced zero neigh-
@n,k(xn)Yk f o r
denote t h e space of alL nuclear n
l i n e a r mappings
into F. P
E
@(%;F) i s called a nuclear
n
homogeneous polynomial
i f there e x i s t a convex baZanced zero neighbourhood
a bounded subset W
( @ k ) k = l IUo
and
B
of
F,
(Xk)L=l GB
E
R1
such t h a t
U
and sequences
in
E,
21
Polynomials on locally convex topological vector spaces
1"k = l A k @"(x)yk k
~ ( x )=
x
f o r every
E.
in
we l e t P 1 , ( " ~ ; ~ ) denote the space of aZZ nucZear n-homogeneous polyE
nomiaZs from Taking
F.
into
1.26(a) we obtain the d e f i n i t i o n of a nucZear Zinear
n=l i n
A ZocaZZy convex space
mapping between ZocaZZy convex spaces. t o be nucZear i f f o r every
a
E
cs(E)
B
there e x i s t s
EB
t h a t the carwnicaZ mapping from
Ea
onto
E
cs(E),BZa,
i s nucZear.
e x i s t s a sequence of
E'
such t h a t E
convex space
(An)n=l
in
p(x)
C n = l ( a n l I @ n ( ~ ) f o r every
and
.tl
1
EA
i s said t o be duaZ nucZear i f
Theorem 1 . 2 7
If E dN("E)
p
x
in
;k("E)
and
on
E
there
PN("E)
E.
A ZocaZZy
is nucZear.
i s a nuclear ZocaZZy convex space and
=
such
(@n)n a n equicontinuous s u b s e t
m
,<
i s said
This i s equi-
v a l e n t t o t h e c o n d i t i o n t h a t f o r e v e r y c o n t i n u o u s semi-norm m
E
nEN
then
p("E).
=
Proof
The second e q u a l i t y f o l l o w s immediately from t h e f i r s t s i n c e
-[x,("E))
= PN(nE).
show t h a t i f Since IL(xl
L
L
,..., xn) I
E
By d e f i n i t i o n c??~(~E)c- J?-(nE)
X(nE)
L
then
E
$("E).
is continuous t h e r e e x i s t s Q a(x,)
...
a(xn)
f o r any
f a c t o r i z a t i o n lemma, we may look upon E
is nuclear t h e r e e x i s t s a
from
EB
onto
(Yk)k C E a ,
x
For any
xl,
Ea
i s nuclear.
( @ k ) F = lc E '
=
in
E
L
cs(E)
~ E C S ( E ) such t h a t xl
,..., xn
E
Hence, by t h e
E.
as a n element o f
%(nEa).
Hence t h e r e e x i s t
in
w e have
Ea
Since
such t h a t t h e c a n o n i c a l mapping (Ak)kE k , ,
such t h a t
lE=lAk$k(x)yk
..., xn
in
B
and so it s u f f i c e s t o
f o r every
x
in
E
where
22
Chapter I
Since
t h i s completes t h e p r o o f .
51.3
TOPOLOGIES ON
SPACES OF
POLYNOMIALS
We now look a t t o p o l o g i e s on t h e v a r i o u s s p a c e s of polynomials we have S i n c e @(lE) = E '
defined i n t h e previous section.
( a l g e b r a i c a l l y ) , we
h a v e v a r i o u s g u i d e s from t h e d u a l i t y t h e o r y o f l o c a l l y convex s p a c e s .
The
most u s e f u l t o p o l o g y from t h e l i n e a r v i e w p o i n t i s t h e s t r o n g t o p o l o g y
-
t h a t i s t h e t o p o l o g y o f uniform convergence on t h e bounded s u b s e t s o f
E.
T h i s t o p o l o g y w i l l b e d e n o t e d by B .
We s h a l l s e e t h a t i t i s not t h e most u s e f u l t o p o l o g y from t h e holomorHowever, i t does s e r v e a p u r p o s e i n o u r development
p h i c p o i n t o f view. and m o t i v a t i o n .
For t h e s a k e o f e f f i c i e n c y we s h a l l always t r y t o d e f i n e
o u r t o p o l o g y on as l a r g e a s p a c e as p o s s i b l e . De finit i on 1.28
topology or the
Let B
E
and
F
be l o c a l l y convex spaces.
topology on @ ("E;F)
M
i s defined t o be the topoZogy of
uniform convergence on t h e bounded subsets o f (PM(nE;F),B) b y t h e semi-norms
The strong
E.
i s a l o c a l l y convex s p a c e and i t s t o p o l o g y i s g e n e r a t e d
I\
(/a,3
t h e bounded s u b s e t s o f
where
c1
ranges over
cs(F)
and
B
ranges over
E.
The f o l l o w i n g a r e e a s i l y proved. P roposi t i on 1.29
space, then
I f
E
(?("E;F),B)
P roposi t i on 1.30
If
l o c a l l y convex space then
i s a normed linear space and
F
i s a Banach
i s a Banach space. E
i s a ZocalZy convex space and
F
i s a complete
23
Polynomials on locally convex topological vector spaces
(~~,,("E;FI,B)
i s a complete ZocaZZy convex space.
we naturally consider the compact open topology
On @Hy(nE;F)
Let
Definition 1.31
E and
compact subsets of
F be ZocaZZy convex spaces.
We denote t h i s topoZogy by
E.
The compact
i s the topoZogy of uniform convergence on t h e
open topoZogy on $IHy("E;F)
T ~ .
(pHy(nE;F),~o) is a locally convex space and its topology is generated by the semi-norms 11 11 where c1 ranges over cs(F) and K ranges a,K
over the compact subsets of E. Since every compact subset of a locally convex space is bounded it follows that B fj =
on FHy(%;F).
3 T~
By the Hahn-Banach theorem we have
~ if and only if the closed convex hull of each bounded subset of E
T
is compact
i.e. if and only if E
is a semi-Monte1 space.
Since the uniform limit of continuous functions on a compact set is continuous the following is true. -Proposition 1.32
E be a ZocaZZy convex space and
Let
ZocaZZy convex space.
Then
space.
(BHy(n~;~) s ~ o )
By restriction B
convex topologies on
F a complete i s a compZete ZocaZZy convex and
T~
define locally
6 ("E;F) . However, we shall need a further
topology on
!?fE;F). This topology, denoted by T ~ , is the strongest topology we It is motivated by the factorization formula, the define on p(nE;F). definition of the strong topology on @("E;F)
when E' is a normed linear
space and certain properties of analytic functionals in several complex variables theory. It is perhaps more useful than T~ or B since it has stronger topological properties but it is more difficult to characterize in a concrete fashion. We first consider polynomials with values in a normed linear space. Let
E be a ZocaZZy convex space and l e t F be a T~ topology on Q("E;F) i s defined a s t h e i n d u c t i v e l i m i t topoZogy i n t h e category of ZocaZly convex spaces and conDefinition 1.33
normed Zinear space.
The
tinuous l i n e a r mappings
(B("E;F)
, T ~ ) =
o ~ P (;F), ~ Ea E CS(E) , t h a t i s lim a Zcs(E)
(P(~E~;F), B ) .
24
Chapter 1 p
Hence a semi-norm V
every neighbourhood P(P) ,<
on
d:
("E;F)
o f zero in C(V) I/pIIV
E
is .ru-continuous if and only i f f o r there e x i s t s
f o r every
P
in
c(V) 2 0
such t h a t
F("E;F).
W e w i l l subsequently s e e t h a t t h i s amounts t o saying t h a t a semi-norm on
i s .ru-continuous i f and only i f it i s ported by t h e o r i g i n and
,>("E)
f o r t h i s reason we c a l l
( 3 (nEa;F) ,B)
Since
t h e ported topology.
Tu
i s always a normed l i n e a r space
i s a bornological space when Banach space
F
i s a normed l i n e a r space.
i s a Banach space and
?(nEa;F)
(
( ?(nE;F)
When
, T ~ )
t i v e l i m i t of Banach spaces, i . e . an u l t r a b o r n o l o g i c a l space. ular,
(
? (nE;F) ,
(nE;F) ,.rW) F
is a
i s an inducIn p a r t i c -
i s then a b a r r e l l e d l o c a l l y convex space, t h a t i s
T ~ )
every closed convex balanced absorbing subset i s a neighbourhood of zero. For a r b i t r a r y
we use t h e weak form of t h e f a c t o r i z a t i o n lemma
F
(lemma 1.18) and d e f i n i t i o n 1.33 t o d e f i n e Definition 1.34
E
Let
Q("E;F)
defined on
and
F
.ru
on
P("E;F).
be l o c a l l y convez spaces.
Then
T*
as
The following elementary r e s u l t shows t h e r e l a t i o n s h i p between t h e topologies we have defined. Proposition 1.35
any p o s i t i v e i n t e g e r
(a) .ru Ibi
B
3
B
.ro
2,
and
For arbitrary locaZ7.y n we have
.ro
on
convex spaces
.
and
?("E;F).
d e f i n e t h e same bounded subsets o f
and hence have t h e same associated bornological topology
E
6(nE;F)
F
and
is
25
Polynomials on locally convex topological vector spaces We now g i v e a number o f e l e m e n t a r y examples r e l a t i n g t h e above t o p o l -
-
ogies
Afterwards, we d e f i n e a
f u r t h e r examples appear i n l a t e r c h a p t e r s .
topology on t h e s p a c e o f n u c l e a r p o l y n o m i a l s . Example 1 . 3 6
i s a n i n f i n i t e dimensional Banach s p a c e and
E
If
a l o c a l l y convex s p a c e , t h e n T
=
T~
2
6
have t h e same bounded sets and hence
a s s o c i a t e d with
T
Example 1 . 3 7
~
T
g'(lE)
If
bounded s u b s e t s o f
T~
i f and o n l y i f
= E'
Example 1 . 3 8
T~
and
Let
Example 1.39
Moreover, i f
If
subsets of
compact s u b s e t o f n
all
u
n
v
n
norm on P(*E) P(P) a
C")
E
=
cs(E)
-
..., 1
E
=
( P ,..., 4.0
d e f i n e d by
do n o t
and l e t
t h e n t h e r e e x i s t compact
such t h a t
K2,
K c K1 x K 2 .
Now e v e r y
IIPnIIK
i s a bounded s u b s e t o f
B
T~
= 0
for
(8( 2 E ) , T o ) .
0 . E and l e t C ntil p o s i t i o n
(0,
then
and
T~
P n ( ( x n ) n , ( y n ) n ) = xnyn,
Thus
=
I",=,
<'("E;F).
i s f i n i t e dimensional and hence
sufficiently large.
Let
?(2E)
and
K1
C"),
There-
and hence are n o t e q u a l .
i s a compact s u b s e t o f
K
and
EN
on
We show t h a t
E =
b e d e f i n e d by
Pn:E + E
B = (Pn):=l.
Using t h e method o f example
i s semi-Monte1 and hence a
E
= B = T~
T~
d e f i n e t h e same bounded s u b s e t s o f
Let
is a
E
E'.
d e f i n e t h e same bounded s u b s e t s of
T
s p a c e , it f o l l o w s t h a t
g 3 1'1
T~
on
= B
.?w
(j' ("E) ,@) i s m e t r i z a b l e and hence b o r n o l o g i c a l .
on ?(("E;F).
= @
R
~
c o n t a i n s a c o u n t a b l e fundamental system o f bounded s e t s
E
it follows t h a t
T~
T
Consequently i f
on
is a
be a c o u n t a b l e i n d u c t i v e l i m i t o f normed l i n e a r
T~,@
Since
> >
i s distinguished.
E
F
a r e l o c a l l y bounded.
,?(nE;F)
b e a normed l i n e a r s p a c e .
F
1.24 w e s e e t h a t 'j, ("E;F).
E
Let
s p a c e s and l e t
and
T
i s t h e b o r n o l o g i c a l topology
T~
i s t h e bornological topology a s s o c i a t e d with
w
is
F
Moreover,
i s a m e t r i z a b l e l o c a l l y convex s p a c e and
E
non-distinguished Frgchet space then
fore
?("E;F).
.
Banach s p a c e t h e n t h e Hence
If
on
To
I P(nun,vn)
,...
)
E
)
E
Let
C").
p
d e n o t e t h e semi-
nth position
- P(o,~,)
a(un,O) = 0
for all
I
for all n
P
E
~ ( 2 ~ 1 .
sufficiently large.
Hence
26
Chapter 1
P(nun,vn) = P(O,vn) P
i n P (2E).
n
and hence
T o
it f o l l o w s t h a t
for all
s u f f i c i e n t l y l a r g e and
S i n c e t h e semi-norm which maps
is
P((O,vn)(
n
for all
p
is a T
P(2E)
E
(Q( ~ E ) , T)
c o n t i n u o u s and B
( P (2E)
is not a
,T~]
T ~ , @
and
E
convex subset of a ZocalZy convex space
and
L
and
P
B
If
.
II
we l e t
$ k , i ~El}
8 N ( n E ) we l e t
E
where t h e i n f i m a are taken over a l l possible representations of IIB(L) IIB(L) hood
and
and
( r e s p . P) V
These
i s a balanced
$,("E)
E
and each and if
is
P(nE).
p N n( E ) , no, li B r e s p e c t i v e l y o n 8 ("E) .
T~
!?(2E). T~
on Y ( ~ E ) .
T~
We now d e f i n e t h r e e t o p o l o g i e s on correspond t o
p(P n ) = n
bounded s u b s e t o f
T~
-
is barrelled Since
i s a b o r n o l o g i c a l s p a c e and
t h e b a r r e l l e d topology a s s o c i a t e d with This r e s u l t extends e a s i l y t o
f o r every
m
I P(nun,vn)
to
c o n t i n u o u s semi-norm on :?(2E).
w e h a v e shown t h a t
We s h a l l s e e l a t e r t h a t
P
p(P) <
IIB(P)
may b e i n f i n i t e .
ItB(P)
are always f i n i t e .
&:,(%)
(respectively
E
of z e r o such t h a t
ITv(L)
However, i f
L
and
P.
i s bounded t h e n
B
Moreover, by d e f i n i t i o n , i f
L
t h e n t h e r e e x i s t s a neighbour-
Q,(nE)) <
and
TIv(€')
<
m.
These a l l o w us t o
give t h e following d e f i n i t i o n . Definition 1.40
(a)
The
IIo
topology on
t h e l o c a l l y eonvex topology generated by subsets of
AN(nE)
flK
PN(nE)I is K ranges over a l l compact
as
iresp.
E.
( b ) the
IIB
topology on J,("E)
( r e s p e c t i v e l y @,(nE)
locaZ2y convex topo2ogy generated by
nB
as
3
)
i s the
ranges over
all bounded subsets of E. ( c ) A semi-norm be
p
on J'",lnE)
IIwcontinuous
(respectiveZy ? N ( n E ) )
if for every neighbourhood V
is said t o in
of zero
21
Polynomials on locally convex topological vector spaces E
c(v) > 0
there e x i s t s p(L) s c(v)nV(L)
nu
11 LII B n 6
Since
of
B
E
it f o l l o w s t h a t for all
d N ( n E ) and '$"(nE) Proposition 1.41 TIo = IIR = B =
K1
ced subset
E
of
Since
~
,
n.
for every
%
3 T
w
on
Then
c K >0
I\L/I
(L) i c:
and a convex balan-
(Kl)n
for every
L
in
n.
i s q u a s i - c o m p l e t e and d u a l n u c l e a r i t s c l o s e d bounded
E
B =
and
T
II 0
= ItR.
E,
+Ei
such t h a t t h e c a n o n i c a l mapping from E' m a, K1 (xk)k,l E E l , ( a k ) k = l a sequence i n
i v e ) and a bounded sequence
@
Now l e t A
nu
Moreover, for each convex
exist
~1 E
and
E ' i s n u c l e a r w e c a n , g i v e n K a convex b a l a n c e d compact subB choose a convex b a l a n c e d compact s u b s e t K1 o f E c o n t a i n i n g K
Since
f o r every
f o r e v e r y convex b a l a n c e d
II R > B
there e x i s t
such t h a t
s e t s a r e compact and hence
set i n
;J~(~E))
be a quasi-complete dual nuclear space.
and for every non-negative
Proof
IIB(P) T
n.
K of E
balanced compact subset $N(nE)
E
Let
on d,("E)
T~
Xo 3
E
continuous serni-norms.
11 Pllgi
and
IIB(L)
P
irespectively
is t h e topology generated by a l l Xu
subset
p ( P ) 6 c(V)IIv(P))
(respectively
i n A.,("E)
L
f o r every
such t h a t
i n E'
K1
i n L ~ ( ~ E )NOW . suppose K.
E
Hence t h e r e i s semi-reflex-
such t h a t
E' K1
K.
There e x i s t s , by t h e Hahn-Banach theorem,
such t h a t
v e r g e s u n i f o r m l y on
(@k)k=l i n
(since
where t h e s e r i e s converges u n i f o r m l y on
L €AN("E).
(XN(nE),no)'
m
i s nuclear. K~
a ( L ) = IIK(L) L
=
I",l
We t h e n have
and
$m,l..
Ia(A)I
.
Q IIK(A)
f o r every
J,m,n where t h e series con-
28
...,kn =1xk 1
=
*
* . 'kn
1
...
@k 1 L(a , . . . , a 1 n kl kn
T h i s completes t h e p r o o f . Combining theorem 1 . 2 7 and p r o p o s i t i o n 1 . 4 1 w e o b t a i n t h e f o l l o w i n g result. C o r o l l a r y 1.42
then
X("E)
=
If
gN(nE)
If
Theorem 1 . 4 3
E
E
is a quasicomplete nuclear and dual nuclear space and
TIo =
T~
for every
n.
is a quasi-complete dual nuclear space then
( L ( ~ E ) , T ~is ) a nuclear space.
K
Proof
If
sequence
(Xm)m=l
f o r every
m
@
in
i s a compact s u b s e t of in
E'.
E and
Hence i f
m
n
E
E
R1
t h e n t h e r e e x i s t s a compact such t h a t
i s any p o s i t i v e i n t e g e r and
then
lrisn- 1
(and by induction)
L
E
&(("E)
29
Polynomials on locally convex topological vector spaces
a compact sequence
t h i s completes t h e proof.
The form o f t h e above i n e q u a l i t y w i l l b e u s e d i n c h a p t e r 3 Proposition 1.44 T
E
Let
on &("E)
= i7 w w
be a quasi-complete nuclear space.
Proof
nu
II~(L)c cn
such t h a t
C>O
Since
= T
IIw
2 T
w
W
of zero, contained i n V
Vn f o r every
L
in
, and
~ ( n ~ ) , n = 1 , ... 2,
it s u f f i c e s t o p r o v e t h e above i n e q u a l i t y t o show
w'
By t h e n u c l e a r i t y o f bourhood
V
i c a l mapping m
($k)k=l in
IILI/
of zero, there
V
Moreover, f o r any convex balanced neighbourhood e x i s t a convex balanced neighbourhood
Then
n.
f o r any p o s i t i v e i n t e g e r
W"
E
E we can choose, g i v e n a convex b a l a n c e d n e i g h -
o f z e r o , a n e i g h b o u r h o o d of z e r o EW + EV
and
m
is nuclear.
L ~ d ( % ) and
Now suppose If
Y1,.
Hence t h e r e e x i s t
. . ,Yn
V
E
L1,
E
co
( x ~ ) c~ V= ~such t h a t
where t h e convergence i s i n
E
s u c h t h a t t h e canon-
WCV
x = rk=lXkQk(x)xk
for all
EV'
IILII
= 1.
V"
then
and h e n c e
=
Now
Ikl,
lL(xk , 1
...,kn . .. , x
kn
'kl )
1
... x
kn
,< 1
.
L(Xk ,. .,Xkn) bkl 1
f o r any c h o i c e o f Hence
x
. .. $k .
,. . . , x
kl
n
kn
in
V
since
x
Chapter 1
30
.,knIhkl ."
kn
I
=
then
Thus nw(L) if 11 LII vn
6 =
cn IlLll vn and since this inequality is trivially satisfied we have completed the proof. m
The preceeding results can be transferred to n homogeneous polynomials by using the inequality II*(L)
for any L B
of
E.
6 IIB(L)
in XS("E)
nn
c n! nB(L)
n A~("E)
=
~ki(~E) and any convex balanced subset (X ZPE) ,no)
In particular, we find
(A i(nE> ,nu)
=
($?N(nE)
=
WNN(nE),no)
and
,nu).
We now summarize results obtained in this way. Proposition 1.45 Let E be a quasi-complete Locally convex space and l e t n be a non-negative i n t e g e r . ( a ) If E i s dual nuclear then
no
= B =
and
('J(~E),T~)i s a nucZear space. ( b ) IS E i s nuclear then 2bN(n~) = 'J'("E and
= T
w
on J"(nE)
.
Moreover, the estimates given in propositions 1.41 and 1.44 are still valid, with minor modifications, for spaces of homogeneous polynomials on the appropriate locally convex spaces.
31
Polynomials on locally convex topological vector spaces 51.4
DUALITY THEORY FOR SPACES OF POLYNOMIALS I n t h i s s e c t i o n w e c o n s i d e r l i n e a r f u n c t i o n a l s on t h e l o c a l l y convex This t o p i c is currently t h e subject
s p a c e s o f polynomials d e f i n e d i n 5 1 . 3 .
o f r e s e a r c h and s h o u l d p l a y an i m p o r t a n t r o l e i n t h e development o f t h e Our p r e s e n t a t i o n of r e s u l t s i s n o t f u l l y com-
subject i n t h e near future.
p r e h e n s i v e b u t h o p e f u l l y o u t l i n e s t h e main developments and p r o v i d e s a g l i m p s e o f f u t u r e developments. We show t h a t c o n t i n u o u s l i n e a r f u n c t i o n a l s on s p a c e s o f polynomials Our main t o o l i n o b t a i n i n g
can themselves b e r e p r e s e n t e d by polynomials. t h i s r e p r e s e n t a t i o n i s t h e Borel t r a n s f o r m . D e f i n i t i o n 1.46
Let
A be a vector space of
.
form on h
The Bore2 transform of
T,BT,
vaZued n-homogeneous
C
E
poZynomiaZs defined on a ZocaZZy convex space
and Zet
i s defined on
T
be a Zinear
{$IEE*I+"EA >
by the formuZa BT($I) If BT
to
then
=
i s a subspace o f
F
T(99
EX
and
ms(Fn)c&
i s an n-homogeneous polynomial.
F
'4
if BT E $ ~ ( ~ E * and )
=
?(nE)
then t h e r e s t r i c t i o n of
For example i f
o r 3N(nE)
then
A
= qa("E)
BT E ? a ( n E 7 ) .
A i s a l o c a l l y convex s p a c e and T i s c o n t i n The Borel t r a n s f o r m w i l l o n l y be u s e f u l i f it i s i n j e c t i v e . T h i s
In t h e c a s e s w e consider uous.
w i l l always b e t h e case i f
p r o p e r t y and
fi
= PN(nE)
or if
E
h a s t h e approximation
'J'("E).
=
P r o p o s i t i o n 1.47
The Bore2 transform i s a vector space isomorphism from
( i ) ( 2 , ( n ~ ),n6)
1
onto D ( ~ E ; )
and
(ii)( @ N ( n ~,no) )
Under t h i s isomorphism the equicontinuous subsets o f espond t o the locaZZy bounded subsets o f (PN(%) ,no)
subsets of P(n(E,.ro) Proof
p("E;)
onto
P(~(E:T~)
(PN(("E),ITB)
corr-
and the equicontinuous
correspond t o the Zocally bounded subsets of
'1. S i n c e b o t h c a s e s a r e proved i n a s i m i l a r f a s h i o n , we o n l y c o n s i d -
er t h e case There e x i s t
(PN(E),i16).
B
i s l i n e a r and i n j e c t i v e .
and
B
an a b s o l u t e l y convex bounded s u b s e t o f
c>O
Let
such t h a t IT(P) I f
c nB(p)
f o r every
P
i n P,("E).
T E (@N(nE),TIbgll E
32
Chapter 1 In p a r t i c u l a r i f
c 11$"1/, 6 c .
$
Hence
tinuous subset of Now suppose
1 then
h
T h i s a l s o shows t h a t t h e image by
PI
and
P("Egt)
E
of
B
E.
If
/I PillBo P
E
T(P)
li=lP1($i).
=
ear o p e r a t o r on 3'N(nE), t h e representation of
c
6
of z e r o i n
V
ll$ill
Moreover, i f
P.
and
?(nEgt).
E,
then w e
T
is a well defined l i n -
T(P)
i s independent o f
One e a s i l y shows t h a t i.e. the definition of
Eb
o f an equicon-
f o r some c l o s e d a b s o l u t e l y m n and P = l i = l $ i , $ i E~l
PN(nE)
f o r some neighbourhood m
B
i s a l o c a l l y bounded s u b s e t o f
(?,("E),TIB)'
convex bounded s u b s e t
let
/ B T ( $ ) / = IT($")[ h
i s bounded on a neighbourhood o f z e r o i n
BT
so it i s continuous.
I/$ //
and
E E I
= 0
then
Pt(Oi) = 0
and
hence
=
Hence
T
cnB(P).
( ? N(nE) ,")
E
and
BT($)
=
T($n) = P I ( $ ) .
i s s u r j e c t i v e and a v e c t o r s p a c e isomorphism.
{P
=
E
v("Ei) ;
)I P \ \ Bo 8
C}
T h i s shows t h a t
B
The above a l s o shows t h a t
f o r any
c>O
and t h i s completes
t h e proof. Proposition 1.48 (?N(nE)
E'
,nu)
The Borel transform i s a vector space isomorphism from
onto
,FE ("El)
f t h e space of n
homogeneous poZynomiaZs on
which are bounded on the equicontinuous subsets of
El).
Under t h i s
isomorphism t h e equicontinuous subsets of subsets of
? ("El)
subsets of
El.
Proof T
E
5
6
(?N(nE) ,II) I correspond t o which are uniformly bounded on :he equicontinuous
The p r o o f i s v e r y similar t o t h e proof of p r o p o s i t i o n 1 . 4 7 .
(?N(nE) ,nu)
and l e t
V
Let
d e n o t e a n a b s o l u t e l y convex neighbourhood of
33
Polynomials on locally convex topological vector spaces 0
in
and
E.
BT
E
c(V)> 0
There e x i s t s
? ("El). 5
uous subset of
such t h a t
This a l s o shows t h a t t h e image by
i s a subset of
(yN("E),na)'
5
B
("El)
of an equicontin-
c o n s i s t i n g of fun-
E'.
c t i o n s which a r e uniformly bounded on t h e equicontinuous s u b s e t s of Now suppose
PI
(nE').
E
We d e f i n e
i s a neighbourhood of zero i n
Moreover, s i n c e isomorphism.
E
and
BT($) = PI($)
as i n proposition 1.47.
T P
E
?N(nE)
If
V
then
t h i s shows t h a t
B
i s a v e c t o r space
The r e s u l t about equicontinuous s e t s a l s o follows from t h e
above. We g i v e
Quite a number of c o r o l l a r i e s can be deduced from t h e above. The f i r s t i s perhaps t h e most i n t e r e s t i n g .
j u s t a few examples.
Since
t h e c o l l e c t i o n of spaces which occur i n t h i s c o r o l l a r y i s r a t h e r i n t e r e s t -
s),
ing, (see chapters 3and Definition 1.49 EL;
we g i v e them a s p e c i a l name.
A locally convex space
E
E
i s fuzzy nuclear i f
and
are both complete infrabarrelled nuclear spaces. A f u l l y n u c l e a r space i s a r e f l e x i v e nuclear space and t h e s t r o n g dual
of f u l l y n u c l e a r space i s f u l l y n u c l e a r .
Every FrGchet n u c l e a r space i s
f u l l y nuclear. Corollary 1.50 @("E),
n
I f
E
is a f u l l y nuclear space then
T~
= T*
a p o s i t i v e integer, i f and only i f F'M(nEA) = ?(nE;j)
B -bounded subsets of p:d(nEL)
are locally bounded.
on
and the
Chapter 1
34 Proof
Since
i s an i n f r a b a r r e l l e d l o c a l l y convex s p a c e t h e equicon-
E
c o i n c i d e w i t h t h e bounded sets and hence
tinuous subsets of
Ek
PM("Eb) = Pt(("E').
I t now s u f f i c e s t o a p p l y theorem 1 . 2 7 , and p r o p o s i t i o n s
1 . 4 5 , 1 . 4 7 and 1 . 4 8 t o complete t h e p r o o f . In particular w e note t h a t
o = Also t h i s shows t h a t
nuclear space.
T
T
w
T~
on
#
?(nE) on
T~
if
i s a Frgchet
E
if
F(nC(N) x CN)
1122,
a r e s u l t which w e h a v e a l r e a d y proved d i r e c t l y (example 1 . 3 9 ) . Corollary 1.51
If E (8(nE)
If E
Corollary 1.53
If E
ng
Q,(nE;)
I
C o r o l l a r y 1.52
FreTchet space then
is a r e f l e x i v e nuclear space then
=
i s an infrabarrelled l o c a l l y convex space then
i s an infrabarreZled
nw
DF s p a c e o r a distinguished
on P N ( n E )
We now look a t some examples i n which t h e Bore1 t r a n s f o r m g i v e s a t o p o l o g i c a l isomorphism.
We f i r s t need some p r e l i m i n a r y r e s u l t s .
E, a Locally convex space, has property E
of
K
subset
E K
such t h a t
EB.
is contained and compact i n
(EB i s t h e v e c t o r s u b s p a c e o f t h e norm whose u n i t b a l l i s
E
If
B).
g e n e r a t e d by B
EB
i s a Banach
S t r i c t i n d u c t i v e l i m i t s o f FrGchet s p a c e s and s t r o n g d u a l s o f
space.
f u l l y nuclear space has property Lemma 1 . 5 4 f,,(nE;F)
9M ("E;F)
If in
B
K E
i s compact i n such t h a t
continuous.
Lemma 1 . 5 5
If E
(EJ)
K
Hence
E, If
K. E
F
(s)
then
nEN.
and any
t h e n t h e r e e x i s t s an a b s o l u t e l y convex
i s compact i n P
In p a r t i c u l a r , every
has property
for any l o c a l l y convex space
i n d u c e t h e same t o p o l o g y on T
(5).
(s).
If the locally convex space =
Proof
hence
and endowed w i t h
B
i s complete t h e n
i n f r a b a r r e l l e d Schwartz s p a c e s have p r o p e r t y
set
if for each cornpact B of
(5)
there e x i s t s an absolutely convex bounded subset
P
E
EB.
Hence
PM(("E;F) t h e n
QHy(nE;F)
and
T , T ~
PIK
and
/I I / B
is
T~
and
$h(nE;F) = PH,,("E;F).
is a f u l l y nuclear space then @Hy(nE) is equal t o
35
Polynomials on locally convex topological vector spaces
t h e cornpZetion of Proof
The completion o f
convex s p a c e
.
~ )
( L 1 ( n E ) , ~ o ) l i e s i n j?Hy(nE) (j'("E)
and q u a s i c o m p l e t e w e c a n c h o o s e
KIC-K,
m
m
(yn)n=l C K 1
(Xn)n=l
E
(Qn)zX1
R1,
such t h a t f o r e v e r y x
x
A(xl,.
. .,xn)
. .,xn
xl,.
1
n
IA(yi
1
. . . . .yi
n
choose a f i n i t e s e t o f i n d i c e s
in
Kl
EK
E;<
with
.
Since
-+
E
i s nuclear.
K1
llQnll
C
1
and
m
m
6 (li=llxil)n
)
I
m.
<
m
and
Hence, f o r any 6 ' 0 ,
w e can
such t h a t
F
1 A . ... X i
<
i s c o n t i n u o u s on
A
. . ,&=lXi+ii(xn)Yi)
A(1.1=1X .1 + . 1( x l ) y i , .
1
-
E
K
E
m
=
A. , . . . , X i
IIA
and l e t
m
=
Now
?UP l , . . . ,in
LiY("E)
E
K
in
where t h e series converges u n i f o r m l y i n
we have f o r any
E
A
E. Since E i s dual nuclear a n a b s o l u t e l y convex compact
K1, such t h a t t h e c a n o n i c a l i n j e c t i o n
E
Hence t h e r e e x i s t
K:
Let
, T ~ ) .
b e a n a b s o l u t e l y convex compact s u b s e t o f
s u b s e t of
f o r any l o c a l l y
and t h u s t o p r o v e t h i s r e s u l t i t s u f f i c e s t o show
E
l i e s i n t h e completion o f
S'HY(nE)
K
,T
(t ("E)
A(yi ,..., yi ) $ i . . . $i ( 1 < 6/2. S i n c e E l i s d e n s e n 1 k 1 k K w e c a n choose a sequence of c o n t i n u o u s l i n e a r forms on E , ( $ i ) i = l , l1
ElK
such t h a t
Ill
F
Q
il
... Q
in
-
1 JI F
il
... 1L i II
< 6/2
n K
Combining t h e s e two i n e q u a l i t i e s we o b t a i n t h e d e s i r e d r e s u l t . If
(E,T)
iated with
i s a l o c a l l y convex s p a c e t h e n t h e Mackey t o p o l o g y a s s o c ( n o t t o b e c o n f u s e d w i t h t h e t o p o l o g y o f t h e M-closure
i s t h e f i n e s t l o c a l l y convex t o p o l o g y on
d u a l as T
(E,T).
E
If t h e Mackey t o p o l o g y a s s o c i a t e d w i t h
t h e n we s a y t h a t
(E,r)
T ~ )
which h a s t h e same c o n t i n u o u s
i s a Mackey s p a c e .
T
coincides with
An i n f r a b a r r e l l e d l o c a l l y
Chapter I
36
convex space is a !.lackey space.
If E i s a f u l l y nuclear space then
Proposition 1.56
is the completion o f a nuclear space and hence it @Hy(nE),~o) is a complete nuclear space. Let s s be the strong topology on
Proof
(nEb)
'
CJ?,,("E)
(proposition 1.47).
, T ~ )
($("E;)
, T ~ )
is the strong
dual of a semi-reflexive space and hence is a barrelled Mackey space. (jJ(nE;),~u) is also a barrelled space and hence a Mackey space. To complete the proof we need only show
'
(?(nEb) Since
' .
, T ~ )
(P(~E;) ,TB)
is semi-reflexive,
, T ~ )
osition 1.48,
(j)("EA)
=
, T ~ )
'
(5'(nEb)
, T ~ )
2
$'M(nE)
FHy(nE),
=
1
=
S;iY(n~).
BY prop-
and this completes the
proof. i s a fully nuclear space then
If E
Proposition 1.57
(!j"nE) ,TJ i f and only i f the If
Proof ((P("E)
,T,);)
(PHy ("E;)
, @(%)
su bounded subsets o f
(P (nE), g
;
T
~
@("E)
;a )
(PHY("E;),
and hence
T ~ )
then by proposition 1.56
(Q("E),T~)
space and thus it is reflexive. Hence
are Locally bounded.
is a barrelled semireflexive
(gHy(nEb),~o)
is also reflexive
and the equicontinuous subsets of the dual coincide with the strongly bounded subsets. The strong topology on (BHy(nE ; ~ ) , T ~ ) ' is the T topology by proposition 1.56 and the equicontinuous subsets are the locally bounded sets by proposition 1.47. Conversely if the su bounded subsets of
p("E) are locally bounded then the bounded subsets of (QHy("E;) , T ~ ) are equcontinuous (propositions 1.47 and 1.56). Hence (PHy(nEk),~o) is infrabarrelled and thus reflexive. By proposition 1.56, ((3(nE) , T ~ ) = (!!Hy(nE;),~o)~ is also a reflexive space and (OYEI,TJ;
=
((pHy("~;),
T ~ I ; ) ~
=
(pHY(n~t),~o~.
This completes the proof. We now look at linear functionals on spaces of homogeneous polynomials
37
Polynomials on locally convex topological vector spaces d e f i n e d on F r e c h e t s p a c e s w i t h t h e a p p r o x i m a t i o n p r o p e r t y . nomials s t i l l a p p e a r and p l a y a n i m p o r t a n t r o l e . uous forms as t h e c o r r e s p o n d i n g t h e o r y f o r developed and indeed t h e g e n e r a l t h e o r y f o r an u n t i l l e d f i e l d .
T~ T~
Nuclear p o l y -
We s t u d y o n l y
T~
contin-
c o n t i n u o u s forms i s n o t y e t c o n t i n u o u s forms i s a l m o s t
One c a n e a s i l y show t h a t t h e v e c t o r s p a c e isomorphisms
o f p r o p o s i t i o n 1.1 y i e l d a t o p o l o g i c a l isomorphism when t h e a p p r o p r i a t e s p a c e s are endowed w i t h t h e compact open t o p o l o g y .
This implies t h e f o l l -
owing r e s u l t .
Lemma 1.58
If E
i s a Frgchet space and F
If
P r o p o s i t i o n 1.59
T
convex space and
E
'
IT(L)I
for all L i n .Y.(E;F) then there e x i s t
E
i s a FrLchet spaee,
(x (E;F) ,T
'
~ )
F
i s an arbitrary locally
satisfies
CIILIly,K ~ E C S ( F ) and
where
( a ) a null sequence ($n)n
( b ) a sequence
i s a locally convex space
n
then f o r any p o s i t i v e integer
K
i s a compact subset of E
( x ~ )i n~ E ,
in
F'
such that
f o r all y i n F and all n and m (c) (hn)n=l E k1 where l n l h n l < l
I$,(y)
I S cY(y)
such that
m
T(L)
=
Cn,lXn+n(L(Xn))
for every Proof
L
i n X(E;F)
The compact s u b s e t
K
. of
E
i s c o n t a i n e d i n t h e a b s o l u t e l y con-
vex h u l l o f a n u l l s e q u e n c e , i . e . t h e r e e x i s t s
as n*
Hence
such t h a t
m
(Xn)n=lCE
where
Xn+o
Chapter 1
38
Thus
I
IT&) Since
c SUP Y(LX,). n
L
E
( L ( x ~ ) ) i~s a n u l l sequence i n
X(E;F),
{ ( Y ~ ) : = ~ ; y,
E
Fy and
y(yn)
-to
as
n -1.
F.
Let
co(Fy)
=
i s a normed l i n e a r
co(Fy)
s p a c e w i t n norm
II (Y,),"=, II
SUP Y(Yn). n
=
By t h e Hahn-Banach theorem w e can e x t e n d co(Fy)
s c supy(fn)
I?'(ifn}n)l Since
m
($n)n=l
T
t o a l i n e a r functional
'? on
such t h a t
n
co(Fy)l
=
R1((F
such t h a t %rfnIn) for all
(fn)n
Hence € o r any
in L
t h e r e exist
) I )
Y where
(Fy)l C F '
E
=
f o r every
$,(y)l
s cy(y)
{fnIn
in
m
E
co(FY). R1,
y
for all
lz=llh,,\ in
F
,< 1,
and a l l
n
In= m
co(Fy).
in
Z(E;F)
we h a v e
and t h i s c o m p l e t e s the p r o o f . Proposition 1.60
space.
If
in 2("E;F) B
E
cs(F)
T
E
Let
(d("E;F)
where
K
E
F
be a Frechet space and
'
,T~)
satisfies
I
I
C
a ZocaZZy convex
IIL//B,Kn f o r every
i s an absolutely convex compact subset o f
then there e x i s t an absolutely convez compact s e t
4
K
E
and
and:
L
39
Polynomials on locally convex topological vector spaces
(a)
for every
K
T(L)
such that
Proof
Nn
a sequence i n
L
, {x
lk
,. . .,xn
k k=l
,
1"
=
A~$~(L(X ,..., x ) ) k=l lk nk
%("E;F).
in
We proceed by induction on n .
Proposition 1.59 covers the case
n=l. Assume the proposition is true f o r the positive integer n . T
E
it a ,
x(n+lE;F).
4
Define T
of lemma 1.58.
? E
on
%("E;
f(E;F))
Let
by the isomorphism, call
( % ( n E ; ( ~ ( E ; F ) , ~ O ) ) , ~ O ) and '
Hence, by induction, there exist
-n
a sequence in K
n
By proposition 1.59, there exists a null sequence in K, (yj)y=l each k there exists a sequence of scalars
and f o r
Chapter I
40
Hence
in Q ~ + ' E ; F ) .
for every L
we may reorder the above to obtain a sequence with the required properties This completes the proof. Proposition 1.61
If
E
i s a FrLehet space w i t h t h e approximation prop-
e r t y , then t h e Bore1 transform,
Proof
T
E
( p ("E)
B , i s a l i n e a r isomorphism from
Since E has the approximation property B is injective. Let
'
, T ~ )
IT(P)
and suppose c/lpIIK
for all
P
E
P ("E)
where K is an absolutely convex compact subset of E . By the polarization formula and proposition 1.60 there exist a relatively compact sequence
such that
41
Polynomials on locally convex topological vector spaces A further application of proposition 1.60 shows that
that the equicontinuous subsets of the form {P
B
is surjective and
(~("E),T~)' correspond to sets of Since (E',T~)'= E and d cK}.
@N(n(Ef,~o));IIPI/KO 8 (nE) = BE(nE) it follows, by proposition 1.48, that the closed convex hull of sets of the form U(@~(~(E~,T~)), @(("E)) E
K compact are a fundamental neighbourhood system at 0 in ( @N(n(Eq,~o];flu) c ranges over all possible sets of positive numbers.
as
K
A fundamental neighbourhood system at
0 in
( 6'(nE),~o)h
by the polars of bounded sets. Since
K compact =
U((~("E),T~)',@
U { P E P PE); KCE
("E))
closed convex hull of 0
I I P Ic IcK} ~
K compact
=
U(((P(~E),T~)',
K compact This completes the proof.
@("E))
closed convex hull of
is given
42
Chapter I
51.5
EXERCISES
The f o l l o w i n g e x e r c i s e s d e v e l o p t o p i c s which w e s h a l l e n c o u n t e r i n
l a t e r c h a p t e r s and a l s o c e r t a i n material which we d i d n o t f i n d c o n v e n i e n t t o include i n t h e t e x t . difficult.
Consequently, some o f t h e s e e x e r c i s e s a r e r a t h e r
A s e r i o u s a t t e m p t a t s o l v i n g them, w i l l , however, p r o v i d e a
good d e a l o f i n s i g h t i n t o t h e t h e o r y f y i n g n o n t r i v i a l problems.
-
even i f o n l y as a means o f i d e n t i -
For t h e r e s e a r c h worker t h e y c o u l d e a s i l y l e a d
t o new t e c h n i q u e s and worthwhile r e s e a r c h p r o j e c t s .
Starred exercises are
commented o n i n Appendix 111. 1.62
Show t h a t
Dim(E) 6 1 o r 1.63* -
If
F =
If
E,F
for all and
show t h a t
1.65
E
If P
i f and o n l y i f e i t h e r
and
i f and o n l y i f
n
E
2
are v e c t o r s p a c e s o v e r
G
Q €!fa(F;G)
that
= X:(mE;F)
m=l,m=O,
i s a n i n f i n i t e d i m e n s i o n a l l o c a l l y convex s p a c e , show t h a t
E
Fa(nE) = @("E) 1.64 -
fa(mE;F)
to}.
a"). P
E,
E
pa(E;F)
and
Q o P E(~).(E;G).
a r e l o c a l l y convex s p a c e s and
F
i s c o n t i n u o u s a t one p o i n t i f and o n l y i f
P
P
E
Pa(E;F)
show
i s everywhere
continuous. 1.66
Replace c o n t i n u o u s by hypocontinuous ( r e s p . Mackey c o n t i n u o u s ) i n
e x e r c i s e 1 . 6 5. 1.67 I f E i s a m e t r i z a b l e l o c a l l y convex s p a c e , F i s a l o c a l l y convex s p a c e and P E p a ( E ; F ) show t h a t P E. @(E;F) i f and o n l y i f I$o P
E
1.68*
Let
E
and
mapping from
E
into
and d e f i n e
Ayl by2
nomial o f d e g r e e Y1'.
. . ,yn+l
+
f o r every
@(E)
and
F
F'.
b e r e a l Banach s p a c e s and l e t F.
...
Let
Aynf(x)
Ayf(x) = f ( x + y ) - f ( x ) inductively.
i f and o n l y i f
in x
in
in
E.
Ayl
...
f
be a continuous
for all
Show t h a t
f
x,y
in
i s a poly-
A ~ ~ + ~ f =( x0 ) f o r a l l
Show t h a t t h i s r e s u l t d o e s n o t e x t e n d t o
Banach s p a c e s o v e r t h e complex f i e l d .
E
43
Polynomials on locally convex topological vector spaces Let
1.69*
be a Frgchet space and suppose
E
P
E
Pa(nE).
Show t h a t
i s continuous i f i t s r e s t r i c t i o n t o a 2nd category subset of
P
is
E
continuous.
3
endow
0=
E =
Let
1.70*
m
,&
Space of
be t h e Dirac d e l t a function a t t h e point E
(
6('a)
=*
I:=,
. €in
(ansa)
Show t h a t t h e bounded s u b s e t s of
bounded and hence deduce t h a t
on @(nJl)
Show t h a t
for a l l
n
and
@Hy(nJl I ) =
#
T~
("a
I ) .
on @(rial)
T~
Let
spaces.
m Show t h a t t h e following a r e equivalent:
+
m
(b)
PPnEl;)
(c)
!?("El;) = PHy(nEl;)
E
E;
In=, $n$n
Em
E
E
a neighbourhood that
= !?Hy(nEb)
=
lm=o Em m
and l e t
m
$1 .
Conclude t h a t for all
T~ = T~
n22.
Ei
is a basis for
admits a continuous norm,
each
Let
('
be a s t r i c t inductive l i m i t of FrGchet Monte1
(a)
($n)n=l P =
E = lim E
?.!
(!?(n$jt),~O a r)e l o c a l l y
1.73* -
m
(ansa) .6n = f m
HY ( 2 B )\
E
1.72" I f E i s a Frgchet nuclear space and ($n)n show t h a t (9n@m)n>m=l i s a b a s i s f o r ( @('E),T,).
__ 1.74
We
st
E
m
Show t h a t
T~).
a.
Let
(fm)m=2 i s a Cauchy sequence i n
and t h a t
!?(*a) ,
R.
-functions of compact support i n
with i t s usual s t r i c t i n d u c t i v e l i m i t topology.
0
8H,(2E)
f o r some
n?2,
for a l l
nEN.
where each
Em
# J,n
E
n.
for a l l
EA
and t h a t
i s a l o c a l l y convex space.
P
'6 ( 2 E )
E
V
of zero such t h a t
(6 (2E) , T ~ )
i s not complete i f
I\$nIIV<
Let
Show t h a t
i f and only i f t h e r e e x i s t s f o r each
n.
Hence show
i s a non-normed metrizable
Eo
l o c a l l y convex space.
1.75
Let
E
=
lm=lEm m
where each
Em
i s a Banach space.
P E 8("E) and f o r each p o s i t i v e i n t e g e r m l e t m m E E j and y E Ej. Show t h a t Pm E
X
lj=l
as
m
+ m
lj=m+l
Let
Pm(x+y) = P(x) ("E)
uniformly on a neighbourhood of each point of
and t h a t E.
where Pm
+
P
Chapter I
44
1.76*
If
i s a m e t r i z a b l e l o c a l l y convex s p a c e and
E
i n t e g e r , show t h a t t h e compact open t o p o l o g y on l o c a l l y convex t o p o l o g y on
If
is a positive
is t h e finest
?! ("E)
which c o i n c i d e s w i t h t h e t o p o l o g y o f
("E)
p o i n t w i s e convergence on e v e r y e q u i c o n t i n u o u s s u b s e t o f 1.77 -
n
!?
(nE).
i s a l o c a l l y convex s p a c e i n which e v e r y n u l l sequence i s a
E
i s t h e s p a c e o f F-valued sequen-
Mackey n u l l sequence, show t h a t P M ( E ; F ) t i a l l y c o n t i n u o u s polynomials from
into
E
F.
denote an uncountable s e t .
1.78 __
Let
A
k-space,
but t h a t P ( ~ ( c ~ )=
p H yn(cA
=
CA
Show t h a t
8 ("cA)
for all
M
i s not a
n.
1.79 Show t h a t t h e f o l l o w i n g two c o n d i t i o n s on a l o c a l l y convex s p a c e E a r e equivalent: (a)
e v e r y compact s u b s e t o f
(b)
(i)
E
i s s t r i c t l y compact;
e v e r y n u l l sequence i n
i s a Mackey n u l l
E
sequence ; (ii)
e v e r y compact s u b s e t of
E
is contained i n t h e
a b s o l u t e l y convex h u l l o f a n u l l sequence. If
1.80 -
i s a l o c a l l y convex s p a c e , show t h a t
E
k-space a s s o c i a t e d w i t h 1.81 If B C !? ("E;F) x
in
E.
P
E
If
E
P,("(ExF))
1.83*
F
i s a Banach s p a c e , show t h a t
s u p I P(x) I < f o r every PEB C o n s t r u c t a counterexample which shows t h a t t h i s r e s u l t i s n o t is
Let X
bounded i f and o n l y i f
T~
true for arbitrary 1.82*
(E',u(E',E)).
i s a F r e c h e t s p a c e and
E
and
E. F
are b o t h F r 6 c h e t s p a c e s o r b o t h
i s s e p a r a t e l y c o n t i n u o u s , show t h a t
P
ayF/L
Show t h a t f o r each
n
s p a c e s and
i s continuous.
b e a c o m p l e t e l y r e g u l a r H a u s d o r f f s p a c e and l e t
t h e s p a c e o f E-valued c o n t i n u o u s f u n c t i o n s on topology.
( E ' , T ~ ) is the
h(X)
be
X with t h e compact open
45
Polynomials on locally convex topological vector spaces (a)
fM ("A (XI)
(b)
PHY(*,&(X))
=
p ( n J , (X))
E = Co(T),
r
uncountable,
1.84"
If
P E !?("E;F) -,
P
from
Co(r)
P F = II ( r l ) ,
r2
Co(r2).)
r
in
rl
u n c o u n t a b l e and
and
is t h e n a t u r a l p r o j e c t i o n
(Lr2
= Lr2.
=
i s Lindelbf.
X
.r/
such t h a t
onto
{ $ J n l $E JC O ( T ) ' } s p a n s a d e n s e
Show t h a t
( C? (nCo(r) 1 , 6).
subspace of
1.85
if
i s paracompact;
show t h a t t h e r e i s a c o u n t a b l e
P(nCo(r2 ) ; F )
E
x
B ( n a ( ~ ) ) if
=
K
Let
2 L ([O,lln+l)
E
i s symmetric w i t h r e s p e c t
K
and suppose
t o its coordinates. [P(x)l(t)
Let
j' ... jol
K tl,.
=
..,tn,t)x(tl).
.. x ( t n ) d t l . .
.dtn
0
f o r every
x P
1.86"
L2([0,1]).
E
If
6
8 (nL2[o,l];L2[o,ll) i s a s e p a r a b l e H i l b e r t s p a c e and
E
t h e r e e x i s t s an such t h a t
1.87" __
x
hP(x) If
Show t h a t
in
a
P
E
@ (nE;E)
C,
a r e Banach s p a c e s , we s a y t h a t
F
weakly compact i f i t maps t h e u n i t b a l l o f compact s u b s e t o f
in
show t h a t
1x1
= 1,
T
6
llpllx.
=
and
E
I / x ( I = 1 and
E,
F.
If
T
E
6 (E;F)
E
E
(E;F)
is
o n t o a r e l a t i v e l y weakly
show t h a t t h e f o l l o w i n g a r e
equivalent (i)
T
i s weakly compact,
(ii)
T"
E
(iii) 1.88" __
8
(F';E')
(the adjoint of
A Banach s p a c e
into
that
E
i s weakly compact,
T**(E") C F . E
i s s a i d t o have t h e polynomial D u n f o r d - P e t t i s
p r o p e r t y i f f o r e v e r y Banach s p a c e E
T)
F
F
t h e weakly compact polynomials from
map weak Cauchy s e q u e n c e s o n t o s t r o n g Cauchy s e q u e n c e s .
Show
h a s t h e polynomial D u n f o r d - P e t t i s p r o p e r t y i f and o n l y i f e v e r y
Banach v a l u e d weakly compact l i n e a r mapping maps weak Cauchy s e q u e n c e s o n t o s t r o n g Cauchy sequences.
Chapter 1
46
1.89* P
E
If
P(F)
i s a n u c l e a r subspace o f a l o c a l l y convex s p a c e
F
show t h a t t h e r e e x i s t s If
P c PN(E)
(v
such t h a t
i s a Banach s p a c e , show t h a t
E
=
E
and
P.
is also a
(@,("E),IIw)
Banach s p a c e .
Let
1.91*
d e n o t e t h e t o p o l o g y on
T~
8 a (nE)
o f uniform convergence
on t h e f i n i t e dimensional compact s u b s e t s of t h e v e c t o r s p a c e E:
phism from 1.92* let
( @ a ( n E ) , ~ f ) ' onto
m
6(nE*o)
( x ~ ) ~b e= an ~ orthonormal s u b s e t o f a H i l b e r t space
be a p o s i t i v e i n t e g e r .
only if
In=1I A n I <
__ 1.93*
If
03
E;
1.94
If
= F;iEAwhere
m
and
n
1.95*
(
8f
~ ) -c0) ,
(p (",&
1.96*
(X))
X
,B)
Show t h a t
xm
E
E
and
@N(mE) i f and
.
Ei = (E',T~) i f and
which c o n t a i n s
a s a subspace
E
{@EF';$I~=O}. msn,
and
E
is a locally
( @ ( m E ) , ~ o ) i s isomorphic t o a complemented sub-
A compact Hausdorff s p a c e
closed subset of that
EL =
F
are p o s i t i v e i n t e g e r s ,
convex s p a c e , show t h a t space of
In=,A
=
and a l s o t h a t
m
o n l y i f f o r each l o c a l l y convex s p a c e
w e have
P
Show t h a t
i s a l o c a l l y convex s p a c e , show t h a t
E
and l e t
.
m
Let
E
Show t h a t t h e Bore1 t r a n s f o r m i s an a l g e b r a i c isomor-
(E*,a(E*,E)).
=
-
i s s a i d t o be d i s p e r s e d i f e v e r y
X
contains an isolated point.
If
X
i s d i s p e r s e d , show
h a s t h e approximation p r o p e r t y . ( i f , ( " ~ ~ ) ,B) =. El
/-
QE
El
A
. . . 8,E l .
n times
51.6
NOTES AND REMARKS Mathematicians began e x p l o r i n g t h e c o n c e p t s o f polynomial and holomor-
p h i c mapping in i n f i n i t e dimensions a t a t i m e when i d e a s and t h e o r i e s such a s t h e t o t a l d e r i v a t i v e , p o i n t s e t topology and normed l i n e a r s p a c e , e t c .
Polynomials on locally convex topological vector spaces
47
were either still in their infancy o r not yet discovered. Moreover, it appears that the search for fundamental concepts in infinite dimensional differential calculus stimulated much of the work which resulted in the satisfactory linear theory that we now know as functional analysis. These pioneers were motivated by many different considerations, and at times were not aware of one another's work.
We provide here a brief outline of the
early development of polynomials, a similar treatment of holomorphic functions is given in 52.6, and refer to the historical survey of A . E . Taylor [680] for further details.
It is generally recognised that the definitive step in the creation of infinite dimensional analysis was taken by V. Volterra in 1887. I n a series of notes [705,706,707,708,709], which later evolved into the book [710], he developed a theory of scalar valued differentiable functions on ,&[a,b] and obtained the following Taylor expansion [705,p.105] for the real-valued analytic function y on &[a,b]
where
@,$
E
&[a,b].
The nth term in the above expansion is an n on ,& [a,b]. mappings.
homogeneous polynomial
Volterra did not, however, specifically discuss polynomial
The next step was taken by I). Hilbert who outlined a theory of holomorphic functions in infinitely many variables at the international congress in Rome in 1908 and published his results the following year, [332]. To Hilbert, each variable was a coordinate evaluation and he used a monomial expansion with absolute convergence on polydiscs as we do in chapter 5. Each holomorphic function had, in his notation, the following Taylor series expansion
48
Chapter 1
=
cc
n l . . .nk
n 1 x1
...
x
"k k
1 I IE
\x31 i / c 3 \ ,
IC 21
I
a b s o l u t e l y on x1 $ 1 , I x2 5 , I t i s c l e a r from t h e above t h a t H i l b e r t had a d e f i n i t e
t h e s e r i e s converging
. .. .
c o n c e p t o f polynomial i n i n f i n i t e l y many v a r i a b l e s . During t h e same y e a r , 1909, M. F r g c h e t p u b l i s h e d h i s f i r s t c o n t r i b u t i o n [240] t o t h e a b s t r a c t t h e o r y o f polynomials i n i n f i n i t e l y many variables.
Motivated by Cauchy's o b s e r v a t i o n t h a t any c o n t i n u o u s r e a l
valued function
o f a r e a l v a r i a b l e which s a t i s f i e d t h e ' e q u a t i o n
f
f(x+y) - f(x) had t o have t h e form
-
f(y) = 0
for a l l
x,y
in
R
f ( x ) = Ax, h e gave a n a b s t r a c t " d i f f e r e n c e " c h a r a c t -
e r i z a t i o n o f r e a l polynomials o f one o r s e v e r a l r e a l v a r i a b l e s ( s e e e x e r c i s e 1.68).
H e t h e n used t h i s c h a r a c t e r i z a t i o n t o d e f i n e r e a l polynomials
depending on a c o u n t a b l y i n f i n i t e number o f v a r i a b l e s . was
RN
H i s domain s p a c e
and on i t h e d e f i n e d , f o r c o n t i n u i t y p u r p o s e s , a m e t r i c which
g i v e s t h e u s u a l c o o r d i n a t e w i s e convergence t o p o l o g y .
The.following year
h e used t h e same method i n [241] t o d e f i n e r e a l polynomials on , & [ a , b ] and showed t h a t a r e a l n-homogeneous polynomial
U
on t h i s s p a c e c o u l d be
r e p r e s e n t e d as (xl,
"f
where
11;~)
. . ., x n ) f (x,) . . . f (xn)dxl. . .dxn, f c &,
[a,b] ,
..
i s a s e q u e n c e o f n-homogeneous polynomials i n
independent o f
f
xl,. 'n and t h e l i m i t i s uniform o v e r t h e compact s u b s e t s of
& [ a , b ] . He a l s o showed t h a t any polynomial c o u l d be r e p r e s e n t e d a s a f i n i t e sum o f homogeneous p o l y n o m i a l s . I n a subsequent p a p e r , [243], h e o b t a i n e d a R i e s z r e p r e s e n t a t i o n theorem f o r b i l i n e a r forms on The n e x t s t e p i s due t o R . GGteaux.
,&
[a,b]
.
H e made v e r y fundamental c o n t r i b -
u t i o n s t o t h e t h e o r y o f i n f i n i t e d i m e n s i o n a l c a l c u l u s ( s e e § 2 . 6 ) , and h i s s i m p l e e l e g a n t s t y l e makes f o r v e r y p l e a s a n t r e a d i n g .
G s t e a u x ' s work
c o n s i s t s e s s e n t i a l l y o f two p a p e r s [252,253], which he w r o t e d u r i n g t h e p e r i o d 1912-1914.
H e d i e d i n 1914, and h i s r e s u l t s were e d i t e d by P . L'evy
and p u b l i s h e d i n 1919 and 1922. KN
( K = IR
or
C),
R2
Gzteaux worked o n l y on t h e s p a c e s
and g [ a , b ] .
H e noted t h a t F r e c h e t ' s d e f i n i t i o n
o f polynomial was i n a d e q u a t e f o r f u n c t i o n s d e f i n e d on v e c t o r s p a c e s o v e r
49
Polynomials on locally convex topological vector spaces t h e f i e l d o f complex numbers and proposed i n s t e a d t h a t a c o n t i n u o u s function p
P
such t h a t
f o r any v e c t o r s
degree
n.
z
P(Xz+ut) and
t
i s a polynomial o f d e g r e e
n
in
and
X
i n t h e domain b e c a l l e d a polynomial o f
H e showed t h a t h i s d e f i n i t i o n c o i n c i d e s w i t h F r & h e t ' s f o r
real
v a l u e d f u n c t i o n s of r e a l v a r i a b l e s and went on t o prove v a r i o u s r e s u l t s such as t h e r e l a t i o n s h i p between t h e homogeneous p a r t s and t h e "G$teauxq' d e r i v a t i v e s o f a polynomial - w i t h h i s d e f i n i t i o n .
The development o f t h e
concept of normed l i n e a r s p a c e and a s s o c i a t e d i d e a s between 1910 and 1925 allowed F r g c h e t t o extend h i s d e f i n i t i o n of r e a l polynomial t o a r a t h e r g e n e r a l s e t t i n g i n [244] and [246]. I n 1931-1932, A . D . Michal, a s t u d e n t o f F r g c h e t , gave a s e r i e s o f l e c t u r e s a t t h e C a l i f o r n i a I n s t i t u t e o f Technology i n which h e o u t l i n e d t h e r e l a t i o n s h i p between symmetric n - l i n e a r forms and homogeneous polynomials. T h i s r e l a t i o n s h i p had been n o t i c e d e a r l i e r f o r b i l i n e a r forms and 2-homogeneous polynomials by M . F r g c h e t [243] and R . Ggteaux [252]. F u r t h e r work on t h e d e f i n i t i o n o f polynomial between Banach s p a c e s was c a r r i e d o u t by A . D . Michal and h i s s t u d e n t s A.H.
C l i f f o r d , R.S. M a r t i n , I . G . Highberg
and A . E . T a y l o r [331,492,493,449,677].
R.S. M a r t i n , i n h i s t h e s i s [449]
proved t h e p o l a r i z a t i o n formula and I . G . Highberg [331] c l a r i f i e d t h e r e l a t i o n s h i p between t h e d i f f e r e n t d e f i n i t i o n s and showed t h a t F r 6 c h e t ' s d i f f e r e n c e method could b e extended t o t h e complex case i f one added t h e h y p o t h e s i s o f G3teaux d i f f e r e n t i a b i l i t y . W . O r l i c z [481,482],
I n d e p e n d e n t l y , S. Mazur and
e s t a b l i s h e d t h e c o n n e c t i o n between t h e n - l i n e a r
approach and t h e now c l a s s i c a l approach of F r g c h e t and G2teaux f o r real Banach s p a c e s , and proved t h e p o l a r i z a t i o n formula.
T h i s ends o u r b r i e f
s k e t c h o f t h e development o f t h e concept o f a b s t r a c t polynomial.
For t h o s e
i n t e r e s t e d , we s t r o n g l y recommend t h e o r i g i n a l s o u r c e s as i n t e r e s t i n g reading
.
We r e t u r n now t o commenting on t h e t e x t and w i l l t r y t o a t t r i b u t e
r e s u l t s t o t h e i r o r i g i n a l sources.
There a r e b a s i c a l l y t h r e e approaches
t o s t u d y i n g polynomials, by c o n s i d e r i n g r e s t r i c t i o n s t o f i n i t e dimensional spaces,by means of t e n s o r p r o d u c t s
and by u s i n g m u l t i l i n e a r mappings.
of t h e s e methods a r e u s e f u l and none should be n e g l e c t e d .
All
The r e s t r i c t i o n
method, a s a l r e a d y n o t e d , was one o f t h e o r i g i n a l methods used and r e a p p e a r s i n o u r work e v e r y so o f t e n .
The t e n s o r p r o d u c t approach i s due
t o R . S c h a t t e n [626] and A . Grothendieck [287].
I n [620], R.A.
Ryan shows
50
Polynomials on locally convex topological vector spaces
t h a t most of t h e r e s u l t s we p r e s e n t can b e o b t a i n e d by t h i s method and t h e
same approach i s a l s o t o be found i n C . P . Gupta [295,296], [214,215,217,218,220,223],
T . A . Dwyer
A.Colojoar?i [138,139] and P . Krze [402].
We
f o l l o w t h e m u l t i l i n e a r approach i n t h i s book. Theorem 1 . 7 i s due t o R.S. M a r t i n [449] and an a l t e r n a t i v e proof i s g i v e n i n L.A. Harris [310].
Example 1 . 8 i s due t o L . Nachbin [SO91 and
example 1 . 9 was d i s c o v e r e d i n d e p e n d e n t l y by O . D .
Kellogg [379], J . G . van
d e r Corput and G . Schaake [169] and S . Banach [ 4 5 ] .
The proof g i v e n h e r e
i s due t o S . k o j a s i e w i c z and can be found i n [ 7 3 ] .
Prop.ositions 1.10 and
1.11 a r e due t o L.A.
found i n t h a t a r t i c l e
Harris [310] and f u r t h e r s i m i l a r r e s u l t s may a l s o b e
and i n [316].
The f a c t o r i z a t i o n lemma (and t h e c o r r e s p o n d i n g r e s u l t f o r holomorphic f u n c t i o n s ) h a s been i m p l i c i t i n t h e works o f many a u t h o r s , e . g . A . Hirschowitz [335], C . E . R i c k a r t [605] and L . Nachbin [ 5 1 4 ] .
A system-
a t i c s t u d y o f t h i s i d e a and i t s consequences i s u n d e r t a k e n i n S. Dineen [190] and E . Ligocka [443].
A l l r e s u l t s using s u r j e c t i v e l i m i t s (see
c h a p t e r 6) depend i n some way on a f a c t o r i z a t i o n p r o p e r t y . due t o S. Mazur and W. O r l i c z [482].
Lemma 1 . 1 9 i s
I t i s a l s o a consequence o f t h e
uniform boundedness p r i n c i p l e f o r polynomial mappings on a Banach s p a c e and more g e n e r a l r e s u l t s on t h e same t o p i c can be found i n J . Bochnak and J . S i c i a k [73], P . Lelong [431] and P . Turpin [687].
J . Bochnak and
J . S i c i a k [73,74] make e x t e n s i v e u s e of t h e "Polynomial lemma o f Leja"
[424] i n p r o v i n g t h e i r r e s u l t s . P r o p o s i t i o n 1 . 2 1 i s w e l l known, and p r o b a b l y due t o J . S a b a s t i a g e Silva.
The r e s u l t of example 1 . 2 4 i s proved f o r a c o u n t a b l e d i r e c t sum of
Banach s p a c e s i n S . Dineen [185] and f o r
33 N s p a c e s
i n S . Dineen [ 1 9 4 ] .
The p r o o f g i v e n h e r e i s modeled on t h o s e g i v e n i n [185] and [194] and r e l a t e d r e s u l t s are t o b e found i n [SO].
Example 1 . 2 5 i s due t o P . J .
Boland and S. Dineen [92] and t h e p r o o f i s s i m i l a r t o t h a t o f t h e p a r t i c N u l a r case C XE") which a p p e a r s i n S. Dineen [185]. The method o f p r o o f ha5 been f u r t h e r developed by L . A . de Moraes [498] i n h e r s t u d y of holomorphic f u n c t i o n s on s t r i c t i n d u c t i v e l i m i t s . Nuclear polynomials on Banach s p a c e s were i n t r o d u c e d by C . P . Gupta
Chapter 1
51
[295,296,297] i n o r d e r t o prove e x i s t e n c e and approximation p r o p e r t i e s of c o n v o l u t i o n o p e r a t o r s on Banach s p a c e s and m o t i v a t e d L . Nachbin [508,509] t o i n t r o d u c e t h e concept o f holomorphy t y p e .
T h i s a l l o w s one t o d i s c u s s
compact, i n t e g r a l , n u c l e a r , H i l b e r t Schmidt, e t c . polynomial and holomorp h i c mappings between l o c a l l y convex s p a c e s - c o n c e p t s which have proved u s e f u l i n l i n e a r f u n c t i o n a l a n a l y s i s and i n p r o b a b i l i t y t h e o r y on l o c a l l y convex s p a c e s ( s e e appendix I ) . P.J.
Boland [79,82,83,84]
i n i t i a t e d and developed t h e t h e o r y o f
holomorphic f u n c t i o n s on n u c l e a r s p a c e s and f o r such s p a c e s n u c l e a r p o l y nomials p l a y a fundamental r o l e ( s e e c h a p t e r s 3 and 5 ) . R.A.
Recent work by
Ryan [620] h a s shown t h e i r importance f o r holomorphic f u n c t i o n s on
F r g c h e t s p a c e s w i t h t h e approximation p r o p e r t y . The s t r o n g t o p o l o g y and t h e compact open t o p o l o g y are d e r i v e d from f u n c t i o n a l a n a l y s i s and p o i n t s e t topology r e s p e c t i v e l y .
The
T,,,
topol-
ogy i s more o r less s p e c i a l t o t h e t h e o r y of holomorphic f u n c t i o n s on l o c a l l y convex s p a c e s , a l t h o u g h some l i n e a r p r o p e r t i e s o f t h i s t o p o l o g y a r e d i s c u s s e d i n R . E . Edwards [229, p.511-5131, [397, p.4001 and J . A . B e r e z a n s k i i [ 5 7 ] .
K . F l o r e t [237], G . Kb'the
This t o p o l o g y was i n t r o d u c e d by
L . Nachbin [509] and was m o t i v a t e d by r e s u l t s of A. Martineau [450,453] on
a n a l y t i c f u n c t i o n a l s ( o f s e v e r a l complex v a r i a b l e s ) s u p p o r t e d by e v e r y neighbourhood o f a compact s e t b u t n o t by t h e compact s e t i t s e l f .
I t may
a l s o b e d e s c r i b e d as t h e t o p o l o g y o f l o c a l convergence. Example 1 . 3 8 i s g i v e n i n S. Dineen [185] f o r a c o u n t a b l e d i r e c t sum o f Banach s p a c e s . s e t s of a
$ 3 2 space i s 83% s p a c e s
[50] and f o r S . Dineen,
The same r e s u l t f o r holomorphic f u n c t i o n s on open subproved by J . A .
Barroso, M . C . Matos and L . Nachbin
by S. Dineen [ 1 9 4 ] .
Example 1.39 a p p e a r s i n
[185].
I n d e a l i n g w i t h n u c l e a r polynomials and t h e Bore1 t r a n s f o r m w e have a t t e m p t e d t o c o r r e l a t e v a r i o u s r e s u l t s on n u c l e a r and d u a l n u c l e a r s p a c e s by P . J .
Boland, [79,82,83,85],
r e s u l t s on f u l l y n u c l e a r s p a c e s by P . J .
Boland and S. Dineen [go] and r e s u l t s on F r g c h e t s p a c e s w i t h t h e approxi m a t i o n p r o p e r t y due t o R . A .
Ryan [620].
T h i s l e a d s t o a more compact
t r e a t m e n t b u t e s s e n t i a l l y t h e o n l y new r e s u l t s i n t h e f i n a l two s e c t i o n s
a r e p r o p o s i t i o n s 1.44 and 1 . 4 8 .
D e f i n i t i o n 1 . 4 0 ( a ) i s due t o P . J .
Boland
52
Polynomials on locally convex topological vector spaces
[ 7 9 ] . The fundamental i n e q u a l i t y needed i n d e f i n i t i o n 1 . 4 0 ( b ) i s a l s o due t o Boland [87]. The d e f i n i t i o n i s , however, new a l t h o u g h s p e c i a l c a s e s have p r e v i o u s l y been c o n s i d e r e d by C . P . Gupta [295] and R . A .
Ryan [620].
P r o p o s i t i o n 1 . 4 1 , c o r o l l a r y 1 . 4 2 and theorem 1 . 4 3 are due t o P . J . Boland [79,83].
P r o p o s i t i o n 1.44 i s proved f o r f u l l y n u c l e a r s p a c e s i n [ 8 7 ] .
The Bore1 ( o r F o u r i e r - B o r e l ) t r a n s f o r m was f i r s t used i n i n f i n i t e dimensional holomorphy by C . P . Gupta [ 2 9 5 ] .
Subsequently, i t h a s been
a p p l i e d by v a r i o u s a u t h o r s , see c h a p t e r s 3 , 5 , 6 and appendix I , i n t h e s t u d y o f c o n v o l u t i o n o p e r a t o r s and d u a l i t y t h e o r y .
Proposition 1.47 i s
due t o C . P . Gupta [295] f o r Banach s p a c e s and t o P . J . r e f l e x i v e n u c l e a r and d u a l n u c l e a r s p a c e s .
Boland [79] f o r semi-
Other p a r t i c u l a r cases o f
p r o p o s i t i o n 1 . 4 7 and 1.48 can b e found i n R . A . Ryan [ 6 2 0 ] .
Fully nuclear
s p a c e s ( d e f i n i t i o n 1 . 4 9 ) were d e f i n e d by P . J . Boland and S. Dineen [go] and f i g u r e p r o m i n e n t l y i n c h a p t e r s 5 and 6 .
C o r o l l a r y 1 . 5 0 and lemma 1.55
a r e proved f o r f u l l y n u c l e a r s p a c e s w i t h a b a s i s i n P . J . Boland and S. Dineen [go] and t h e l a t t e r r e s u l t h a s r e c e n t l y been extended t o q u a s i complete d u a l n u c l e a r s p a c e s by J . F . Colombeau, R . Meise and B. P e r r o t
Lemma 1.54 i s due t o S. Dineen [190].
[153].
P r o p o s i t i o n s 1 . 5 6 and 1.57 a r e due t o P . J . Boland and S . Dineen [go] when
E
h a s a b a s i s , and t o P . J .
Boland [87] i n t h e g e n e r a l c a s e .
p r o o f g i v e n h e r e o f p r o p o s i t i o n 1.57 i s new. 1 . 5 9 , 1.60 and 1 . 6 1 are a l l due t o R.A.
Ryan.
The
Lemma 1.58 and p r o p o s i t i o n s They appeared i n a p r e l i m -
i n a r y d r a f t o f h i s t h e s i s [620], b u t were r e p l a c e d by more e l e g a n t and p e r h a p s s l i g h t l y less g e n e r a l r e s u l t s i n t h e f i n a l v e r s i o n .
Chapter 2 HOLOMORPHIC MAPPINGS BETWEEN LOCALLY CONVEX SPACES
I n t h i s c h a p t e r , we g i v e t h e v a r i o u s d e f i n i t i o n s o f holomorphic mappi n g s between l o c a l l y convex s p a c e s , which we s h a l l u s e as well as t h e d i f f e r e n t t o p o l o g i e s on t h e s e s p a c e s o f mappings.
I n many c a s e s , i t i s o n l y
n e c e s s a r y t o c o n s i d e r Banach s p a c e v al u ed mappings and t h e r e s u l t s may be extended t o a r b i t r a r y v e c t o r v al u ed mappings q u i t e e a s i l y . The compact open t o p o l o g y and t h e
T
w
t opology e a s i l y e xte nd from
polynomials t o holomorphic f u n c t i o n s on l o c a l l y convex s p a c e s , b u t t h e s trong topology does n o t g e n e r a l i s e i n a s u i t a b l e fashion.
While t h e
T~
t o p o l o g y p l a y s a n i m p o r t an t r o l e i n o u r s t u d y , it doe s n o t , i n g e n e r a l , have good t o p o l o g i c a l p r o p e r t i e s .
We i n t r o d u c e t h e
-r6
topology which may b e
d e s c r i b e d as t h e t o po l o g y s u p p o r t ed by t h e c o u n t a b l e open c o v e r s .
We prove
e le m e n t a r y p r o p e r t i e s o f t h e s e t o p o l o g i e s and g i v e some sim ple examples.. The s i g n i f i c a n c e o f t h e s e examples and counterexamples i s c l a r i f i e d i n l a t e r chapters.
The r emai n i n g c h a p t e r s a r e d evote d t o a d e e p e r s t u d y o f
t h e s e t o p o l o g i e s on c e r t a i n classes of l o c a l l y convex s p a c e s and on s p e c i a l domains.
W e a l s o d e f i n e germs o f holomorphic f u n c t i o n s .
These are o f
i n t r i n s i c i n t e r e s t and a l s o p l a y a r o l e i n d u a l i t y t h e o r y . 52.1
G ~ T E A U X HOLOMORPHIC MAPPINGS
De finit i on 2.1 A subset U of a vector space E i s said t o be f i n i t e l y open i f UnF i s an open subset of t h e Euclidean space F f o r each f i n i t e dimensional subspace F of E. The f i n i t e l y open s u b s e t s o f ogy
tf.
The b a l a n c ed
neighbourhoods o f z e r o .
tf
E
define a translation invariant topol-
neighbourhoods o f z e r o form a b a s i s f o r t h e
On a l o c a l l y convex s p a c e
(E,T),
t h a n any o f t h e t o p o l o g i e s we have p r e v i o u s l y c o n s i d e r e d on Tf
3 TM 3 Tk 3 T.
53
T~
E,
is f i n e r i.e.
tf
Chapter 2
54 Pefinition 2.2
A function E
a vector space Gateaux or
f
defined on a f i n i t e l y open subset F
with values i n a l o c a l l y convex space
-
b EE
aE U,
G-holomorphic i f for each
of
i s said t o be
Q, E F '
and
U
t h e complex
valued f u n c t i o n of one complex variable A
$of(a+Xb)
i s holornorphic i n some neighbourhood of zero. s e t of a l l
G-holomorphic mappings from
U
HG(U;F)
We l e t
into
denote t h e
F.
Hartog's theorem i n f i n i t e dimensions says t h a t s e p a r a t e l y holomorphic f u n c t i o n s on
UxV
(UCK", V C C m )
a r e holomorphic.
Hence
f:UCEjF
i s G-holomorphic i f and only i f @ o f ) U n G i s a holomorphic function of
Q, i n
s e v e r a l complex v a r i a b l e s f o r each subspace
G
of
F'
and each f i n i t e dimensional
Consequently, one can u s e any of t h e equivalent
E.
f i n i t e dimensional conditions (e.g. Taylor s e r i e s expansions, Cauchy Riemann equations, e x i s t e n c e of t h e t o t a l d e r i v a t i v e ) i n t h e d e f i n i t i o n of G-holomorphicity. A t t h i s s t a g e , one may wonder why we demanded a l o c a l l y convex range
space i n d e f i n i t i o n 2 . 2 .
a:
v e c t o r space over
We could a l s o g i v e an analagous d e f i n i t i o n with a However, w e would then run i n t o
a s t h e range space.
d i f f i c u l t i e s i n showing t h a t a G2teaux holomorphic function has a Taylor s e r i e s expansion about each p o i n t s i n c e t h i s r e q u i r e s a convergence s t r u c t u r e on t h e range space and t h e f i n i t e open topology on
i s so f i n e t h a t
Even with a l o c a l l y
very few f u n c t i o n s would have t h e d e s i r e d expansion. convex range space w e s t i l l have t o be c a r e f u l .
F
We f i r s t show t h a t G-holo-
morphic functions a r e continuous f o r t h e f i n i t e open topology. Lemma 2 . 3
I f
E
i s a v e c t o r space,
i s a l o c a l l y convex space and
i s given t h e f i n i t e open topology. I t i s e a s i l y seen t h a t
i s a f i n i t e l y open subset of
f &HG(U;F)
F U
Proof
U
tf
then
f
i s continuous when
i s t h e inductive l i m i t topology, i n
t h e category of t o p o l o g i c a l spaces, given by t h e i n c l u s i o n mappings where
G
function
ranges over a l l f i n i t e dimensional subspaces of f
defined on a
tf
open subset
U
of
E
E.
G-E
Hence a
i s continuous i f and
only i f i t s r e s t r i c t i o n t o t h e f i n i t e dimensional s e c t i o n s of uous.
E,
U
a r e contin-
Since an a n a l y t i c f u n c t i o n of s e v e r a l complex v a r i a b l e s i s continuous
t h i s completes t h e proof. A
We now look a t Taylor s e r i e s expansions of Gateaux holomorphicfunctions.
Holomorphic mappings between locally convex spaces
55
If U i s a f i n i t e l y open subset of a vector space E, F i s a ZocaZly convex space and fEHG(U;F) then f o r each 5 E U there Proposition 2.4
e x i s t s a unique sequence of homogeneous polynomiak from
E into
A
F,
such t h a t
f o r all y i n some
Proof
where
Let
p,
P (x) m,S,@
neighbourhood of zero.
tf
E E U be fixed.
For each positive integer m,
@
EF' and
is chosen so that is independent of
By lemma 2.3, f is continuous and we may use Riemann's definition of the integral to define Pm,E(X)
=
dh .
The limit (of the Riemann sums) may not exist in F
but will always exist
A
in F , the completion of F. It will lie in F if F is sequentially complete o r if the closed convex hull of each compact subset of F is compact. F o r this reason, we sometimes place a completeness condition on the range space. Since f is continuous, Pm,S,@ (XI all 6 and x.
=
@(Pm,S(x))
for
Since the restriction of @of to any finite dimensional section of U is a holomorphic function of several complex variables, it follows that the function x -+ P (x) lies in @a(m~> f o r every positive integer m m,S,@ and each @ in F'. Let L be the associated symmetric m-linear m,S,@ mapping on E.
56 If
Chapter 2 xl,
...,xm
L
: Em
Let
m, 5
t h e n by t h e p o l a r i z a t i o n formula (theorem 1 . 5 )
E
E
-
h
F
b e d e f i n e d by 1
E
By t h e Hahn-Banach theorem Let
1
Lm, 5
i z (mE;F)
El...€
and
P n m,{
(1"i = lE.X.) 1 1
= A Lm,< E $,("E;?).
Pm,<
b e a t f - b a l a n c e d neighbourhood o f z e r o such t h a t
V
example, t a k e
V
{ x E E ; S + A X E U , I A ~ & ~ II)f.
=
a compact s u b s e t o f
S+{Ax;IA/&p} C U .
such t h a t
B ECS(F) let
If
Y E F ~ . By l e m m a 2 . 3
p>l
B = { $ E F ' ; l$(y)(sB(y) B i s c o n t i n u o u s and w e have
f
sup B(f(S+Ax))
sup
=
I$of(S+Ax)
I A I&P,OEBB
I A l W
I
(for
S + { X X ; I A ~ S ~i}s
then
XEV
and hence t h e r e e x i s t s
U
<+VCU
<
=
m
for all
.
Hence
f o r every p o s i t i v e integer
Hence f(S+x)
=
LmZo m
m.
T h i s shows t h a t
P
(x)
m, 5
f o r every
x
in
V.
Using t h e u n i q u e n e s s o f T a y l o r series expansions i n one complex v a r i a b l e we
57
Holomorphic mappings between locally convex spaces
see t h a t t h e sequence
m
i s u n i q u e l y d e t e r m i n e d by
('m, S)m=O
The f i n i t e d i m e n s i o n a l t h e o r y a l s o shows t h a t partial derivative of
f
at 5
i n the direction
f.
is the
Pm,s (x)
m
th
x, -and following
c l a s s i c a l terminology, P
The c o r r e s p o n d i n g
m, 5 m
m! l i n e a r form
Our expansion now becomes
L
i s d e n o t e d by
=
'mf(S) ___ (x) m!
m, 5
f(S+x)
proof.
.
dmf(S) ~
m!
T h i s completes t h e
I n p r o v i n g p r o p o s i t i o n 2 . 4 , w e h a v e a l s o shown t h e f o l l o w i n g : P r o p o s i t i o n 2 . 5 (Cauchy i n e q u a l i t i e s )
i s a balanced subset of
B
and
cs(F)
in
If f
such t h a t
and every non-negatiue integer
HOLOMORPHIC
12.2
E
Definition 2.6
MAPPINGS
Let
E
BETWEEN
and
a f i n i t e l y open subset of E.
F
HG(U;F),
E
c+pBCU
LEU;
p
LOCALLY
A function
5
in
~A { O }
m
CONVEX
SPACES
be locally convex spaces and l e t
if it i s G-hoZomorphic and f o r each
E
then for every
f:U U
+
F
U
be
i s calZed hoZornorphic
t h e function
converges and defines a continuous function on some r-neighbourhood of zero. We Zet U
into
H(U;F)
denote t h e vector space of a12 hoZomorphic functions from
F.
We u s u a l l y c o n s i d e r f u n c t i o n s d e f i n e d on open s u b s e t s o f
E
and i n
t h i s case, b e c a u s e o f t h e u n i q u e n e s s o f t h e T a y l o r series expansion and t h e
f a c t t h a t t h e f i n i t e open neighbourhoods of z e r o a r e a b s o r b i n g , a G-holomor-
Chapter 2
58
p h i c f u n c t i o n i s holomorphic i f and only i f it i s c o n t i n u o u s . The f o l l o w i n g o b s e r v a t i o n i s e a s i l y proved and f r e q u e n t l y a p p l i e d . Lemma 2 . 7
U
If
is an open subset of a ZocalZy convex space f
a Zocally convex space and naof
E
for every
H(U;Fa)
A continuous fu n ct i o n
HG(U;F)
E
f
then
E
i s bounded).
f
is
i f and only i f
w i t h v a l u e s i n a normed l i n e a r spa c e i s
f
l o c a l l y bounded ( i . e . each p o i n t i n t h e domain o f whose image under
H(U;F)
F
cs(F).
in
a
E,
f
h a s a neighbourhood
The co n v erse i s f a l s e i n g e n e r a l b u t it
i s t r u e f o r G-holomorphic f u n c t i o n s as o u r n e x t r e s u l t shows.
Lemma 2 . 8
U
I f
i s an open subset of a Zocally convex space
a normed Zinear space a d
f
E
HG(U;F)
f
then
E
H(U;F)
E,
F
i f and only if
is f
i s ZocaZZy bounded. Let
Proof
5
E
U
be a r b i t r a r y .
hood of z e r o such t h a t
5+VCU
By p r o p o s i t i o n 2 . 5 , f o r a l l
If
5,+5
a > a
.
as
a + -
m
and and
Choose f(5+V) O < S <1,
t h e n we can ch o o s e
Hence we h av e
T h i s completes t h e p r o o f .
V
ct 0
a convex ba la nc e d neighbouri s a bounded s u b s e t o f
F.
w e have
such t h a t
5-5,
E
6V
for all
59
Holomoiphic mappings between locally convex spaces S i n c e e v e r y l o c a l l y bounded polynomial i s c o n t i n u o u s , we a l s o have shown t h e f o l l o w i n g : C o r o l l a r y 2.9
E
Let
and E
U
be an open subset o f
U
and every p o s i t i v e i n t e g e r
F
be arbitrary ZocaZZy convex spaces. f
and suppose
is ( m ~ F; ) .
Zmf(E)
m,
E
8(mE;e)
E
A
Corollary 2.10
Let
E
be open i n
f
and
E
F
and
E
H(U;F).
Let 5
Then f o r every dmf(E)
and
E
be arbitrary ZocaZly convex spaces,
HG(U;F).
f
Then i f
in
U
i s ZocaZly bounded i t Zies i n
H(U;F). Apply lemmata 2.7 and 2.8
Proof
H
We Zet
LB
(U;F)
denote the vectop space o f a l l G-hoZomorphic ZocalZy
bounded mappings defined on t h e open subset U E
with vaZues i n t h e ZocaZZy convex space
MLB(U;F) C H ( U ; F )
f o r any
and
U,E
o f t h e ZocaZZy conuex space We have j u s t s e e n t h a t
F.
We now look a t t h e r e v e r s e
F.
inclusion.
Lemma 2 . 1 1
is an open subset of a ZocaZZy convex space E and f o r every ZocaZZy convex space F then E i s a normed U
If
H(U;F) = H LB (U;F)
l i n e a r space.
I t suffices t o take
Proof
i d e n t i t y mapping (from
F=E
and to n o t e t h a t t h e r e s t r i c t i o n o f t h e
into itself) t o
E
l o c a l l y bounded i f and o n l y i f
U
i s always c o n t i n u o u s b u t i s
i s a normed l i n e a r s p a c e .
E
There are, however, s e v e r a l n o n - t r i v i a l examples o f p a i r s o f s p a c e s and
F
i n g s from
into
E
F
coincide.
One can o b t a i n some i n f o r m a t i o n a b o u t t h i s
problem by e x t e n d i n g H a r t o g s ' theorem t o l o c a l l y convex s p a c e s .
f;U+G
E
for w h i c h , t h e holomorphic and t h e l o c a l l y bounded holomorphic mapp-
where
U
i s an open s u b s e t o f
ExF,
E,F
convex s p a c e s , i s separateZy hoZomorphic i f f o r each y + f(x,y)
i s holomorphic and f o r each
i s holomorphic.
y
in
F
and
G
x
in
A function
being l o c a l l y E
the function
t h e function x
+
g(x,y)
H a r t o g s ' theorem i n s e v e r a l v a r i a b l e s i m p l i e s t h a t s e p a r -
a t e l y holomorphic f u n c t i o n s a r e G-holomorphic. P r o p o s i t i o n 2.12
Let
E
and
F
be ZocaZZy convex spaces and suppose
60
Chapter 2
every separately holomorphic function defined on an open subset of i s holomorphic.
Proof f(x,$)
Let =
Then HLB(U;F) = H(U;F)
f
E
@(f(x)).
H(U;F).
We define
f o r every open subset
f; UxFb
+
B:
ExF'
U of
B E.
by the formula
f is obviously separately holomorphic and by our hypoth-
esis, it is holomorphic. Since Q: is a locally compact space f is a locally bounded function. Hence, if 5
E
U, we can find a neighbourhood of
5, VE, and a neighbourhood of zero in Fb, Bo,
where
B
is a bounded
Hence sup \@(f (x)) I < m xev for every $ in F' and by Mackey's theorem f(Vg) isc a bounded subset of F. This complets the proof. subset of F,
Example 2.13
Bo such that \ \ f \ \ v E
=
M <
-.
If E is a Frgchet space and
F
is a DF
space (the
strong dual of F is a Frgchet space), then HLB(U;F) = H(U;F) open subset U
for any
of E. This follows from proposition 2.12 since it is
known that separately holomorphic functions defined on open subsets of the product of Frgchet spaces are holomorphic. Example 2.14 If E is a $ 3 2 space and F is an 3 d space then HLB (U;F) = H(U;F) for any open subset U of E. This also follows from proposition 2.12 and the fact that separately holomorphic functions defined on open subsets of a product of $ 3 2 spaces are holomorphic. We do not know if the same result holds for
a3@ spaces although we
do know that separately continuous polynomials defined on a product of PTkyL spaces are continuous. We now look at functions which are holomorphic analogues of the hypocontinuous and the M-continuous polynomials defined in section 1.2 Definition 2.15
A function
ZocalZyconvex space
f defined on an open subset
U of a
E
with values i n a l o c a l l y convex space F i s said it i s G-holomorpkic and continuous on the compact subsets of E. We l e t HHY(U;F) denote the vector space of a l l hypoanalyt i c mappings from U i n t o F. t o be hypoanalytic i f
Some authors give a slightly more general definition of hypoanalytic functions - they consider functions which are G-holomorphic and bounded on the compact subsets of U. The following example shows that this can lead to a strictly larger class of functions.
61
Holomotphic mappings between locally convex spaces Example 2.16 (en)n
E
Let
be an i n f i n i t e dimensional H i l b e r t space and l e t
be a sequence of mutually orthogonal u n i t v e c t o r s i n
I : (E,u(E,E'))
+
/I 11)
(E,
E.
Then
maps compact s e t s onto bounded s e t s b u t i s not
hypoanalytic s i n c e t h e sequence
(en)n
i s weakly b u t not s t r o n g l y
conver-
convergent. If a
E
i s a k-space ( i n p a r t i c u l a r , i f
,33fM,space)
then
HHY(U;F)
f
A function
l o c a l l y convex space
i s metrizable o r i f
E
f o r any open subset
U
of
E
is
E
and
F.
any l o c a l l y convex space Definition 2.17
H(U;F)
c
defined on a
T~
open subset
of a
U
i s said t o be Mackey or S i l v a holomorphic i f
(EJ)
i t i s G-holomorphic and M-continuous. We l e t space of a l l Mackey holomorphic mappings from
%(U;F)
denote t h e vector
U
F.
into
The following r e s u l t gives an a l t e r n a t i v e d e f i n i t i o n of Mackey holomorphic f u n c t i o n s .
We omit t h e p r o o f .
Proposition 2.18
U
Let
be a
open subset of the l o c a l l y convgx
T~
space E and l e t F be a 1ocaZZ.y convex space. following are equivalent:
b)
for each 5 exists E>O
U
f(t+EB)
such t h a t
5
f o r each
E
U
g
and
E
If H(U;F)
(E,T) f o r any
H(V;E),
rdmf(5)
where
In p a r t i c u l a r , space o r a holds f o r
838
C
i s a superinductive space then , T
open subset
%(U;F)
= H(U;F)
space and
F
U
if
of
U
H (U;F) G
then the
B
V
and g(O)= 5, the f u n c t i o n on some neighbourhood of zero i n CC.
of zero i n
E
of E there i s a bounded subset of F,
and each bounded subset
5 E U and each m i n N ,
c ) for each d)
E
If f
E
E
@,("E;?).
i s a neighbourhood fog
T ~ = T
i s hoZaorphic
and hence
%(U;F)
and any l o c a l l y convex space
F.
is an open subset of a Frzchet
is arbitrary.
We do not know i f t h i s r e s u l t
3% spaces.
There are s e v e r a l o t h e r t y p e s of holomorphic f u n c t i o n s t o b e found i n the literature.
=
We s h a l l introduce them i f t h e need a r i s e s .
Our main
Chapter 2
62
i n t e r e s t l i e s i n t h e s t u d y o f holomorphic f u n c t i o n s and a l l o t h e r f u n c t i o n s p a c e s are i n t r o d u c e d s o l e l y t o h e l p o u r s t u d y i n t h i s d i r e c t i o n .
The d i f f -
e r e n t k i n d s of holomorphic f u n c t i o n s we have d e f i n e d s a t i s f y t h e f o l l o w i n g inclusions. E
Let E.
open subset of
and
F
U
be l o c a l l y convex spaces and
an
The following inclusions hold
An i m p o r t a n t q u e s t i o n which w i l l a r i s e i n t h i s book and which i s s t i l l undergoing a c t i v e r e s e a r c h i s t h e f o l l o w i n g :
f o r what
t h i s q u e s t i o n f o r polynomial mappings.
U,E
and
F
are
We have a l r e a d y looked a t
some (or a l l ) o f t h e above i n c l u s i o n s p r o p e r ?
For t h e moment, w e c o n s i d e r o n l y a
few s i m p l e examples. Example 2.19 %(U;F)
= H
If
(U;F)
has property
E
(s)
f o r any open s u b s e t
HY p o l y n o m i a l s i n c h a p t e r 1 can b e e x t e n d e d .
Example 2 . 2 0 of
A G-holomorphic f u n c t i o n
with val ues i n
E
and
F
of
E.
U
f
i s a r b i t r a r y , then The p r o o f g i v e n f o r
d e f i n e d on a n open s u b s e t
U
i s hypoanalytic i f e i t h e r of t h e following condit-
F
ions hold: a)
f i s bounded on compact sets and "m d f ( c ) E 8Hy(mE;F) f o r e v e r y 5 i n and e v e r y p o s i t i v e i n t e g e r
b)
s e p a r a b l e and criterion (i.e.
Since
E
(E,o(E,E'))
T
and
T~
d e f i n e t h e same conver-
E).
i s l o c a l l y convex and h y p o a n a l y t i c i t y i s a l o c a l prop-
e r t y , w e may suppose w i t h o u t l o s s o f g e n e r a l i t y t h a t anced and t h a t a)
Let
F
K C U
Hence there e x i s t s
is
s a t i s f i e s t h e Mackey convergence
E
g e n t sequences i n Proof
m.
i s bounded on compact s e t s ,
f
U
i s convex and b a l -
U
i s a normed l i n e a r s p a c e .
b e compact. X>1
such t h a t
We may suppose t h a t
XKCU.
K
i s balanced.
The Cauchy i n e q u a l i t i e s imply
63
Holornotphic mappings between locally convex spaces
lim sup( 1 mn! 3 f(0) By hypothesis ___ E PH,,(("E;F) and hence f is the uniform limit on K n! of a sequence of continuous polynomials. Thus f is continuous on K and
that
f
E
HHY(U;F). b)
By (a) we may suppose that
Since E
f is an m-homogeneous polynomial.
is weakly separable, the compact subsets of E are metrizable and
hence it suffices to show that f is sequentially continuous. Let be a null sequence in E . such that
(An\
There exists a sequence of scalars,
and
+ +m
m
(x~)~=~
Anxn
+
as n-.
0
Since f i s bounded on com-
pact sets
I An(
is a bounded subset of F. Now n
-t m
+ a
and this implies f(xn)
+
0 as
and completes the proof.
We now look at holomorphic versions of the Factorization Lemma proved for polynomials in the first chapter. The situation is much more complicat-
ed due to the fact the polynomials are always defined on the entire space and continuity at a single point implies continuity at all points. These properties are not necessarily true of arbitrary G-holomorphic functions. Here the topological and geometric properties of the set U and the global continuity properties of the function f have to be taken into consideration. E
Let
Theorem 2.21
be a l o c a l l y convex space and l e t
F be a normed
U i s a connected open subset of E and f E H(U;F) then there e x i s t s an a E cs(E) such t h a t f o r any XEU, YEE for which Zinear space.
If
a(y) = 0 and Ix
ue have
+
f(x+y) Proof
OCX,
xy; =
(*I
f(x). is convex and balanced. Condition (*)
We first suppose that U
is then satisfied if x and x+y
E
U.
Since F
is a normed linear space
there exists an a in cs(E) such that Ba = IxEE;~(x)< 11 C U and = M <=. By the Cauchy inequalities it follows that Ba
64
Chapter 2
"m
Ild)l/
m! Ba ials we have
E
U
for all m
in E
for every x x,x+y
M
6
a(y) = 0
and
"m
m
f(x+y)
=
and all
Cm=o
y
in E
for which a(y)
=
0. Hence if
then
f(o)
m!
and by the factorization lemma for polynom-
(x+y) = 1" m=O
This completes the proof when U
!L@l (x) m!
=
f(x3.
is convex and balanced. For arbitrary U
we choose S E U and V a convex balanced open set such that S + V C U . By the above there exists an a in cs(E) such that if x,y E E , x E V , x+y E V and a(y) = 0 then f(S+x) = f(€,+x+y). Moreover, if x,y E E , XEV, S+x+y E U, a(y) = 0 and (*) function of one complex variable
-
A
f(S+x+Ay)
is satisfied, then we may consider the
- f(S + X I
This function is constant, by the above, on some neighbourhood of zero, and hence it is constant on the connected interval
[0,1]. Hence
f(F+x+y)
=
f(S+x). Let Uo = {xsUlif YEE, a(y) = 0 and (*) is satisfied then f(x) = f(x+y)l. We have just shown that Uo is a nonempty open subset of U.
If xg
E
Uo
+
x
E
U
as
8
+
-, YEE and
{x+AylObh$lj C U , then)
since E i s a topological vector space, x8+Ay E U for all 9 sufficiently large and all A E [0,1]. Hence f(x+y) = lim f(xe+y) = lim f(x8) = f(x)
e-
Thus Uo is a non-empty open and closed subset of U . ected, this implies U=Uo and completes the proof.
e-
Since U
is connru
Our next step motivated by the polynomial case would be to define f on n a ( U ) . There are, however, several difficulties which cannot be surmounted without certain modifications. Without condition
(*)
N
f may not
be well defined.
It is possible to surmount this problem, in the general situation, by considering domains spread over E and using a pullback operator or by restricting oneself to pseudo-convex open sets. To simplify our presentation, we confine ourselves to convex open sets. A second difficulty arises from the fact that ila(lJ)
is not necessarily an open subset of E
65
Holomorphic mappings between locally convex spaces However,
w i l l always b e a
17 (U)
we c a n a s k , assuming
tf
and c o n s e q u e n t l y Ea N i s w e l l d e f i n e d , whether o r n o t f i s a holomor-
N
f
open s u b s e t of
N
The s e t o f p o i n t s o f c o n t i n u i t y o f
phic function.
may n o t b e a l l o f
na(U).
f
w i l l b e nonempty b u t
Example 2 . 2 2 i l l u s t r a t e s t h i s d i f f i c u l t y .
d i f f i c u l t y i s overcome by p l a c i n g e x t r a c o n d i t i o n s e i t h e r on
na
mappings
Example 2.22
We d e n o t e by
H(C)
t h e s p a c e o f e n t i r e f u n c t i o n s of one
Let
complex v a r i a b l e endowed w i t h t h e compact open t o p o l o g y . b e d e f i n e d by f u n c t i o n on
in
a
subset of
H(C)cl
f o r any
cs(H(C))
a
E
such t h a t
suplf(z)l
=
cs(H(E)). B
fn(z) fn(0) = 0,
where
I z1sR
n 2
=
Since If
O t h e r w i s e , t h e r e would
is a closed
z+z
=
R > 2.
let
+...
+ z
(2R) + . . . +
2 fn(2R) = 4R - 2R
n
(2R)
.
n
2
.
(Z
for all n
2
-2R)
n +
and
-.
F ( f + f n ) = ( f + f n ) ( f ( 0 ) + f n ( O ) ) = f (2R)+fn(2R) = 2 F ( f ) = f(2R) = 2R-4R # 0 w e h a v e shown t h a t no such a e x i s t s .
f(z)
U
2R-z
Then
i s convex and b a l a n c e d (or even pseudo-convex)
o f Theorem 2 . 2 1 i s s a t i s f i e d f o r a l l
(*)
C
i s an e n t i r e
{fEH(C);F(f) = 0 )
=
as
0.
+.
Without l o s s o f g e n e r a l i t y we may assume
For e a c h p o s i t i v e i n t e g e r
Let
F
F!H(C)
does n o t f a c t o r as a holomorphic
F
We claim t h a t
H(E),. a(f)
Then
I t is e a s i l y seen t h a t
F(f) = f ( f ( 0 ) ) . H(C).
func ti on through e x i s t an
Ea
and w e g i v e v a r i o u s s u f f i c i e n t c o n d i t i o n s .
This
o r on t h e
x
and
then condition
and we o b t a i n t h e
y
following f a c t o r i z a t i o n r e s u l t . Proposition 2.23 f
E
Proof
be a l o c a l l y convex space and l e t
F
be a
is a convex balanced open subset of E and r) then there e x i s t s an a i n c s ( E ) and f E HG(na(U);F) such
H(U;F)
that
E
Let
normed l i n e a r space.
If
U
J. .
f = fona.
We d e f i n e
ru
f
on
na(U)
by
.V
f(x) = f(2)
if
2E
U
and
66
Chapter 2 rrl
n,(x)
Theorem 2 . 2 .
= x.
f = f e n . Since f .-./ HG(na(U);F). N
shows t h a t
f"
i s w e l l d e f i n e d and by c o n s t r u c t i o n
i s a G-holomorphic f u n c t i o n , it f o l l o w s t h a t
W e now g i v e a s u f f i c i e n t c o n d i t i o n f o r t h e c o n t i n u i t y o f P r o p o s i t i o n 2.24
a directed s e t
If D
f
cs(E)
i s a ZocaZZy convex space and
E
4
of semi-norms which d e f i n e the topology of a in D then
contains
E
nO1
and
i s an open mapping f o r every
li.e.
if
such t h a t
f
E
H(U;F)
then there e x i s t s an
f = faofla)
to lie in
D.
Hence
na(U)
there exists a and
since
Ra
llfll
B
i s a n open s u b s e t o f
HG(na(U);F)
6
l o c a l l y bounded s i n c e
C U
fa
E
U
of
H(TIu(U);F) E
and
We first n o t e t h a t i n t h e p r o o f o f theorem 2 . 2 1 w e may choose
p r o p o s i t i o n 2.23, f a 0
and
F.
any normed l i n e a r space Proof
D
in
01
for any convQx balanced open subset
f
D m.
and
such t h a t
p>O
Hence
w
Ilfa//
BB,s,P
i s open, i t f o l l o w s t h a t
fa
and t h e r e e x i s t s , by
Ea
Now
= f O 1 olla.
i s a normed l i n e a r s p a c e .
F
in <
such t h a t
f
is
Hence f o r each
BB,S,P
d B B , S,P
a
) =
=
5 in IxEE; B(5-X)
IlfiiB
10
and,
B,S,P
i s l o c a l l y bounded and so b e l o n g s
t o H(nO1(U);F). T h i s c o m p l e t e s t h e p r o o f s i n c e t h e o p p o s i t e i n c l u s i o n i s obvious. The above p r o p o s i t i o n c o v e r s t h e case where each s p a c e and y i e l d s t h e f o l l o w i n g examples.
Let
Example 2 . 2 5
nnZl m
Eu
i s a Banach
En where each En i s a Banach s p a c e . n EO1 = II. E . f o r a l l n . I f f E H(E;F) and F i s a normed 1=1 I l i n e a r s p a c e , then t h e r e e x i s t s a p o s i t i v e i n t e g e r n and f n E H(E ; F ) an such t h a t f = f n o Rn where IIn i s t h e c a n o n i c a l p r o j e c t i o n o f E o n t o E
=
Let
Ea
n
.
As a p a r t i c u l a r example, we see t h a t
u ,
H((CN)
=
I
neN H((Cn).
67
Holomotphic mappings between locally convex spaces
Let X be a completely regular Hausdorff topological space
Example 2.26 and let E
=
& (X).
Then we can choose our directed set D
such that
Ea Y &,(K), K compact in X, for each a in D . Hence each E, is a Banach space and we obtain a factorization result for normed linear space valued holomorphic functions defined on convex balanced open subsets of
.
&(XI
Ea, is given in the following proposition. This condition also arises in various other A further sufficient condition, this time on the spaces
parts of infinite dimensional holomorphy, for instance analytic continuation and we show (theorem 2.28) that it is satisfied by any Banach space.
In
fact, we prove a more general result which we shall use later. This, then, I n more general
gives an alternative proof for examples 2.25 and 2.26.
factorization theorems it is applied to prove results unobtainable by using proposition 2.24. Let
E be a l o c a l l y convex space and l e t F be a normed linear space. I f there e x i s t s a directed family D of semi-norms which define t h e topology of E and each a i n D s a t i s f i e s the following condition: Proposition 2.27
f
i f
E
where
H(U,;F)
i s an open subset of
U,
then t h e s e t of points o f c o n t i n u i t y o f
Ea
f i s open and
ctosed. Then
H(U;F)
=
u
H(n,(U);F)
C~ED
f o r any convex balanced open subset
Proof
u
~ E C (SE l
U of
H(lla(U);F)
E.
This follows immediately from theorem 2.21.
Theorem 2.28 let
=
Let
U be a connected open subset o f t h e Banach space
F be a normed l i n e a r space and l e t f E HG(U;F). SEU and every p o s i t i v e integer m then f
f o r some
Proof
pf(E) H(U;F).
If E
Without l o s s of generality, we may assume that U
balanced set and that 5 = 0.
Let
E
E,
6(mE;/\F)
is a convex
Chapter 2
68
Am
Since each
i s continuous,
d f(0)
6
Vn = U s i n c e n=l say V some Vn,
i s a closed subset of
Vn
f
i s G-holomorphic.
,
has nonempty i n t e r i o r .
"0
V C V
no
.
n
1-m=O zm
L
f
qeU "0
and
i s a convex
V
then lemma 1.13 implies
2 f (0)
SUP
L
-V 2
Thus
If
n+VcV
Hence
Ilf Ill
and
By t h e Baire category theorem
balanced neighbourhood of zero such t h a t that
U
2no.
=
i s l o c a l l y bounded and continuous a t t h e o r i g i n .
By using t h e
Taylor s e r i e s expansion of G-holomorphic functions, we s e e t h a t
f o r any
0
in
f o r any
n
and any
that
i s continuous a t
f
have shown t h a t
and any
U
f
0
E
in
x
in
U.
Since
8.
H(U;F).
By lemma 1.19,
E.
Pf(0)
E
(P(nE;$)
By t h e f i r s t p a r t of o u r p r o o f , i t follows (3
was a r b i t r a r i l y chosen i n
U
we
This completes t h e proof.
6332
Since Frzchet spaces and
spaces a r e superinductive l i m i t s of
Banach spaces, one can e a s i l y prove a r e s u l t s i m i l a r t o theorem 2.28 f o r such spaces. So f a r we have been d e s c r i b i n g f a c t o r i z a t i o n r e s u l t s which use contin-
uous semi-norms on t h e domain space. solving t h e Levi problem on s o r t of f a c t o r i z a t i o n .
C e r t a i n s i t u a t i o n s ( f o r example, i n
&3mspaces
with a b a s i s ) r e q u i r e a d i f f e r e n t
We g i v e some r e s u l t s i n t h i s d i r e c t i o n .
A topological space
X
i s a L i n d e Z Z f space i f every open cover o f
contains a countable subcover.
X
i s s a i d t o be h e r e d i t a r y LindeZb'f i f
X
69
Holomotphic mappings between locally convex spaces every open subset of
i s Lindelbf.
X
533%
Separable FrGchet spaces and
spaces a r e examples of h e r e d i t a r y Lindelb'f l o c a l l y convex spaces. Proposition 2.29
F f
be a hereditary Lindelb'f l o c a l l y convex space,
H(U;F)
IJ
Proof
a metrizable l o c a l l y convex space
then there e x i s t f
U),
and
H(nf(U);F)
E
IIf(U)
in
5
t h e r e e x i s t s an
U
where
<
f
The semi-norms l o c a l l y convex topology on TIf
E.
Let
a5
U
cs(E)
If
(which E
onto
and
and
f
f = fonf.
such t h a t
5
5
it contains a countable sub-
g e n e r a t e a semi-metrizable
denote t h e a s s o c i a t e d m e t r i z a b l e
denote t h e q u o t i e n t mapping from
c o n s t r u c t i o n nf(U)
Ef
E. Ef
Ba (1)= { x E E ; ~(x) < 1 ) .
(a5 , )mn = l Ef
E
E
from
i s an open subset o f
I1 115+B a g ( l ) + Ba5(1) cu and Since { 5 + B a _ (1)IEEu i s an open cover of
c.
space and l e t
nf
a continuous s u r j e c t i o n
such t h a t
For every
a convex balanced open subset of
U
a normed l i n e a r space and E
depends on f
E
Let
i s an open s u b s e t of
Ef.
E
onto
Ef.
By
on
We now d e f i n e
i n t h e usual manner and s i n c e it i s l o c a l l y bounded, it l i e s i n
nf(U)
H(nf(U);F).
This completes t h e proof. Corollary 2.30 E
F
and
If U
space
i s a Banach space, then H(U;F)
Proof
is a convex balanced open subset of a
A bjq
=
u
~ E C (E) S
H(na(U);F).
space i s a h e r e d i t a r y Lindelb'f space and a l s o a DF-space.
The r e s u l t now follows e a s i l y by using t h e c o n s t r u c t i o n of p r o p o s i t i o n 2.29 and t h e following property of
DF
spaces:
if
m
i s any sequence o f
continuous semi-norms on a DF-space, then t h e r e e x i s t s a continuous semi-
norm
a
an 6 c na
on
E
and a sequence of p o s i t i v e r e a l numbers
for all
m
( c ~ ) ~such = ~t h a t
n.
Corollary 2.30 may be strengthened i n t h e c a s e of e n t i r e functions (see e x e r c i s e 2.105). Our f i n a l example f i r s t a r o s e i n f i n d i n g a counterexample t o t h e Levi problem.
The proof i s q u i t e d i f f e r e n t from t h o s e j u s t given and v a r i a t i o n s
of t h e technique used w i l l appear i n chapter 5 . Example 2.31
Let
r
denote an uncountable d i s c r e t e s e t .
If
Chapter 2
70 x = (x )
CL CiEr
c0(r)
E
( x E c o ( r ) ; s ( x ) cI',) /If11
and If
J
m
where
...,a n1
{al,
=
rl
f o r any s u b s e t
M <
=
s(x) = {acr;xa # O } .
we let
of
co(rl)
Let
Now suppose
I'.
i s t h e open u n i t b a l l of
B
r
i s any f i n i t e s u b s e t o f
=
f
E
(co(r),
H(2B;C)
11 1 1 ) .
t h e n , by u s i n g a monomial
e x p a n s i o n i n s e v e r a l v a r i a b l e s , w e see t h a t f(z) If
e,
has i t s
zero,lthen
where
IkENjw(k)zk
=
M2
z
for a l l
in
ith c o o r d i n a t e e q u a l t o 2 sup{)f(z)I2; z =
1
i=l
1
2Bn c o ( J )
and a l l o t h e r c o o r d i n a t e s
Ziei,\Zij=
1
il
all
I 1
i s n o r m a l i s e d Haar measure on { z i e i , z . =1} f o r i = l , .. . , n . iBm zm = e , m = l , ..., n, it f o l l o w s t h a t
dzi
By u s i n g the change of v a r i a b l e
Since
Jcr
where
N(')
Hence
{k;w(k)#O}
was a r b i t r a r y , i t f o l l o w s t h a t
= { $ : r + N, $ ( a ) = O
r l ={azr\ 3 s u b s e t o f r. I t
(x,A),
x
E
2B
and
EN'^),
k
E
2B,
co(r)
c o n t i n u o u s , we have shown t h a t E
B.
and
w(k)#O
is e a s i l y seen t h a t
x+Ae
i s a dense subspace o f
(xa)aer
a
in
r}.
is countable.
Let
able
f o r a l l e x c e p t a f i n i t e number o f
if
a
E
laer 1
i s a count-
ufor
f(x+Xea) = f(x)
r\rl.
Since
( i n t h e norm t o p o l o g y ) ( ("a
rl
k(a)#O}.
JCT , J f i n i t e C o ( J ) and f i s
= f((xaIaErl)
for a l l
By u s i n g t h e p r i n c i p l e o f a n a l y t i c c o n t i n u a t i o n ( i n s e v e r a l
71
Holomotphic mappings between locally convex spaces complex v a r i a b l e s ) one c a n e a s i l y e x t e n d t h e above p r o o f t o show t h e f o l l owing : if
then t h e r e e x i s t s a countable subset &
such t h a t onto
co (r I+ and
i s a convex b a l a n c e d open s u b s e t o f
U
f = f on
co(rl).
rl
where
'r
rl
of
r
and
f
E
f
E
H(U;K)
H(U n Co(i-l);E)
i s t h e c a n o n i c a l s u r j e c t i o n of c o ( r )
Many o f t h e above f a c t o r i z a t i o n theorems can be extended t o pseudoconvex domains (and t h i s e s s e n t i a l l y means t o a l l open s e t s ) by v i r t u e of t h e following r e s u l t : if and
U
U
i s a pseudo-convex open s u b s e t o f t h e l o c a l l y convex s p a c e
contains an a - b a l l , a
f i n i t e l y open s u b s e t o f
,E
E
and
cs(E),
then
n
(U)
E
i s a pseudo-convex
U = Uil(II,(U)T.
T h i s r e s u l t i s u s e d i n studying pseudo-convex domains, h o l o m o r p h i c a l l y convex domains and domains of holomorphy i n l o c a l l y convex s p a c e s . F a c t o r i z a t i o n r e s u l t s f o r Mackey holomorphic f u n c t i o n s a r e r e q u i r e d i n Chapter 6 .
The c o n c e p t s and methods needed t o p r o v e t h e s e r e s u l t s w i l l b e
given later. 52.3
LOCALLY CONVEX TOPOLOGIES ON SPACES OF HOLOMORPHIC FUNCTIONS Two t o p o l o g i e s a r e u s u a l l y c o n s i d e r e d on t h e s p a c e o f Mackey holomor-
phic functions;
t h e t o p o l o g y o f u n i f o r m convergence on t h e f i n i t e dimen-
s i o n a l compact s u b s e t s of t h e domain and t h e t o p o l o g y o f uniform convergence on t h e s t r i c t l y compact s u b s e t s o f t h e domain.
Since w e s h a l l not use
a n y r e s u l t s d e r i v e d by u s i n g t h e s e t o p o l o g i e s , we c o n f i n e our i n t e r e s t i n t h e s e t o p o l o g i e s t o some e x e r c i s e s a t t h e end of t h e c h a p t e r .
On t h e
s p a c e o f h y p o a n a l y t i c f u n c t i o n s , t h e n a t u r a l t o p o l o g y i s t h e compact open topology. Definition 2.32
Let
U
be an open subset of the l o c a l l y convex space
E and l e t F be a l o c a l l y convex space. The compact open topology ( o r t h e topology of uniform convergence on the compact subsets of U l on HHY(U;F) i s t h e l o c a l l y convex topology generated by t h e semi-norms
12
Chapter 2
ranges over the compact subsets of
K
where
F.
continuous semi-norms on Naturally
U
and
ranges over the
$
We denote t h i s topoZogy by
T
induces a l o c a l l y convex topology on
T~
again c a l l t h e compact open topology and denote by
T
~
~
.
which we
H(U;F) .
This i s t h e most
n a t u r a l topology t o consider on spaces o f holomorphic f u n c t i o n s .
We f i n d ,
however, it does n o t always possess very u s e f u l p r o p e r t i e s and f o r t h i s reason w e introduce t h e
topology.
T~
This topology has good topological
p r o p e r t i e s b u t can be d i f f i c u l t t o d e s c r i b e i n a concrete fashion and i t s r e l a t i o n s h i p with t h e
topology may not always be c l e a r .
T~
duce a f u r t h e r topology,
T
w’
We a l s o i n t r e
whose d e f i n i t i o n was motivated by c e r t a i n
p r o p e r t i e s o f a n a l y t i c f u n c t i o n a l s i n several complex v a r i a b l e s . topology i s intermediate between t h e
T
0
and t h e
f u l l y it w i l l s h a r e t h e good p r o p e r t i e s o f both
topologies.
T~
and
T~
This
r o l e appears t o be a s a t o o l i n proving r e s u l t s about
T~
T
6 and
Hope-
but i t s main (see
T~
f o r i n s t a n c e chapter 5 ) . D e f i n i t i o n 2.33 F
and l e t
Let
U
be an open subset of a ZocaZZy convex space
be a normed Linear space.
to be ported by t h e compact subset V,KCVcU,
The
T~
C(V)>O
there e x i s t s
topoZogy on
H(U;F)
K
A semi-norm
of
U
p
on H ( U ; F )
E
i s said
i f , f o r every open s e t
such t h a t
i s t h e LocaZZy convex topology generated by
a22 seminorms ported by compact subsets of U . D e f i n i t i o n 2.34
Zet
F
D e f i n i t i o n 2 .3 5
and l e t
Let
u
be an open subset of a ZocaZZy convex space and
be a ZocaZZy convex space.
F
Let
U
We d e f i n e
T~
on
H(U;F)
by
be an open subset of a ZocaZly convex space
be a normed l i n e a r space.
A semi-norm
p
on
H(U;F)
E
i s said
13
Holomorphic mappings between locally convex spaces
t o be (Vn);=l’
T&
P(f)
The
6
C
I1 f l l V
n
topology on H ( U ; F )
T&
the
U,
continuous i f for each increasing countable open cover o f there e x i s t s a p o s i t i v e i n t e g e r no and c , O such t h a t f o r every
f
in
H(U;F).
0
i s t h e l o c a l l y convex topology generated by
continuous seminoms.
Definition 2.36 F
and l e t
be an open subset of a l o c a l l y convex space
U
Let
be a l o c a l l y convex space.
(H(U;F)
We d e f i n e
T&
on H ( U ; F )
E
by
=
,T&)
The g e n e r a l r e l a t i o n s h i p between t h e t h r e e t o p o l o g i e s j u s t d e f i n e d i s g i v e n i n t h e f o l l o w i n g lemma.
Lemma 2 . 3 7
E.
F
and
be l o c a l l y convex spaces and l e t
On H ( U ; F )
we have
T
~
~
T
~
~
We may suppose, w i t h o u t l o s s o f g e n e r a l i t y , t h a t
Proof
l i n e a r space.
Since
follows t h a t on
E
Let
open subset o f
H(U;F)
T~ 2 T
0
.
IlfllK
<
llfllv
Now suppose
p
is a
exists
no
such t h a t
such t h a t
c>O
shows t h a t
T
p
is
T&
containing of
K
Since
U.
i s a neighbourhood o f
Vn
p(f)
cll€lIvno
F
be an
~
i s a normed K
it
c o n t i n u o u s semi-norm
w
which i s p o r t e d by t h e compact s u b s e t
d e n o t e an i n c r e a s i n g c o u n t a b l e open cover o f can choose
v
f o r every
U
T
f o r every
f
U. K
Let
(Vn)i=l
i s compact w e
K.
Hence t h e r e
in
H(U;F).
This
c o n t i n u o u s and completes t h e p r o o f .
Our n e x t r e s u l t shows t h a t
T&
has good t o p o l o g i c a l p r o p e r t i e s when
t h e r a n g e s p a c e i s a Banach s p a c e ( a s l i g h t l y less g e n e r a l r e s u l t h o l d s when t h e r a n g e s p a c e i s a normed l i n e a r s p a c e ) . a l t e r n a t i v e description of t h e Proposition 2 . 3 8
Let
U
T&
T h i s r e s u l t a l s o g i v e s an
topology.
be an open subset o f t h e l o c a l l y convex space
E and l e t F be a Banach space. Then ( H ( U ; F ) , T & ) is an inductive l i m i t o f Frgchet spaces and hence i t is barrelled, bornological and ultrabornol-
ogical .
Chapter 2
74 Proof
For each i n c r e a s i n g c o u n t a b l e open c o v e r o f
l e t HU(U;F) = I f EH(U;F); llfll < m Vn t o p o l o g y g e n e r a t e d by t h e semi-norms
m e t r i z a b l e l o c a l l y convex s p a c e .
n).
all
L') = (Vn);=l,
U,
We endow
.
p n ( f ) = Ilflb
HV(U;F) w i t h t h e
H,(U;F)
is then a
I t i s i n f a c t a 8 r & h e t space s i n c e
F
i s a Banach s p a c e and l o c a l l y bounded G-holomorphic f u n c t i o n s a r e holomor-
phic.
We c l a i m
H(U;F) = U$Ho(U;F)
c o u n t a b l e open c o v e r s o f i s open and 0
Wn
Since
f
H(U;F)
x m Hb(U;F).
( H ( U ; F ) , T ~ )=
follows t h a t
H(U;F) w e l e t
E
Wn
=
CXEU; [lf(x)(\ < n3
We now l e t
w e h a v e proved our claim.
i n d u c t i v e l i m i t t o p o l o g y on i.e.
f
ranges over a l l increasing
i s an i n c r e a s i n g c o u n t a b l e open c o v e r o f
= (Wn)mnZ1
Hw(U;F)
E
If
U.
where
T~
d e f i n e d by a l l t h e s p a c e s
Since
Hv(U;F)
U.
denote the H,(U;F),
i s a F r g c h e t s p a c e it
( H ( U ; F ) , T ~ ) i s a n u l t r a b o r n o l o g i c a l s p a c e and hence it i s
b a r r e l l e d and b o r n o l o g i c a l . We complete t h e p r o o f by showing t h a t
mapping from
Hw(U;F)
into
a b l e open c o v e r CJ o f Let not
p
of
denote a
(Vn):=l
U,
/lfnlIVn$ Let
Wn
=
~
.
Since t h e i d e n t i t y
i s c o n t i n u o u s f o r any i n c r e a s i n g count-ii
and a sequence
p(fn) > n
for all
(1
n
CXEU; ( ( f m ( x )
denotes t h e i n t e r i o r o f
W,"
T
>
7
6' H(U;F).
+,
(U;F) (fn)iz1
i s continuous.
(fn)EZ1
for all Since
Wn.
ml.
in
Let
]]fmIIVQ
H(U;F)
is
for a l l
n.
such t h a t
= (W")"
where n n=l 1 f o r a l l min, it
s i p llfmlb ,<
BY c o n s t r u c t i o n
i s a bounded s u b s e t o f
p(fn) > n
fact that
p
n.
f o l l o w s t h a t D i s an i n c r e a s i n g c o u n t a b l e opt% c o v e r o f 'IH Hence
Suppose
Then t h e r e e x i s t s an i n c r e a s i n g c o u n t a b l e open c o v e r
,
=
1 and
=
c o n t i n u o u s semi-norm on
T i
continuous.
T&
H(U;F)
we have
U
T.
Ho(U;F).
Hence
7.
=
7&
U.
n
Hence f o r every n .
This Fontradicts t h e and we have completed
t h e proof. Since 'o,b with
H(U;F)
and T~
T~
and
w,b and T
T
w
a r e n o t i n g e n e r a l b o r n o l o g i c a l t o p o l o g i e s , we l e t
d e n o t e t h e b o r n o l o g i c a l t o p o l o g i e s on
T
w
respectively.
H(U;F)
associated
(Note t h a t t h e t o p o l o g y induced on
by t h e b o r n o l o g i c a l t o p o l o g y a s s o c i a t e d w i t h t h e compact open
topology o f
HHY(U;F) need n o t b e
o g i e s t h a t can b e p l a c e d on
H(U;F)
T ~ , ~ T ) h. e r e a r e a l s o f u r t h e r t o p o l -
-
such a s t h e t o p o l o g y o f uniform
convergence of t h e f u n c t i o n and i t s f i r s t
n
d e r i v a t e s on t h e compact
75
Holomotphic mappings between locally convex spaces
...,
U, n = 1 , 2 ,
s u b s e t s of
but s i n c e we s h a l l not u s e t h e s e topologies
we w i l l not go i n t o any f u r t h e r d e t a i l s . A g r e a t p o r t i o n of t h i s book i s concerned with f i n d i n g conditions on
U,E
and
F
which imply e i t h e r
=
T~
T
w
, T~
= T~
or
= T
T~
(together
6
The remainder of t h i s s e c t i o n
with t h e i m p l i c a t i o n s of t h e s e c o n d i t i o n s ) .
is devoted t o a number of b a s i c f a c t s , concerning t h e s e t o p o l o g i e s , which we s h a l l f r e q u e n t l y u s e and t o a few examples and counterexamples which w i l l prove u s e f u l i n l a t e r c h a p t e r s .
lVe f i r s t n o t e t h a t t h e compact open topology i s a sheaf topology, i . e .
it i s l o c a l l y defined. T&
We do not know i f t h i s i s t r u e f o r t h e
T
and t h e
w
topologies and any r e s u l t s i n t h i s d i r e c t i o n would c e r t a i n l y h e l p t h e
general development of t h e t h e o r y .
The l o c a l c h a r a c t e r o f t h e compact open
topology i s contained i n t h e following lemma. Lemma 2 . 3 9 (Ui)iEI
Let
U
U
an open cover o f (H(U;F),T~) i n t o
from
and
F
a locally convex space.
IIisI(H(Ui;F),~o)
i s t h e r e s t r i c t i o n of f (where f ' U i onto a subspace of IIiEIH(Ui;F)
H(U;F) Proof
I t suffices t o note t h a t
K
only i f t h e r e e x i s t s a f i n i t e subset of a compact subset of
E,
be an open subset of a l o c a l l y convex space
f o r each
UL.
1
j,
which maps
to
.
Uil
f
to
The mapping (f(Ui)isI
i s an isomorphism o f
i s a compact subset of I,
Ll,
..., Ln
such t h a t
K
and
U
i f and
(Kj)Y=l,
=U3,1Kj.
K.
1
I t i s obvious t h a t a s i m i l a r r e s u l t holds f o r hypoanalytic f u n c t i o n s . We now show t h a t (H(U;F),ro)
( p ( m E ; F ) , ~ o ) i s a closed complemented subspace of f o r any open subset U of t h e l o c a l l y convex space E , any
complete loca?.ly convex space (H(U;F),ro) spaces then
F
m.
Hence i f
( 6 ( m E ; F ) , ~ o ) must a l s o have t h e same property. U
i s an open subset of a ZocalZy convex space E i s a complete locaZZy convex space then ( 6 (mE;F) ,TO) i s a cZosed If
compZemented subspace o f Proof
and any p o s i t i v e i n t e g e r
has any property i n h e r i t e d e i t h e r by subspaces o r by q u o t i e n t
Proposition 2 . 4 0
and
F,
Since
(H(U;F),T~) for any p o s i t i v e integer m.
(H(U;F),T~) = (H(U-S,F),T~) f o r any
5
E
E
we may suppose
76
Chapter 2 As u n i f o r m convergence on t h e compact s u b s e t s o f
O E U.
is equivalent t o
E
uniform convergence on t h e compact s u b s e t s o f some neighbourhood o f z e r o f o r
elements o f
(mE;F)
it f o l l o w s t h a t
t h e compact open t o p o l o g y . A
dm -
H(U;F)
:
m!
f
( H ( U ; F ) , T ~ ) i n d u c e s on @(mE;F)
Now c o n s i d e r t h e mapping
-
H(U;F) ;;mf ( 0 )
_ _ _ _ _ f -
m!
A
T h i s i s a l i n e a r mapping and s i n c e it i s a p r o j e c t i o n from
p(mE;F)
dm - (P)(O) = P f o r every P i n m! H(U;F) o n t o @ (mE;F): To complete t h e
p r o o f w e must show t h a t it i s a c o n t i n u o u s p r o j e c t i o n . convex b a l a n c e d open s u b s e t o f balanced s u b s e t o f
f o r every
in
E
such t h a t
V C U .
Let If
denote a
V
i s a compact
K
t h e n , by t h e Cauchy i n e q u a l i t i e s
V
and hence t h e p r o j e c t i o n i s c o n t i n u o u s .
cs(F)
This
completes t h e p r o o f . For a r b i t r a r y
w e do n o t have any u s e f u l r e p r e s e n t a t i o n o f t h e
U
@
t o p o l o g i c a l complement o f
(%;F)
in
H(U;F)
b u t we s h a l l see, i n t h e
n e x t c h a p t e r , t h a t t h e T a y l o r s e r i e s r e p r e s e n t a t i o n o f holomorphic f u n c t i o n s g i v e s u s a means o f i d e n t i f y i n g a u s e f u l t o p o l o g i c a l complement when
We now p r o v e t h e a n a l o g u e o f p r o p o s i t i o n 2.40 f o r t h e
balanced.
xS
topologies.
T
is
U
w
and
Our p r o o f i s f o r Banach s p a c e v a l u e d mappings b u t t h e same
r e s u l t f o r a n a r b i t r a r y complete l o c a l l y convex r a n g e s p a c e c a n be proved i n a similar f a s h i o n . Proposition 2.41
and
F
If
U
((3 ( m E ; F ) , ~ w ) is a cLosed complemented
i s a Banach space then
subspace of (H(U;F),-cw ) and of ( H ( U ; F ) , T ~ ) . I n particular induce the same topoZogy on 8 (mE;F). Proof
We f i r s t show t h a t
denote a
T~
T~
i n c r e a s i n g c o u n t a b l e open c o v e r o f N
and
c o n t i n u o u s semi-norm on
b a l a n c e d neighbourhood o f z e r o i n integer
and
C>O
E
is an open s u b s e t of a l o c a l l y convex space
such t h a t
E. U
T~
c o i n c i d e on
H(U;F)
and l e t
The sequence
l$'(mE;F). V
T
w
and
T6
Let
p
d e n o t e a convex m
(UnnV)n=l
i s an
and h e n c e t h e r e e x i s t a p o s i t i v e
71
Holomorphic mappings between locally convex spaces
for a l l
f
E
H(U;F).
'i;
Hence
Ptopology I PPE;F)on
is a
continuous semi-norm on
T , ~
4
=
8(
i)/p(mE;F) m ~ ; ~f o) r a l l
it follows t h a t
T~
H(U;F) and
and s i n c e induce t h e same
T~
m.
The above a l s o shows t h a t t h e mapping given i n p r o p o s i t i o n 2.40 i s a continuous p r o j e c t i o n f o r both
T~
and
T ~ .
We now look a t t h e l o c a l l y bounded o r equicontinuous s u b s e t s of H(U;F). D e f i n i t i o n 2.42 E
F
and l e t
be an open subset of the locally convex space
U
Let
be a ZocaZly convex space.
locally bounded i f f o r every c,Vg,
Lemma 2.43
A subset
2
of H(U;F)
is
there e x i s t s a neighbourhood of
E
where
Proof
F.
A locally bounded subset of
(H(U;F),T6)
and
F
U
an open subset of
are l o c a l l y convex spaces, i s a bounded subset of
We may assume, without l o s s of g e n e r a l i t y , t h a t Let r) be a l o c a l l y bounded subset of
continuous semi-norm.
wn and l e t
H(U;F),
*
l i n e a r space. T~
in U
such t h a t
i s a bounded subset of
E
5
=
IXEU;
For each p o s i t i v e i n t e g e r
(IfCx)((
Vn = I n t e r i o r (Wn).
c n f o r every f Since
3
i s a normed
F
H(U;F), n
and
p
a
let
i n 31
i s l o c a l l y bounded
(Vn)n
i s an
78
Chapter 2
i n c r e a s i n g c o u n t a b l e open c o v e r of
Hence t h e r e e x i s t s
U.
and
C>O
N,
a p o s i t i v e i n t e g e r such t h a t
cllfll
p (f) s
f E
f
in
H(U;F)
and t h i s completes t h e p r o o f .
sup p ( f ) s C.N
Hence
f o r every
vN
3
If U i s an open subset of a ZocaZZy convex space E, F is a l o c a l l y convex space and every bounded subset of (H(U;F),ro) is l o c a l l y bounded then T ~ , T and ~ T & have the same bounded subsets in C o r o l l a r y 2.44
H(U;F). In particular
etc. i n place o f
C o r o l l a r y 2.45
then
and
(H(U),-ro)
H(U;F) = H(U;Fu)
H(U;C), YIY(U;E), e t c . F
be ZocaZly convex spaces.
Let
Fa = ( F , o ( F , F ' ) ) .
where
.3
= (@of)@EB
lies i n
H(U).
Thus
:y
Consequently
in
Vg
f(Vg)
containing
U
5
f @
i s a l o c a l l y bounded f u n c t i o n .
H(U;Fu).
E
in
(H(U),ro)
Hence f o r each such t h a t
sup
then
U
F'.
The
Hence
5 E U there V5 < a.
exists
and once more
i s a bounded s u b s e t of
f(Vg) Hence
and s o by
I($of((
i s a weakly bounded s u b s e t o f @ E B F
by Mackey's theorem it f o l l o w s t h a t f
f o r every
C
i s a bounded s u b s e t o f
o u r h y p o t h e s i s , it i s l o c a l l y bounded. a n open s e t
and l e t
and by Mackey's theorem i t i s
F
i s a weakly bounded s u b s e t o f
s t r o n g l y bounded.
i s a normed
F
i s a compact s u b s e t o f
K
If
@cf(K) = @ ( f ( K ) ) i s a bounded s u b s e t o f f(K)
F' B
be t h e u n i t b a l l o f
B
I f the
are ZocaZZy bounded f o r every open subset U
We may assume w i t h o u t l o s s o f g e n e r a l i t y t h a t
l i n e a r space. set
E
Let
bounded subsets of
Proof
.
T
I f t h e r a n g e s p a c e i s t h e f i e l d o f complex numbers w e w r i t e
Notation H(U), HHy(U)
of E
is t h e bornoZogicaZ topoZogy associated with
T~
F, i . e .
i s a holomorphic f u n c t i o n and
f
t h i s completes t h e p r o o f s i n c e t h e c o m p o s i t i o n o f holomorphic f u n c t i o n s i s holomorphic and so we always h a v e Example 2.46
Let
s p a c e and l e t subset,
3,
F of
U
H(U;F)
C
b e a n open s u b s e t o f a m e t r i z a b l e l o c a l l y convex
We claim t h a t any
b e a normed l i n e a r s p a c e . H(U;F)
H(U;F,).
i s l o c a l l y bounded.
SEU such t h a t f o r e v e r y open s e t
V,
,€ E
T~
bounded
I f not, then t h e r e e x i s t s
VCU,
w e have
sup I l f l ) fE
3
=
a.
79
Holomotphic mappings between locally convex spaces Hence we can choose IIfn(Sn)II > n
S,,
n.
for a l l
5,
U,
E
+
Since
5
n
as
and
+ m,
(fn)n
c3such
that
i s a compact subset of
{5n}nUIC)
U
t h i s i s impossible and we have proved our claim. The remaining examples given i n t h i s s e c t i o n d e a l with holomorphic and on l o c a l l y 8 3 % spaces, Banach spaces, aN x C")
f u n c t i o n s on
convex spaces which do not admit a continuous norm.
These examples a r e
elementary i n s o f a r a s t h e proofs a r e r a t h e r d i r e c t .
However, they a r e of
i n t e r e s t s i n c e they show t h e divergence between l i n e a r and holomorphic f u n c t i o n a l a n a l y s i s and a l s o because many of t h e examples and methods encountered h e r e have e x p l i c i t l y and i m p l i c i t l y motivated t h e development These examples a l s o provide a good
of t h e theory a s o u t l i n e d i n t h i s book.
i n t u i t i v e guide t o t h e t y p e of behaviour we may look f o r i n d e l i c a t e situations. Example 2.47 F
Let
U
be an open subset of a
b e a normed l i n e a r space.
We show t h a t t h e
space
and l e t
E
bounded s u b s e t s of
a r e l o c a l l y bounded (we have already proved t h i s r e s u l t f o r homo-
H(U;F)
geneous polynomials i n chapter 1 ) . ity, that
We may assume, without l o s s of genera
i s a convex balanced open subset of
U
bounded subset
T~
83 T~
3
of
H(U;F)
E
and we show t h a t t h e
i s l o c a l l y bounded a t t h e o r i g i n .
Let
be a fundamental system of convex balanced compact s u b s e t s of
(BJn
L-
E.
As we have previously noted, i t s u f f i c e s t o f i n d a sequence of p o s i t i v e
r e a l numbers,
(in),., such t h a t
B1
f E 3
B = kl 1
If
11 IIA
1.1=1A.B. 1 1
=
6>0
choose and
we l e t
f
M < m .
L(6) = L + 6Bk+l.
such t h a t
in
inequalities.
Now suppose
H(U;F) Hence
L(S1)CU
B1 B2L(61)
where
c(B1)
11
<
m.
'i;=l~n%
il> 0 such
...,
that
Hence
AIBICU.
i2, ik have been chosen
is. a compact s u b s e t of
such t h a t
61>0
B 2 '1,
for a l l
f~3-
i s compact, we can choose
Since
L
sup Ilf
U
Let
E
and >O
sup fE3
((fl(
S M +
be a r b i t r a r y .
and next choose
B1
so t h a t
I,,=,
--
-MI.
2"
We f i r s t
and
i s a l s o a compact subset of
B2, U.
i s derived by using t h e Cauchy
B1>
1
Hence
Chapter 2
80
continuous semi-norm on
___ dnf(o)
Let
n! If
f
E
H(U;F)
and s i n c e
H(U;F)
we can f i n d a p o s i t i v e i n t e g e r
and 6 > 0
s2
T
-bounded and
61,
B2>1
fin d f(O)/n!.
then
n. <
0,
is
be t h e symmetric n - l i n e a r form a s s o c i a t e d with
f o r any non-negative i n t e g e r
62
3
such t h a t
N
m
f o r each
n
and
6
we can choose
so t h a t
Hence, by induction, we can choose a sequence of p o s i t i v e real numbers,
3 and
i s a l o c a l l y bounded family o f f u n c t i o n s . T
6
d e f i n e t h e same bounded s u b s e t s o f
This implies t h a t H(U;F).
T ~ ,T~
Since B 8 3 b s p a c e s
a r e h e r e d i t a r y Lindelzf spaces and contain a fundamental sequence of compact s e t s it follows t h a t every open subset of a 3 3 M s p a c e contains
81
Holornorphic mappings between locally convex spaces a fundamental system of compact s e t s . Since
T6
that
T~
643%
spaces a r e
=
This, i n t u r n , implies t h a t
i s a metrizable, and hence a bornological l o c a l l y convex space. i s a l s o a bornological topology on H(U;F) we have i n f a c t shown
(H(U;F),ro)
T
-
w
-
on
6'
H(U;F).
F i n a l l y we remark t h a t open s u b s e t s of
k-spaces and s o
i s a Frgchet space i f
(H(U;F),ro)
F
is
a Banach space. Example 2.48
Let
Banach space m
E
is a
T
0
c0
be an open subset of an i n f i n i t e dimensional
U
Let
E.
5
continuous semi-norm on
be t h e u n i t b a l l of
If
E.
H(U).
H(U) Hence
which i s not continuous f o r t h e (H(U) ,
T ~ )#
(H(U) ,.cw)
f o r any open
o f any i n f i n i t e dimensional Banach space.
U
Example 2.48
Let
CN
E = CN
((? (2E) , T ~ ) .
( @ (2E) , T ~ )#
s e t of
B
( t h e space of n u l l sequences of complex numbers) then
compact open topology on subset
and l e t
U
E
x
C(N).
x
a").
Hence
We have a l r e a d y seen t h a t (H(U),ro) # (H(U),ru)
Example 1.39 shows, a l s o , t h a t
T
f o r any open sub~
#
,T
~~
on ,
~H(U).
For our next example we need a concept which f r e q u e n t l y a r i s e s i n i n f i n i t e dimensional holomorphy
-
t h e concept of very s t r o n g s e q u e n t i a l '
convergence - b u t which does not a r i s e i n l i n e a r f u n c t i o n a l a n a l y s i s .
Since
t h e dual concept - very weak s e q u e n t i a l convergence - w i l l a l s o be needed l a t e r , we t a k e t h e opportunity of giving i t s d e f i n i t i o n h e r e .
Further
information on t h e s e concepts i s o u t l i n e d i n t h e e x e r c i s e s .
D e f i n i t i o n 2.50
A sequence
( x ~ )i n~ a locally convex space
said t o be very strongZy convergent if sequence o f scalars x
Xnxn+ 0
in
E
as
n-
E
is
f o r every
The sequence is said t o be nontrivial i f
n # 0 f o r each n. A sequence i s obviously very s t r o n g l y convergent i f and only i f f o r
each
p
p(xn) = 0
in
cs(E)
for a l l
there exists a positive integer,
n 2 n(p).
n(p),
such t h a t
A metrizable l o c a l l y convex space
a n o n t r i v i a l very s t r o n g l y convergent sequence i f and only i f
E
E
admits
does n o t
-
82
Chapter 2
admit a c o n t i n u o u s norm.
un
(0, ...,1 , O ...)
=
Definition 2.51
CN
For example, i n
t h e sequence
i s a n o n t r i v i a l v e r y s t r o n g l y convergent sequence.
nth position
For example, i n
E
( x ~ )i n~ a l o c a l l y convex space
A sequence
t o be very weakly convergent if A n xn + 0 i n of non-zero scalars ( A ~ ) ~ .
E
as
i s said
f o r some sequence
M
u = (0 ,..., 1,0 ,...) i s n o t a n L- n t h p o s i t i o n In f a c t , i f E i s any Fre‘chet s p a c e which
t h e sequence
v e r y weakly convergent sequence. i s n o t a Banach s p a c e t h e n
c o n t a i n s a sequence which does n o t converge
E’
v e r y weakly.
Let
Example 2 . 5 2
b e a l o c a l l y c o n v e x s p a c e which c o n t a i n s a non-
E
( x ~ ) ; = ~ . Let
t r i v i a l v e r y s t r o n g l y convergent sequence
E.
For each
If
f
f
f(x+w) = f ( x )
Since
00
such t h a t
n > n
for all
a
f + If(ny+x )
uous semi-norm on
n H(E)
a b a r r e l l e d t o p o l o g y on semi-norm on
f o r every
x
and
w
and
-
for all
a(xn) = 0 p(f)
E
C,
all
defines a
f(ny)(
?
H(E)
n,.
it f o l l o w s t h a t
Hence f
in
and hence a
-co
B
be a
H(E), T~
i.e.
n
2
no
p
is a
= 0.
p
f(ny+xn) = H(E).
-c6
nu and s i n c e is a
-i6
The
continT~
is
continuous
and a l l
f
fn(xny + xn) # f n ( x n y )
i s bounded on t h e
T~
c o n t i n u o u s semi-norm on o,b H(E). We b e g i n by showing t h a t
n
such t h a t
in
B.
for all
p
‘I
bounded s u b s e t o f
f(iy+xn) = f(hy)
for all
I f t h i s were n o t t r u e , t h e n by
u s i n g subsequences i f n e c e s s a r y , w e can choose that
n
f o r every p o s i t i v e i n t e g e r
there exists a positive integer h
cs(E)
.(w)
H(E).
bounded s u b s e t s of Let
CL E
such t h a t
E
i s f i n i t e f o r every
W e now improve t h i s r e s u l t by showing t h a t
H(E).
in
i s a v e r y s t r o n g l y convergent sequence t h e r e e x i s t s a p o s i t -
(Xn)n=l i v e i n t e g e r n, function
w e c o n s i d e r t h e sum
H(E)
such t h a t
f(ny)
belong t o
t h e n , by t h e F a c t o r i z a t i o n Lemma, t h e r e e x i s t s an
H(E)
E
in
y#O
n.
An E C and f E B such n For each p o s i t i v e i n t e g e r n l e t
83
Holomorphic mappings between locally convex spaces
By t h e i d e n t i t y theorem f o r f u n c t i o n s o f one complex v a r i a b l e we may
s e l e c t a sequence o f complex numbers, gn(Xh) # 0.
Now
hn
For each i n t e g e r
H(E)
E
f o r each
n,
n.
Hence
and each
and s i n c e m
sequence o f complex numbers, for all
n
IXAl
(wn)n=l,
Ifn(X;y+wnxn)I
>
n
let
C
w E
6
% , n
hn(0) # h n ( l ) , for all
we c a n choose a
I hn(wn) I
such t h a t n.
such t h a t
Since
I
I
n+ fn(X,!,y)
>
( x ~ )i s~ a
v e r y s t r o n g l y c o n v e r g e n t sequence, (w x ) i s a n u l l sequence. Since n n-n IhAl s - f o r a l l n it f o l l o w s t h a t XAy+wnxn -t 0 as v and hence n2 03 K = I01 u IA;Y+wnxnln=l i s a compact s u b s e t o f E . As IlfnllK > n f o r
all
n
t h i s contradicts the fact that
B
is a
H(E).
Hence t h e r e e x i s t s a p o s i t i v e i n t e g e r
f(Xy)
for all
n
We now show t h a t
n
5
p
all 1
0'
C
is not a
otherwise, i . e . t h a t
p
T~
n
and a l l
f
bounded s u b s e t o f
such t h a t in
T h i s shows t h a t
B.
c o n t i n u o u s semi-norm on
T
i s p o r t e d by t h e compact s u b s e t
H(E).
$,(y)
# 0,
+ o ( ~ n )= 0
for all
n,
$,(y)
of
K
u s i n g subsequences i f n e c e s s a r y we c a n choose a s e q u e n c e i n such t h a t
f(Xy+xn) =
= 0
Suppose By
E.
E', for all
n>O
$ n ( ~) # 0 i f and o n l y i f n=m. Let m $,,, = $o$m n f o r any p o s i t i v e i n t e g e r s n and m . $n,m E P("+'E) and n n P($n,m) = Im $ o ( y ) $ m ( x m ) ~ f o r a l l n and lil. I f V i s any neighbourhood and f o r
of
H(E).
ll~olIvm < m'
O
m($,(Y)
I
follows t h a t
# 0.
$,(y)
Let
V
=
ll$mllvm <
II$,1IVm.
llgoIIvm.
6
c(V) > 0
such that.
Choose a n a r b i t r a r y neighbourhood and
II$ I l n
c(V,,,)
positive integers
then there e x i s t s
K
in
f
m,n
p
Hence
Taking Since
ml+o(y)I 6 Hence
-.
nth
Vm
II$oIlx.
is not
{f € H ( E ) ; p ( f ) ,< l}.
p ( f ) 6 c(V) llfllv Vm
of
K
w
V
such t h a t
mnl~o(Y)lnl$m(xm)l 6
r o o t s and l e t t i n g
we get
n-
was a n a r b i t r a r y neighbourhood o f T h i s cannot hold f o r a l l
T
f o r every
m
K
it
since
continuous.
i s a convex b a l a n c e d
T
(and h e n c e
T
w
.)
Chapter 2
84
bounded subset
of
Since
E.
neighbourhood of zero (because that neither spaces.
(H(E) ,
p
nor
T ~ )
i s not a
V
i s not
T T
w
(and hence not a
T ~ )
continuous) we have shown
(H(E) , T ~ ) a r e i n f r a b a r r e l l e d l o c a l l y convex
The above can e a s i l y be modified t o show t h a t t h e same r e s u l t
holds f o r
H(U:F\,
an a r b i t r a r y open subset of
U
E,
and
F
any l o c a l l y
convex space.
12.4
GERMS OF
HOLOMORPHIC
FUNCTIONS
We now introduce t h e space of holomorphic germs on a compact subset of Apart from i t s c l o s e r e l a t i o n s h i p with spaces of
a l o c a l l y convex space.
holomorphic f u n c t i o n s defined on open sets, t h e space
of germs i s a l s o an
important t o o l i n developing a s a t i s f a c t o r y d u a l i t y theory.
The problems
t h a t a r i s e i n studying t h e topological vector space s t r u c t u r e of t h e space of germs a r e of a d i f f e r e n t kind from those which a r i s e i n function space theory and t h i s d i f f e r e n c e a r i s e s p r i m a r i l y from t h e d i f f e r e n c e between p r o j e c t i v e and inductive l i m i t s . Let
K
be a l o c a l l y convex space.
relation which
f
4
u
be a compact subset o f a l o c a l l y convex space
f
where g
and
N
g
On
H(V;F) V 3K V open
We denote by
H(K;F)
and l e t
F
we define the equivalence
i f there e x i s t s a neighbourhood
are both defined and
E
W
K
of
on
flw = g l w .
the r e s u l t i n g vector space o f equivalence classes
If f is and the elements of H(K;F) are called holomorphic germs on K. an F-valued holomorphic f u n c t i o n defined on an open subset of E which contains
K
then we a l s o denote by
determined by
f.
f
t h e equivalence c l a s s i n
The natural topology on
H(K;F)
H(K;F)
i s given by t h e
l i m (H(V;F),rcw) ( t h e inductive l i m i t being taken i n t h e --f V 3 K Vopen category of l o c a l l y convex s p a c e s ) .
inductive l i m i t
for
I f F i s a normed linear space we l e t Hm(V;F) = {f E H(V;F);I) f l I V < m j V open in E and on t h i s space we define a topology by means
of the norm F
11 11 v.
i s a Banach space.
see that
H"(V;F)
i s a normed l i n e a r space which i s complete i f
Using t h e same equivalence r e l a t i o n s h i p we e a s i l y
85
Holomorphic mappings between locally convex spaces
=
H(K;F)
V3 K V open
If K i s a compact subset o f a l o c a l l y convex space
Lemma 2 . 5 3
E and
F i s a normed l i n e a r space then
V3K V open
V3K V open
Proof If V is an open subset of E which contains K then the natural injection from Hm(V;F) into (H(V;F) ,T,,,) - is continuous and hence the identity mapping from lirn (Hm(V;F), 11 ) into lirn (H(V;F) , T ~ )
IIv
-+
VDK V open is also continuous. Conversely, if p lim
(Hm(V;F),
IIv)
11
then for each V
3
V> K V open is a continuous semi-norm on open, V 2 K
there exists
c(V)>
0
-----f
V 2K V open such that p(f)
c(V) IIfllV for every f in Hm(V;F). If then llfllv = m and the same inequality holds. Hm(V;F) ,<
f E H(V;F)\ Hence the restriction of p to H(V;F)
is a
ported by the compact subset K
Thus p
norm on
lim
+
(H(V;F),T~)
of V.
T
continuous semi-norm is also a continuous semi-
and this shows that the two topologies coincide
V> K V open
on H(K;F)
and completes the proof.
It follows that H(K;F) is a bornological space if F is a normed linear space and an ultrabornological (and hence a barrelled) space if F is a Banach space.
If E
is a metrizable space and F
is a Banach space
then H(K;F)
will be a countable inductive limit of Banach spaces and hence a bornological DF-space. Thus we see that the space of germs will always have some good topological properties since it is an inductive limit and indeed the main topological problems connected with H(K;F) are those generally associated with inductive limits (as opposed to those connected with projective limits) such as completion, description of the continuous semi-norms (sometimes we only need a description of sufficiently many
86
Chapter 2
continuous semi-norms) and a c h a r a c t e r i z a t i o n of t h e bounded s e t s . encounter a l l of t h e s e problems i n l a t e r c h a p t e r s . s e l v e s t o c h a r a c t e r i z i n g bounded s e t s when E = limE
Let
E
We s h a l l
Here we confine our-
i s metrizable.
be an i n d u c t i v e l i m i t of l o c a l l y convex spaces.
The
4
a
inductive l i m i t i s said t o be regular i f each bounded subset of E contained and bounded i n some
is
Ea.
Lemma 2 . 5 4 W
let B
=
Let K be a compact subset o f a l o c a l l y convex space be a convex balanced open subset of E . Then
{f EH(K+W); ( ( f ( \ K + ,W < 1) i s a cZosed subset o f
Proof
Let
{falaEA be a convergent n e t i n
H(K)
which l i e s i n
show t h a t
{fa)aEA i s a Cauchy n e t i n K+W
L CK+pW.
By using t h e Cauchy i n e q u a l i t i e s we s e e t h a t
(H(K+W) , T ~ ) .
'm, K , L
p,
If
L
O
H(K)
and a l s o on
We
i s a compact
such t h a t
(H(K+W),
YEL
f o r every nonnegative i n t e g e r
m.
{ f a l a E A C B t h e Cauchy i n e q u a l i t i e s imply t h a t
Hence, f o r some
B.
XEK YEL
d e f i n e s a continuous semi-norm on
Since
and
H(K).
subset of
then t h e r e e x i s t s a r e a l number
E
00
and a l l
a,B E A ,
T ~ ) .
Hence
87
Holomorphic mappings between locally convex spaces this
t e n d s t o z e r o as in
a+=
H(K).
ci,B
since
+ m
By t h e above, t h e r e e x i s t s a
u n i f o r m l y on t h e compact s u b s e t s of every non-negative i n t e g e r m h f o l l o w s t h a t s m f ( F ) = d g(C) g
Proof
Since
in
Since
K+W.
m
for all
+
such t h a t
B
as
f fa
+
g
Pm,K,L(f) = Pm,K,L(g)
and a l l H(K)
fa
5
of
L
K.
in
K+W
it
Hence
f
and
and t h i s completes t h e p r o o f .
i s a compact subset of a rnetrizable locally i s a regular inductive Z i m i t .
H(K)
i s m e t r i z a b l e it c o n t a i n s a c o u n t a b l e fundamental
E
for
K
If
convex space then
g
and e v e r y compact s u b s e t
d e f i n e t h e same e q u i v a l e n c e class i n
Proposition 2.55
Now suppose
O
neighbourhood system a t z e r o ,
(Vn)n,
and hence
\In)
H(K) = l i m (Hm(K+iTn),ll --f
n
i s a bornological
DF
space.
Hence each bounded s u b s e t o f
c o n t a i n e d i n t h e c l o s u r e o f a bounded s u b s e t o f some
is
Hm(K+Vn). S i n c e t h e
Hm(K+Vn) i s a l s o a c l o s e d s u b s e t o f
closed u n i t b a l l of
H(K)
H(K), by lemma
2 . 5 4 , t h i s completes t h e p r o o f A semi-norm c h a r a c t e r i z a t i o n o f t h e bounded s u b s e t s o f
H(K)
is given
i n t h e following p roposition Proposition 2.56
convex space
E.
K
Let A subset
be a compact subset of a rnetrizabze ZocalZy B
of
H(K)
i s bounded i f and onZy i f it
s a t i s f i e s each of the foZlowing conditions: ( a ) f o r each continuous semi-norm
(b) i f
03
( x ~ ) ~and = ~ (x;):=~ m
p
on
H(0)
are two convergent sequences i n
m
(ynInz1
and (yA)n=l are nu22 sequences i n E, (kn);=l a s t r i c t l y increasing sequence of p o s i t i v e integers and xn+yn
Proof
=
xA+y'
n
for a22
n
K, is
then
W e f i r s t show t h a t t h e semi-norms g i v e n i n
(*)
and
(**)
are
Chapter 2
88
continuous with r e s p e c t t o t h e i n d u c t i v e l i m i t topology
T
on
H(K).
Using t h e above n o t a t i o n w e l e t
f o r every
f
Since and
n
in (H(K)
i s a b a r r e l l e d l o c a l l y convex s p a c e and b o t h
T)
induce t h e
H(0)
T~
t o p o l o g y on
8
(nE)
H(K)
f o r each p o s i t i v e i n t e g e r
it s u f f i c e s t o show each o f t h e above semi-norms i s f i n i t e .
Let such t h a t
f
E
f
H(K
be a r b i t r a r y .
c a n be i d e n t i f i e d w i t h an element o f
M = sup lf(x+4y) XEK,YEV
Since
p
C(V) > 0
i s a c o n t i n u o u s semi-norm on
such t h a t
p(g)
n>N
o f zero
Hm(K+4V). L e t
(H(O),T)
t h e r e e x i s t s a constant
c C(V) / / g / / v f o r e v e r y g i n H(0).
Hence
is continuous.
(*)
Now choose a p o s i t i v e i n t e g e r For a l l
V
I .
and each semi-norm o f t h e form
n>N.
There e x i s t s a neighbourhood
N
such t h a t
yn
and
yn '
E
V
for a l l
89
Holomorphic mappings between locally convex spaces
and hence
Thus each semi-norm o f t h e form
set of
H(K)
i s c o n t i n u o u s and any bounded sub-
(**)
s a t i s f i e s c o n d i t i o n s ( a ) and ( b ) . i s a s u b s e t o f H(K) which s a t i s f i e s (fa)aar Using semi-norms o f t h e form (*) we see t h a t
Conversely, suppose ( a ) and ( b ) .
$"fa (x) (-
n!
1
a~y,x~Kn , arbitrary
i s a bounded s u b s e t o f
Since
H(0).
t h e r e e x i s t s a neighbourhood o f
f o r every If
a XEK,
in
r,
x
and
YEW
in aE.T
0
H(0)
in
and a l l
K
E,W,
i s r e g u l a r ( p r o p o s i t i o n 2.55) and
such t h a t
&O
n.
let
To complete t h e p r o o f , it s u f f i c e s t o show t h a t t h e r e e x i s t s a neighbourbood x,x'
V E
o f zero i n
K, y,yf E V
E,
with
VCW,
such t h a t
x+y = x ' + y '
f,(x)(y)
and e v e r y
a in
= fa(x')(y')
r.
for all
90
Chapter 2
If not, there exist sequences in E ,
xn+yn
such that
two sequences in m
(yn)n=l
and
(Y;);=l,
= x’+y’ f o r all
n
K,
n
n
m
03
( x ~ ) ~ =and ~
( x A ) ~ = ~ ,two null
and a sequence
(an):=1
in
r
and
Now choose inductivelz a strictly increasing sequence of positive integers m
(kn)n= 1
such that
2 nSn > n+2M
for all n.
Let
Since
(faIaEr
satisfies condition ( b ) ,
sup q(fa)
<
m.
c1
On the other hand,
This is a contradiction, and completes the proof Condition (a) says that bounded subsets of H(K)
satisfy Cauchy
K while condition (b) says that the Taylor series expansions
estimates on
satisfy certain coherence properties. If K
satisfies certain connected-
ness conditions, then condition (a) is sufficient. That some conditions on K are necessary for boundedness is shown by the following example. Example 2.57
Vn Let
=
Let
IzEE; z
=
E
= C.
x+iy, x <
F o r each positive integer
-1
E
H(V
n
u Wn)
let
and let tun = I z E C ; z=x+iy, x
n + -2
fn
n
be identically one on Vn
1 1 n+z
> __
and identically zero on Wn
91
Holomorphic mappings between locally convex spaces 1 K = { --} n ~ ul { O }
If
n.
for all
then
The sequence
i s a compact s u b s e t o f
K
{fn};=l
and
C
fn
E
H(K)
s a t i s f i e s c o n d i t i o n (a) o f p r o p o s i t i o n
2.56 b u t i s n o t bounded by p r o p o s i t i o n 2.55. We s h a l l u s e p r o p o s i t i o n 2 . 5 6 and i t s method o f p r o o f a g a i n i n a l a t e r
chapter.
The f o l l o w i n g r e s u l t f o l l o w s e a s i l y from t h e c o r r e s p o n d i n g r e s u l t
f o r holomorphic f u n c t i o n s on open s e t s .
a compact subset of a l o c a l l y convex space i s a cornpZete ZocaZly convex space, then ( 8 ( m E ; F ) , ~ , ) is a
P roposi t i on 2.58 E
and
F
K
If
is
closed complemented subspace o f H(K;F)
f o r every p o s i t i v e i n t e g e r
I n d i s c u s s i n g holomorphic f u n c t i o n s on a compact s u b s e t l o c a l l y convex s p a c e complete.
A
If not, l e t
E
b a l a n c e d open s u b s e t o f into
6
d e n o t e t h e completion o f
E.
If
of a
K
w e may always suppose t h a t t h e s p a c e
E
is
E
i s a convex
V
t h e n t h e n a t u r a l r e s t r i c t i o n from
m.
Hm(K+V)
H m ( K + V ~ E ) i s a b i j e c t i v e i s o m e t r y and hence t h e s p a c e s
H(KE)
H(KA) are isomorphic as l o c a l l y convex s p a c e s , ( K E ( r e s p e c t i v e l y E t h e s e t K c o n s i d e r e d as a s u b s e t o f E ( r e s p e c t i v e l y g ) ) .
Kg)
and is
We s h a l l a l s o need h y p o a n a l y t i c germs i n l a t e r c h a p t e r s and t h e d e f i n i t i o n s are e x a c t l y as one would e x p e c t .
If
K
u
i s a compact subset of a l o c a l l y convex space
ence r e l a t i o n s h i p f
N
and
v
by g
and
F
is a
HHY(V;F) 7')e define t h e equivalKCV, V open fNg i f there e x i s t s a neigkbourhood W of K
l o c a l l y convex space then on
on which both
E
are defined and coincide.
open
We l e t
/
and t h e elements of t h i s vector space are caZled F-valued hypoanaZytic germs
on K.
On
Yly(K;F)
we d e f i n e t h e l o c a l l y convex i n d u c t i v e l i m i t t o p o l o g y
W e have used an i n d u c t i v e l i m i t o f t o p o l o g y on
H(K;F).
T
topologies t o define t h e natural
I t i s n o t known i f t h e
as a p r o j e c t i v e l i m i t o f t h e s p a c e s
H(K;F).
T
topology can be recovered
T h i s problem i s r a t h e r impor-
t a n t and a p p e a r s t o b e i n t i m a t e l y r e l a t e d t o t h e l o c a l i z a t i o n problem
92
Chapter 2
is
mentioned p r e v i o u s l y :
T
a l o c a l topology?
w
i n q u i r y , w e c a n d e f i n e a "new'' t o p o l o g y on
a l o c a l l y convex s p a c e and T
.
D e f i n i t i o n 2.59 F
and
U
If
H(U;F),
a n open s u b s e t o f
U
a l o c a l l y convex s p a c e , which w e d e n o t e by
F
I t i s conjectured t h a t
Following t h i s l i n e o f
and
T~
always c o i n c i d e .
T
E
i s an open subset o f a locaZly convex space
i s a Zocally convex space then the
defined as the projective l i m i t o f
topoZogy on H(U;F)
T~
is
H(K;F)
where
on
H(U;F).
I n c h a p t e r s i x , we s h a l l
and
coincide.
K
ranges over the
compact subsets of U. One e a s i l y sees t h a t
T ~ '
~
TS
g i v e examples o f s i t u a t i o n s i n which
52.5
EXERCISES
2.60
Show t h a t
HG(U;F)
dimensional space E .z; (E"). 2.61*
T
w
f o r any open s u b s e t
and a n y l o c a l l y convex s p a c e
E
Show t h a t
H(U;F)
=
T
(E,tf)
U
o f an i n f i n i t e
i f and o n l y i f
F
i s a t o p o l o g i c a l v e c t o r s p a c e i f and o n l y i f
E
has a countable algebraic b a s i s . 2.62 E
onto
If
and
E
TI : ( E , t f )
2.63
If
f o r any 2.64* E.
f o r every 2.65*
@
i s a l i n e a r mapping from
fl
Let
E
i s a n open mapping.
a r e v e c t o r s p a c e s and
E
E
and
A
E
Show t h a t
f
E
it(mE;F)
show t h a t
n6m.
b e Banach s p a c e s and l e t
F
HG(U;Fi). in
(F,tf)
y1 ,..., yn
and
E E
---t
F
E,
in
Let f
and
E
x
Let
a r e v e c t o r s p a c e s and
F
show t h a t
F
H(U;F;)
U
b e a n open s u b s e t o f
i f and o n l y i f
@ o f EH(U)
F. and
F
b e Banach s p a c e s and
U
a connected open s u b s e t
93
Holomorphic mappings between locally convex spaces of
E.
2.66 -
E.
If
by
f(U)
f : U
Let
f(V) C F '
Let f
2.67* -
F*
-f
and suppose
@
0
f EH(U)
f o r some nonempty open s u b s e t
H(U;F)
E
F
and
E
f o r every
of
V
b e Banach s p a c e s and l e t
Let
show t h a t t h e c l o s e d v e c t o r s u b s p a c e o f
and
E
F
b e Banach s p a c e s .
such t h a t
f(Vx)
F
A mapping
generated
F
g e n e r a t e d by
is said
f : E-tF
i f t h e r e e x i s t s a neighbourhood
x
and
F
f EH(U,; Fb).
b e a n open s u b s e t o f
U
i s equal t o t h e c l o s e d v e c t o r subspace o f
t o b e compact a t t h e p o i n t
in
@
Show t h a t
U.
i s a r e l a t i v e l y compact s u b s e t o f
F.
If
of
Vx
f
x
H(E;F)
E
show t h a t t h e f o l l o w i n g are e q u i v a l e n t : f
i s compact a t e v e r y p o i n t o f
(b)
f
i s compact a t some p o i n t o f
(c)
t h e r e e x i s t s a compact s u b s e t
(a)
f(E)
E, E,
K
of
F
such t h a t
i s c o n t a i n e d i n t h e v e c t o r s p a c e spanned by
K,
(d)
t h e t r a n s p o s e mapping
2.68* Suppose
Let fn
f * ( @ ) = $of)
dnf(0)
i s compact f o r a l l
A
(e)
E
=
H(En)
E
f*; (F',To)
(where
m
where each
En
f o r each
n
I:=,
(H(E),ro)
n.
U
i s a l o c a l l y convex s p a c e .
En
and l e t
If t h e r e e x i s t s a neighbourhood
-f
i s continuous,
gn
E
o f zero i n
8 E
Ek)
such t h a t
f o r each
n.
Ilgnllu
+
for all
n
2.69
Show t h a t , t h e c o m p o s i t i o n o f holomorphic ( r e s p e c t i v e l y
show t h a t
F =
gnfnE
c
m
H(E).
h y p o a n a l y t i c ) f u n c t i o n s i s holomorphic ( r e s p e c t i v e l y
M-holomorphic,
M-holomorphic, h y p o a n a l y t i c ) .
2.70
If
i s a l o c a l l y bounded holomorphic f u n c t i o n ,
f
c o n t i n u o u s holomorphic f u n c t i o n and
gof
i s a Mackey
g
i s d e f i n e d , show t h a t
gof
is a
( c o n t i n u o u s ) holomorphic f u n c t i o n . 2.71 -
Let
U
be a n open s u b s e t o f a l o c a l l y convex s p a c e
b e a l o c a l l y convex s p a c e .
If
f
HG(U;F)
and e i t h e r
E
or
and l e t
E F
is
F
Chapter 2
94
separable, show t h a t
f
HHY(U;F)
E
if
gof
E
H(U)
f
HG(E) show t h a t
If
E
i s a Frgchet space and
and only i f
f
i s a Bore1 measurable f u n c t i o n .
2.72*
E
f o r every
g f
in
H(F).
H(E)
E
if
2.73* L e t U be an open subset o f a Banach space E and l e t F be a Banach space with a b a s i s , I f f i s a mapping from U i n t o F
then
f(x)
The mapping of
m
=
f
fn(x)en
where
fn : U
+
C.
i s s a i d t o be normal i f , f o r each compact subset
U
and each p o s i t i v e
fn
E
6,
there exists a positive integer
n(K)
K
such
that
If
H(U)
n
for a l l
show t h a t
f
H(U;F)
E
i f and only i f
f
is a
normal mapping. Let
__ 2.74*
in
be a l o c a l l y convex space and l e t
E
(a)
If
E
i s a Frgchet space o r a83fLvl space show t h a t
l z = l ( $ n y ~ H ( E ) i f and only i f f o r every (b)
If of
x
in
($n)n
show t h a t
bounded s e t s
8
such t h a t
i s a connected
being
as
E
H(E)
n-tm
i f and
contains a fundamental system of
E
( t h e vector span o f
EB B
in 6
HG(U;F)
in
f
f
E
%(U;F).
normed by t h e gauge E
is
TM
complete.
complete l o c a l l y convex show t h a t
f
E
%(U;F)
and every p o s i t i v e i n t e g e r
complete l o c a l l y convex spaces, and
U
is a
U
If
continuous show t h a t
where
T!,
f c HG(U;F)
T~
E
B
then we say t h a t
open subset of t h e
T
space E , F i s a Banach space and if 'dhf ( x ) E 6 M ( n E ; F ) f o r some x 2.76*
0
i s a n u l l sequence and t h e vector span ($n)n i s f i n i t e dimensional.
of B ) i s complete f o r every
F
+
1"n=l( @n )"
I f t h e l o c a l l y convex space
2.75* -
U
@,(x)
E.
E = (E,o(E,E'))
only i f
If
($n)n be a sequence
E'.
T~
open subset of f
ExF, E
n. and
i s s e p a r a t e l y Mackey
95
Holomoiphic mappings between locally convex spaces __ 2.77
If
i s an open s u b s e t of a l o c a l l y convex s p a c e
U
l o c a l l y convex s p a c e , show t h a t f o r each
f
in
HG(U;F)
E
and
is a
F
t h e following are
equivalent: (a)
f E H(U;F)
(b)
f o r each
(respectively in
5
HLB(U;F)),
U
is a n e q u i c o n t i n u o u s ( r e s p e c t i v e l y l o c a l l y bounded) f a m i l y of
mappings. 2.78 -
Let
b e a Banach s p a c e and l e t
E
sequences i n
2.79 -
If
(WE) ,
T~)
f
H( II Ea)
E
where each
If
J
fon
and &(Y), that
If
m
(0,)
n
.
of
A1
i s a l o c a l l y convex s p a c e , show A
and
II
%(afAEa) __ 2.80*
L(($n)n) =
f
i s t h e n a t u r a l p r o j e c t i o n from A1 i s m e t r i z a b l e f o r each a i n A show t h a t
where
A1
Ea
X
E
H( JI Ea) a d1
TI Ea a EA
such t h a t
onto
aZAIEa.
=
and
Y
a r e c o m p l e t e l y r e g u l a r Hausdorff s p a c e s
and
& , (X)
each w i t h t h e compact open t o p o l o g y , are i n f r a b a r r e l l e d , show HLB(,&(X); QCY);)
2.81 __
Ea
~ E A
t h a t there e x i s t s a f i n i t e subset f =
b e d e f i n e d by
i s w e l l d e f i n e d and holomorphic.
L
Show t h a t
-
be t h e s p a c e o f a l l n u l l
endowed w i t h t h e sup norm t o p o l o g y .
E'
L : co(E')
Let
co(E')
Show t h a t
and any open s u b s e t
=
H(U;F)
=
of
& ,
U
H(A(x);
,&(Y)~l-
H (U;F) f o r any l o c a l l y convex s p a c e F HY (X) where X i s a paracompact t o p o l o g i c a l
space. __ 2.82
Show t h a t
any open s u b s e t
U
H(U;F) = %(U;F) of
@,
(X)
where
f o r any l o c a l l y convex s p a c e X
i s a Lindel'df s p a c e .
F
and
96
Chapter 2
If
2.83*
r
is an uncountable discrete set and F
is a separable
Banach space, show that
where the topology on co(r)
is given by the sup norm and
r 'cr T ' countable 2.84* -
F
Let U be a
T~~
open subset of a locally convex space and let
be a locally convex space. Let
T~
denote the topology on $(U;F)
of uniform convergence on the finite dimensional compact subsets of U.
is metrizable if and only if E
(%(U;F),rF)
contains
a countable fundamental system of finite dimensional compact sets and F is metrizable. Show that $(U;F)
T~
induces on each
T~
bounded subset of
the topology of pointwise convergence.
Show that
(%(U;F),T~) is a semi-Monte1 space if and only if F is a semi-Monte1 space. Show that ($(U;F); T ~ ) is quasi-complete if and only if F is quasi-complete. is an arbitrary set, show that the bounded subsets of locally bounded. 2.86 -
If E
is a complete locally convex space and the
subsets of H(E 2.87*
the T~ U of E.
Let E and T
are locally bounded, show that E =
T~
bounded
is barrelled.
En where each En is a Banach space. Show that bounded subsets of H(U) coincide for every open subset
We say that a countable increasing cover 3 = (Fn)n=l of a topological space U is k-dominating if each F, is closed and each compact subset of U is contained in some 'n. Let U be an open subset of 2.88*
50
97
Holomorphic mappings between locally convex spaces a l o c a l l y convex s p a c e all
n}.
H
"
(U)
t h e semi-norms
E
and l e t
Hg(U)
I f cHG(U);
=
llfllF<
pn(f) =
Ildg .
Let
n
(H(U),T
6K
)
l i m H3(U)
=
2+
r a n g e s o v e r a l l c o u n t a b l e i n c r e a s i n g k-dominating c o v e r s o f T
6K
is t h e bornological topology a s s o c i a t e d w ith
(H(U),T
i s a n u l t r a - b o r n o l o g i c a l space i f
)
6K
-
for
i s endowed w i t h t h e l o c a l l y convex t o p o l o g y z e n e r a t e d by
U
T
0
on
U.
where
.3
Show t h a t
H(U).
Show t h a t
i s a k-space.
2.89 On H(U), U a n open s u b s e t o f a l o c a l l y convex s p a c e , l e t T P d e n o t e t h e t o p o l o g y o f p o i n t w i s e convergence. I f = (Fn):=l is a
3
countable i n c r e a s i n g cover o f
c o n s i s t i n g o f c l o s e d sets and
U
H3(U)
is
t h e a s s o c i a t e d m e t r i z a b l e s p a c e o f holomorphic f u n c t i o n s , show t h a t
is t h e bornological space associated with
l i m H 3(U)
(H(U) , T ~ ) . (
3
-3+ r a n g e s o v e r a l l p o s s i b l e c o v e r s o f t h e above t y p e ) .
If
2.90 and
i s a n open s u b s e t o f a m e t r i z a b l e l o c a l l y convex s p a c e
U
i s a Banach s p a c e , show t h a t
F
E
(H(U;F),T&) i s a r e g u l a r i n d u c t i v e
l i m i t of Frgchet spaces. If t h e l o c a l l y convex s p a c e
2.91*
c o n v e r g e n t sequence, show t h a t
Ei
E
contains a nontrivial very strongly
c o n t a i n s a sequence which i s n o t v e r y
weakly c o n v e r g e n t . __ 2.92
If
i s a s e p a r a b l e l o c a l l y convex s p a c e i n which e v e r y sequence
E
i s v e r y weakly c o n v e r g e n t , show t h a t 2.93
Let
E
be a F r g c h e t s p a c e .
EL
a d m i t s a c o n t i n u o u s norm.
Show t h a t t h e f o l l o w i n g are e q u i v a l -
ent : (a)
E
c o n t a i n s a n o n t r i v i a l v e r y s t r o n g l y c o n v e r g e n t sequence,
(b)
E
c o n t a i n s a s u b s p a c e isomorphic t o
(c) (d)
i s a quotient of
CN,
E'
a'
e v e r y c o n t i n u o u s semi-norm on
E
has an i n f i n i t e dimensional
kernel. Show t h a t ( a ) and (d) a r e n o t e q u i v a l e n t f o r a r b i t r a r y l o c a l l y convex spaces.
Chapter 2
98
Let
2.94*
~
Let
E,
E A
{xgE;
=
3 (x~):=~ E
.
c o n v e r g e n t sequence}. f
E
A
b e a m e t r i z a b l e l o c a l l y convex s p a c e w i t h completion
E
such t h a t
i s a v e c t o r subspace o f
E,
Show t h a t
E.
( x ~ - x ) i~s a v e r y s t r o n g l y
-
A
E.
If
show, by u s i n g t h e f a c t o r i z a t i o n lemma o r o t h e r w i s e , t h a t t h e r e
H(E),
d
e x i s t s a unique
f
in
H(E,)
such t h a t
flE = f .
Generalise t h i s r e s u l t
t o a r b i t r a r y l o c a l l y convex s p a c e s .
~
Let
2.95 (H(U),-ro)
b e a n open s u b s e t of a l o c a l l y convex s p a c e .
U
i s a l o c a l l y m-convex a l g e b r a .
m u l t i p l i c a t i v e semi-norm on
~
Let
2.96*
f
Let
E
H(U;F).
F-valued holomorphic germs a t rJ
f(x)
show t h a t
N
f
If
2.97"
T~
Show t h a t e v e r y
=
{germ o f
If in
0 f
?: E)
at
T~
continuous
continuous.
be a n open s u b s e t of a Banach s p a c e
U
a Banach s p a c e .
is
H(U)
Show t h a t
U
-f
E
H(0;F)
F
and l e t
be
( t h e s p a c e of
i s d e f i n e d by
x
translated t o the origin)
HG(U;H(O;F)). i s a compact s u b s e t o f a l o c a l l y convex s p a c e
K
i s a l o c a l l y convex s p a c e , show t h a t
H(K;F)
and
E
F
i s a Hausdorff l o c a l l y convex
space. Show t h a t
__ 2.98*
H(K)
i s n e v e r a s t r i c t i n d u c t i v e l i m i t when
is a
I<
compact s u b s e t of a l o c a l l y convex s p a c e . If
__ 2.99*
space
E
neighbourhood
f o r every 2.100*
i s a convex b a l a n c e d s u b s e t of a m e t r i z a b l e l o c a l l y convex
K
show t h a t
5 If
V
in U
B CH(K)
i s bounded i f and o n l y i f t h e r e e x i s t
of z e r o and two p o s i t i v e numbers
K
and e v e r y n o n - n e g a t i v e i n t e g e r
c
and
C
2.101"
F' B
'2
If
such t h a t
m.
i s a n open s u b s e t of a Banach s p a c e , show t h a t
a d u a l Banach s p a c e ( i . e . show t h a t t h e r e e x i s t s a Banach s p a c e that
a
Hm(U)
F
is
such
Hm(U)).
K
i s a compact s u b s e t o f a m e t r i z a b l e l o c a l l y convex s p a c e ,
99
Holomorphic mappings between locally convex spaces show t h a t
m -convex a l g e b r a .
is a l o c a l l y
H(K)
Show t h a t a convex b a l a n c e d s u b s e t of
2.102
o f a l o c a l l y convex s p a c e
is a
E,
T&
i f it a b s o r b s e v e r y e q u i c o n t i n u o u s s u b s e t o f 2.103*
Let
integer
n
rn
let
an open s u b s e t
U
H(U).
b e a l o c a l l y convex space.
E
H(U),
neighbourhood o f z e r o i f and o n l y
d e n o t e t h e t o p o l o g y on
For each n o n - n e g a t i v e g e n e r a t e d by t h e semi-
H(E)
norms
where
K
r a n g e s o v e r t h e compact s u b s e t s o f
bounded s u b s e t s o f
B
ranges over t h e
E.
rn =
Show t h a t
and
E
T
~
+f o r~
some (and hence f o r a l l )
n
i f and o n l y i f
each bounded s u b s e t o f
E
i s c o n t a i n e d i n t h e c l o s e d convex h u l l o f a
compact s e t . A subset
2.104"
K
of a l o c a l l y convex s p a c e
i s s a i d t o be s t r i c t l y
E
compact i f t h e r e e x i s t s a convex b a l a n c e d bounded s u b s e t that
K
i s c o n t a i n e d and compact i n
EB.
On
%(E)
B
let
of rS
E
t o p o l o g y o f uniform convergence on t h e s t r i c t l y compact s u b s e t s of Show t h a t
(%(E),rS)
2.105"
If
f
t h e r e e x i s t s a neighbourhood
52.6
E.
i s a semi-Monte1 s p a c e .
i s a n e n t i r e f u n c t i o n on t h e
every p o s i t i v e i n t e g e r
such
denote t h e
n.
V
a3w
o f z e r o such t h a t
space
/If
lLv
E,
show t h a t
is f i n i t e for
NOTES AND RtPIARKS
The h i s t o r i c a l n o t e s o f A . E . T a y l o r [680] a r e p r o b a b l y t h e most comprehensive g u i d e t o t h e development o f i n f i n i t e dimensional holomorphy up t h e mid-nineteen f o r t i e s a v a i l a b l e and w e have made e x t e n s i v e u s e o f t h e s e n o t e s i n 11.6 and s h a l l do s o a g a i n i n t h i s s e c t i o n .
In another
p a p e r , [681], A . E . T a y l o r documents t h e r o l e of a n a l y t i c i t y i n o p e r a t o r and s p e c t r a l t h e o r y and h i s h i s t o r y of t h e d i f f e r e n t i a l i n t h e n i n e t e e n t h
Chapter 2
100
and t w e n t i e t h c e n t u r i e s [682] c o n t a i n s much o f i n t e r e s t c o n c e r n i n g t h e growth o f i n f i n i t e d i m e n s i o n a l a n a l y s i s .
E . H i l l e and R.S. P h i l l i p s ( [ 3 3 4 ] ,
s e c t i o n 3.16 and c h a p t e r 26) a l s o p r o v i d e a summary o f t h e fundamentals known a t t h a t t i m e
-
1958
-
t o g e t h e r w i t h a s h o r t b i b l i o g r a p h y on t h e sub-
j e c t ( c h a p t e r 26 was w r i t t e n i n c o l l a b o r a t i o n w i t h M.A.
Zorn).
D. Pisanelli
[578] g i v e s a s u r v e y o f most o f t h e d i f f e r e n t c o n c e p t s o f holomorphic f u n c t i o n t h a t are c u r r e n t l y i n u s e and documents t h e i r o r i g i n . V.I. Averbukh and O . G . Smolyanov ( [ 4 0 ] , § 2 ) g i v e a comprehensive a c c o u n t of t h e development o f t h e concept of t h e d i f f e r e n t i a l i n t o p o l o g i c a l v e c t o r spaces.
They a r e m a i n l y concerned with the r e a l t h e o r y b u t n a t u r -
a l l y t h i s h a s many consequences f o r t h e complex c a s e .
An i n t e r e s t i n g p o i n t
t h a t emerges from [40] i s t h a t t h e s e a r c h f o r a d e f i n i t i o n o f d i f f e r e n t i a b l e f u n c t i o n between r e a l l i n e a r t o p o l o g i c a l v e c t o r s p a c e s h a s been i n p r o g r e s s more o r l e s s c o n t i n u o u s l y f o r t h e l a s t s e v e n t y y e a r s w i t h many d i f f e r e n t d e f i n i t i o n s b e i n g proposed - t h e a u t h o r s l i s t t w e n t y - f i v e i n [40] - and s i n c e a number o f d e f i n i t i o n s were n o t s e e n t o b e e q u i v a l e n t and i n d e e d , v a r i o u s a u t h o r s were n o t aware o f one a n o t h e r ' s work, t h i s l e d t o a c e r t a i n
a mount o f chaos - o r a t l e a s t a p p a r e n t chaos - which h a s o n l y been r e c t i f i e d i n [40].
F o r t u n a t e l y f o r d i f f e r e n t i a t i o n o v e r complex s p a c e s we
have power series e x p a n s i o n s and t h i s h a s l e d t o a f a i r l y unanimous a c c e p t a n c e o f Frgchet d i f f e r e n t i a b i l i t y as t h e s t a n d a r d d e f i n i t i o n w i t h minor r o l e s b e i n g p l a y e d by GGteaux holomorphic, Mackey o r S i l v a holomorp h i c and h y p o a n a l y t i c f u n c t i o n s . The names c h i e f l y a s s o c i a t e d w i t h t h e development o f r e a l i n f i n i t e d i m e n s i o n a l d i f f e r e n t i a l c a l c u l u s a r e ( s e e [40] f o r more d e t a i l e d i n f o r mation) V . V o l t e r r a , J . Hadamard, M . F r g c h e t , R . Ggteaux, P . L&y,
A.D.
Michal, L . M . Graves, E.W. Paxson, J . G i l d e Lamadrid, J . Sebasti'go e S i l v a , H.H.
Keller and G . Marinescu.
I n p a r t i c u l a r , we would l i k e t o mention
J . Hadamard, M. F r g c h e t and A . D . Michal.
Hadamard was one o f t h e f i r s t t o
r e c o g n i s e t h e importance o f V o l t e r r a ' s work, h e e x e r c i s e d a d e e p i n f l u e n c e
on h i s s t u d e n t s FrGchet and Gsteaux and was always v e r y i n s i s t a n t on l i n e a r i t y b e i n g i n c o r p o r a t e d i n t o any d e f i n i t i o n o f t h e d e r i v a t i v e .
M . F r g c h e t and A . D . Michal b o t h s p e n t a c o n s i d e r a b l e p o r t i o n of t h e i r l i v e s developing t h e t h e o r y
-
r o u g h l y f o r t y y e a r s by F r g c h e t - and opened many
avenues s t i l l being explored. b e found i n P . L&y
([442],
Readable a c c o u n t s o f t h e r e a l t h e o r y a r e t o
b o t h t h e 1922 and 1950 e d i t i o n s ) , A . D . Michal
101
Holomoiphic mappings between locally convex spaces [491], V.I. Averbukh and O . G .
Smolyanov [39], J . P . Penot [567], A.
Frb'hlicher and W . Bucher [247] and S. Yamamuro [718]. Surveys of a more s p e c i a l i s e d n a t u r e but which a r e more r e l e v a n t t o t h e t o p i c s discussed i n t h i s book a r e L . Nachbin [518], M . Schottenloher B i e r s t e d t and R . Meise [70], P . Krce [405] and S . Dineen [198].
[642], K.D.
We now p r e s e n t our b r i e f h i s t o r y . We have a l r e a d y noted i n 51.6 t h a t t h e d e f i n i t i v e s t e p s i n t h e c r e a t i o n of r e a l and complex i n f i n i t e dimensional d i f f e r e n t i a l c a l c u l u s were taken by V . V o l t e r r a (1887) and D . H i l b e r t (1909) r e s p e c t i v e l y . P r i o r t o t h e s e , t h e works of J . Bernoulli on t h e curve of quickest descent, L . Euler on t h e c a l c u l u s of v a r i a t i o n s (see J . Hadamard [298]) and H. Von
Koch [393], on i n f i n i t e systems of d i f f e r e n t i a l equations may a l s o be mentioned a s of p a r t i c u l a r importance f o r l a t e r developments.
D. Hilbert
[332] discussed a n a l y t i c continuation and t h e composition of holomorphic f u n c t i o n s but h i s work i s not easy t o read, and a t times r a t h e r vague. Next came R. G2teaux [251,252,253] who looked a t both r e a l and complex analysis
-
a s opposed t o M. Frgchet who only s t u d i e d t h e r e a l c a s e
s e t out t o c l a r i f y and extend t h e works of Frgchet and H i l b e r t . he succeeded b r i l l i a n t l y .
-
and
In t h i s
I n [252], he defined complex polynomials and
n o t i c e d t h e i r r e l a t i o n s h i p with b i l i n e a r and q u a d r a t i c forms, defined t h e "G$teaux" d e r i v a t i v e , extended t h e Cauchy i n t e g r a l formula and t h e Cauchy i n e q u a l i t i e s , proved an i d e n t i t y theorem and v a r i o u s convergence theorems f o r holomorphic functions, e s t a b l i s h e d t h e now standard correspondence between d e r i v a t i v e s of a holomorphic function and t h e homogeneous polynomials i n i t s Taylor series expansion and gave various r e s u l t s about a n a l y t i c continuation and power s e r i e s expansions - a l l i n an i n f i n i t e dimensional s e t t i n g .
H i s o t h e r paper [253] i s a l s o q u i t e i n t e r e s t i n g and
contains b r i e f o u t l i n e s of a number of t o p i c s which h i s untimely death prevented him from developing.
CN,
&[a,b]
and
G2teauxIs work was confined t o t h e spaces
.Q2.
In 1920, S . Banach gave, i n h i s t h e s i s , t h e axioms f o r a complete normed l i n e a r space (a Banach space) and i n 1923, N.Wiener
[715], who
i n c i d e n t a l l y gave independently t h e same axioms t h r e e months a f t e r Banach,
102
Chapter 2
n o t i c e d t h a t t h e Cauchy i n t e g r a l formula e x t e n d s t o Banach v a l u e d holomorp h i c f u n c t i o n s o f one complex v a r i a b l e and i n t h i s s e t t i n g , many o f t h e c l a s s i c a l r e s u l t s such as M o r e r a ' s theorem, A b e l ' s theorem and t h e r e s i d u e theorem are a l s o v a l i d .
L . FantappiGIs long a r t i c l e [234] appeared i n 1930
I n i t h e proposed t h a t a c o n t i n u o u s f u n c t i o n aBanach s p a c e be c a l l e d holomorphic i f
f
d e f i n e d on a domain
f o $ i s holomorphic ( a s a f u n c t i o n
o f one complex v a r i a b l e ) f o r a n y holomorphic f u n c t i o n @ t h e complex p l a n e i n t o
D.
in
D
from a domain i n
F a n t a p p i g ' s work s e r v e d as m o t i v a t i o n f o r A . E .
T a y l o r 116751 and was developed and p u t i n a modern c o n t e x t u s i n g t o p o l o g i c a l v e c t o r s p a c e s by J . S e b a s t i r o e S i l v a [649,650,651,652,653]
(see a l s o
D. P i s a n e l l i [578]).
I n t h e n i n e t e e n t h i r t i e s , an i n t e n s i v e s t u d y o f t h e whole f i e l d o f a n a l y s i s a n d geometry i n a b s t r a c t s p a c e s was c a r r i e d o u t by A . D . Michal and h i s s t u d e n t s R . S . M a r t i n , A . H . C l i f f o r d , I . E . Highberg and A . E . T a y l o r .
In
1932, R . S . M a r t i n [449] developed t h e t h e o r y o f holomorphic mappings f o r Banach s p a c e s u s i n g a power series approach ( s e e a l s o A . D . Michal and A.H. C l i f f o r d [492]).
The f i n a l s t e p towards t h e c u r r e n t d e f i n i t i o n of holomor-
p h i c mapping between Banach s p a c e s was t a k e n i n d e p e n d e n t l y by L.M. Graves i n 1935 [275] and A . E . b r i e f , p.649-653,
T a y l o r i n 1937 [ 6 7 5 ] .
Graves' t r e a t m e n t i s r a t h e r
and i n it h e n o t e s t h a t Ggteaux d i f f e r e n t i a b i l i t y p l u s
c o n t i n u i t y are e q u i v a l e n t t o F r g c h e t d i f f e r e n t i a b i l i t y ( d e f i n e d f o r normed l i n e a r s p a c e s by F r g c h e t i n 1925, "2451) when d e a l i n g w i t h f u n c t i o n s between complex normed l i n e a r s p a c e s and a l s o t h a t t h i s l e a d s t o a s a t i s f a c t o r y t h e o r y o f holomorphic f u n c t i o n s between normed l i n e a r s p a c e s . T a y l o r ' s work i s much more e x p l i c i t .
H e d e f i n e s a holomorphic f u n c t i o n as
a c o n t i n u o u s f u n c t i o n whose one d i m e n s i o n a l s e c t i o n s are holomorphic.
This
d e f i n i t i o n i s a n a t u r a l outgrowth o f t h e work o f G2teaux and b r i n g s t o g e t h e r t h e i d e a s o f many o f h i s p r e d e c e s s o r s .
H e proved t h e f o l l o w i n g
result: If
f
i s a mapping between normed l i n e a r s p a c e s , t h e n t h e f o l l o w i n g
are equivalent a)
f
i s c o n t i n u o u s and h a s a Ggteaux d e r i v a t i v e a t each
point; b)
f
h a s a F r g c h e t d e r i v a t i v e a t each p o i n t ;
c)
f
h a s a power s e r i e s expansion which converges
103
Holomorphir mappings between locally convex spaces u n i f o r m l y i n a neighbourhood o f each p o i n t . T a y l o r goes on t o g e n e r a l i s e Riemann's theorem on removable s i n g u l a r i t i e s , M i t t a g - L e f f l e r ' s theorem, L i o u v i l l e ' s theorem and t h e CauchyRiemann e q u a t i o n s .
He a l s o shows [675,p.292] an awareness o f t h e f a c t t h a t
t h e r a d i u s o f uniform convergence i s n o t e q u i v a l e n t t o t h e r a d i u s o f a n a l y t i c i t y and t h i s i s t h e f i r s t s t e p i n a n e x t e n s i v e s t u d y o f t h e r a d i u s o f boundedness which w e u n d e r t a k e i n c h a p t e r 4 .
I n a f u r t h e r p a p e r [678],
A . E . T a y l o r d i s c u s s e s a number o f i n t e r e s t i n g examples o f holomorphic
f u n c t i o n s on
s p a c e s and g i v e s a g e n e r a l i z a t i o n of H a r t o g s ' theorem on
kP
separate analyticity.
The P o l i s h s c h o o l o f f u n c t i o n a l a n a l y s i s (S. Banach,
S. Mazur, W . O r l i c z , e t c . ) a l s o c o n t r i b u t e d d u r i n g t h i s p e r i o d ( s e e c h a p t e r 1) and a l t h o u g h t h i s t r a d i t i o n a l
interest
i n non-linear functional
a n a l y s i s was m a i n t a i n e d u n t i l r e l a t i v e l y r e c e n t l y ( s e e A . Alexiewicz and W . O r l i c z [ 7 ] , W .
[563,564,565]),
f o r instance
Bogdanowicz [76] and A . P e l c z y n s k i
it was l a r g e l y overshadowed by t h e r a p i d development of
t h e l i n e a r theory.
T h i s predominance o f l i n e a r f u n c t i o n a l a n a l y s i s may b e
t r a c e d t o t h e i n f l u e n c e o f S. Banach's book [44], p u b l i s h e d i n 1932, a l t h o u g h Banach h i m s e l f hoped t o d e v e l o p t h e n o n - l i n e a r t h e o r y i f we a r e t o j u d g e from h i s comment [ 4 4 , p . 2 3 1 ] on complex v e c t o r s p a c e s
"Ces espaces c o n s t i t u e n t Ze point de de'part de l a the'orie des op&ations
Zingaires cornpZexes e t d ' m e cZasse, encore pZus
vaste, des op6rations anazytiques, qui prgsentent une g'enerali s a t i o n des fonctions anaZytiques ordinaires icf. p . ex. L . Fantappi;,
I . funz?:onaZi a n a l i t i c i , G i t t a d i Caste220 1 9 3 0 ) .
Nous m u s proposons d 'en exposer Za thebrie dans un autre vo ZTme
".
See a l s o t h e p r e f a c e t o [ 4 4 ] . M . A . Zorn [723,724,725] made a number o f i m p o r t a n t c o n t r i b u t i o n s i n
t h e m i d - f o r t i e s and w e refer t o h i s r e s u l t s l a t e r i n t h e t e x t .
The n i n e -
t e e n f i f t i e s saw t h e a p p e a r a n c e of A . G r o t h e n d i e c k ' s memoir [287] on t o p o l o g i c a l t e n s o r p r o d u c t s and n u c l e a r s p a c e s and t h i s h a s had and w i l l c o n t i n u e t o h a v e a c o n s i d e r a b l e i n f l u e n c e on t h e development o f i n f i n i t e d i m e n s i o n a l holomorphy.
During t h i s p e r i o d , J . S e b a s t i z e S i l v a a l s o
developed h i s t h e o r y o f i n f i n i t e d i m e n s i o n a l holomorphy and H . J .
Bremermann
[103,104,105] proved v a r i o u s r e s u l t s on pseudo-convex domains and t u b u l a r
104
Chapter 2
domains o f holomorphy i n Banach s p a c e s . T h i s b r i n g s u s t o t h e modern p e r i o d , t h e l a s t s i x t e e n y e a r s , d u r i n g which time most o f t h e work we d i s c u s s h a s been d i s c o v e r e d .
The con-
t r i b u t i o n s o f t h e various authors during t h i s period are d e t a i l e d i n t h e f i n a l s e c t i o n o f each c h a p t e r . The concept o f Mackey holomorphic f u n c t i o n i s u s u a l l y a t t r i b u t e d t o J . S e b a s t i a o e S i l v a [652], a l t h o u g h an e a r l i e r e q u i v a l e n t d e f i n i t i o n (con-
d i t i o n 2 . 1 8 ( d ) ) i s due t o L . F a n t a p p i g [234].
The t h e o r y o f Mackey holo-
morphic f u n c t i o n s f i n d s i t s most n a t u r a l e x p r e s s i o n w i t h i n t h e language of b o r n o l o g i e s - s e e f o r i n s t a n c e D . P i s a n e l l i [578], D . Lazet [423] and J . F . Colombeau [141]. E . H i l l e [333].
L o c a l l y bounded holomorphic f u n c t i o n s were i n t r o d u c e d by I t i s d i f f i c u l t t o locate t h e o r i g i n of hypoanaliticity
( d e f i n i t i o n 2.15) b u t t h e name u s u a l l y a s s o c i a t e d w i t h t h e r e a l analogueh y p o c o n t i n u i t y i s J . L . K e l l e y who a l s o i n t r o d u c e d t h e concept of k-space ( s e e , f o r i n s t a n c e , S. W i l l a r d [716] o r Z . Semadeni [654]).
Examples 2.16,
2.19 and 2 . 2 0 t o g e t h e r w i t h f u r t h e r examples o f a similar k i n d are g i v e n i n S. Dineen [190] ( s e e a l s o c h a p t e r 5 ) .
The c o n n e c t i o n between H a r t o g s '
theorem and l o c a l l y bounded holomorphic f u n c t i o n s ( p r o p o s i t i o n 2 . 1 2 , examples 2 . 1 3 and 2.14) a p p e a r s i n [190] and i n f i n i t e dimensional g e n e r a l i z a t i o n s o f H a r t o g s ' theorem a r e d i s c u s s e d i n t h e comments on e x e r c i s e 2.76. F a c t o r i z a t i o n r e s u l t s (most o f which depend on L i o u v i l l e ' s theorem c o n c e r n i n g bounded e n t i r e f u n c t i o n s ) are i m p l i c i t i n A. Hirschowitz [335],
C . E . R i c k a r t [605] and L . Nachbin [514]. S. Dineen [188,189,190,191]
and
E . Ligocka [443], developed i n d e p e n d e n t l y a g e n e r a l t h e o r y o f f a c t o r i z a t i o n
which we d i s c u s s i n more d e t a i l i n 16.2 and a p p l i e d i t t o such t o p i c s as t h e Levi problem, RungeS theorem, H a r t o g s ' theorem, holomorphic c o m p l e t i o n , Zorn's theorem and t o p o l o g i e s on s p a c e s o f holomorphic f u n c t i o n s .
Factor-
i z a t i o n methods h a v e a l s o been a p p l i e d t o t h e Levi problem by Ph. Noverraz [540,544,546],
S. Dineen [183,186],
B. J o s e f s o n [358], M . S c h o t t e n l o h e r
[640], J . F . Colombeau and J . Mujica [154], J . Mujica [506],to t h e c o n s t r u c t i o n o f t h e envelope o f holomorphy by M . S c h o t t e n l o h e r [640] and P. Berner [59,60], i n s t u d y i n g meromorphic f u n c t i o n s and t h e Cousin I problem by V . Aurich
[35] and S. Dineen [192], t o t h e t h e o r y of c o n v o l u t i o n o p e r a t o r s
by P. Berner [ 6 2 ] and M.C. Matos [463,467] and i n s t u d y i n g l o c a l l y convex t o p o l o g i e s on s p a c e s of holomorphic f u n c t i o n s by P . Berner [61], S. Dineen
105
Holomotphic mappings between locally convex spaces
[194], and Ph. Noverraz [552]. Theorem 2 . 2 1 and proposition 2.33 are given in [190;. Example2.22and proposition 2.24 aredue to L. Nachbin [514] and a generalization can be found in S. Dineen [190]. subset U
in (EN and
A. Hirschowitz [335] gives an example of an open
f &H(U)
which admits a local but not a global
factorization. This example shows that condition (*) of theorem 2 . 2 1 is not sufficient to obtain a global factorization result. The difficulties posed by Hirschowitz's example were overcome by P. Berner [59] and M. Schottenloher [640]. Examples 2.22, 2.25 and 2.26 are due to L. Nachbin [514], and the proof of example 2 . 2 2 appears in [193].
Proposition 2.27 is
due to S. Dineen [190]. Theorem 2.28 is due to M.A. Zorn [724].
Generalizations to other
locally convex spaces are due to Ph. Noverraz [536,537], D. Pisanelli [578], A. Hirschowitz [341], D. Lazet [423], J.F. Colombeau [141], E. Ligocka [443]
and S. Dineen [190].
A locally convex space E
for which the con-
clusion of proposition 2.29 is valid is called an w-space (S. Dineen
[184],
[190]). E . Grusell [292,293] and M. Schottenloher [645] give examples of locally convex spaces which are not w-spaces. Corollary 2.30 is due to J.F. Colombeau and J. Mujica [154], who use it together with the solution to the Levi problem for Hilbert spaces (L. Gruman [290]) to solve the Levi problem in strong duals of Frgchet nuclear spaces (&3?:spaces). Example 2.31, which arose in constructing a counterexample to the Levi problem, is due to B. Josefson [358] andthe proof given here is taken from
S. Dineen [193].
The inequality of example 2.31 is used in A. Renaud
([604, proposition 4) to obtain a vector-valued Schwarz lemma.
Other
factorization results may be proved by considering locally bounded holomorphic functions and in such cases one gets results for non-normed range spaces [190]. The topological vector space structure of H(U),
U an open subset of
Cn, has been studied by a number of authors including A. Grothendieck [285], G. Kothe [396] and A. Martineau [451]. The compact open topology was first investigated on spaces of holomorphic functions in infinitely many variables by H. Alexander [S] and L . Nachbin [509]. They found, however, that it did not possess very good topological properties and s o , motivated by properties of analytic functionals in several complex variables
Chapter 2
106
due t o A. Martineau [450], L . Nachbin I5091 i n t r o d u c e d t h e ( d e f i n i t i o n s 2 . 3 3 and 2 . 3 4 ) .
The
topology
T~
t o p o l o g y ( d e f i n i t i o n s 2.33 and 2.34)
T&
was f i r s t i n t r o d u c e d f o r holomorphic f u n c t i o n s on s e p a r a b l e Banach s p a c e s by G . Coeur;
[128,129] i n c o n n e c t i o n w i t h problems o f a n a l y t i c c o n t i n u a t i o n
and t h e g e n e r a l d e f i n i t i o n i s due t o L. Nachbin [510]. and 2.41 are g i v e n i n S. Dineen [185]. known and a p r o o f i s g i v e n i n K - D . c o r r e s p o n d i n g problems f o r t h e
T~
P r o p o s i t i o n s 2.38
Lemma 2.39 a p p e a r s t o b e w e l l
B i e r s t e d t and R . Meise [ 6 9 ] . and
T~
The
topologies, i . e . are t h e s e
l o c a l o r s h e a f t o p o l o g i e s ? are p e r h a p s t h e most i n t e r e s t i n g c u r r e n t open problems i n t h e g e n e r a l t h e o r y o f l o c a l l y convex s p a c e s of holomorphic f u n c t i o n s i n i n f i n i t e l y many v a r i a b l e s .
A r e l a t e d open problem i s t o
c h a r a c t e r i z e t h e open sets i n which
and
l6
7
continuous m u l t i p l i c a t i v e
l i n e a r f u n c t i o n a l s are p o i n t e v a l u a t i o n s (by e x e r c i s e 2 . 9 5 t h e the
T
continuous m u l t i p l i c a t i v e linear f unc t i o n a l s c o i n c i d e ).
T~
and This
problem arises i n s o l v i n g t h e Levi problem and i n c o n s t r u c t i n g t h e envelope R e s u l t s d e a l i n g w i t h t h i s problem are t o be found i n
of holomorphy.
H. Alexander [S], G . Coeur:
641,6431, J . M .
[129,131], M. S c h o t t e n l o h e r [632,633,634,635,
I s i d r o [352] and J . Mujica [502,503,504,505].
Lemma 2.43,
c o r o l l a r y 2.44 and example 2.46 are proved i n S. Dineen [185]. 2 . 4 5 and example 2.47 a r e t o be found i n S . Dineen [194].
For
Corollary
832
( s t r o n g d u a l s o f F r g c h e t Schwartz) s p a c e s , example 2.47 i s due t o J . A . Barroso, M . C .
Matos and L . Nachbin [ 5 0 ] .
Reference [194] i s devoted t o t h e
t h e o r y o f holomorphic f u n c t i o n s on &$hl ( s t r o n g d u a l s o f Fr‘echet Monte1 s p a c e s ) and i t i s i n t e r e s t i n g t o n o t e t h a t a l t h o u g h many r e s u l t s on 8 3 8 s p a c e s ( f o r i n s t a n c e , example 2.47) can be extended t o holomorphic f u n c t i o n s on83K;spaces, t h e methods o f proof a r e q u i t e d i f f e r e n t .
OnB&f
s p a c e s , one u s e s f r e q u e n t l y t h e p r o p e r t y t h a t such s p a c e s are i n d u c t i v e l i m i t s o f Banach s p a c e s w i t h compact l i n k i n g mappings i n t h e c a t e g o r y o f
t o p o l o g i c a l s p a c e s ( t h i s i s n o t t r u e o f &J-h’L
s p a c e s ) w h i l e on
8 3%
s p a c e s , one u s e s t h e f a c t t h a t such s p a c e s a r e k-spaces and h e r e d i t a r y Lindelb’f s p a c e s .
Example 2.48 i s due t o L . Nachbin [SO91 and example 2.49
can b e found i n S . Dineen [185]. Very s t r o n g l y convergent s e q u e n c e s ( d e f i n i t i o n 2.50) were i n t r o d u c e d by S . Dineen [184] i n s t u d y i n g holomorphic completions ( s e e 5 4 . 4 ) .
Subsequent-
l y t h e y were used by S . Dineen [185] i n counterexample 2.52 and i n [190] t h e y were a p p l i e d t o g e t h e r w i t h v e r y s t r o n g l y convergent n e t s and v e r y weakly c o n v e r g e n t sequences ( d e f i n i t i o n 2 . 5 1 ) i n o t h e r areas o f i n f i n i t e
107
Holomorphic mappings between locally convex spaces dimensional holomorphy.
F u r t h e r a p p l i c a t i o n s a r e t o b e found i n P . Berner
[58,59] and S. Dineen and Ph. Noverraz [ 2 0 5 ] . T
0
= T
0
A on H(C )
example 2 . 5 2 ,
i f and o n l y i f
A
J.A.
Barroso [46] shows t h a t
i s c o u n t a b l e and t h i s , t o g e t h e r w i t h
completes t h e p i c t u r e o f t h e r e l a t i o n s h i p between t h e d i f f e r -
e n t t o p o l o g i e s on
A
H(C*),
a r b i t r a r y ( s e e a l s o c h a p t e r 5, J . A .
Barroso
and L . Nachbin [53] and V . Aurich [ 3 3 ] ) . Holomorphic germs on compact s u b s e t s o f Banach s p a c e s were f i r s t i n v e s t i g a t e d by S.B. Chae [119,120] and A . Hirschowitz [339,343].
Subse-
q u e n t l y , J . Mujica [503] developed t h e t h e o r y on m e t r i z a b l e l o c a l l y convex spaces ( s e e a l s o R.R. 3. Mujica [41], K-D.
Baldino [ 4 3 ] , A . J . M .
Wanderley [714], P . A v i l e s and
B i e r s t e d t and R . Meise [69,70] and E . Nelimarkka [525])
Lemma 2.54 and p r o p o s i t i o n 2.55 are due t o J . Mujica [503].
Earlier,
S.B. Chae [120] and A . H i r s c h o w i t z [339] had g i v e n incomplete p r o o f s f o r t h e Banach s p a c e case.
In p r o v i d i n g a complete p r o o f , A. Hirschowitz
i n t r o d u c e d semi-norms s i m i l a r t o t h o s e which a p p e a r i n p r o p o s i t i o n 2.56 ( s e e a l s o S. Dineen [ZOO]).
The q u e s t i o n o f whether o r n o t c o n d i t i o n (b)
o f p r o p o s i t i o n 2.56 i s redundant i s answered n e g a t i v e l y by example 2.57 (due t o R.M. W.R.
Aron) and h a s l e d t o some i n t e r e s t i n g i n v e s t i g a t i o n s by
Z a m e [721], K. Rusak [617] and J . T . Rogers, J r . and W.R.
f i n i t e dimensions and by R . L .
Z a m e [609] i n
S o r a g g i [664,665,666] i n i n f i n i t e dimensions.
Rogers and Zame show t h a t c o n d i t i o n (b) i s n o t n e c e s s a r y f o r compact subs e t s of
which have o n l y a f i n i t e number o f connected components and
(c
t h i s r e s u l t does n o t extend t o h i g h e r dimensional s p a c e s .
For a r b i t r a r y
l o c a l l y convex s p a c e s , s u f f i c i e n t c o n d i t i o n s f o r t h e removal o f (b) a r e o b t a i n e d by p l a c i n g r a t h e r t e c h n i c a l l o c a l c o n n e c t e d n e s s c o n d i t i o n s on t h e compact s e t . The
T~
t o p o l o g y was i n t r o d u c e d by S.B. Chae [120] and it h a s a l s o
been s t u d i e d by J . Mujica [SO31 and K . D . chapter 6).
B i e r s t e d t and R . Meise [70], ( s e e
This Page Intentionally Left Blank
Chapter 3
HOLOMORPHIC FUNCTIONS ON BALANCED SETS
I n many b r a n c h e s o f f u n c t i o n t h e o r y t h e r e a r e c o l l e c t i o n s o f s e t s which a r i s e n a t u r a l l y and p o s s e s s p r o p e r t i e s which e n s u r e t h a t t h e y p l a y an i m p o r t a n t r o l e - f o r e x a m p l e c o n v e x s e t s i n f u n c t i o n a l a n a l y s i s and e f f i l g s e t s i n p o t e n t i a l theory.
I n t h e t h e o r y o f s e v e r a l c o m p l e x v a r i a b l e s many
d i f f e r e n t k i n d s o f s p e c i a l s e t s a r i s e - pseudoconvex s e t s , p o l a r s e t s , p o l y n o m i a l l y convex s e t s , p o l y d i s c s and S t e i n manifolds. B a l a n c e d open s e t s a r i s e a s t h e n a t u r a l domain o f convergence o f t h e Taylor s e r i e s expansion a t t h e o r i g i n f o r holomorphic f u n c t i o n s .
I n t h i s c h a p t e r we c o n s i d e r a l l t h e
h o l o m o r p h i c f u n c t i o n s on a b a l a n c e d o p e n s e t U a n d show t h a t t h e T a y l o r s e r i e s e x p a n s i o n s l e a d t o a t o p o l o g i c a l decomposition of H(U) chapter 2.
f o r t h e d i f f e r e n t t o p o l o g i e s we d i s c u s s e d i n
We t h e n u s e t h i s d e c o m p o s i t i o n t o d e d u c e t o p -
-0logica1 properties of H(U)
and t o e x t e n d r e s u l t s a b o u t
s p a c e s o f homogeneous p o l y n o m i a l s t o s p a c e s o f h o l o m o r p h i c functions.
Our main t o o l s a r e a s s o c i a t e d t o p o l o g i e s and
Schauder d e c o m p o s i t i o n s and s i n c e t h e s e t o p i c s a r e n o t v e r y w e l l known we d e v o t e t h e f i r s t s e c t i o n t o t h e i r e x p o s i t i o n . The t h e o r y o f h o l o m o r p h i c f u n c t i o n s on b a l a n c e d o p e n s e t s may b e r e g a r d e d a s a l o c a l t h e o r y a n d i f we c o u l d show t h a t T~
and T& were l o c a l
(or sheaf) topologies then the
r e s u l t s o f t h i s c h a p t e r would e x t e n d t o a r b i t r a r y open s e t s .
109
110
Chapter 3
1 3 . 1 . ASSOCIATED
TOPOLOGIES
DECOMPOSITIONS I N
AND
GENERALISED CONVEX
LOCALLY
SPACES
I n l i n e a r f u n c t i o n a l a n a l y s i s we e n c o u n t e r t w o m u t u a l l y exclusive but not exhaustive types of properties
-
those
p r e s e r v e d under l o c a l l y convex i n d u c t i v e l i m i t s and t h o s e preserved under p r o j e c t i v e limits.
For example t h e l o c a l l y
convex i n d u c t i v e l i m i t o f b a r r e l l e d l o c a l l y convex spaces i s b a r r e l l e d
(respectively bornological) (respectively bornological)
and t h e p r o j e c t i v e l i m i t o f complete l o c a l l y convex s p a c e s i s complete
(respectively nuclear)
(respectively n'uclear).
blany o t h e r p r o p e r t i e s a r i s e a s a c o m b i n a t i o n o f p r o p e r t -
i e s from t h e s e two t y p e s , as semi-reflexivity infrabarrelledness
f o r example r e f l e x i v i t y i s d e f i n e d
(preserved under p r o j e c t i v e l i m i t s ) plus (preserved under l o c a l l y convex inductive
l i m i t s ) and t h e combined p r o p e r t y i s n o t p r e s e r v e d under A g e n e r a l method used i n showing t h a t
e i t h e r kind of l i m i t . a l o c a l l y convex s p a c e , associate with
(EJ)
has a given property is t o
(E,T),
a n o t h e r l o c a l l y convex s p a c e ( F , T ' ) with
t h e g i v e n p r o p e r t y a n d t h e n t o show t h a t
(E,T)
= (F,T').
For
e x a m p l e we c a n a s s o c i a t e w i t h a g i v e n l o c a l l y c o n v e x s p a c e i t s completion and a l s o i t s a s s o c i a t e d b o r n o l o g i c a l s p a c e ,
A
s y s t e m a t i c t h e o r y h a s been developed f o r p r o p e r t i e s which are p r e s e r v e d u n d e r l o c a l l y c o n v e x i n d u c t i v e l i m i t s a n d we now d i s c u s s t h i s t h e o r y and a p p l y i t i n l a t e r s e c t i o n s t o t h e theory o f holomorphic functions. Definition 3.1
i s a f a m i l y ?of (a)
A Q f a m i l y of
ZocaZZy c o n v e x t o p o l o g i e s
l o c a l l y convex topologies such t h a t
3 i s s t a b l e under l o c a l l y convex i n d u c t i v e Z i m i t s ,
( b ) on any v e c t o r s p a c e t h e f i n e s t ZocaZly c o n v e x t o p o l o g y
belongs t o 3 . L e t ( E , T ) b e a Z o c a Z l y c o n v e x s p a c e and l e t Q f a m i l y of
ZocalZy c o n v e x t o p o l o g i e s .
3 be a
The ZocaZly c o n v e x
i n d u c t i v e Z i m i t o f a l l t o p o l o g i e s on E w h i c h l i e i n s a n d a r e f i n e r than
T
also l i e s i n
3.
We d e n o t e t h i s t o p o l o g y by
and c a l Z i t t h e 3 t o p o l o g y a s s o c i a t e d w i t h
T.
T~
111
Holomorphic functions o n balanced sets The t o p o l o g y
c a n a l s o b e c h a r a c t e r i z e d as t h e s o l u t i o n 3 t o a universal problem. S p e c i f i c a l l y i f (E,T) i s a l o c a l l y T
is
convex space and 3 i s a Q-family t h e n
t h e unique
;C
t o p o l o g y on E s u c h t h a t a n y c o n t i n u o u s l i n e a r m a p p i n g f r o m (F,T!) into E,
~ ~ € f a3c t o,r s
For c e r t a i n Q-families
through
(EJ.rg).
t h e r e a r e a l s o o t h e r known
c h a r a c t e r i z a t i o n s which can be u s e f u l . Let b denote t h e family o f a l l bornological
Example 3 . 2 topologies.
Then b i s a Q - f a m i l y and f o r a n y l o c a l l y convex
topology,
we l e t
T ,
topology.
If
T~
denote the associated bornological
( E J ~ )i s a l o c a l l y c o n v e x s p a c e a n d B i s a
c l o s e d c o n v e x b a l a n c e d b o u n d e d s u b s e t o f E t h e n we l e t E B d e n o t e t h e v e c t o r s u b s p a c e o f E g e n e r a t e d by B and normed s o that B i s its closed unit b a l l . ( E , T ~ )=
I t i s w e l l known t h a t
l i m
7E B
where B r a n g e s o v e r a l l c l o s e d convex b a l a n c e d bounded s u b s e t s of E. Example 3 . 3
Let ub d e n o t e t h e f a m i l y o f a l l u l t r a -
b o r n o l o g i c a l l o c a l l y convex t o p o l o g i e s .
Then ub i s a Q - f a m i l y .
F o r a n y l o c a l l y c o n v e x s p a c e ( E , T ) we l e t
T
a s s o c i a t e d u l t r a b o r n o l o g i c a l t o p o l o g y on E .
-
example 3 . 2 ,
ub
denote t h e
We h a v e , a s i n
( E , T ~ ~= ) l i m EB B where B r a n g e s o v e r a l l c o m p l e t e b a l a n c e d convex bounded subsets of E. Example 3 . 4
A l o c a l l y convex space,
(E,T) is c a l l e d a
K e l l e y s p a c e i f a n y f i n e r l o c a l l y convex t o p o l o g y on E h a s l e s s compact s e t s . topology.
An u l t r a - b o r n o l o g i c a l
topology is a Kelley
The c o l l e c t i o n o f a l l l o c a l l y convex K e l l e y
t o p o l o g i e s i s a Q - f a m i l y a n d we l e t
T~
denote the Kelley
topology a s s o c i a t e d w i t h t h e l o c a l l y convex topology
T .
Chapter 3
112
The b a r r e l l e d a n d i n f r a b a r r e l l e d t o p o l o g i e s a l s o f o r m Q - f a m i l i e s a n d f o r a g i v e n l o c a l l y c o n v e x s p a c e ( E , T ) we l e t and T d e n o t e t h e a s s o c i a t e d b a r r e l l e d ( t o n n e l e ) and t i infrabarrelled topologies respectively. An a l t e r n a t i v e
T
description of these topologies, induction,
by means o f t r a n s f i n i t e
i s given i n proposition 3.5.
The c o l l e c t i o n o f a l l
b a r r e l l e d and b o r n o l o g i c a l l o c a l l y convex t o p o l o g i e s ,
bt,
a l s o a Q - f a m i l y a n d we d e n o t e b y
and
T
b a r r e l l e d topology associated with
the bornological
bt T.
For any Hausdorff l o c a l l y convex s p a c e ( E , T )
-
is
we h a v e t h e
following i n c l u s i o n s between t h e various a s s o c i a t e d topologies, where a
b means t h a t a i s f i n e r t h a n b ,
We c o m p l e t e o u r d i s c u s s i o n o f a s s o c i a t e d t o p o l o g i e s b y
showing t h a t t h e b a r r e l l e d topology a s s o c i a t e d w i t h a complete topology i s a l s o complete. Proposition 3.5
I f
( E , T ) is a c o m p l e t e l o c a l l y c o n v e x s p a c e
t h e n (E,-rt) i s a l s o a complete l o c a l l y convex space. Proof
We f i r s t c o n s t r u c t a n o r d e r e d f a m i l y o f l o c a l l y c o n v e x
Let
t o p o l o g i e s on E . number and (E,T,)
=
l i m B Z
T
Now s u p p o s e a i s a n o r d i n a l
~ T= .
h a s b e e n d e f i n e d f o r a l l o r d i n a l s B
T~
(E,T ) B
d e f i n e d by a l l
-
i.e.
.
Let
Ta i s t h e p r o j e c t i v e l i m i t t o p o l o g y
T ~ , B < ~ . We d e f i n e - r a a s t h e t o p o l o g y w h i c h h a s
a neighbourhood b a s e a t z e r o c o n s i s t i n g o f a l l T a c l o s e d convex balanced absorbing subsets of E.
By t r a n s f i n i t e i n d u c t i o n
i s defined f o r every ordinal a.
Since t h e c a r d i n a l i t y of t h e
T
s e t of a l l convex balanced absorbing subsets o f E i s less than o r equal t o
*IE1
it f o l l o w s t h a t t h e r e e x i s t s an o r d i n a l
number 8 s u c h t h a t
T~
=
'eel
for all
el 2
0.
Since a locally
convex s p a c e i s b a r r e l l e d i f and o n l y i f each c l o s e d convex balanced absorbing s e t i s a neighbourhood o f zero it follows that
( E , T ~ ) i s a b a r r e l l e d l o c a l l y convex s p a c e .
S i n c e T t ->
T
113
Holomorphic functions on balanced sets
t h e above c o n s t r u c t i o n shows t h a t
T
> T ~ . t -
w e a k e s t b a r r e l l e d t o p o l o g y on E f i n e r t h a n
T
is the t it follows t h a t As T
We now s h o w b y t r a n s f i n i t e i n d u c t i o n t h a t
= Tee (E,T") i s a c o m p l e t e l o c a l l y c o n v e x s p a c e f o r e a c h o r d i n a l n u m b e r a. Tt
By h y p o t h e s i s ( E , T ~ ) = ( E , T ) i s c o m p l e t e . is complete for a l l
c1<
al.
Now s u p p o s e
(E,T )
Since the projective l i m i t of
complete l o c a l l y convex spaces i s complete t h e space (E, Ta ) =
lim "<"l
c -
1
(E,T")
is complete.
Let (x,) B E
.
B
b e a Cauchy
) Hence (xg) i s a l s o a Cauchy n e t i n t h e "1 B E B c o m p l e t e s p a c e (E,T ) and s o c o n v e r g e s t o an e l e m e n t x i n "1 Now l e t V b e a TC1 - n e i g h b o u r h o o d o f z e r o i n E . With(E,Tal). 1 closed o u t l o s s o f g e n e r a l i t y we may s u p p o s e t h a t V i s a T "1 convex balanced s u b s e t o f E . Since ( x ) is T - Cauchy B E B "1 t h e r e e x i s t s P o i n B s u c h t h a t x B 1 - x B 2 E V f o r a l l B ~ , B ~B ~ . Since x x i n ( E , T ) a s B2and V i s T closed 82 "1 "1 it follows t h a t xB x E V for all B 13,. H e n c e x -+ x i n B T h i s shows t h a t ( E , T " ~ ) i s c o m p l e t e . ( E , T a l ) a s -8 By
n e t i n (E.T
-
-.
-
-
t r a n s f i n i t e i n d u c t i o n (E,.ra) i s complete f o r each o r d i n a l number a and h e n c e ( E , T ~ )= (E,rt) i s a c o m p l e t e l o c a l l y c o n v e x space.
This completes t h e proof.
A m o d i f i c a t i o n o f t h e above proof
yields the following
r e s u l t and a l s o g i v e s an a l t e r n a t i v e d e s c r i p t i o n o f t h e associated infrabarrelled topology. Proposition 3.6
L e t ( E , T ) be a l o c a l l y c o n v e x s p a c e and
b e , r e s p e c t i v e l y , t h e b a r r e l z e d and i n f r a l e t T~ and T~ b a r r e l l e d t o p o l o g i e s a s s o c i a t e d w i t h T. I f ( E , T ) i s compZete ,
then (E,ri) ( E , T ~ )and
is a l s o c o m p l e t e . (E,.ri)
I f
( E , T ) i s quasi-complete
are both quasi-complete.
sequentiaZZy complete then (E,.rt)
I f
then
(E,T) i s
and ( E , T ~ )a r e b o t h s e q u e n t -
i a l l y complete. We now l o o k a t v a r i o u s k i n d s o f S c h a u d e r d e c o m p o s i t i o n s .
These are g e n e r a l i z a t i o n s o f t h e c o n c e p t o f a b a s i s and i n d e e d a l o c a l l y convex s p a c e h a s a Schauder b a s i s i f and o n l y i f it
Chapter 3
114
has a Schauder decomposition c o n s i s t i n g of one dimensional subspaces. Definition 3.7
conoex space
(E,
[Enjn
of a ZocaZZy i s a S c h a u d e r d e c o m p o s i t i o n of ( E , T ) i f
A s e q u e n c e of s u b s p a c e s T)
e a c h x i n E can be w r i t t e n i n a u n i q u e way as x =
-
C m x ) w h e r e xn n = l 11
l i m m--.,
urn : E
u m (:zl
E,
E
E,
xn) =
If t h e s e q u e n c e of m a p p i n g s ,
;zl
xn ( i . e .
a 2 2 n and if t h e p r o j e c t i o n s
zm
xn a r e aZZ c o n t i n u o u s .
n =1
(um),",,,
f o r m an e q u i c o n t i n uous f a m i Z y t h e n we s a y t h a t t h e d e c o m p o s i t i o n i s e q u i -
If E c o n t a i n s a f u n d a m e n t a 2 f a m i Z y of s e m i - n o r m s ,
Schauder.
'1,
such t h a t p(E" xn) = Yl p ( x n ) f o r e v e r y p i n n=l
and e v e r y
C" Xn i n E t h e n we s a y t h a t t h e d e c o m p o s i t i o n i s a b s o b u t e . The n =1 decomposition i s s a i d t o be s h r i n k i n g i f ( ( E ) ' ) " - 1 i s a nB n Schauder d e c o m p o s i t i o n f o r E h .
I t i s e a s i l y s e e n t h a t e v e r y a b s o l u t e S c h a u d e r decompos-
i t i o n i s an equi-Schauder decomposition.
s e t of aZZ s e q u e n c e s of c o m p t e x n u x b e r s
We l e t (an)
A
denote t h e
such t h a t n
l i m s i p lanll'n 5 1. If i s a Schauder d e c o m p o s i t i o n nfor ( E , T ) t h e n we s a y { E n I n i s an A-Schauder d e c o m p o s i t i o n i f W
Xn E E and imply anxn E . The d e c o m p o s i t i o n n=l n=l i s an A - a b s o Z u t e d e c o m p o s i t i o n i f f o r e v e r y p E c s ( E ) and (an)E=1
A
E
t h e semi-norm ;(c"
continuous. Proposition 3.8
n=1
xn) = xm n=l
a fundamentaZ s y s t e m of semi-norms = SU nP P,(Un(x))
f o r e v e r y a i n A and x
Proof
is
A S c h a u d e r d e c o m p o s i t i o n {En}n of a ZocaZZy
convex space (E,T) i s equi-Schauder P,(x)
bnI P ( X n )
i f and onZy i f t h e r e e x i s t s A for
(E,-r)such t h a t
(*I i n E.
We f i r s t s u p p o s e t h a t { E n ) ,
i s an equi-Schauder
-
d e c o m p o s i t i o n o f E a n d p i s a c o n t i n u o u s s e m i - n o r m on ( E , T ) .
-
Let q ( x ) as n
= sup p(un(x)) f o r every x i n E. Since un(x) x n and p i s c o n t i n u o u s i t f o l l o w s t h a t q ( x ) i s f i n i t e
115
Holomorphic functions on balanced sets f o r every x i n E. w h e r e m 5 n we h a v e q(x) = Let
SIIP
B = {x E E;
Then B = { x
E
E;
S i n c e u (u x)=u ( x ) f o r e v e r y x i n E m m n
q(un(x)) f o r every x i n E. q ( x ) 5 11 and
8
= {x; p(x) 5 1).
p(un(x)) 5 1 f o r a l l n}
=
n n Ix e E ;
=
Qun-'
p ( u n ( x ) ) 5 1)
($1.
-
-
i s an e q u i c o n t i n u o u s f a m i l y o f mappings q u n - l i s a neighbourhood o f zero i n E and hence q i s a continuous
S i n c e (un),
semi-norm on E .
Since un(x)
i n E t h i s shows t h a t q 2 p .
x as n
As p r a n g e s
m
(5)
f o r every x
over cs(E) q ranges
o v e r a fundamental f a m i l y of semi-norms which s a t i s f y ( * ) . This completes t h e proof i n one d i r e c t i o n . C o n v e r s e l y i f E c o n t a i n s a fundamental f a m i l y o f semi-
*,
n o r r n s , ( ~ ~ ) w ~ h~i c h s a t i s f y ( * ) t h e n f o r e v e r y a i n A . I x E E ; p a ( x ) 5 11 = a n d h e n c e (un),"=,
Ba
=G
un-'
(B,)
i s an equicontinuous f a m i l y o f mappings.
This completes t h e proof. Proposition 3.9
A S c h a u d e r d e c o m p o s i t i o n ~ f 'a b a r r e Z Z e d
ZocaZZy c o n v e x s p a c e is an e q u i - S c h a u d e r Proof
Let {En),
decovposition.
be a Schauder decomposition f o r t h e
b a r r e l l e d l o c a l l y c o n v e x s p a c e (E,T).
If p i s a continucus
semi-norm on E l e t q ( x ) = s x p p ( u ( x ) ) f o r e v e r y x i n E .
n
As
i n t h e p r e v i o u s p r o p o s i t i o n q i s a semi-norm on E ,
B = {x EE;
u
q ( x ) 5 11 = O { x E E ; p ( u n ( x ) ) 5 1 ) .
i s a c o n t i n u o u s l i n e a r mapping
Let
Since each
I x E E ; p ( u n ( x ) ) 5 11 i s a
n c l o s e d s u b s e t o f E and hence B i s a c l o s e d convex b a l a n c e d absorbing subset of E.
S i n c e E is b a r r e l l e d B i s a n e i g h b o u r -
hood o f z e r o and h e n c e q i s a c o n t i n u o u s semi-norm on E .
As
i n t h e p r o o f o f t h e p r e c e d i n g p r o p o s i t i o n i t now f o l l o w s t h a t q L P a n d q ( x ) = S,UP q ( u n ( x ) ) . An a p p l i c a t i o n o f p r o p o s i t i o n 3 . 8 now c o m p l e t e s t h e p r o o f . The a b o v e method o f p r o o f i s a l s o u s e d i n t h e f o l l o w i n g proposition.
Chapter 3
116 Proposition 3.10
i s a n /J1-SchazAer d e c o m p o s i t i o n for t h e b a r r e Z Z e d ZocaZly c o n v e x s p a c e ( E , r ) t h e n {En}, i s a n A-absoZute decomposition. Proof
Let p
E
If
cs(E) and l e t
E
4.
The s e q u e n c e
(n 2 a n r 2 a l s o l i e s i n a a n d s o i f 1" x E E then n a x n n n=l n n=l 2 also lies i n E. H e n c e En a x la' i s a bounded s u b s e t c f E . n I? n = l - 1 2 T h e r e f o r e s x p p ( n a x ) = M
ifl
%
%
shown t h a t p i s a s e m i - n o r m on E . S i n c e B = { X EE ; p ( x ) 5 1 ) n n m = l x m E E ; ;z1 l a m l p i x , ) 5 11 i s a c l o s e d c o n v e x
=n{c-
b a l a n c e d a b s o r b i n g s u b s e t o f E and E i s a b a r r e l l e d s p a c e i t 2,
f o l l o w s t h a t B i s a neighbourhocd o f z e r o and hence p i s a c o n t i n u o u s s e m i - n o r m on E .
This completes the proof.
We now l o o k a t a s s o c i a t e d t o p o l o g i e s f o r s p a c e s w i t h a n A-absolute decomposition. I f i s an j - c b s o Z u t e d eco mp o s itio n Proposition 3.11 for ( E , T ) t h e n { E n , ~ b } nis a Z s o an $ - a b s o l u t e d e c o m p o s i t i o n
for
(E>Tb).
Proof of
For each A i n
(E,T).
p is a
T
p
xn) =
'L
[4zl
A
Let B = (x ) A s r
d e n o t e an a r b i t r a r y bounded s u b s e t
r
l e t xA =
;Z1
x:
c o n t i n u o u s s e m i - n o r m o n E a n d (a , )
norm on E .
:zl
kzl
n21an/ p(xn)
T
E
(E,T).
4
If
then
n continuous semi-
is a a n d {n2 a x A } n n n,X I f q i s a rb c o n t i n u o u s semi-norm b o u n d e d s u b s e t o f (E,T) o n E t h e n t h e a b o v e s h o w s t h a t s u [ n 2 l a n l q ( x n1 ) ] = M 1 < n x a 1 and h e n c e s p xn) 5- M1 $ = 1 < m . Hence t h e
Hence syp
2,
n
2
is also a
in
,K
ifl
x
+
-
-
lanl
q(Xn)
o as A m.
w
-
-*
i s b o u n d e d on t h e
so i s rb continuous.
bounded s u b s e t s o f ( E , T ~ )a s A
M <
.
semi-norm q ( C m n = 1X n ) t h a t x:
=
l a n l p[x:)
in(E,Tb)
T
T h i s a l s o shows
i f xA
o in
T o c o m p l e t e t h e p r o o f we m u s t s h o w t h a t
PI
xm
If p is a
T
Zx i n ( E , T ~ )a s n -m. m=l m c o n t i n u o u s semi-norm on E and x =
&zl
x m €E then
117
Holomorphic functions on balanced sets
i s a bounded s u b s e t o f
(E,Tb). q(x
-
as n
i!yl
If q is a
-
n2 q ( n 2
Xm) = q ( m c"= n Xm) = 1
-.
(E,'I)
and o f
c o n t i n u o u s semi-norm on E t h e n
Tb
zrn
m=n xm) __+
O
This completes t h e proof.
The s i t u a t i o n f o r t h e a s s o c i a t e d b a r r e l l e d t o p o l o g y i s s l i g h t l y morc c o m p l i c a t e d .
T h i s i s d u e t o t h e f a c t t h a t we
d o n o t , i n g e n e r a l , know i f a S c h a u d e r d e c o m p o s i t i o n ,
i s a l s o a S c h a u d e r decompos-
(E,T),
f o r a l o c a l l y convex s p a c e ,
i t i o n f o r the associated barrelled space (F,Tt). a p o s i t i v e answer t o t h i s problem i f t o p o l o g i e s on E i . e . and only i f o(E',E).
(E,'I)'
(E,T)'
if
=
'I
and
'I
We o b t a i n
are compatible
t ( E , T ~ ) 'a n d t h i s o c c u r s i f
i s q u a s i - c o m p l e t e f o r t h e weak t o p o l o g y
On t h e o t h e r h a n d we d o h a v e t h e f o l l o w i n g u n i q u e i s a Schauder decovposition
n e s s r e s u l t ; i f {En), and i f each E
for (E,r)
i s a b a r r e l l e d s p a c e t h e n t h e r e e x i s t s a t most
n one b a r r e l l e d t o p o l o g y on E which c o i n c i d e s w i t h and h h i c h h a s {En}
n
T
on e a c h En
a s a Schauder decomposition.
A l t h o u g h we d o n o t know i f s u c h a t o p o l o g y e x i s t s t h e a b o v e s a y s t h a t when i t d o e s i t m u s t b e t h e a s s o c i a t e d barrelled topology.
he p r o v e a l e s s g e n e r a l r e s u l t which i s
adequate f o r our applications. Proposition 3.12
~ e {t E ~ be ) ~a r ~ R - . d e e o m p o s i t i o n for (E,T)
and suppose each En i s a b a r r e l l e d l o c a l l y convex space.
I f
t h e r e e x i s t s G b a r r e l l e d t o p o l o g g T~ o n E w h i c h c o i n c i d e s w i t h 'I on each. Erl and i.f { E n } n i s c S c h a u d e r d e c o m p o s i t i o n fcr ( E , T ~ )t h e n
Tl
=
Tt
a x d a f u n d c m e n t a l s g s t e n : of sen;<-norms
f o r ( E 3 r t ) i s g i v e n by G Z Z s e m i - n o r m s , following conditions; (a)
PIE,
is ~ - e o n t i n ~ o u s
( b ) p ( , ? l l x n ) = ;:l Cm x i n E. n=l n Proof
P , w h i c h satisfy t h e
p(xn) f o r every
By h y p o t h e s i s
'I
5
T~
5
' I ~ and
hence
Chapter 3
118
=
"En
T
=
.
T
S i n c e {En}n i s a S c h a u d e r decomposit follows t h a t it is also a
i t i o n f o r ( E , T ) and ( E , r l )
S i n c e I E n I n i s an
Schauder d e c o m p o s i t i o n f o r (E..rt). A-decomposition
f o r (E,T) it i s a l s o an 2-decomposition f o r
a n d ( E , T ~ )a n d h e n c e b y p r o p o s i t i o n 3 . 1 1 i t i s a n
(E,rt)
d - a b s o l u t e and i n p a r t i c u l a r an a b s o l u t e d e c o m p o s i t i o n f o r both of these topologies.
Hence b o t h o f t h e s e t o p o l o g i e s a r e
g e n e r a t e d by semi-norms which s a t i s f y ( a ) and ( b ) .
On t h e
o t h e r hand i f p i s a semi-norm on E which s a t i s f i e s ( a ) and p(x) 5 1 ) is a
(b) then V = {x E E ;
closed c o n v e x b a l a n c e d
a b s o r b i n g s u b s e t o f E a n d h e n c e p i s a -rt c o n t i n u o u s s e m i norm.
Hence
' I ~ =
r t and
T~
i s t h e l o c a l l y c o n v e x t o p o l o g y on
E g e n e r a t e d by a l l t h e semi-norms on E which s a t i s f y ( a ) and
(b). We now l o o k a t d e c o m p o s i t i o n s o f t h e s t r o n g d u a l . Proposition 3.13
E
a
a n d BI' = { n 2 C m a m=n 'm)n,X
n
= TIE
n
i s bounded
E
El
i s an
Since
decomposition it follows t h a t
I f T E E ' we l e t T
I f T8
are bounded s u b s e t s
f o r every positive integer n .
f o r e a c h 8 i n s o me d i r e c t e d s e t R a n d B
projection o f B i n E then A n 1 IiTenIIB = s u p I T e ( x n ) 5 -2 I l ~ ~ l l Bf t o r a l l n * n a an Hence, s i n c e En i s a complemented s u b s p a c e o f E , T i
-
i f Te
B
& ( w e may a n d s h a l l a s s u m e w i t h o u t l o s s o f
B ' = ( n 2 an ' n ) n , a of E. B"
T) I .
a
g e n e r a l i t y t h a t an 2 1 f o r a l l n ) . A-absolute
decomposition f o r
(x )aET.b e a bounded s u b s e t o f E and l e t
Let B
a = (a,),
{ E n I n is a n & a b s o l u t e
I n is an ~ - a b s o Z u t e d e c o m p o s i t i o n for ( E ,
{ (En);
(E,T) t h e n Proof
I f
o i n (E o in
I
) I
nB
as 0
-
m
n
Since
is the
f o r each n
E l .
B
T h i s shows t h a t i s a Schauder decomposition f o r To c o m p l e t e t h e p r o o f i t s u f f i c e s t o n o t i c e t h a t
E l .
B
Holornorphic functions on balanced sets
119
C"
n=1
< C"
n=
f o r every T i n E'.
A n d - a b s o Z u t e d e c o m p o s i t i o n of a ZocaZZy
Corollary 3.14
convex space i s a shrinking d e c o m p o s i t i o n . F u r t h e r p r o p e r t i e s o f a s s o c i a t e d t o p o l o g i e s and S c h a u d e r decompositions a r e given without proof a s t h e need a r i s e s . For f u r t h e r d e t a i l s we r e f e r t o t h e N o t e s a n d Remarks a t t h e e n d of t h i s chapter,
Apart from t h e immediate a p p l i c a t i o n ' s o f t h e above r e s u l t s w h i c h we g i v e i n t h i s c h a p t e r we w i l l a l s o f i n d t h a t t h e techniques developed t o o b t a i n t h e s e r e s u l t s a r e useful i n s t u d y i n g h o l o m o r p h i c f u n c t i o n s on c e r t a i n n u c l e a r s e q u e n c e spaces.
13.2
EQUI-SCHAUDER DECOMPOSITIONS O F ( H ( U ; F )
,T).
In t h i s s e c t i o n U w i l l d e n o t e a b a l a n c e d open s u b s e t o f a l o c a l l y c o n v e x s p a c e E and F w i l l d e n o t e a Banach s p a c e (Most o f t h e r e s u l t s ,
quasi-complete
however, can be e x t e n d e d t o a r b i t r a r y
l o c a l l y convex s p a c e s ) .
P r o p o s i t i o n 3.15
If U is a b a l a n c e d o p e n s u b s e t o f a Z o c a l Z y c o n v e x s p a c e E , F i s a Banach s p a c e , f E H ( U ; F ) and a =
(an)*
A then g = 1" a n=o n
E
Let K C
Proof
s
E
bT e Uc o m p a c t .
H(U;F) We may s u p p o s e w i t h o u t l o s s
o f g e n e r a l i t y t h a t K i s b a l a n c e d , t h e n t h e r e e x i s t A > 1 and V ,
a c o n v e x b a l a n c e d n e i g h b o u r h o o d o f z e r o , s u c h t h a t A(K+V) C U and l
l
IlfIIA(K+v) = ~
\
<
"'
By t h e C a u c h y i n e q u a l i t i e s
l (K+V) A < M for a l l n.
S i n c e (an),
E
3
t h e r e e x i s t s C > o such t h a t
120
<
Chapter 3 n
1+A :zo(x)
C.M
<m.
Thus g i s a l o c a l l y b o u n d e d f u n c t i o n on U a n d s i n c e i t i s G-holomorphic i t
in H(U;F).
is
Proposition 3.16
This completes t h e proof.
I f U i s a baZanced o p e n s u b s e t of a ZocaZZy
c o n v e x s p a c e E , F is a Banach s p a c e and f a H ( U ; F )
then the
TayZor s e r i e s e x p a n s i o n o f f a t t h e o r i g i n c o n v e r g e s t o f i n (H(U;F)
*
For each positive integer m l e t
Proof =
"m
,T6)
{x E U ;
n
2 d f o )
the interior of V i t i o n shows t h a t
m
.
(W
Since
5 m f o r a l l n l a n d l e t Wm d e n o t e 2 "
(n ) n = l
Ed t h e
preceding propos-
i s an i n c r e a s i n g c o u n t a b l e open c o v e r
)" m m=l
of U.
Now l e t p b e a
exist
a positive integer m
c o n t i n u o u s semi-norm on H ( U ; F ) .
T~
for every f i n H(U F).
\I
(x)
a n d C>o s u c h t h a t p ( f ) < C
Ilfll
There
wm 0
IIence f o r a l l n w e h a v e
^.
J = O
j!
d j f (0) j!
Ic
Fl'n+l
"Wmo
m0
j2
c
*
Therefore p(f
- j=o
djf(0) )j!
0
as n
-
and t h i s completes t h e p r o o f . Theorem 3 . 1 7
L e t U b e a baZanced o p e n s u b s e t of a ZocaZZy
c o n v e x s p a c e E and Z e t F be a Banach s p a c e . {&"E;
m
F ) ,~~l~~~
d e c o m p o s i t i o n for Proof
Then
i s a n , # - d e c o m p o s i t i o n and an A - a b s o Z u t e (H(U;F)
,T
By p r o p o s i t i o n 2 . 4 1
complemented s u b s p a c e o f
6) (P("E;F),T,)
(H(U;F)
, T )~
is a closed
and s o p r o p o s i t i o n
121
Holomorphic functions on balanced sets m
3.16 implies that {!?(nE;F),~w}n=O
for (H(U;F),T&).
is a Schauder decomposition
Proposition 3.15 implies that it is an
,$-decomposition
and since
T~
is a barrelled topology
proposition 3.10 implies that it is an A - a b s o l u t e decomposition.
This completes the proof.
We now obtain the same result for the
-c0
and
T~
topologies and
their associated bornological topologies. L e t U b e a baZanced o p e n s u b s e t of a
Proposition 3.18
l o c a l l y c o n v e x s p a c e E , l e t F be a Banach s p a c e , l e t K b e a Then t h e semi-norm compact s u b s e t of U - a n d l e t ( a n ) n E 8 . n f (0 p(f) = I I W b K
Efola,l
is
T
Proof
continuous on H ( U ) .
We may assume without loss o f .generality that K
is a balanced subset o f U . compact subset o f U and n lanl 5 ( I +T A)
for all n 1. n
For any f in H(U;F)
.
we have by the Cauchy inequalities
for every f in H ( U ; F ) Theorem 3.19
0
Choose X > l such that X K is a a positive integer such that
and this completes the proof.
L e t U b e a b a l a n c e d o p e n s u b s e t of a ZocaZZy
c o n v e x s p a c e E and l e t F b e a Banach s p a c e . Then {' 6 (nE ;F) ,T ~ } : = ~ i s a n 8 - d e c o m p o s i t i o n and an d - a b s o Z u t e
d e c o m p o s i t i o n f o r (H(U;F) , T ) . By proposition 3.16, since T 6 > T the Taylor Proof 0' series expansion at the origin o f a holomorphic function
converges t o the function in the compact open topology. Since (@("E;F) , T ~ )is a closed complemented subspace o f n (H(U;F) , T ~ ) (proposition 2.40) this s h o w s that I(?( E;F) ,
OD
T ~ } ~ = ~
is a Schauder decomposition for (H(U;F) , T ~ ) . Proposition 3.15 implies that it is a n A -decomposition and proposition 3 . 1 8 shows that it is a n d - a b s o l u t e decomposition. This
Chapter 3
122 completes t h e proof. Corollary
3.20
L e t U b e a b a l a n c e d o p e n s u b s e t o f a ZocaZZy
c o n v e x s p a c e E and l e t F be a Banach s p a c e . i s and-decomposition
and an 4 - a b s o l u t e
( H ( U ; F ) , T ~ , ~i ) f e a c h p ( " E ; F )
Then { P ( n E ; F ) l z = o
d e c o m p o s i t i o n for
i s given t h e bornoZogical
t o p o l o g y a s s o c i a t e d w i t h t h e compact o p e n t o p o l o g y . Proof
Apply p r o p o s i t i o n
3.11 and theorem 3 . 1 8 .
L e t U be a b a l a n c e d o p e n s u b s e t of a
Proposition 3.21
l o c a l l y c o n v e x s p a c e E , Z e t F b e a Banach s p a c e , be a
T
w
Zet p
c o n t i n u o u s semi-norm on H ( U ; F ) and l e t ( a )
n n
Then t h e semi-norm
E
A.
Proof
Suppose p i s p o r t e d by t h e compact b a l a n c e d s u b s e t K
o f U.
We s h o w t h a t p i s a l s o p o r t e d b y t h e same c o m p a c t s e t .
%
Let V b e a n e i g h b o u r h o o d o f K which l i e s i n U .
ChooseX> 1
and a b a l a n c e d neighbourhood o f z e r o W such t h a t K A(K+W)CV i+x n s u c h t h a t la 5 (-1 for all
I
Choose a p o s i t i v e i n t e g e r n n > no.
2
T h e r e e x i s t s a p o s i t i v e n u m b e r C(W) s u c h t h a t Hence, f o r e v e r y f o r e v e r y f i n H(U;F) p ( f ) 5 C(W) llf
IIK+W
.
f i n H ( U ; F ) , we h a v e
%
and p i s a
T~
c o n t i n u o u s semi-norm on H(U;F).
Theorem 3 . 2 2
L e t U b e a baZanced o p e n s u b s e t o f a ZocaZZy
c o n v e x s p a c e E and Z e t F be a Banach s p a c e . n
Cp(
E; F), T
m
~
}
i s =a n ~ A-decomposition ~
decomposition f o r (H(U;F)
Then
and a n / j - a b s o Z u t e
,T~).
Proof
By p r o p o s i t i o n 3 . 1 6 , s i n c e T 6 > T the Taylor w' series expansion a t t h e o r i g i n o f a holomorphic function
converges t o t h e function i n t h e m
T
w
t o p o l o g y . BY P r o P o s i t i o n
2.41 {@(nE;F),~w)n=O i s a Schauder decomposition f o r I t i s an,J'-decomposition
by p r o p o s i t i o n
3 . 2 1 shows t h a t i t i s a n d - a b s o l u t e
(H(U;F);:)
3.15 and p r o p o s i t i o n
decomposition.
This
123
Holomoiphic finctions on balanced sets completes t h e proof. m
( p ( n E ; F) , T ~ n} = o is m / j ' - d e c o r n p o s i t i o n and an
Corollary 3 - 2 3
2 - a b s o Z u t e d e c o m p o s i t i o n f o r (H(U;F) , T ~ , ~ ) . Proof
Since T
w
-<
T
w,b <
Tc6
and
TwI p ( n E ; F )
=
T
i n d u c e s t h e T~ t o p o l o g y on w,b The r e q u i r e d r e s u l t now f o l l o w s b y a p p l y -
f o r a l l n it follows t h a t T P("E;F)
f o r a l l n.
i n g p r o p o s i t i o n 3 . 1 1 and t h e o r e m 3 . 2 2 . The e x i s t e n c e o f a n d - a b s o l u t e d e c o m p o s i t i o n o f H(U;F)
by
s p a c e s o f homogeneous p o l y n o m i a l s i s a d e q u a t e f o r a l l o u r applications i n t h i s chapter.
However we s h a l l n e e d ,
in
c h a p t e r 4 , a s l i g h t l y s t r o n g e r r e s u l t which a p p e a r s t o h o l d o n l y f o r t h e compact open and t h e
T
w
topologies.
L e t U b e a b a l a n c e d o p e n s u b s e t of a
Proposition 3.24
l o c a l l y c o n v e x s p a c e E , l e t F b e a Banach s p a c e and l e t p b e a T c o n t i n u o u s semi-norm on H ( U ; F ) , T = T 0 o r T w . T h e n t h e r e e x i s t s A > 1 such t h a t
is a l s o a Proof
T
c o n t i n u o u s semi-norm
If p is a
T~
on H ( U ; F ) .
c o n t i n u o u s s e m i - n o r m on H(U;F)
p ( f ) 5 C llfIIK f o r e v e r y f i n H(U;F) such t h a t AK i s a subset of U.
2,
a n d h e n c e p i s T~
continuous.
and
t h e n we may c Q o o s e X > 1
For e v e r y f
=&zo
d " f 0 E
n!
This completes t h e proof
H ( U ;F)
for
t h e compact open t o p o l o g y . Now s u p p o s e p i s a
T
w
c o n t i n u o u s s e m i - n o r m on H(U;F)
i s p o r t e d by t h e compact b a l a n c e d s u b s e t K o f U . s u c h t h a t X K i s a compact s u b s e t o f U .
W a balanced neighbourhood o f K such t h a t
L e t C(W)>o b e s u c h t h a t p ( f ) 5 C(W) I f Em n=o
anfo n!
E
H(U;F)
Choose X > 1
Let V d e n o t e an open
b a l a n c e d s u b s e t o f U which c o n t a i n s A K .
H(U;F).
which
T h e r e e x i s t a > l and AK C a A W C V C U .
\ ( f \ l Wf o r e v e r y f i n
124
Chapter 3
2
c . Ilfll" .
Hence p i s p o r t e d b y X K a n d s o i t i s
continuous.
T~
This
completes t h e proof. We c o m p l e t e t h i s s e c t i o n b y c o n s i d e r i n g s p a c e s o f g e r m s . Theorem 3.25
If K is a baZanced compact s u b s e t o f a ZocaZZy
c o n v e x s p a c e E and F is a Banach s p a c e t h e n {6'("E;F)
,T
lm
w n=o
decomposition for
is a n & - d e c o m p o s i t i o n and a n B - a b s o l u t e H(K;F). Proof
I f f EH(K;F) t h e n f €H(U;F)
subset U of E.
f o r some b a l a n c e d o p e n (H(U;F),T ) as U r a n g e s
S i n c e H(K;F)=="
w
o v e r a l l b a l a n c e d open s u b s e t s o f E c o n t a i n i n g K and s i n c e t h e Taylor s e r i e s converges in(H(U;F),T
w
)
(theorem 3 . 2 2 )
it
follows t h a t t h e Taylor s e r i e s of E a t t h e o r i g i n converges t o f i n H(K;F).
By p r o p o s i t i o n 2 . 5 8
complemented s u b s p a c e o f H(K;F)
( p ( n E ; F ) , ~ w )i s a c l o s e d m
a n d h e n c e I ~ ' ( " E ; F ) , T ~ i~s~ = ~
a Schauder decomposition o f H(K;F).
P r o p o s i t i o n 3.15 shows
t h a t it i s a n $ - d e c o m p o s i t i o n and an a p p l i c a t i o n o f p r o p o s i t i o n 3 . 1 0 c o m p l e t e s t h e p r o o f s i n c e H(K;F)
is a
b a r r e l l e d l o c a l l y convex s p a c e . 53.3 A P P L I C A T I O N S O F G E N E R A L I S E D D E C O M P O S I T I O N S T O T H E STUDY OF H O L O M O R P H I C F U N C T I O N S O N BALANCED OPEN S E T S
The r e s u l t s o f t h e t w o p r e v i o u s s e c t i o n s a r e a p p l i e d t o H(U;F)
where U i s a b a l a n c e d open s u b s e t o f a l o c a l l y convex
s p a c e and F i s a Banach s p a c e .
The f i r s t p a r t o f t h i s s e c t i o n
i s devoted t o topologies associated with t h e
'cU
topology.
Our
first r e s u l t motivated t h e introduction of associated
t o p o l o g i e s i n t h e t h e o r y o f holomorphic f u n c t i o n s on l o c a l l y convex s p a c e s .
125
Holomorphic functions on balanced sets L e t U b e a b a l a n c e d o p e n s u b s e t of a Z o c a l Z y
Theorem 3 . 2 6
c o n v e x s p a c e E and l e t F b e a Banach s p a c e . = T
T6
w,t
=
T
w,bt =
'Iw , u b .
S i n c e ( H ( U ; F ) ,T ) i s a n u l t r a b o r n o l o g i c a l s p a c e i t 6 T i s t h e b a r r e l l e d topology associated
Proof
s u f f i c e s t o show t h a t with
On H ( U ; F )
each n .
6
By p r o p o s i t i o n
T ~ .
2.41
T~
and
T
6
agree on6)(nE;F)
for
An a p p l i c a t i o n o f p r o p o s i t i o n 3 . 1 2 now c o m p l e t e s t h e
p r o o f s i n c e {P("E;F) ( H ( U ; F ) , T ~ )a n d
i s an.3 -decomposition
,T,}:=~
f o r both
(H(U;F),-r ) b y t h e o r e m s 3 . 1 7 and 3 . 2 2 .
6
P r o p o s i t i o n 3 . 1 2 a l s o shows t h a t
T~
i s t h e f i n e s t topology
f o r w h i c h we h a v e a b s o l u t e c o n v e r g e n c e o f t h e T a y l o r s e r i e s expansion and which c o i n c i d e s w i t h T
w
on s p a c e s o f
Formally t h i s i s expressed as
homogeneous p o l y n o m i a l s . follows.
L e t U b e a baZanced o p e n s u b s e t of a ZocaZZy c o n v e x s p a c e E and Z e t F b e a Banach s p a c e . The Proposition 3.27
t o p o Z o g y o n H ( U ; F ) i s g e n e r a t e d b y aZZ s e m i - n o r m s ,
T~
p , which
s a t i s f y t h e foZZowing c o n d i t i o n s ;
is
( b ) 'I@("E;F)
continuous.
T
T h e f o l l o w i n g lemma i s a n i m m e d i a t e c o n s e q u e n c e o f t h e existence of an,8-absolute
An a n a l o g o u s
decomposition.
r e s u l t f o r t h e compact open t o p o l o g y i s a l s o t r u e . Lemma 3 . 2 8 L e t U b e a baZanced o p e n s u b s e t of a ZocaZZy c o n v e x s p a c e E and l e t F b e a Banach s p a c e . L e t be a T (respectively T , T & ) bounded n e t i n H ( U ; F ) . Then fa
w
-0
w ,b
- -
a s a-
m
i n (H(U;F),rw) (respectively
( H ( U ; F ) , T ~ , ~ ) (, H ( U ; F ) , T ~ ) ) if and o n l y i f An o a s c1 = i n (@("E;F) d f,(O) In!
,T
w
I f o r e v e r y non-
negative i n t e g e r n. This means,
in particular,
s a m e t o p o l o g y on t h e 3.26 implies,
T~
that
T ~ , T ~ , a ,n d
'c6
induce t h e
bounded s u b s e t s o f H(U;F).
among o t h e r t h i n g s , t h a t T~
and
T~
Theorem
define the
same c o n v e x b a l a n c e d c o m p l e t e b o u n d e d s u b s e t s o f H ( U ; F ) .
126
Chapter 3
U s i n g lemma 3 . 2 8 we show t h a t t h e same r e s u l t h o l d s f o r compact b a l a n c e d convex s e t s .
L e t U be a baZanced o p e n s u b s e t of a
Proposition 3.29
ZocatZy c o n v e x s p a c e E and l e t F be a Banach s p a c e . c o n v e x b a l a n c e d compact s u b s e t s o f ( H ( U ; F ) (H(U;F)
with
i s t h e KelZey t o p o l o g y a s s o c i a t e d
) c o i n c i d e and 6 on H ( U ; F ) .
T
Since
s u f f i c e s t o show t h a t a n y c o n v e x
,T
‘cw
Then t h e
, T ~ )and
Proof
T
> T it 6 w
6
b a l a n c e d compact s u b s e t K o f ( H ( U ; F ) , T ~ )i s
compact. By 6 bounded s u b s e t o f T
theorem 3 . 2 6 K i s a complete balanced T 6 I f ( f a l a c r i s a n e t i n K t h e n i t c o n t a i n s a T~ H(U;F). convergent subnet. By lemma 3 . 2 8 t h i s s u b n e t i s a l s o T 6 convergent. Hence K i s a T c o m p a c t s u b s e t o f H ( U ; F ) . Since -i6
6 i s an u l t r a b o r n o l o g i c a l topology i t i s a l s o a Kelley
t o p o l o g y and h e n c e
T
6 = Tw , K .
One c a n a l s o show t h a t topology associated with
T
w
T is the infrabarrelled w,b on H(U;F). Thus we s e e t h a t t h e r e
are,
i n g e n e r a l , t w o t y p e s o f t o p o l o g i e s t h a t we may a s s o c i a t e
with
T
w
.
On t h e o n e h a n d t h e r e a r e t h e a s s o c i a t e d b a r r e l l e d ,
u l t r a b o r n o l o g i c a l , b a r r e l l e d and b o r n o l o g i c a l , t o p o l o g i e s a l l o f which a r e e q u a l t o
T~
and K e l l e y
and t h e a s s o c i a t e d
i n f r a b a r r e l l e d and b o r n o l o g i c a l t o p o l o g i e s which a r e e q u a l t o I t i s an o p e n q u e s t i o n w h e t h e r o r n o t t h e s e t w o ‘w,b’ t o p o l o g i e s c o i n c i d e i . e . is T w , b = T6?. Theorem 3 . 2 6 a n d p r o p o s i t i o n 3 . 2 9 i n d i c a t e t h a t t h e y a r e v e r y c l o s e t o one another.
The f o l l o w i n g r e s u l t g i v e s n e c e s s a r y a n d s u f f i c i e n t
c o n d i t i o n s u n d e r w h i c h t h e s e t o p o l o g i e s c o i n c i d e a n d we s h a l l i n t h i s and l a t e r c h a p t e r s , e n c o u n t e r v a r i o u s s u f f i c i e n t conditions for t h e i r equality. Proposition 3.30
L e t U be a baZanced o p e n s u b s e t of a
l o c a l l y c o n v e x s p a c e and l e t F be a Banach s p a c e .
The
f o Z Z o w i n g a r e e q u i v a l e n t on H ( U ; F ) ; (a) (b)
T w , b = T6 ‘c6 and T d e f i n e t h e same bounded s e t s
(c) (d)
T~ T
w
and and
w
T~ T&
d e f i n e t h e same compact s e t s , i n d u c e t h e same t o p o Z o g y o n T
w
bounded s e t s
127
Holomorphic functions on balanced sets T T
w
i s a barrelled topology
,b
i s t h e f i n e s t l o c a l l y convex topology f o r which t h e
W,b
Taylor s e r i e s expansion a t t h e o r i g i n converges a b s o l u t e l y and w h i c h i n d u c e s t h e T @ t o p o l o g y o n P ( n E ; F ) f o r e v e r y positive integer n, i f T~ E (@(:E ;F) ,T 1 1 f o r every non-negative i n t e g e r n W dnf o and T n ( . e ) converges f o r every f = Zm n=O n!
:lo
( a ) , ( b ) , ( e ) and ( f ) a r e e q u i v a l e n t by t h e o r e m 3.26
Proof
and p r o p o s i t i o n 3 . 2 7 . and B i s a
T
(a)*(c)
w
bounded t h e n t h e r e e x i s t s a (f,),,
b y lemma 3 . 2 8 .
If
b o u n d e d s u b s e t o f H(U;F) w h i c h i s n o t
-
6
c o n t i n u o u s semi-norm p and
6
such t h a t p(fn)
a sequence i n B , -
T
(c) holds T
m
as n
I m.
U { o } i s T~ c o m p a c t b u t n o t T b o u n d e d . 'he s e t { f " / J p ( f n ) } n = l 6 T h i s c o n t r a d i c t i o n shows t h a t ( c ) - ( b ) . ( c ) and ( d ) a r e e q u i v a l e n t b y lemma 3 . 2 8 .
Now s u p p o s e ( a ) h o l d s a n d t h e
sequence CTn}nAsatisfies t h e conditions of ( g ) .
5=o m
I
ITn (-1 n! By p r o p o s i t i o n 3 . 2 7 3.15
:Io
<
m
2zo
f o r every f =
By p r o p o s i t i o n E
H(U;F)
n!
I
d e f i n e s a -c6 a n d h e n c e a
T
c o n t i n u o u s s e m i - n o r m on H ( U ; F ) .
i t f o l l o w s t h a t ,Zzo
( H ( U ; F ) , T ~ , ~a)n~d h e n c e ( a )
p(f1 =
IT,
1-(
Tn
E
w.b
(9)
Conversely i f (9) i s s a t i s f i e d then ( H ( U ; F ) ; T ~ , ~ ) ' = (H(U;F),r6)'
and s i n c e
T
i s a Mackey
,b topology ( i t i s i n f r a b a r r e l l e d ) t h i s implies t h a t and (g) 4 ( a ) .
W
T
~
=
,T &~
This completes t h e proof.
Some a n a l o g u e s o f t h e a b o v e r e s u l t s c a n a l s o b e p r o v e d f o r t h e compact open t o p o l o g y .
The r e s u l t s , h o w e v e r ,
are not as
c o m p l e t e i n t h i s c a s e s i n c e ( ~ ( , , E ; F ) , T ~i )s n o t i n g e n e r a l a b a r r e l l e d l o c a l l y convex s p a c e . Proposition 3.31
We g i v e o n e e x a m p l e .
L e t U b e a b a l a n c e d o p e n s u b s e t of a
l o c a l l y c o n v e x s p a c e E and l e t F be a Banach s p a c e . f o l l o w i n g a r e e q u i v a l e n t on H(U;F);
The
128 (a) (b)
Chapter 3 T
is, a ~ b a r r e Z Z e d t o p o l o g y ,
~
(i)
($'(nE;F),~o,b)
( i i ) i f Tn
E
Em n = o Tn (
(i) (ii)
,T
inf(0) ) <
0
then
;=,
,b
f o r e a c h n and
) I
for e a c h f = nC = o
n!
i n H(U;F) (c)
i s barreZZed f o r each i n t e g e r n ,
(f"(nE;F)
m
Tn
E
(H(U;F)
Pf(0) n!
, T ~ , I~ , )
( 6 ' ( n ~ ; F ) , ~ o , b is ) barreZZed f o r each i n t e g e r n ,
is t h e f i n e s t ZocaZZy c o n v e x t o p o Z o g y on for w h i c h t h e T a g l o r s e r i e s c o n v e r g e s and w h i c h i n d u c e s on Q ( n ~ ; ~t h)e T ~ t o,p o Z~o g y for e a c h n . T
,b H(U;F) 0
-
We now i n t r o d u c e a weak f o r m o f c o m p l e t e n e s s
-
s e r i e s completeness
Taylor
which a l l o w s us t o extend v a r i o u s
t o p o l o g i c a l p r o p e r t i e s o f s p a c e s o f homogeneous p o l y n o m i a l s t o h o l o m o r p h i c f u n c t i o n s on b a l a n c e d o p e n s e t s . Definition 3.32
L e t E and F be ZocaZZy c o n v e x s p a c e s and
Z e t U be a b a l a n c e d o p e n s u b s e t o f E . space ( H ( U ; F ) , - c )
is T . S .
T
The ZocaZZy c o n v e x
complete (T.S.
i f t h e f o Z l o w i n g c o n d i t i o n is s a t i s f i e d ;
s e q u e n c e o f homogeneous poZynomiaZs, nE m = o p ( P n ) < m for e a c h Em E H(U;F). n = o 'n
Pn
TayLor s e r i e s )
=
m
( P n ) n = o is a
i f E
$' ( " E ; F ) ,
and
c o n t i n u o u s semi-norm p , t h e n
T
We h a v e a l r e a d y s e e n e x a m p l e s o f T . S .
completeness.
is T.S.
F o r example theorem 2 . 2 8 s a y s t h a t H ( U ; F )
T
P
i f U i s a b a l a n c e d open s u b s e t o f a Banach s p a c e and t h e topology o f pointwise convergence. Let
T~
and
T~
and suppose
i s a l s o T.S.
complete 'I
P
is
d e n o t e two l o c a l l y c o n v e x t o p o l o g i e s on H(U;F) >
T T~
T
2.
I f H(U;F)
complete.
i s T.S.
T~
complete then it
The f o l l o w i n g r e s u l t d e s c r i b e s a
s i t u a t i o n i n which t h e converse h o l d s .
L e t U b e a b a l a n c e d o p e n s u b s e t of a ZocaZZy c o n v e x s p a c e and l e t F b e a Banach s p a c e . I f T i s a ZocaZZy m R ~ an ,8 - a b s o Z u t e c o n v e x t o p o z o g y o n H ( U ; F ) and (6(E ; F ) , T ) ~ = is Lemma 3 . 3 3
decomposition f o r H ( U ; F ) and o n l y i f H ( U ; F ) Proof
then H(U;F)
is T . S .
Suppose H ( U ; F )
T,,
i s T.S.
T
complete i f
compzete
i s T.S.
T~
complete.
Let
m
(Pn)n=o
129
Holomorphic functions on balanced sets b e a s e q u e n c e o f homogeneous p o l y n o m i a l s ,
Pn€ @(nE;F)
,
and
< m f o r e v e r y T c o n t i n u o u s s e m i - n o r m p on s u p p o s e nz m =o p(pn) i s a T and hence a T bounded H(U;F). The s e q u e n c e { P 1 b n n m n s u b s e t o f H(U;F). S i n c e I @ ( E ; F ) , T ~ } ~i s= ~a l s o a n d - a b s o l u t e
d e c o m p o s i t i o n f o r ( H ( U ; F ) , T ) ( p r o p o s i t i o n 3 . 1 1 ) we h a v e b zm p ( p n ) < m f o r e v e r y T b c o n t i n u o u s semi-norm on H ( u ; F ) . n=o This Hence Em PncH(U;F) and H(U;F) i s T . S . T c o m p l e t e . n=o completes t h e p r o o f . Corollary 3.34
L e t U be a b a l a n c e d o p e n s u b s e t o f a l o c a l l y
c o n v e x s p a c e and l e t F be a Banach s p a c e . T.S.
T~
(respectively
T.S.
T
w
(respectiuelyr
0 ,b Proposition 3.35
Then H ( U ; F ) i s
it i s
) c o m p l e t e i f and o n l y i f
Sb
) comptete.
L e t U be a b a l a n c e d o p e n s u b s e t of a
ZocaZZy c o n v e x s p a c e and l e t F be a Banach s p a c e . i s T.S.r complete t h e n r = ' c 6 on H ( U ; F ) . w
Proof
Let
W, b We s h o w t h a t r w a n d
(fa)aEr denote a T
T~
d e f i n e t h e same b o u n d e d s e t s .
bounded s u b s e t o f H(U;F).
homogeneous p o l y n o m i a l s form a n 4 - a b s o l u t e ( H ( u ; F ) , ~ ~ ) if to l l o w s t h a t E m s;p n=o c o n t i n u o u s semi-norm p on H ( U i F ) . complete it follows
n!
<
m
for every
<
i s T.S.
Sincz!H(U;F) E
Since the
decomposition f o r
p(anfa(o))
t h a t E m dnfC'-n(o) n=o n!
sequence (a ) in r. m n ; Hence q (dnfan(o))
If H ( U ; F )
H(U;F)
T
w
T~
f o r any
f o r e v e r y -r6 c o n t i n u o u s s e m i - n o r m q
and any sequence ( a n ) n i n r . I t i s now e a s y t o s e e -n SLIP q ( d f a ( o ) ) < m f o r a n y T & c o n t i n u o u s s e m i - n o r m t h a t Cm n=o n! q and t h i s completes t h e p r o o f .
on H(U;F)
Our n e x t r e s u l t s h o w s t h e c o n n e c t i o n b e t w e e n T . S . completeness and completeness. Proposition 3.36
Let U be a balanced
o p e n s u b s e t of a l o c a l l y
c o n v e x s p a c e E and l e t F b e a Banach s p a c e . c o n v e x t o p o l o g y o n H (U ;F) s u c h t h a t { p ( n E ; F ) d e c o m p o s i t i c n for H ( U ; F ) (H(U;F)
(e.g.
Let
, T I is
T
be a l o c a l l y an
4- a b s o l u t e
T = T ~ , T ~ , ~ , T ~ o , r T T~~ ,) ~.
Then
,T ) i s c o m p l e t e f r e s p e c t i v e l y q u a s i - c o m p l e t e , (p(nE;F)
,TI
is complete ( r e s p e c t i v e l y quasi-complete, s e q u e n t i a l l y complete) f o r e v e r y n and H ( U ; F ) i s T . S . T c o m p l e t e . s e q u e n t i a l l y c o m p l e t e ) i f and o n l y i f
130
Chapter 3
Proof
The c o n d i t i o n s a r e o b v i o u s l y n e c e s s a r y .
are sufficient.
cases a r e handled i n a s i m i l a r fashion.
f
Cauchy n e t i n ( H ( U ; F ) , T ) . Then
Let
--
-
).er
n!
H(U;F).
n! f o r each n .
We may s u p p o s e p ( f )
f o r a l l a,,
zk
I-ience
n=o
Let p be a
~ r a,
ao,
6
-
Pn)
p[d"fa(o) n!
T
Pn
E
@(nE;F) a s
c o n t i n u o u s s e m i - n o r m on p(
=
(fa)aEr be a i s a Cauchy n e t i n
;i"fa(o)
( @ ( " E ; F ) , ~ )f o r e a c h n a n d h e n c e a n f a ( o ) a
We p r o v e t h e y
We c o n s i d e r o n l y t h e c o m p l e t e c a s e , t h e o t h e r
anfo) n! for
every
ao.
5
for a l l a
E
a
and e v e r y
In particular
positive integer k. m
anf
n=o
(O)
n!
1
+
F
f o r a l l k and s o
zm p ( p n ) < f o r e v e r y T c o n t i n u o u s semi-norm p . Since n=o m E H(U;F). i s T.S..r c o m p l e t e t h i s i m p l i e s t h a t f P n H(U;F) -n The a b o v e a l s o shows t h a t p ( d f a ( o ) - Pn) 5 for all n! T h i s c o m p letes a n d h e n c e f a f a s a m . a 1. a the proof.
- -
=AFo
izo
O u r a i m now i s t o show t h a t
( H ( U ) , Tw ) i s c o m p l e t e when-
e v e r U i s a b a l a n c e d open s u b s e t o f a m e t r i z a b l e l o c a l l y convex s p a c e .
S i n c e ( H ( U ) , T ~ )i s c o m p l e t e f o r a n y o p e n s u b s e t
U o f a m e t r i z a b l e l o c a l l y convex s p a c e p r o p o s i t i o n
i m p l i e s t h a t H(U) i s T.S..co b a l a n c e d open s e t U . show t h a t
( @ ( n E ),
T ~ )i
3.36
a n d h e n c e T . S . T ~c o m p l e t e f o r a n y
H e n c e t o p r o v e t h i s r e s u l t we must s c o m p l e t e for any p o s i t i v e i n t e g e r n .
F i r s t we n e e d some p r e l i m i n a r y r e s u l t s w h i c h a r e a l s o o f independent i n t e r e s t . Proposition 3 . 3 7
Let
u
b e a n o p e n s u b s e t of a ZocaZZy
convecc k - s p a c e . T h e n ( H ( U ) , r o ) is a semi-Monte2 s p a c e ( i . e . t h e r,-bounded s u b s e t s of H(U) a r e r e Z a t i v e Z y c o m p a c t ) .
131
Holomorphic functions on balanced sets Proof
Let & ( U )
d e n o t e t h e c o n t i n u o u s complex v a l u e d
f u n c t i o n s on U endowed w i t h t h e c o m p a c t o p e n t o p o l o g y . B be a s u b s e t o f H(U).
Now ( H ( U )
, T ~ )i
Let
s a closed subspace o f
and h e n c e B i s a c l o s e d bounded ( r e s p e c t i v e l y compact)
,&(U)
subset of
i f and o n l y i f i t i s a c l o s e d bounded
(H(U),.ro)
( r e s p e c t i v e l y compact) s u b s e t o f , & ( U ) . e x p a n s i o n s we s e e t h a t a n y
T
0
By u s i n g T a y l o r s e r i e s
b o u n d e d s u b s e t o f H(U) i s e q u i -
c o n t i n u o u s on t h e c o m p a c t s u b s e t s o f U a n d h e n c e a n a p p l i c a t i o n o f Ascoli’s t h e o r e m c o m p l e t e s t h e p r o o f . Corollary 3.38 (H(U)
, T ~ )i
Proof
I f U i s an o p e n s u b s e t o f a 2 3 h Z s p a c e t h e n
s a Frgchet-Monte1 s p a c e .
A a j r s p a c e i s a k - s p a c e a n d h e n c e (H(U) Example 2 . 4 7 shows t h a t
Montel s p a c e .
(H(U)
,To)
, T ~ )
i s a semi-
i s a Frgchet
s p a c e and t h i s c o m p l e t e s t h e p r o o f . Corollary 3.39
If
u
i s an open s u b s e t o f a m e t r i z a b l e
Z o c a Z l y c o n v e x s p a c e t h e n ( H ( U ) , - r o ) i s a semi-Monte1 s p a c e . The a b o v e r e s u l t s a n d s i m i l a r M o n t e l t y p e t h e o r e m s c o u l d a l s o b e p r o v e d by u s i n g S c h a u d e r d e c o m p o s i t i o n s .
Some o f
t h e s e a r e t o be found i n t h e e x e r c i s e s a t t h e end o f t h i s chapter. We now n e e d a l i n e a r r e s u l t w h i c h w i l l a l s o b e u s e f u l i n chapter 6. P r o p o s i t i o n 3.40
Let
T
~
T, *
and
T~
be t h r e e Hausdorff
l o c a l l y convex t o p o l o g i e s on a v e c t o r space E such t h a t (a) (b)
L
2 T ~ ; i s a b o r n o l o g i c a l DF s p a c e f o r e q u i v a l e n t l y a c o u n t a b l e i n d u c t i v e l i m i t of normed l i n e a r s p a c e s ) w i t h a countable fundamental s y s t e m o f c l o s e d convex balanced bounded s e t s ( B n ) n ; (c)
‘cl
T~
(E,rl)
( E , r 2 ) i s a b a r r e l l e d locaZZy c o n v e x s p a c e ;
B n is ‘ c 3 compact f o r all n . (d) T h e n T~ = T 2 * Proof A fundamental system o f neighbourhoods o f zero i n ( E , r l ) i s g i v e n by s e t s o f t h e form Cw
n =1
XnBn
= {Em
n=l
X
x
*
n n’
x
n
E
B
n
and m a r b i t r a r y ) where h
n
Chapter 3
132 is positive for all n EN.
5
ill
Let V =
denote the algebraic closure of V i n E,
%
V =
{ X EE ;
X > o and l e t
AnBn,
i.e.
S i n c e Bn
Ax E V f o r o 5 X < 1 ) .
i s a compact s u b -
AnBn i s a l s o a c o m p a c t s u b s e t s e t o f (E,-r3) i t f o l l o w s t h a t Z k n =1 o f ( E , T ) and h e n c e a c l o s e d s u b s e t o f (E,.r2) f o r e v e r y
3 positive integer k. %
such t h a t x integer
and hence x
AV
f!
k.
#
Now l e t x
F o r e a c h k c h o o s e $,
I
k and +k(&l AnBn) 5 1. S i n c e of ( E , r 2 ) it fcllows t h a t {$k)k
1
'L
Then t h e r e e x i s t s X > 1 k XnBn f o r every
V.
,d
k
E(E,T~)' such t h a t $k(x) m
A
=
i s an a b s o r b i n g s u b s e t
XnBn
i s a p o i n t w i s e bounded and
hence a r e l a t i v e l y weakly ccmpact s u b s e t of
(E,r2)'.
+
If
i s a l i m i t o f a weakly convergent subnet of t h e sequence {+k}k = and so t h e n $ ( x ) = A and ] $ ( V ) l 5 1. Hence
vT2 C ( ~ + EV )f o r a b s o r b i n g and
vT2
every
neighbourhood.
Since
0 .
vT2 i s
convex balanced
c l o s e d it i s a neighbourhood o f zero i n
T~
and s o every
(E,r2)
>
E
5
T~
neighbourhood o f zero c o n t a i n s a
T h i s shows t h a t
~
-
and c o m p l e t e s t h e
=
T~
T
proof. Propositicn 3.41 L e t E be a r n e t r i z a b z e L o c a l l y c o n v e x s p a c e ( i . e . Tw clnd l e t n b e a p o s i t i v e i n t e g e r . On B(nE) , T =~ T o,t i s t h e b a r r e l l e d topoZogy a s s c c i a t e d w i t h T ~ ) . ( @ ( n E ) , ~ w i) s a b o r n o l o g i c a l D F s p a c e w i t h Proof fundamental system o f bounded sets
Bm = {P
8
E
(nE);
IIP
lIvm -<
1 ) w h e r e Vm r a n g e s o v e r a
fundamental neighbourhood system of zero i n E c o n s i s t i n g of c l o s e d convex balanced s e t s . compact s u b s e t o f topology
T
'
u - T O, t L i n proposition 3.40. T
01
=
T
By C o r o l l a r y 3 . 3 8 B m i s a
( P ( n E ) , ~ o ) . Since T
~
.
L e t T~
=
T~
T
~
is a barrelled T,
~
= , T~
~
a n d -c3 =
T
0
Since a l l the requirements are s a t i s f i e d
and t h i s conpletes t h e p r o o f .
O , t
Corollary 3.42
If E i s a m e t r i z a b l e l o c a l z y c o n v e x s p a c e t h e n ( @ l n E ) , Tw ) i s a compZete l o c a l l y c o n v e z s p a c e f o r e v e r y nor1 n e g a t i v g i n t s y e r n . Proof
S i n c e (@("E) , T o )
t h a t @(nE)
i s c o n : p ~ e t ep r o p o s i t i o n 3 . 5 i m p l i e s
endowed w i t h t h e a s s o c i a t e d b a r r e l l e d t o p o l o g y i s
a l s o complete.
The a s s o c i a t e d b a r r e l l e d t o p o l o g y i s
p r o p o s i t i o n 3 . 4 1 and t h i s completes t h e p r o o f .
T
by
133
Holomorphic functions on balanced sets As a f u r t h e r c o r o l l a r y we o b t a i n a g e n e r a l i z a t i o n t o
h o m o g e n e o u s p o l y n o m i a l s o f t h e w e l l known l i n e a r c h a r a c t e r ization of distinguished metrizable spaces. Corollary 3.43
If E i s a m o t r i z a b l e l o c a l l y c o n v e x s p a c e t h e n
t h e following a r e e q u i v a l e n t ; (a) ( p ( " E ) p 0 ) ( r a e s p e c t i v e l y (6'("E) , @ ) ) i s a b a r r e l l e d l o c a l l y convex space, (b)
(@("E) , T ~ ) ( r e s p e c t i v e l y ( 8 ( n E ) , @ ) ) is a b o r n o l o g i c a l ZocaZ 7y c o n v e x s p a c e , (@("E), T o )
(c)
( r e s p e c t i v e l y ( Q ( ~ E, )B ) ) is an u l t r a -
bornological l o c a l l y c o n v e x s p a c e . Proof
I C suffices to notice that
on @ ( n E ) a n d , s i n c e T
= Bb
= 6,
= 'ub
'c0
2 6 5
T
~
,
T = T o,b = 'o,t we ~ , a l s o have
'
=
o,ub
*
I f U is a b a i a n c e d o p e n s u b s e t o f a m e t r i z -
-C -orollary 3.44
a b l e Z o c a l l g c o n v e x s p a c e t h e z (ti(u),'cw) and ( H ( U ) , r ) a r e 6 b o t h c o m p i e t e ZocaZZy c o a v e x s p a c e s . Proof
I t s u f f i c e s t o a p p l y p r o p o s i t i o n 3 . 3 6 and c o r o l l a r y
3.42.
i s coniplete
I f U i s a b a l a n c e d open s e t and ( H ( U ) , ' c w )
[H(U),'c ) i s a l s c c o m p l e t e , we may p r o v e t h i s i n t h r e e 6 d i f f e r e n t ways;
then
since
T
i s t h e b a r r e l l e d topology associated with
6
'c
w
( t h e o r e m 1 . 2 6 2 we may a p p l y p r o p o s i t i o n 3 . 5 w h i c h s a y s t h a t t h e t a r r e l l e d topology associated with a complete l o c a l l y convex t o p o l o g y i s a l s o c o m p l e t e , if
(H(U),r
w
)
i s complete then H(U)
( p r o p o s i t i o n 3 . 3 6 ) and h e n c e s i n c e , T.S.
T~
complete.
Since
T~
and
T
w
i s T.S. 'c6
2
'cw
complete
T ~ , ,H ( U )
agree on@("E)
is
for all
n we may t h e n a p p l y p r o p o s i t i o n 3 . 3 6 t o c o m p l e t e t h e proof, if
(H(U),.rw)
i s complete then H ( U )
and h e n c e a l s o T.S.-c
i s T.S.rw c o m p l e t e
c o m p l e t e ( c o r o l l a r y 3 . 3 4 ) and an
w,b a p p l i c a t i o n o f proposition 3 . 3 6 completes t h e proof.
I f U i s an open s u b s e t o f a q u a s i - n o r m a b l e m e t r i z a b l e
Chapter 3
134
s p a c e t h e n i t i s k n o wn t h a t ( H ( U ) , - c U )
is complete.
However
t h e g e n e r a l problem f o r open sets i n m e t r i z a b l e spaces i s s t i l l open.
We r e t u r n t o t h i s q u e s t i o n i n c h a p t e r 6 .
R e s u l t s s i m i l a r t o t h e a b o v e may a l s o b e p r o v e d f o r h o l o m o r p h i c germs b y u s i n g t h e same t e c h n i q u e s .
In t h i s
manner w e o b t a i n t h e f o l l o w i n g r e s u l t s . Proposition 3.45
L e t K be a compact b a l a n c e d s u b s e t of a
l o c a l l y c o n v e x s p a c e E and l e t F b e a Banach s p a c e .
Then
H(K;F)
is a c o m p l e t e ( r e s p e c t i v e l y q u a s i - c o m p l e t e , s e q u e n t i a l l y n c o m p l e t e ) ZGCaZZy c o n v e x s p a c e i f and onlyv if ($( E ; F ) , r U ) is comp 7 e t e l r e s p e c t i v e 2 y q u a s i -camp t e t e , s e q u e n t i a l Z y comp 1 e t e 1 f o r a l l n and for any s e q u e n c ~o f homogeneous p o l y n o m i a t s m ( P ~ ) ~ p=n ~E , P ( ~ E ; F ) ,n z="o p ( p n ) < m f o r e a c h c o n t i n u o u s s e m i n 0 y . m p on H ( K ; F ) i m p Z i e s Zm n = o 'n
E
H(K;F).
Corollary 3.46 I f K i s a compact b a l a n c e d s u b s e t of a m e t r i z a b l e l o c a l l y convex space E t h e n H ( K ) i s a complete l o c a l l y convex space. B Y c o r o l l a r y 3 . 4 2 ( ~ ( " E ) , T ~ i)s c o m p l e t e f o r a l l n . Proof If
m
(Pn)n=o i s a s e q u e n c e o f homogeneous p o l y n o m i a l s , m
Pn E ? l ( n E ) , a n d =;, p(Pn) < m f o r each continuous semi-norr Since p o n H ( K ) t h e n CPn);=o i s a bounded sequence i n H ( K ) . H(K)
is a regular inductive l i m i t
( p r o p o s i t i o n 2.55)
there
e x i s t s a n e i g h b o u r h o o d V o f K and X > 1 s u c h t h a t m 1 S,UP l l P n l l A V = M m * hence IIPr!IIV 5. M p < co a n d s o Zm Pn E H ( K ) . An a p p l i c a t i o n o f p r o p o s i t i o n 3 . 3 6 now C O m F l e t e s n=o the proof.
iZC
hZC
I n t h e a b o v e c o r o l l a r y we u s e d t h e r e g u l a r i t y o f H ( K ) w h i c h was p r o v e d f o r a r b i t r a r y c o m p a c t s u b s e t s o f a m e t r i z a b l e space i n chapter 2.
T h i s r e s u l t can a l s o b e proved independ-
e n t l y f o r b a l a n c e d c o m p a c t s e t s by u s i n g S c h a u d e r decomposi t i o n s ; s p e c i f i c a l l y o n e u s e s t h e semi-norms
where
( x ~ r )a n~g e s o v e r a l l s e q u e n c e s w h i c h t e n d t o K ,
135
Holomotphic functions on balanced sets
I n c h a p t e r 5 , which d e a l s w i t h holomorphic f u n c t i o n s o n n u c l e a r s p a c e s , we s h a l l s e e t h a t r e g u l a r i t y a n d c o m p l e t e ness o f s p a c e s o f germs a r e e q u i v a l e n t i n a number o f nonYe g i v e h e r e a n e x a m p l e o f a s p a c e o f
trivial situaticns.
germs which i s n o t r e g u l a r .
L a t e r r e s u l t s w i l l show t h a t i t
i s a l s o n o t complete. Exaniple 3 . 4 7
cn
we i d e n t i f y
a").
Let E =
For each p o s i t i v e i n t e g e r n
with t h e subspace o f
spanned by t h e f i r s t
g")
Let 0 d e n o t e t h e o r i g i n i n
n coordinates.
b e a c o n t i n u o u s semi-norm on H ( 0 ) .
@(')
and l e t p
Ye c l a i m t h e r e e x i s t s a n
C n = o f o r some H(0) and f l u neighbourhood U o f z e r o i n Cn t h e n p ( f ) = 0 . We may s u p p o s e i n t e g e r n such t h a t i f f
E
:Io
by theorem 3.25 t h a t p ( f ) = Since p
I p(kE)
is
T~
and hence
p ( m ) f o r every f i n H(0).
n.
T~
continuous
(E i s a
3zw
he c a n f i n d f o r e v e r y i n t e g e r k a n o t h e r p o s i t i v e k i n t e g e r k ' s u c h t h a t i f P E 6(E ) a n d P I tk ' = 0 t h e n p ( P ) = H e n c e i f o u r c l a i m i s n o t s a t i s f i e d t h e n we c a n f i n d a
space)
sequence o f homogeneous p o l y n o m i a l s , P
=
0
and p(P.)
#
0.
Let Q
j l,j 1 i s a T o b o u n d e d s u b s e t o f H(
j
a:
m
( P j ) j = l , such t h a t
jP
2
=
p(pj)
for all j .
{Q.)" 3 j=1
s i n c e e v e r y compact s u b s e t
CC ( N ) i s c o n t a i n e d a n d c o m p a c t i n C n f o r s o m e p o s i t i v e integer n. H e n c e , s i n c e T~ = T w on H( C(N)) ( e x a m p l e 2 . 4 7 ) . ,
of
m
IQjljZl i s a boupded s u b s e t o f H(0).
But p ( Q . ) = j 3 claim.
for all j
and t h i s c o n t r a d i c t i o n proves o u r X
n For each i n t e g e r n l e t f , ( ( ~ ~ ) = ~ i) q q -
.
Since fn = o i n
f o r a l l n t h e a b o v e shows t h a t I f 1 i s a bounded Q: n n s u b s e t o f H ( @ ) . I f H(0) was a r e g u l a r i n d u c t i v e l i m i t t h e n t h e r e w o u l d e x i s t a n e i g h b o u r h o o d W o f z e r o i n CE ('1 on
H(O
w h i c h e a c h f n was d e f i n e d a n d b o u n d e d .
1
(--,
o
. . .
.,o,
)
1
we c o n c l u d e t h a t H(O =
v3 0 V open
W f o r all n. (i:
( H m ( V ) , 11
By
OUT
construction
Since t h i s i s impossible
(N))
[Iv)
is not a regular inductive l i m i t .
0 .
136
Chapter 3 We now c o n s i d e r
a r b i t r a r y E)
linear
and on H ( E )
f u n c t i o n a l s on HN(E) ( f o r
f o r E a m e t r i z a b l e l o c a l l y convex
We t h e n c o m b i n e t h e s e
space with t h e approximation property. r e s u l t s t o show t h a t
T
= T
o n H ( E ) when E i s a F r g c h e t
W
n u c l e a r s p a c e a n d ( H ( E ) , T ~ ) '2~ H(OEI@)a l s o u n d e r t h e s a m e conditions.
These r e s u l t s g e n e r a l i s e r e s u l t s a l r e a d y proved
f o r homogeneous p o l y n o m i a l s
i n chapter 1 (proposition 1.61).
The p r o o f s u s e S c h a u d e r d e c o m p o s i t i o n s and e s t i m a t e s p r e v i o u s l y obtained i n proving t h e corresponding r e s u l t polynomials.
f o r homogeneous
The r e s u l t s p r e s e n t e d h e r e a r e r e l a t i v e l y r e c e n t
as these topics are currently the object of research.
We
d i s c u s s h e r e two d i f f e r e n t s i t u a t i o n s a n d i t i s p r o b a b l e t h a t a g e n e r a l t h e o r y which c o v e r s b o t h s i t u a t i o n s s i m u l t a n e o u s l y w i l l appear i n the not too d i s t a n t f u t u r e .
A similar theory
f o r balanced open s e t s has n o t y e t been developed. Definition 3.48
L e t E b e a ZocaZZy c o n v e x s p a c e .
If V i s a
c o n v e x b a l a n c e d o p e n s u b s e t of E we d e f i n e m
HN(V) = c f
E
H(V); inf(o)CfN("E)
f o r e a c h n and
( H i ( V ) , a ) is a Banach s p a c e . V
HN(E), t h e nuclearly e n t i r e
f u n c t i o n s on E, is d e f i n e d a s { f E H ( E ) ; d n f ( o ) & N ( n E )
all n
and for e a c h compact s u b s e t K of E t h e r e e x i s t s an o p e n s u b s e t V of
E,
KCV,
such t h a t a V f f ) <
m),
The r o t o p o Z o g y on H N ( E ) is g e n e r a t e d by a l l semi-norms
d"f
m
T K ( f )
=
n=o
T
T
~
(
( 0 )
n!
)
a s K r a n g e s o v e r t h e compact s u b s e t s of E . A semi-norm p on H ( E ) is s a i d t o be n u c o n t i n u o u s if t h e r e N
e x i s t s a compact s u b s e t K o f E s u c h t h a t for e v e r y o p e n s u b s e t V of E c o n t a i n i n g
p(f)
K there e x i s t s c(V)>o such t h a t
2 c ( V ) n v ( f ) for e v e r y
We l e t H ~ ( o )=
5
O E V . oDen
f i n HN(E).
( H ~ ( v ) , T ~and ) c
a t h~ i s ~t h e s p a c e of
conirejt balanced
n u c l e a r h o l o m o r p h i c germs a t 0 . I t i s immediate t h a t
IT
0
<
-
T
w
on HN(E) a n d H N ( 0 )
i s an
137
Holomorphic functions on balanced sets u l t r a b o r n o l o g i c a l l o c a l l y convex space. Proposition 3.49 {
L e t E be a ZocaZZy c o n v e x s p a c e
m
(? N ( n E ) , 1 ~ ~ i ls ~a n = . 8 -~d e c o m p o s i t i o n and an A - a b s o 2 u t e
decomposition f o r (HN(E)
, T ~ )
a n J - d e c o m p o s i t i o n and a n
and H N ( 0 ) .
4 -absoZute
{
6 N(
n
E)
,
T
m ~
i~ s =
}
~
decomposition f o r
(HN(E) ,no)
We l e a v e t h e p r o o f o f t h i s p r o p o s i t i o n t o t h e r e a d e r . I t p r o c e e d s i n a s i m i l a r manner t o t h a t u s e d f o r t h e t o p o l o g i e s on H ( U ; F )
T
and
0
(theorem 3 . 1 9 and p r o p o s i t i o n 3.18)
also uses the following equality;
'cW
and
i f B is a subset of E ,
a
i s a p o s i t i v e r e a l number and n i s a p o s i t i v e i n t e g e r t h e n n 71 71 (PI for a l l P E P N ( n ~ ) . aB ( P I = a B P r o p o s i t i o n 3.50
L e t E b e a ZocaZZy c o n v e x s p a c e .
transform, B , i s a Zinear isomorphism from (H ( E ) N
(HN(E)
, T ~ )
onto
) and u n d e r t h i s i s o m o r p h i s m e q u i c o n t i n u o u s s u b s e t s
H ( o ( 1 ~, T o )
of
The Bore2
,T
~
)
correspond t o subsets o f
H(O(E
which are
,To))
d e f i n e d and bounded on some n e i g h b o u r h o o d o f z e r o i n ( E ! , T ~ ) . Proof
f o r (HN(E)
, T ~ )p
r o p o s i t i o n 3 . 1 3 i m p l i e s t h a t {(@,(
is and-absolute
E
decomposition f o r (HN(E) ,v0);i.
( 8 N ( n E ) , ~ o ) 1f o r a l l n . m
BT = ;=o
BT,.
m IT^)^)^=^
Hence a n y .
We d e f i n e BT b y t h e f o r m u 2 a
IT(f)l 5 crK(f) f o r every f i n
( E ) t h e n t h e p r o o f o f p r o p o ~ i t i o n 1 ~ 4s7h o w s t h a t
N f o r a l l n. Since Em (BT,) 5
io i s a n e i g h b o u r h o o d ~ i p== 2~c < this m
of zero i n i m p l i e s BT
n = o '+KO and t h e image by B o f an e q u i c o n t i n u o u s s u b s e t o f i s a bounded s u b s e t of Hm(V) Also,
for
Morever i f K i s a compact s u b s e t
integer n.
o f E and c >o a r e such t h a t
( E ' , T ~ ) .
E)
B Y p r o p o s i t i o n 1 . 4 7 B T ~E P ( " ( E * , T ~ ) )
every non-negative H
n
c a n b e r e p r e s e n t e d a s nE m = o Tn w h e r e
T E (HN(E),so)l Tn
m
n o ) }n=o i s a n A - a b s o l u t e d e c o m p o s i t i o n
Since { (@N(nE),
TKO(BTn) 5 c (El, T E H(0)
~ )a n d
(H (E) ,no) ' N f o r some n e i g h b o u r h o o d V o f 0 i n
s i n c e t h e Bore1 t r a n s f o r m i s an isomorphism
o n e a c h s p a c e o f n - h o m o g e n e o u s n u c l e a r p o l y n o m i a l s , we h a v e shown t h a t B i s a w e l l d e f i n e d i n j e c t i v e l i n e a r m a p p i n g . C o n v e r s e l y l e t A = CgE H
m
(KO);
izo -,[lI
-< 11
where K i s
138
Chapter 3
a compact s u b s e t o f E .
F o r e a c h g i n A t h e r e e x i s t s , by
proposition 1 . 4 7 , a unique sequence o f l i n e a r f u n c t i o n a l s , Tn E ( 8 N ( n E ) , R ~ )I , s u c h t h a t BTn = "n d g(O),,! for a l l (Tnl;=OI
n and lTn(P)
1
2
CT
K
(P ) f o r e v e r y P i n
@N(nE).
If f
E
HN(E)
m
Tn) = g a n d s o t h e B o r e 1 t r a n s f o r m i s a b i j e c t i v e onto H(O ). Since the l i n e a r mapping from ( H ( E ) , n o ) '
Now B(;=,
N
( E 1 , T O )
a b o v e a l s o shows t h a t A i s t h e i m a g e u n d e r B o f a n e q u i continuous subset of (H
N
(E),ao)l
we h a v e c o m p l e t e d t h e p r o o f
L e t E b e a ZocaZZy c o n v e x s p a c e and Z e t V b e a f i n i t e Z y open s u b s e t o f E l c o n t a i n i n g t h e o r i g i n .
We Z e t
I f T i s a ZocaZZy c o n v e x equicontinuous subset A of V l . t o p o l o g y o n E l we l e t H ( 0 ) d e n o t e t h e s p a c e o f germs 5 (E',T) a r i s i n g from t h e usual equivalence r e l a t i o n s h i p i n U H ( V ) , V r a n g i n g o v e r aZZ c o n v e x baZanced n e i g h b o u r h o o d s
v
5
o f zero i n ( E ~ , T ) . Proposition 3.51
L e t E b e a ZocaZZy c o n v e x s p a c e .
The
Bore1 t r a n s f o r m i s a t i n e a r i s o m o r p h i s m f r o m ( H N ( E ) , v u ) ('(E
,TO)
onto
) and u n d e r t h i s i s o m o r p h i s m e q u i c o n t i n u o u s s u b s e t s
of ( H N ( E ) , ~ u ) l c o r r e s p o n d t o s u b s e t s of H 5 ( 0 ( E ' 7 o )) w h i c h a r e d e f i n e d and bounded o n t h e e q u i c o n t i n u o u s s u b s e t s of some neighbourhood o f z e r o i n Proof
( E l
,to).
Use t h e same m e t h o d a s i n t h e p r e c e d i n g p r o p o s i t i o n
and t h e e s t i m a t e s g i v e n i n p r o p o s i t i o n 1 . 4 8 .
Note t h a t a s V
r a n g e s o v e r t h e convex b a l a n c e d open neighbourhoods o f t h e compact s u b s e t K o f E , V o
ranges over t h e closed equicontinuous
s u b s e t s of t h e i n t e r i o r of
KO.
The a b o v e p r o p o s i t i o n s y i e l d a number o f i n t e r e s t i n g c o r o l l a r i e s s i n c e two t o p o l o g i e s o n a l o c a l l y c o n v e x s p a c e a r e e q u a l i f a n d o n l y i f t h e y h a v e t h e same d u a l and d e f i n e t h e same e q u i c o n t i n u o u s s u b s e t s i n t h i s d u a l . Corollary 3.52
I f E i s a Fre'chet MonteZ s p a c e t h e n
139
Holomorphic functions on balanced sets IT
=
IT
on H N ( E ) .
w
If E i s a fully n u c l e a r s p a c e t h e n T = IT = n = T o n H ( E ) if H ( V ) = H(V) and T~ bounded 0 w w 5 s u b s e t s of H(V) a r e l o c a l l y bounded f o r e a c h o p e n s u b s e t V Corollary 3.53
of E A . I f o u r hypotheses are s a t i s f i e d then p r o p o s i t i o n s 3.50
Proof
__.
and 3 . 5 1 imply t h a t
IT
0
=
IT
w
on HN(E).
o f p r o p o s i t i o n 1 . 4 1 we s e e t h a t T~
=
By u s i n g t h e e s t i m a t e on HN(E).
IT
Since
T < T < IT on H ( E ) f o r any l o c a l l y c o n v e x s p a c e E i t f o l l o w s 0 w w N By p r o p o s i t i o n t h a t a l l o f t h e s e t o p o l o g i e s a g r e e on H N ( E ) .
1 . 4 1 HN(E) i s a d e n s e s u b s p a c e o f H(E) a n d h e n c e a l l o f t h e s e t o p o l o g i e s a g r e e on H ( E ) .
This completes t h e proof.
c o r o l l a r y 3 . 5 3 w e r e c o v e r a number o f
As a consequence o f
r e s u l t s previously proved by o t h e r methods; e . g . H(E) w h e r e E i s a a 3 t s p a c e ( e x a m p l e 2 . 4 7 )
and
T
=
0
#
T~
T
w
on
on
T~
H ( E x E ' ) when E i s a n i n f i n i t e d i m e n s i o n a l F r G c h e t n u c l e a r s p a c e B ( e x a m p l e 2 . 4 9 ) a n d a l s o o b t a i n t h e f o l l o w i n g new r e s u l t . Corollary 3.54
on H ( E )
I f E is a F r g c h e t n u c l e a r s p a c e t h e n
-
O
Proof
Since
n
c (6( E) ,
B
;i,i=oi s
T ~ )
H ~ ( o ( , ~T o')
w
1.
a Schauder decomposition f o r
( H ( E ) , T ~ ) ~( c; o r o l l a r y 3 . 1 4 a n d t h e o r e m 3 . 1 9 )
any T
c a n b e r e p r e s e n t e d as nZ = o Tn w h e r e Tn
,To)
m
i t i o n 1 . 6 1 BTn n.
T
The BoreZ t r a n s f o r m is a l i n e a r t o p o l o g i c a l
isomorphism from (H(E),T ) ' onto _c
=
L e t E b e a FrGchet s p a c e w i t h t h e a p p r o x i m a t i o n
Theorem 3 . 5 5
property.
T
E
gN(n(E',To))
E
(@("E)
E
I .
f o r every non-negative
(H(E),T~)' By p r o p o s integer
I f L i s a compact s u b s e t o f E and c > o are s u c h t h a t
I T ( f ) l 5 c llflL f o r e v e r y f i n H ( E ) t h e n , b y p r o p o s i t i o n 1 . 6 0 , t h e r e e x i s t s K compact s u c h t h a t aKO(BTn) 5 c f o r a l l n . S i n c e KO i s a n e i g h b o u r h o o d o f z e r o i n ( E ' , r o ) a n d - 1 Cm nSKO(BTn) 5 c yn = 2 c < t h i s i m p l i e s t h a t BTE H N ( 0 ) n=o and t h e image by B o f a n e q u i c o n t i n u o u s s u b s e t o f ( H ( E ) , T ~ ) ' m
i s a b o u n d e d s u b s e t o f H (V) f o r s o m e n e i g h b o u r h o o d V o f 0 i n N (E' , T o ) . m
C o n v e r s e l y l e t A = {(P,).=.E
m
H N ( ~ ) ; $ " o r K 0 ( P n ) 5. 1 ) w h e r e K
140
Chapter 3
i s a convex b a l a n c e d compact s u b s e t o f E .
If
W
(Pn)n=o F A
then f o r each non-negative i n t e g e r n t h e r e e x i s t s a unique T~
E
( P ( n ~, )T
~ )
s u c h t h a t BTn = Pn a n d
lTn(P)
every P i n Q("E).
Since
it follows t h a t T =
i=o Tn E ( H ( E ) , T ~ ) 'a n d W
I
5 lIPIIK f o r
BTE A.
This a l s o
shows t h a t A i s t h e image o f a n e q u i c o n t i n u o u s s u b s e t o f ( H ( E ), T ~ )
Since
I (@("E) l
~ o )
W
i s an
d -decomposition
for
( H ( U ) , T ~ )p r o p o s i t i o n s 1 . 6 1 a n d 3 . 1 3 i m p l y t h a t m
i s a n d - d e c o m p o s i t i o n f o r ( H ( E ) ,T ) (E T ~ 1) n u ) l n Z o 0 B ' S i n c e (H(E),-co) i s a semi-Monte1 s p a c e i t s s t r o n g dual
(8, ("
Hence,
(H(E),.ro)f; i s a b a r r e l l e d space. I P N ( n ( E ' , T o ) ) ,n
Wn
by p r o p o s i t i o n 3 . 4 9 ,
i s a n d -decomposition
f o r two b a r r e l l e d
t o p o l o g i e s on H (0) and p r o p o s i t i o n 3 . 1 2 s a y s t h a t t h e s e N t o p o l o g i e s must c o i n c i d e . H e n c e ( H ( E ) , T 0 ) 6' 'L H N ( 0 ) a l g e b r a i c a l l y and t o p o l o g i c a l l y . This completes t h e proof.
Lemma 3.56
L e t E be a q u a s i - c o m p l e t e n u c l e a r and dual
nuclear space.
Then H N ( 0 )
= H(0)
( a l g e b r a i c a Z l y and
topo1ogicaZZ.y). Proof:
'Ce a l w a y s h a v e H N ( 0 ) C H ( 0 ) .
By t h e o r e m 1 . 2 7
pN(nE) = 8 ( n E ) f o r e a c h n o n - n e g a t i v e i n t e g e r n and moreover T
= n
on
w
B(nE)
by p r o p o s i t i o n 1 . 4 4 .
Again by p r o p o s i t i o n
1.44 w e can choose f o r every neighbourhood W o f z e r o a n o t h e r neighbourhood V of zero such t h a t
i=o llpn/Iw 2 W
W
"W(Pn)
50
I &o
llpnll".
lience H (0) = H ( 0 ) a l g e b r a i c a l l y and t o p o l o g i c a l l y . Note a l s o N t h a t a s u b s e t o f H N ( 0 ) i s c o n t a i n e d a n d b o u n d e d i n s o m e HZ(V) i f a n d o n l y i f i t i s c o n t a i n e d a n d b o u n d e d i n some Hm(W), V and W b e i n g neighbourhoods
Corollary 3.57
of 0 i n E.
If E i s a F r d c h e t n u c l e a r s p a c e t h e n
( H ( E ) # T o ) i2 H ( o E ! ) * B
141
Holomorphic functions on balanced sets Corollary 3.58 =
T~
'c6
E i s a Fre'chet n u c l e a r s p a c e t h e n
I f
on H(BJ i f and o n l y if
a
v30, V open
(H"(V)
,I1
!Iv) is a r e g u l a r
i n d u c t i v e l i m i t where O E E '
6'
S i n c e E i s a F r g c h e t s p a c e T~
Proof if
(H(E),.ro)
= T~
o n H(E) i f a n d o n l y
i s an i n f r a b a r r e l l e d l o c a l l y convex space.
A
l o c a l l y convex s p a c e i s i n f r a b a r r e l l e d i f and o n l y i f s t r o n g l y bounded s u b s e t s o f t h e d u a l are e q u i c o n t i n u o u s . (H(E),.,)'@
1 H(0
Now
and t h e e q u i c o n t i n u o u s s u b s e t s of
)
'B ( H ( E ) , T ~ ) 'a r e t h e s u b s e t s o f H(0) w h i c h a r e c o n t a i n e d and b o u n d e d i n H"(V)
f o r some n e i g h b o u r h o o d V o f z e r o i n F'
B
So ( H ( E ) , - r o ) i s
i n f r a b a r r e l l e d i f and o n l y i f e a c h bounded
s u b s e t o f H ( 0 ) i s c o n t a i n e d a n d b o u n d e d i n H"(V) neighbourhood V o f zero, ( H m ( V ) , 11
l i m
__f
OEV,
Ilv)
i.e.
f o r some
i f and o n l y i f
is a regular inductive l i m i t .
V open VCE
E,
Example 3 . 5 9
Example 3 . 4 7 shows t h a t H(0)
inductive l i m i t i f 0 that
T
0
#
on H ( Q
T~
N
E
.
is not a regular
By c o r o l l a r y 3 . 5 8 t h i s s h o w s
We h a v e a l r e a d y o b t a i n e d t h i s r e s u l t
),
by a d i f f e r e n t method proposition
C "1
(example 2 . 5 2 ) .
Example 2 . 5 2 and
3 . 5 8 a l s o show t h a t i f E i s a F r G c h e t n u c l e a r
s p a c e which d o e s n o t a d m i t a c o n t i n u o u s norm t h e n H(oEI
B
1
=
V open VCEi B
is not a regular inductive l i m i t . §
3 . 4 SEMI-REFLEXIVITY
A N D N U C L E A R I T Y F O R SPACES O F
H O L O M O R P H I C FUNCTIONS Our f i r s t r e s u l t
f o l l o w s from a g e n e r a l theorem concern-
i n g s e m i - r e f l e x i v e l o c a l l y convex s p a c e s which have e q u i Schauder decompositions. Proposition 3.60
L e t U be a balanced open s u b s e t o f a
l o c a l l y c o n v e x s p a c e E , Z e t F be a Banach s p a c e and l e t
Chapter 3
142 T
'o,b'
semi-refZexive
T
w'
T
w,b'
~
) on
H(U;F).
i f and onZy i f
(P("E;F)
Then ( H ( U ; F ) , r ) i s
, T I is s e m i - r e f Z e x i u e
f o r e a c h n o n - n e g a t i v e i n t e g e r n and H ( U ; F ) is T . S . r - e o m p Z e t e . ( H ( U ; F ) , T ~ )i s semi-
I n p a r t i c u l a r t h i s means t h a t r e f l e x i v e i f and o n l y i f Corollary
(H(U;F),T,,~) i s reflexive.
I f U i s a baZanced open s u b s e t of a
3.61
FrGchet s p a c e and F i s a Banach s p a c e t h e n ( H ( U ; F ) , r ) , where T ~ } , is s e m i - r e f Z e x i v e i f and onZy i f . ( p ( n ~ ; , ~T)) TE{T T 0' w' i s s e m i - r e f Z e x i u e for e a c h n o n - n e g a t i v e i n t e g e r n . iVe now s h o w t h a t
(H(U),T ) i s a n u c l e a r space i f U i s
an a r b i t r a r y open s u b s e t o f a q u a s i - c o m p l e t e
dual nuclear
Since the projective l i m i t of nuclear spaces i s
space.
n u c l e a r and t h e compact open t o p o l o g y i s a l o c a l t o p o l o g y i t s u f f i c e s t o prove t h i s r e s u l t f o r convex balanced open s e t s . This r e s u l t
can e a s i l y b e proved f o r e n t i r e f u n c t i o n s by u s i n g
T a y l o r s e r i e s e x p a n s i o n s a n d t h e f o l l o w i n g lemma w h o s e p r o o f
We l e t
i s already contained i n t h e proof of theorem 1 . 4 3 . s = sup
Note t h a t s i s f i n i t e by S t i r l i n g ' s
1/n'
(n!) formula.
m
Lemma 3 . 6 2
L e t K a n d ( x ~ ) ~be= r~e s p e c t i v e z y a compact
baZanced s u b s e t and a compact s e q u e n c e i n t h e ZocaZZy c o n v e x s p a c e E. L e t L d e n o t e t h e c Z o s e d c o n v e x baZanced huZZ of m
(Sxn)n=l
*
in E' If ( A n ) n = l E: 2 , and II+IIK 5 ; = l I ~ n I I+(x,)l f o r every t h e n f o r e a c h p o s i t i v e i n t e g e r m t h e r e e x i s t a s e q u e n c e in m
m
i1, ( h m , n ) n = l ) m
such t h a t Proof
I1pIIK
m
z J i t h ;=l
c
m
;=l
b m , J= c=;l
m
I A ~ , ~ IP(X;) I I
IjJ)
m
, and
f o r every P
(x",,,"=l E
C
6""~).
By i n d u c t i o n o n e s e e s ( a s i n t h e o r e m 1 . 4 3 )
f o r every L i n
ofLS(mE).
An a p p l i c a t i o n o f t h e p o l a r i z a t i o n
L
143
Holomoiphic functions on balanced sets formula e a s i l y completes t h e proof. A sequence
-
(x ) i n a l o c a l l y convex space i s s a i d t o be n n o as n m f o r every n
r a p i d l y decreasing i f npx positive integer p.
L e t U b e a c o n v e x b a l a n c e d o p e n s u b s e t of a
Lemma 3.63
q u a s i - c o m p l e t e d u a l n u c l e a r s p a c e E and l e t K b e a ( c o n v e x b a l a n c e d ) compact s u b s e t of U . There e x i s t a f i n i t e dimensional subspace F o f E , compact s u b s e t of F , U o an o p e n s u b s e t of F ,
m
KO a
€tlr
m
( x ~ ) a ~ r=a p~i d l y d e c r e a s i n g s e q u e n c e i n E and W a n e i g h b o u r hood of z e r o i n E s u c h t h a t
Let V d e n o t e a convex b a l a n c e d n e i g h b o u r h o o d o f z e r o
Proof
S i n c e any compact s u b s e t o f a
i n E such t h a t K + V C U . quasi-complete
dual n u c l e a r space i s contained i n t h e convex
h u l l o f a r a p i d l y d e c r e a s i n g s e q u e n c e we c a n c h o o s e ( y n I n a r a p i d l y d e c r e a s i n g s e q u e n c e i n E whose c l o s e d c o n v e x h u l l Choose N a p o s i t i v e i n t e g e r s u c h t h a t contains K. 2 4 l o a n y n € V f o r a l l n > N w h e r e a = II s / b . Let F d e n o t e t h e s u b s p a c e o f E s p a n n e d b y { y l , N
3
a n d l e t K O = {;=l nz m =l
l a n ] 5 1 and
i = la n y n
By o u r c o n s t r u c t i o n K Uo
=
KO
Let An
1
7(VT\F).
+
=
1 2a(N+n)'
integer n.
since
("n);=N+1
m
Uo
E
. . .
,yN}
such t h a t
K}.
i s a compact s u b s e t o f F .
Let
i s an open s u b s e t o f F and K o C Uo.
and x
n
2
= 2a(N+n) yN+n f o r each p o s i t i v e
lhnI
5
F1a - g -= l 1n
= -2 a - -I T= ~- 2 s
and
Chapter 3
144
cKo + 14. v CK
+
4
-
+
V +
1 12
n
ii
4
+
1 3
V +
1 V C K 12
t h i s c o m p l e t e s t h e p r o o f i f we l e t W =
+ V C U 1 12 v*
We s h a l l
We now p r o v e t h e m a i n r e s u l t o f t h i s s e c t i o n .
(H(U)
assume t h a t
, T ~ )i
open s u b s e t o f E n .
s a FrGchet n u c l e a r space i f U i s an
T h i s i s a w e l l known f i n i t e d i m e n s i o n a l
r e s u l t a n d i s g i v e n i n a number o f books on f u n c t i o n a l analysis.
I t i s a l s o a s p e c i a l c a s e o f a r e s u l t which w i l l
be proved,
independently o f t h e following r e s u l t ,
in
chapter 5. Theorem 3 . 6 4
L e t U be a quasi-complete
dual nuclear space.
T h e n ( H ( U ) , . r o ) is a n u c l e a r s p a c e if U is an o p e n s u b s e t o f E . Proof
We may s u p p o s e t h a t U i s c o n v e x a n d b a l a n c e d . m
be a compact s u b s e t o f U and l e t K O ,
Let K
m
( x ~ ) a~n d= ~ Uo’(Xn)n,l, W h a v e t h e same m e a n i n g w i t h r e s p e c t t o K and U a s i n lemma 3 . 6 3 . m
(Bn)n=l (H(Uo),
By t h e n u c l e a r i t y o f
€2,
w e can f i n d
an equicontinuous s u b s e t o f
such t h a t
yo)
llfllK
a n d ($n);=l
(H(Uo),ro)
0
2
~ = ~ l @ n ll $ n ( f ) l m
for
every
in
H(Uo)’
On m u l t i p l y i n g e a c h B n b y a c o n s t a n t i f n e c e s s a r y we may s u p p o s e t h a t t h e r e e x i s t s a r e l a t i v e l y c o m p a c t s u b s e t K1
o f Uo,
145
Holornophic functions on balanced sets KO C K 1 , H(Uo)
m
m
=
ll$lk 2
Since
8Z1
'B=~
*
L e t K~
I$n(f) I
such t h a t
(XnI
enxn; m
T1 s
5
1
IXnI
[en] 5
5 $=1 l X n l
f o r a l l n and e v e r y f i n
5 IlfIjK
for all nI.
[$(xn)[ for every 4 i n
lemma 3 . 6 2
E l
and
implies that
f o r e a c h f d e f i n e d a n d h o l o m o r p h i c on a n e i g h b o u r h o o d o f K 2
RE1 m
where i n C;=l K1
of
m
Bnxn;
;Z1
+ K 3 C Uo + K 3
lon( 5 +
u.
Y E
+
W
KJ
If $
i s a n e l e m e n t o f H(K3+W).
f o r a l l x i n Uo and y i n K3
+
+ W then
-
i s an element o f H ( U o ) . ; Y €K3
1 f o r a l l n ) = K3.
W a n d h e n c e K1 If
Now l e t f E H ( U ) .
T(f)
( x m,n ) mm , n = 1 i s c o n t a i n e d
l a m l n ] 5 ( i ) m f o r a l l m and
E
$(fy)
Since
K3
i s a compact s u b s e t
f
d e f i n e d by f ( x ) = f ( x + y ) Y then the function
H'(Uo) E
C
h=o a,
Note t h a t
Y
^n
f(x)(y) = f(x+y) n!
+ W and t h e s e r i e s converges
u n i f o r m l y w i t h r e s p e c t t o x o v e r t h e compact s u b s e t s o f Uo .it follows t h a t
146
Chapter 3
Since
2
zZl m
Sup
lBnl
4
If(x+y)l
x€K1
IcakJml
Cm
k=l
- $
f o r a l l n,m
-1
5
zm
a l l m and
a n d k we h a v e shown
YEK3
IlfII, where of
;I1
I a
,?j=l
1
I$,(f)
6n
6n
m
<
(H(U),ro)'.
m
and ($n)n=l
i s an equicontinuous subset
This completes t h e proof.
Corollary 3.65
If U is an open s u b s e t of a 3 3 Q s p a c e t h e n
( H ( U ) , r o ) is a F r & c h e t nuclear s p a c e .
3.5
EXERCISES
3.66* -
A convex balanced absorbing subset o f a v e c t o r space
is called a b a r r e l ,
be d-barrelled (resp.
A l o c a l l y convex topology
(resp. d-infrabarrelled)
'I
is said t o
if e v e r y b a r r e l
e v e r y b a r r e l which a b s o r b s bounded s e t s ) which i s a
countable i n t e r s e c t i o n of neighbourhoods of zero i s a neighbourhood o f zero. barrelled Let
T~~
(resp.
and
T
di
Show t h a t t h e c o l l e c t i o n o f a l l d -
d-infrabarrelled)
t o p o l o g i e s form a Q - f a m i l y .
d e n o t e t h e d - b a r r e l l e d and d - i n f r a b a r r e l l e d
topologies associated w i t h r respectively.
If U i s a balanced
open s u b s e t o f a l o c a l l y c o n v e x s p a c e E and F i s a Banach s p a c e show t h a t w,dt
T6 = 3.67 -
Let ( E , T )
Suppose T
and
T ~ , ~ = T
w,di
on H(U;F).
be a l o c a l l y convex space.
Letrl
=
T .
h a s b e e n d e f i n e d for a l l a s t r i c t l y l e s s t h a n t h e
o r d i n a l number
ao.
Let
(E,T
)=+* ( E , T ~ ) . ac a
Let
denote
T~ 0
147
Holomoiphic functions on balanced sets t h e t o p o l o g y on E i n h e r i t e d f r o m ( ( E , T a
)IBJ1@.
t h e r e e x i s t s a n o r d i n a l number y s u c h t h a t yo
Show t h a t
2 y.
(E,
)
T
Show t h a t
0 =
T
Y
for all
T
YO
i s an i n f r a b a r r e l l e d l o c a l l y
YO
convex s p a c e and t h a t associated with
3.68
Let
T
T.
(E,T)
YO
i s t h e i n f r a b a r r e l l e d topology
b e a l o c a l l y convex s p a c e which c o n t a i n s
a c o u n t a b l e fundamental s y s t e m o f bounded s e t s . (E,Tdi)
i s a DF space.
3.69"
L e t qh?-.p,
endowed w i t h t h e
1 5 p
2c l
5 q
norm.
Show t h a t
< + m , denote t h e 2 space P Show t h a t qLp i s a b a r r e l l e d s p a c e
i f and o n l y i f p = q .
3.70*
Let
be a Schauder decomposition f o r t h e
l o c a l l y convex s p a c e E and l e t ( p a ) a E r be a fundamental system o f semi-norms o f E .
let^' d e n o t e t h e l o c a l l y
convex t o p o l o g y g e n e r a t e d by t h e semi-norms
q (x) = sup p (u ( x ) ) a s a ranges over r . Show t h a t n a n (E 3 i s an e q u i - S c h a u d e r d e c o m p o s i t i o n f o r ( E , T ' ) a n d t h a t n n T I i s t h e w e a k e s t l o c a l l y c o n v e x t o p o l o g y on E , f i n e r t h a n T,
which c o i n c i d e s w i t h
T
Schauder decomposition.
3.71
If
on e a c h E
n
and h a s {En}n a s an e q u i -
i s a S c h a u d e r d e c o m p o s i t i o n f o r ( E , T ~ )and
( E , T ~ ) show t h a t
and
'cl
T*
h a v e t h e same a s s o c i a t e d
b a r r e l l e d topology. 3.72 -
Let
3.73*
Let
(EJ)
b e a l o c a l l y convex s p a c e .
i s weak* q u a s i - c o m p l e t e
i f and o n l y i f
Show t h a t
E l
(E,T)' = (E,-rt)l.
CEnIn b e a s h r i n k i n g d e c o m p o s i t i o n f o r t h e l o c a l l y
convex s p a c e E .
Show t h a t { ( E n ) l B I n i s a n e q u i - S c h a u d e r
d e c o m p o s i t i o n f o r E'
6
.
Chapter 3
148
n Show t h a t I @ H y ( E;F)
3.74
and an , ? - a b s o l u t e
i s an$ -decomposition
decomposition f o r (HHY(U;F),~o)
whenever U
i s a b a l a n c e d open s u b s e t o f a l o c a l l y convex s p a c e E and F i s a q u a s i - c o m p l e t e l o c a l l y convex s p a c e . 3.75* -
I f U i s an open b a l a n c e d s u b s e t o f a l o c a l l y convex n s p a c e E show t h a t I @ ( n E ) , ~ o , t I Z = oa n d ( 6 ( E ) , T ~I nW, = ~o ~
a r e & d e c omp o s i t i o n s a n d
4 - a b s o 1u t e
d e compo s i t i o n s f o r
(H(U) , T ~ , a~n d) (H(U) , T ~ , r~e s ~p e )c t i v e l y . I f E i s a DF s p a c e show t h a t T
3.76*
0
,t
=
6
=
T
f o r a n y p o s i t i v e i n t e g e r n and h e n c e d e d u c e t h a t
on/P(nE)
w T
0,t
=
T
6
on H(U) f o r a n y b a l a n c e d o p e n s u b s e t U o f E . I f E i s a Frechet
3.77* T
=
T&
s p a c e and F = ( E ' , T
on H ( F ) .
0
)
show t h a t
I f E i s an i n f i n i t e d i m e n s i o n a l r e f l e x i v e Banach s p a c e 1 a n d F = ( E , o ( E , E ' ) ) show t h a t (@( F ) , T ~ ) ' 2 ( E l ) * a n d t h a t
3.78*
' 0
,ub
#
3.79
T~
on H ( F ) .
Show t h a t H(E) i s T . S .
T~
complete b u t not T . S .
T~
c o m p l e t e when E = t N x C " ) .
3.80*
Let T denote t h e topology of pointwise convergence. P I f E a n d F a r e l o c a l l y c o n v e x s p a c e s s u c h t h a t H ( E ) a n d H(F)
are T.S.
T
P
c o m p l e t e and s e p a r a t e l y c o n t i n u o u s p o l y n o m i a l s
o n E x F a r e c o n t i n u o u s show t h a t s e p a r a t e l y c o n t i n u o u s h o l o m o r p h i c f u n c t i o n s on E x F a r e h o l o m o r p h i c i f a n d o n l y i f H(E x F) i s T . S . 3.81 that
T
P
complete.
I f E i s a quasi-complete (H(E),ro)
is Frschet i f
i n f r a b a r r e l l e d s p a c e show
and o n l y i f E is t h e s t r o n g
d u a l o f a F r e c h e t Montel s p a c e . 3.82
I _
I f E i s an i n f r a b a r r e l l e d c o m p l e t e D F s p a c e show t h a t
( H ( E ) , T ~ )i s b a r r e l l e d i f a n d o n l y i f E i s a M o n t e l s p a c e .
149
Holomotphic functions on balanced sets
~
Show t h a t a s e q u e n t i a l l y c o m p l e t e b a r r e l l e d s p a c e w i t h
3.83*
a b a s i s is complete.
3.84*
I f E i s a quasi-complete n u c l e a r and dual n u c l e a r
s p a c e show t h a t t h e c o m p a c t o p e n t o p o l o g y o n H ( E ) g e n e r a t e d by a l l semi-norms
is
o f t h e form
a s K ranges o v e r t h e compact s u b s e t s o f E . Let 0 b e t h e o r i g i n i n t h e Banach s p a c e E a n d l e t B
3.85*
Show t h a t t h e t o p o l o g y on H(0)
be t h e unit b a l l of E.
is
g e n e r a t e d by t h e semi-norms
where
-
(an)n ranges over a l l sequences of
\anllh
s c a l a r s such t h a t
-.
o as n
3.86”
Let 0 b e t h e o r i g i n i n t“),
of H(0)
i s g e n e r a t e d by a l l
show t h a t t h e t o p o l o g y
semi-norms
o f t h e form
m
PK(f) = c n=o where
n n
s u b s e t s o f CE 3.87 -
and i f ,
Un
in C
If
n
-
r a n g e s o v e r a1
(a )
“1
o as n
.
sequences such t h a t and K r a n g e s o v e r t h e compact
( f n I n i s a s e q u e n c e i n H(O),
f o r each n, such t h a t f
0 the origin in
C (N) ,
t h e r e e x i s t s a neighbourhood of zero
I
‘n
= o all m
’> n s h o w t h a t { f n I n
i s a very s t r o n g l y convergent sequence i n H ( 0 ) .
3.88*
Let U b e a b a l a n c e d o p e n s u b s e t o f a q u a s i - c o m p l e t e
l o c a l l y convex s p a c e E .
For each compact s u b s e t K of E
l e t K = { x E E ; I P ( x ) l 5 IIPIIK f o r e v e r y P i n 6 ( E ) I .
150
=Ku
6
Let
Chapter 3
.
I?
6
Show t h a t
i s a b a l a n c e d open s u b s e t o f E .
K compact
If
f
=
zm
n=o
E
n.
H ( u ) show t h a t
-
d"f0 n!
zm
n=o r
c
l
A
a n d d e f i n e s a h o l o m o r p h i c f u n c t i o n f on U . mapping f
(H(U),T~)
E
isomorphism.
I f T& =
w
( t h e boundary o f U i n E)
(H(U),.ro) i s a l i n e a r
E
a n d ,€
t h e r e e x i s t s a n f i n H(U) 3.89" -
f
-
on H(U) show t h a t f o r e a c h €,
'I
A
Show t h a t t h e
A
%
c o n v e r g e s and
E
n
U,
2 n s u c h t h a t f ( € , )n
_ _ _ f
5 as n
L e t E b e a Banach s p a c e w i t h u n i t b a l l B .
P E P ~ ( ~ E l e )t
IIPIk = a B ( P ) .
show t h a t P - Q
E
If P E P ~ ( ~ E a n)d Q
aU m
as n
m
E
m.
If E
@N(mE)
(?N(n+mE) a n d
I f U i s a b a l a n c e d open s u b s e t o f a m e t r i z a b l e
3.90"
l o c a l l y c o n v e x s p a c e show t h a t
3.91*
T
= T
0
,t
o n H(U).
By u s i n g t h e f a c t t h a t t h e r e e x i s t d i s c o n t i n u o u s
p o l y n o m i a l s on CC
I
,
I uncountable,
show t h a t
(H(C
I
) , T ~ )
i s n o t a semi-Monte1 s p a c e . (H(CC N )
a (H(En),~O). n
=*
Show t h a t
3.93
L e t E and F be l o c a l l y convex s p a c e s and l e t U be a
, T ~ )
b a l a n c e d open s u b s e t o f E . TE{T~, T
T.S.
T
~
T , ~ } ,
Show t h a t
(H(U;F),T),
i s semi-Monte1 i f and o n l y i f
(H(U;F),r)
is
c o m p l e t e a n d ( @ ( n E ; F ) , ~ )i s s e m i - M o n t e 1 f o r e a c h
non-negative 3.94
=
integer n.
Let E b e a l o c a l l y convex s p a c e .
Show t h a t
(H(U),.ro)
i s c o m p l e t e f o r e v e r y open s u b s e t U o f E i f and o n l y i f ( P ( n E ) , ~ o )i s complete f o r each non-negative H(V) i s T . S .
T~
subset V of E.
i n t e g e r n and
c o m p l e t e f o r each convex b a l a n c e d open
15 1
Holomorphic functions on balanced sets L e t {En},
3.95*
that
b e a s e q u e n c e o f Banach s p a c e s .
Show
m
(H(iZ1 E , ) , T ~ ) i s a semi-Monte1 s p a c e i f and o n l y i f e a c h
En i s f i n i t e d i m e n s i o n a l .
3.96* ^n
d f(x)
I f E i s a l o c a l l y c o n v e x s p a c e a n d fEHN(E) show t h a t
n
E @ ~ (
E) f o r e v e r y x i n E a n d e v e r y p o s i t i v e i n t e g e r n .
Show, b y c o u n t e r e x a m p l e , t h a t t h e a b o v e c o n d i t i o n o n fsH(E)
i s n o t s u f f i c i e n t t o i n s u r e t h a t it l i e s i n HN(E).
Show a l s o t h a t H ( E ) N
o f H(E).
3.97*
is a translation invariant subalgebra
L e t E b e a l o c a l l y convex s p a c e and F a normed
l i n e a r space.
A f u n c t i o n f s H(E;F)
i s s a i d t o be o f
e x p o n e n t i a l t y p e i f t h e r e e x i s t a c o n t i n u o u s semi-norm a Ilf(x)II 5 C exp ( c a ( x ) )
on E a n d p o s i t i v e n u m b e r s c , C s u c h t h a t f o r every x i n E .
Let
Exp(E;F) d e n o t e t h e s e t of a l l
holomorphic f u n c t i o n s o f e x p o n e n t i a l t y p e from E i n t o F . Show t h a t f = C n=o
d"f0 n!
E
Exy(E;F)
i f and o n l y i f t h e r e
e x i s t s a c o n t i n u o u s semi-norm a on E s u c h t h a t
3.98 f
E
I f E i s a Banach s p a c e and f
Exp(E;C) =
E
H(E) show t h a t
~ x p ( E )i f a n d o n l y i f t h e r e s t r i c t i o n o f f t o
each one dimensional subspace o f E i s a f u n c t i o n o f expone n t i a l type. 3.99 -
I f E i s a l o c a l l y c o n v e x s p a c e show t h a t t h e
mapping f
=
C
n=o
d"f0 n.
E
Exp(E)
m
nC = o d n f ( o )
E
H(OE)
is a linear bijection. Using t h e above, o r o t h e r w i s e , d e s c r i b e a l o c a l l y convex t o p o l o g y on E x p ( E ) s o t h a t t h e a b o v e b i j e c t i o n i s a l i n e a r
152
Chapter 3
t o p o l o g i c a l isomorphism. Let E b e a Banach s p a c e and l e t f and g b e holomorphic i s an e n t i r e I f h = f, f u n c t i o n s o f e x p o n e n t i a l t y p e on E .
3.100
f u n c t i o n on E show t h a t h
E
Exp(E)
g
.
Let E b e a l o c a l l y convex s p a c e .
3.101 H (E) N
An e l e m e n t f o f
i s s a i d t o b e of n u c l e a r e x p o n e n t i a l t y p e i f t h e r e
e x i s t s a convex b a l a n c e d open s u b s e t V of E s u c h t h a t ^n
l i m s u p a,,(d
f(o))
'/n
<
rn,
Let ExpN(E) d e n o t e t h e s p a c e o f a l l holomorphic f u n c t i o n s o f n u c l e a r e x p o n e n t i a l t y p e on E . f
=
C
E
n=o
ExpN(E)
Show t h a t t h e m a p p i n g
-
rn
nC = o i n f ( o )
E
HN(OE)
is a linear bijection 3.102*
Let V and U b e open s u b s e t s o f t h e l o c a l l y convex
spaces E and F r e s p e c t i v e l y .
Let
-b e TI
a continuous linear
mapping from E i n t o F s u c h t h a t a(V) i s a compact s u b s e t o f U.
L e t R;Hrn(U)
__f
Hrn(V) b e d e f i n e d b y R ( f ) = f o n l v .
Show t h a t R i s a c o m p a c t m a p p i n g .
U s i n g t h i s r e s u l t show
t h a t H(K) i s a 8 3 . 2 s p a c e w h e n e v e r K i s a c o m p a c t s u b s e t o f
a FrEchet-Schwartz 3.103"
space.
I f U i s an open s u b s e t o f a l o c a l l y convex s p a c e E
a n d F i s a s e m i - M o n t e 1 s p a c e show t h a t ( H H y ( U ; F ) , T o ) i s a semi-Monte1 s p a c e . 3.104*
that
I f E i s a q u a s i - c o m p l e t e d u a l S c h w a r t z s p a c e show
(H(U),ro)
U of E.
i s a Schwartz s p a c e f o r any open s u b s e t
153
Holomorphic Jitnctions on balanced sets 53.6
NOTES A N D REMARKS The c o n c e p t o f Q - f a m i l y
J . Schmets
[627]
(definition 3.1)
i s due t o
(see also chapter 2 of the lecture notes
o f J . Schmets [628]
and Ph.
Noverraz
[553,556]) and
d e v e l o p e d n a t u r a l l y f r o m t h e r e s u l t s o f e a r l i e r a u t h o r s on particular associated topologies.
Y.
Komura
first t o discuss associated topologies.
[ 3 9 4 ] was t h e
He was i n t e r e s t e d
o n l y i n t h e a s s o c i a t e d b a r r e l l e d topology and proved proposition 3.5.
An a l t e r n a t i v e p r o o f o f t h i s p r o p o s i t i o n
u s i n g t h e axiom o f c h o i c e i n p l a c e o f t r a n s f i n i t e induction i s due t o M. A.
Roberts
topology.
[608]
Kennedy ( L e c t u r e , D u b l i n ,
1979).
The c o r r e s p o n d i n g r e s u l t s f o r t h e i n f r a b a r r e l l e d
t o p o l o g y and f o r t h e q u a s i - c o m p l e t e complete cases ( p r o p o s i t i o n 3 . 6 )
and s e q u e n t i a l l y
are due t o K .
Noureddine
The a s s o c i a t e d u l t r a - b o r n o l o g i c a l t o p o l o g y i s
[532].
s t u d i e d by H .
Buchwalter i n [108],
t o p o l o g y by K .
Noureddine i n
the barrelled-bornological
[533] and t h e o - b a r r e l l e d
a - i n f r a b a r r e l l e d topologies by K .
Noureddine and J .
and
Schmets
General r e s u l t s f o r Q-families are given i n
[535].
J . Schmets [627,628].
i n t r o d u c e d by H . [53411
December
also studies the associated barrelled
Kelley s p a c e s (example 3.4)
Buchwalter [lo71
(see also K.
were
Noureddine
*
S c h a u d e r d e c o m p o s i t i o n s o f Banach s p a c e s were f i r s t d e f i n e d by M . M .
Grinblyum
[283]
(see B.L.
Sanders
and e x t e n d e d t o l i n e a r t o p o l o g i c a l s p a c e s by C.W. and J . R . proof
R e t h e r f o r d [483].
The o n l y r e s u l t w e u s e w i t h o u t
( i n t h e proof of p r o p o s i t i o n 3.60)
B.L.
S a n d e r s [ 6 2 4 ] a n d T.A.
N.J.
Kalton
[370,371]
[624]) McArthur
Cook
[168].
i s due t o We r e f e r t o
for further details.
T h e c o n c e p t s o f a b s o l u t e d e c ompo s i t i o n ,
/Ir - d e c o m p o s i t i o n
a n d 4 - a b s o l u t e d e c o m p o s i t i o n a r e new a n d a r e i n t r o d u c e d h e r e as a s u i t a b l e t e c h n i q u e f o r t r e a t i n g h o l o m o r p h i c f u n c t i o n s on b a l a n c e d open s e t s .
Propositions 3.10,
3.11 and
Chapter 3
154
3 . 1 3 a r e new w h i l e a s t r o n g e r f o r m o f p r o p o s i t i o n b e f o u n d i n Ph. N o v e r r a z
3 . 1 2 may
[553,556].
Schauder decompositions were i n t r o d u c e d i n t o i n f i n i t e d i m e n s i o n a l holomorphy by S . Dineen and a l l t h e 63.2 and a number o f t h o s e i n § 3 . 3 are t o
results of b e found i n
[185].
These r e s u l t s were motivated by e a r l i e r
r e s u l t s c o n c e r n i n g h o l o m o r p h i c f u n c t i o n s on Banach s p a c e s (S.
Dineen
material in
[177], R.
Aron
[17]!.
The a r r a n g e m e n t o f t h e
i s , h o w e v e r , new a n d m o r e c o h e r e n t t h a n t h a t g i v e n
[185]. The a p p l i c a t i o n o f a s s o c i a t e d t o p o l o g i e s , i n
conjunction with
Schauder decompositions,
t o t h e study of
h o l o m o r p h i c f u n c t i o n s on l o c a l l y convex s p a c e s i s due t o Ph.
Noverraz
proposition i n S. [509].
[553,556] where he p r o v e s theorem 3.26 and 3.29.
Dineen
Propositions 3.29,
3.30,
3.31 a r e given
and l e m m a 3.28 i s due t o L .
[185]
I n v i e w o f t h e o r e m 3 . 2 6 we may a s k i f
Nachbin T
=
,t U a b a l a n c e d open s u b s e t o f a l o c a l l y convex s p a c e . Proposition 3.41,
e x e r c i s e 3.76and
example o f a non-complete To
and S.
Ponte
,t
#
T~
To,ub
= T~
Komura's
Recently J . M .
[395]
Ansemil
+
[ l o ] h a v e s h o w n t h a t -c0
weak t o p o l o g y ,
on H ( U ) ,
Monte1 s p a c e shows t h a t
even on E ' .
E an i n f i n i t e d i m e n s i o n a l
T~
c o r o l l a r y 5.26 a l l give
a p o s i t i v e answer f o r s p e c i a l cases but Y . we may h a v e
0
Tg on P ( ' E ) , 3 ub r e f l e x i v e Banach s p a c e w i t h t h e
a n d h e n c e we d o n o t ,
i n g e n e r a l , have
o n H(U).
The c o m p l e t e n e s s o f
(H(U;F),r
w
) has been i n v e s t i g a t e d
b y many a u t h o r s a n d t h e r e s u l t p r e s e n t e d h e r e 3 . 4 4 ) may b e f o u n d i n S .
Dineen
[ZOO].
i n a s e r i e s o f r e s u l t s which a p p e a r i n S . R.
Aron [ 1 7 ] ,
P.
Aviles
S.B.
Chae
[120],
and 3. Mujica [ 4 1 ] .
(corollary
I t is the latest Dineen
[177,185],
J . M u j i c a [SO31 a n d Aspects o f t h e completeness
question w i l l a r i s e i n each o f t h e remaining c h a p t e r s . Taylor
s e r i e s c o m p l e t e n e s s was i n t r o d u c e d b y S . D i n e e n [ 1 8 5 ] .
155
Holomoiphic functions on balanced sets Lemma 3 . 3 3 i s new a n d a g e n e r a l r e s u l t o f t h e s a m e k i n d f o r $-absolute
d e c o m p o s i t i o n s c a n e a s i l y b e s t a t e d and p r o v e d .
C o r o l l a r y 3.34
and p r o p o s i t i o n 3 . 3 5 a r e new.
i s given i n S.
Dineen
P r o p o s i t i o n 3.36
[ 1 8 5 ] w h e r e o n e may a l s o f i n d a
'I
w
analogue of proposition 3.36. The c l a s s i c a l Montel t h e o r e m s a y s t h a t c l o s e d
( H ( U ) , T ~ ) ( U a n o p e n s u b s e t o f Cn) a r e
bounded s u b s e t s o f compact. result
A number o f d i f f e r e n t g e n e r a l i z a t i o n s o f t h i s
(known c o l l e c t i v e l y a s M o n t e l t h e o r e m s )
holomorphic
for
f u n c t i o n s o f i n f i n i t e l y many v a r i a b l e s h a v e
appeared i n t h e l i t e r a t u r e .
The v a r i e t y o f r e s u l t s a r e
o b t a i n e d by v a r y i n g t h e u n d e r l y i n g l o c a l l y convex s p a c e s , t h e c o n c e p t o f d i f f e r e n t i a b i l i t y a n d t h e t o p o l o g y on t h e corresponding space o f holomorphic
functions.
Most o f t h e
p r o o f s r e q u i r e an a p p l i c a t i o n of A s c o l i ' s theorem.
f i r s t r e s u l t o f t h i s k i n d i s due t o D. $3,8spaces
Pisanelli
[571] f o r
and t h i s i s a p a r t i c u l a r c a s e o f c o r o l l a r y 3.38.
F u r t h e r Montel theorems are t o b e found i n D. [576,578,582], [149]
The
D.
Lazet
[423], J . F .
Pisanelli
Colombeau a n d D .
Lazet
( t h i s a r t i c l e c o n t a i n s p r o p o s i t i o n 3 . 3 7 and
c o r o l l a r i e s 3.38 and 3 . 3 9 ) , S . Dineen
[185,194]
J.F.
Colombeau
[141] and
( s e e a l s o e x e r c i s e s 2.84
and 3 . 1 0 3 ) .
A
number o f t h e above a u t h o r s a l s o p r o v e i n f i n i t e dimensional versions of t h e c l a s s i c a l V i t a l i theorem. [462] d i s c u s s e s l o c a l l y convex spaces which s a t i s f y
Elatos
M.C.
a "Montel" p r o p e r t y and shows t h a t t h e y a r e r e l a t e d t o l o c a l l y convex s p a c e s which s a t i s f y t h e c o n c l u s i o n o f Z o r n ' s theorem (theorem 2 . 2 8 ) .
of
Ascoli s t y l e characterizations
compact s e t s a r e due t o L .
'I
w
R.
[120],
Aron [ 1 7 ] ,
M.C.
Nachbin
Matos [461]
[SO91 ,S.B. C h a e
and J . A .
Barroso
[47,48]. Propositions 3.40, 3.43,
3.44,
3.41,
3.45 and c o r o l l a r i e s 3 . 4 2 ,
and 3.46 a r e due t o S.
Dineen
[ZOO].
3 . 4 7 i s new a n d r e l a t e d t o a n e x a m p l e o f R . i n R. L.
Soraggi
[664]
.
Example
Aron g i v e n
156
Chapter 3 A n a l y t i c f u n c t i o n a l s on l o c a l l y c o n v e x s p a c e s
a r e u s u a l l y r e p r e s e n t e d e i t h e r as f u n c t i o n s o f e x p o n e n t i a l t y p e o r as holomorphic germs a t t h e o r i g i n . natural linear topological
There i s a
(but unfortunately not algebraic)
isomorphism between t h e s e r e p r e s e n t a t i o n s ( s e e e x e r c i s e s 3.93,
3 . 9 4 and 3 . 9 5 ) . The e x p o n e n t i a l t y p e r e p r e s e n t a t i o n i s u s e f u l i n
s t u d y i n g c o n v o l u t i o n o p e r a t o r s ( s e e a p p e n d i x I ) w h i l e we h a v e f o u n d t h e g e r m a p p r o a c h u s e f u l when i n v e s t i g a t i n g topological p r o p e r t i e s of holomorphic f u n c t i o n s .
Since the
r e s u l t s h e r e on a n a l y t i c f u n c t i o n a l s were o r i g i n a l l y p r o v e d u s i n g T a y l o r s e r i e s e x p a n s i o n s a b o u t t h e o r i g i n we a r e e s s e n t i a l l y u s i n g t h e o r i g i n a l method.
Sometimes
however t h e S c h a u d e r d e c o m p o s i t i o n a p p r o a c h c a n b e more e f f i c i e n t - as i n theorem 3.55. Holomorphic f u n c t i o n s o f n u c l e a r t y p e ( d e f i n i t i o n 3 . 4 8 ) were i n t r o d u c e d b y C . P . G u p t a [ 2 9 5 , 2 9 6 ]
L.
Nachbin
[511].
and
The f i r s t i n f i n i t e d i m e n s i o n a l
r e p r e s e n t a t i o n theorem f o r a n a l y t i c f u n c t i o n a l s by holomorphic germs i s due t o P . J .
(HN(U),no)A
2
H(Uo)
Boland
whenever U i s a convex balanced open
subset of a 8 3 Q space. proposition
[ 8 5 ] who p r o v e d t h a t
This i s a stronger r e s u l t than
3.50 f o r ~ 8 3 Q s p a c e s . P r o p o s i t i o n s 3.49,
3 . 5 1 and c o r o l l a r y 3 . 5 2 a r e new. t o P.J.
Boland and S . Dineen
i s given i n 15.4.
3.50,
C o r o l l a r y 3.54 i s due
[go] and an a l t e r n a t i v e p r o o f
Theorem 3 . 5 5 i s due t o R .
Ryan
[620],
L e m m a 3 . 5 6 i s new w h i l e c o r o l l a r i e s 3 . 5 3 a n d 3 . 5 7 a r e proved i n P . J .
Boland and S .
Dineen
[QO]
assumption t h a t E has a Schauder b a s i s . a n d e x a m p l e 3 . 5 9 may b e f o u n d i n P . J . [91] and S . Dineen
[202].
under t h e a d d i t i o n a l Corollary 3.58
Boland and S . Dineen
Further r e p r e s e n t a t i o n theorems
f o r a n a l y t i c f u n c t i o n a l s on a Banach s p a c e a r e due t o J.M.
Isidro
[351] while t h e classical theory f o r functions
o r one complex v a r i a b l e i s due t o A . G.
Kb'the [ 3 9 6 ] a n d C . L .
d a S i l v a Dias
Grothendieck [661].
A.
[285],
Martineau
157
Holomorphic functions on balanced sets [451] i n v e s t i g a t e s t h e case of s e v e r a l v a r i a b l e s . P r o p o s i t i o n 3.60 and c o r o l l a r y 3 . 6 1 are due t o S.
[ 1 8 5 ] . T h e o r e m 3 . 6 4 was f i r s t p r o v e d f o r e n t i r e
Dineen
n u c l e a r s p a c e s by P .
f u n c t i o n s on q u a s i - c o m p l e t e see also E.
Kelimarkka [526],
i n d e p e n d e n t l y , t o a r b i t r a r y open s e t s by P . J . and L . by L .
!Vaelbroeck
Perrot
[83],
Boland
[86]
Our p r o o f i s c l o s e t o t h a t g i v e n
[713].
l a e l b r o e c k , who a l s o p r o v e s
lemma 3.63.
proof o f theorem 3 . 6 4 i s given by J . F . B.
Boland
and a f t e r w a r d s e x t e n d e d ,
A further
Colombeau and
[16@,161] and f o r f u l l y n u c l e a r s p a c e s w i t h a
b a s i s we p r o v i d e a n a l t e r n a t i v e p r o o f i n c h a p t e r 5 (corollary 5 . 2 2 ) .
Applications o f theorem 3.64 t o l i f t i n g
theorems f o r l i n e a r mappingsare t o be found i n W. B.
Kaballo
and t o t h e c l a s s i f i c a t i o n o f S t e i n a l g e b r a s i n
[363]
Kramm [398,399]. E x t e n s i o n s o f theorem 3 . 6 4 t o A and s n u c l e a r t y and
t o nuclear bornologies a r e given i n K.D.
B.
Gramsch and R .
[69,70], [152],
Meise
L . Waelbroeck
J.F.
[67], [713],
Colombeau and B .
Bierstedt,
K.D.
B i e r s t e d t and R .
bleise
J.F.
Colombeau a n d R .
Meise
Perrot
[157,159,160,16
For example t h e f o l l o w i n g r e s u l t i s proved i n
l e t E be a quasi-complete (H(U;F),-r )
Sgtz 1.12 of
(E',-r0)
[152];
l o c a l l y convex space, then
i s an s n u c l e a r
i f and only i f
,1651.
s p a c e f o r any open s u b s e t U o f E
and F a r e b o t h s n u c l e a r s p a c e s ( s e e
[67]).
An a p p r o a c h t o t h e m a t h e m a t i c a l
foundations of
quantum f i e l d t h e o r y u s i n g n u c l e a r i t y and i n f i n i t e dimensional holomorphy i s given i n P . 411, B.
413,414,415,416,4171
Perrot
[158]
and J . F .
e x e r c i s e 3.104.
Colombeau and
Colombeau [ 1 4 5 ] ) .
C o r o l l a r y 3.65 i s due t o P . Schwartz property f o r
Kr6e [ 4 0 6 , 4 @ 7 , 4 @ 8 , 4 0 9 ,
(see also J . F . Boland
[82,83].
The
( H ( U ) , - r ) i s d i s c u s s e d i n o u r n o t e s on
This Page Intentionally Left Blank
Chapter 4
HOLOMORPHIC FUNCTIONS ON BANACH SPACES
Banach s p a c e s a n d n u c l e a r s p a c e s p l a y a n i m p o r t a n t r o l e i n l i n e a r f u n c t i o n a l a n a l y s i s and a l s o i n c l a s s i c a l a n a l y s i s b y way o f a p p l i c a t i o n .
This chapter i s devoted t o t h e study of
h o l o m o r p h i c m a p p i n g s b e t w e e n Banach s p a c e s a n d i n c h a p t e r 5 we d i s c u s s h o l o m o r p h i c f u n c t i o n s on n u c l e a r s p a c e s .
As o n e
w o u l d e x p e c t , s i n c e e v e r y n u c l e a r Banach s p a c e i s f i n i t e dimensional,
t h e s e two t o p i c s p r o c e e d a l o n g q u i t e d i f f e r e n t
l i n e s b u t b o t h c o n f i r m t h a t i n f i n i t e dimensional holomorphy l e a d s t o c o n c e p t s and r e s u l t s which a r e o f i n t e r e s t i n thems e l v e s and q u i t e d i f f e r e n t from what one would e x p e c t from t h e underlying f i e l d s . I n t h i s c h a p t e r we f i n d t h a t t h e r e i s a r i c h i n t e r a c t i o n between t h e t h e o r y o f holomorphic f u n c t i o n s and t h e g e o m e t r y
of Banach s p a c e s .
By t h e g e o m e t r y o f B a n a c h s p a c e s , a t o p i c
t h a t h a s undergone r a p i d development i n t h e l a s t f i f t e e n y e a r s , we mean t h e s t u d y o f g e o m e t r i c p r o p e r t i e s o f t h e u n i t b a l l s u c h a s s m o o t h n e s s , t h e e x i s t e n c e of e x t r e m e p o i n t s , d e n t -
a b i l i t y , uniform c o n v e x i t y , sequentiaZ compactness e t c . I f E i s a Banach s p a c e t h e n t h e c o m p a c t o p e n t o p o l o g y
11 Ik
o n H(E) i s g e n e r a t e d b y subsets of E.
as K r a n g e s o v e r t h e compact
Our m o t i v a t i n g p r o b l e m i s t h e f o l l o w i n g ;
do
t h e r e e x i s t a n y o t h e r s e m i - n o r m s o n H(E) w h i c h h a v e t h e f o r m
I / Ik
f o r some s u b s e t A o f E?.
it w i l l always be
‘c6
semi-norm w i l l n o t b e
I f s u c h a semi-norm
11 IIA
T
w
continuous.
Since
11 IIA
llfla <
0
for e v e r y f i n H ( E ) . 159
the
i s a semi-
norm i f i t i s f i n i t e we a r e l o o k i n g f o r n o n - r e Z a t i v e Z y A such t h a t
exists
c o n t i n u o u s and i f A i s n o t pre-compact
compact This problem has
Chapter 4
160
l e d t o much o f t h e r e s e a r c h we r e p o r t i n t h i s c h a p t e r . I n t h e f i r s t s e c t i o n we d i s c u s s a f e w g e n e r a l p r o p e r t i e s o f h o l o m o r p h i c mappings b e t w e e n Banach s p a c e s . Some o f t h e s e a r e u n r e l a t e d t o t h e t o p o l o g i c a l p r o b l e m b u t are of i n t e r e s t i n themselves. 54.1
ANALYTIC
EQUALITIES
A N D INEQUALITIES
The t h e o r y o f h o l o m o r p h i c f u n c t i o n s o f one o r s e v e r a l complex v a r i a b l e c o n t a i n s a number o f i n t e r e s t i n g and u s e f u l e q u a l i t i e s and i n e q u a l i t i e s and it i s n a t u r a l t o e x t e n d t h e s e t o i n f i n i t e l y many v a r i a b l e s .
Such g e n e r a l i z a t i o n s a r e o f
i n t e r e s t i f t h e y s a t i s f y a t l e a s t one o f t h e f o l l o w i n g criteria; a ) t h e y r e q u i r e new n o n t r i v i a l p r o o f s
(and a s t u d y o f t h e s e
i n t u r n may l e a d t o i m p r o v e d a n d e v e n new f i n i t e d i m e n s i o n a l results), b) t h e y l e a d t o a p p l i c a t i o n s n o t c o v e r e d by t h e c o r r e s p o n d i n g f i n i t e dimensional r e s u l t s , c ) t h e y g i v e r i s e t o a c l a s s i f i c a t i o n problem f o r l o c a l l y convex s p a c e s , d ) t h e y l e n d t h e m s e l v e s t o new i n t e r p r e t a t i o n s w h i c h i n t u r n s u g g e s t new c o n c e p t s a n d p r o b l e m s o r non-existent
( w h i c h may e v e n b e t r i v i a l
i n f i n i t e dimensions).
We p r e s e n t h e r e e x t e n s i o n s o f t h r e e w e l l known r e s u l t s from t h e t h e o r y o f one complex v a r i a b l e ;
S c h w a r z ' s lemma, t h e
maximum m o d u l u s t h e o r e m a n d t h e C a u c h y - H a d a m a r d
formula.
S i n c e t h e s e e x t e n s i o n s w i l l n o t b e r e q u i r e d l a t e r w e do n o t give a comprehensive account.
For both Schwarz's
lemna and
t h e maximum m o d u l u s t h e o r e m we n e e d t h e c o n c e p t o f a n extreme point. Definition 4.1
Banach s p a c e .
L e t K b e a c o n v e x s u b s e t of a c o m p l e x A p o i n t e of
K is
161
Holomorphic functions on Banach spaces -1 < X
( a ) a r e a l e x t r e m e p o i n t i f { e + Ax;
implies x
=
0 ,
( b ) a c o m p l e x e x t r e m e p o i n t i f { e + Ax;
implies x
5 1)CK
o 5
1x1
5 l}CK
= 0.
I t i s c l e a r t h a t e v e r y r e a l extreme p o i n t i s a complex I f e v e r y p o i n t o f norm o n e i s a r e a l e x t r e m e
extreme p o i n t .
p o i n t o f t h e c l o s e d u n i t b a l l o f E then E i s c a l l e d a rotund L
o r a s t r i c t l y corivex Bar.ach s p a c e . convex i f 1 < p <
P (X,O,p)
is s t r i c t l y
f o r any f i n i t e measure s p a c e (X,a,p).
m
If
e v e r y p o i n t o f modulus 1 i s a complex extreme p o i n t o f t h e c l o s e d u n i t b a l l o f E t h e n we s a y E i s a s t r i c t l y c - c o n v e x 1 L ( 0 , l ) i s a s t r i c t l y c - c o n v e x Eanach s p a c e Banach s p a c e . which i s n o t s t r i c t l y convex. Now l e t D = ( z E C ; I z I variable says that If(z)l some z
if f
E
S c h w a r z ' s lemma i n o n e
< 1).
H(D;D)
and f ( o ) = o t h e n
5 I z i f o r a l l z E D and moreover i f I f ( z E D then
)I
lzol
=
for
f ( z ) = cz f o r a l l z i n D where c i s a
c o n s t a n t o f modulus 1.
We u s e t h e f i r s t p a r t o f t h i s r e s u l t
t o p r o v e t h e f o l l o w i n g lemma, w h i c h i s a l s o u s e f u l t h e maximum m o d u l u s t h e o r e m ,
i n extending
and e x t e n d t h e second h a l f t o
mappings b e t w e e n Banach s p a c e s .
Proof
If
If(z)l
= 1 for
some z
E
D t h e n t h e one
d i m e n s i o n a l maximum m o d u l u s t h e o r e m i m p l i e s t h a t f i s a c o n s t a n t mapping i n which c a s e t h e above r e s u l t H e n c e we may a s s u m e f E H ( D ; D ) .
z
-'
Z
-
hence
-az
1
mapping z g(o)
-
-a
=
0 .
If(z)
T h e M'dbius t r a n s f o r m a t i o n
(la1 < 1 ) maps D o n t o D a n d g(z)
=
By S c h w a r z
- f(o)l 5
is trivial.
c1
to
0 .
Hence t h e
f(z)-f(o) belongs t o H(D;D) 1 - f o f (z)
and
162
Chapter 4
L e t E a n d F b e Banach s p a c e s w i t h o p e n u n i t baZZs
Theorem 4 . 3
Let f
U and V z - e s p e c t i v e l y .
i s o m e t r y from E o n t o F .
E
H ( U ; v ) and s u p p o s e d f ( o ) i s a n f ( x ) = d f ( o ) ( x ) for a 2 2
Then
x
i n U and i n p a r t i c u Z a r f i s an i s o m e t r y from U o n t o V . We f i r s t n o t e t h a t b y r e p l a c i n g f b y d f ( o ) - l o f
Proof
we may a s s u m e t h a t E = F a n d d f ( o ) = I , w h e r e I i s t h e
We f i r s t s h o w t h a t f ( o ) =
i d e n t i t y map o n E .
o t h e r w i s e , t h e n b y t h e Hahn-Banach theorem, nJ
$IEE',
II$II =
1, such t h a t $ ( f ( o ) )
for all z ED.
g(z1 = $of( zfO l g (o)I
2
V(0)II )
5 l-lg'(o)12,
and f ( o ) =
0 .
Ilf(o)
=
0 .
Suppose
there exists
11.
Let
S i n c e gsH(D;D)
we g e t
( s e e example 2.31).
Now f i x EED\{o}
and l e t
T o c o m p l e t e t h e p r o o f i t s u f f i c e s t o show w i s i d e n t i c a l l y
5
zero f o r every E.
Our f i r s t s t e p i n p r o v i n g t h i s i s t o s h o w Ib( + Xw ( x ) I I 5 I I x I I for a l l x i n U a n d h e D .
5
x
E
U\{o)and
$I
II$(I I 1 ,
E E l ,
f o r m u l a h ( o ) = $I(-)
X
IIX
II
Let
be given.
and h ( z ) =
1 7
Defjne h by t h e zx +of(-) i f ZED\\CO}.
llxll
S i n c e f ( o ) = o a n d d f ( o ) = I i t f o l l o w s t h a t hEH(D;D). By l e m m a 4 . 2 ,
l e t t i n g z = 511xII, we h a v e
163
Holomorphic functions on Banach spaces
(since the function
5 As
l-t t
i s decreasing)
IIXII.
41 was a n a r b i t r a r y e l e m e n t o f t h e u n i t b a l l o f E’ i t f o l l o w s
by t h e Hahn-Banach t h e o r e m , t h a t
Ilx+Xw5(x)
11
5 1.
(The p r o o f
w o u l d now b e c o m p l e t e i f E was a s t r i c t l y c - c o n v e x s p a c e ) . Now l e t L d e n o t e t h e a l g e b r a o f a l l b o u n d e d l i n e a r m a p p i n g s f r o m
d
Hm(U) i n t o i t s e l f . c ’ 3
mapping f r o m H (U) u n i t b a l l of
i s a Banach s p a c e a n d I ‘
into itself,
= $ ( x + ~ w( x ) )
5
seen t h a t k E H ( D ; D ) .
G i v e n $ e H m ( U ) , I(JIIIu(
Now l e t L : Hm(U)
L(@) = $o(I
11
-
:~1 [ “ ‘ x +1~ w 5 ( x ) ) -$ ( X I 1 I +
1
y
1
1 7
yields
(**)-
Hm(U) b e d e f i n e d b y t h e f o r m u l a By
w5).
5
1,
I t is easily
f o r a l l AED.
A f u r t h e r a p p l i c a t i o n o f lemma 4 . 2 w i t h z =
I$(x)
the identity
is a r e a l extreme p o i n t o f t h e
(see exercise 4.52).
and x E U l e t k ( A )
,
1
I ’ 2 T(L - 1’111 = s u p
(**I 1 1 I + ( x ) 5 T [ + ( X + ~ W 5 ( X ) )- + ( x ) I l
IC.EH~(U)
llJI1lu 5
XEU
1 9
- 1. Hence L = I ’
a n d f o r a n y ~ E E ’ we h a v e
By t h e Hahn-Banach
theorem w
5
0 = L(8)
-
8 = &8ow 2 5
’
i s i d e n t i c a l l y z e r o a n d as we
have a l r e a d y noted t h i s completes t h e p r o o f . O t h e r g e n e r a l i z a t i o n s o f S c h w a r z ’ s lemma a r e a l s o a v a i l a b l e and t h e s e t o g e t h e r w i t h t h e above have a p p l i c a t i o n s t o Banach a l g e b r a t h e o r y .
In p a r t i c u l a r they y i e l d a
g e n e r a l i s e d B a n a c h - S t o n e t h e o r e m f o r J * - a l g e b r a s a n d a new p r o o f o f t h e Russo-Dye t h e o r e m . We now l o o k a t t h e maximum m o d u l u s t h e o r e m .
There a r e a
164
Chapter 4
n u m b e r o f d i f f e r e n t f o r m s o f t h e maximum m o d u l u s t h e o r e m d i s c u s s e d i n t h e l i t e r a t u r e a n d h e r e we c o n f i n e o u r s e l v e s t o the following;
i f f c €i(U), h h e r e U i s a c c n n e c t e d open
s u b s e t o f t, t h e n e i t h e r I f ( z ) l h a s n o maximum o n U o r f i s a c o n s t a n t mapping.
We l o o k a t e x t e n s i o n s o f t h i s r e s u l t
by c o n s i d e r i n g Banach v a l u e d h o l o m o r p h i c m a p p i n g s d e f i n e d on open s u b s e t s o f E .
One c a n e a s i l y s h o w t h a t i f f a H ( U ; F ) ,
U a connected open s u b s e t of
Ilf(z)
11
h a s a maximum t h e n
C and F a Banach s p a c e , and
1IfII i s a c o n s t a n t .
Hence t h e
problem r e d u c e s t o showing t h a t t h e c o n s t a n t mappings are t h e This is not
o n l y holomorphic mappings o f c o n s t a n t modulus.
t r u e i n g e n e r a l , even f o r f i n i t e d i m e n s i o n a l s p a c e s , as t h e f o l l o w i n g example shows. Let f
-zz,
: D
s u p norm t o p o l o g y . but
Ilf(z)
11
Theorem 4 . 4 E-valued
=
f(z) = (l,z),
where
. 2 . i s C with the
z 2m
Note t h a t f i s n o t a c o n s t a n t mapping
1 f o r a l l z a D. L e t E b e a B a n a c h space.
Each ho2omorphic
m a p p i n g f d e f i n e d on a c o n n e c t e d o p e n s u b s e t of
C
f o r u h i e h l l f ( z ) l l h a s a rncz-immz ic c o n s t a n t if a n d o n l y i f E
is a s t r i c t Z y c - e o n v e x B a n a c h s p a c e . Proof
F i r s t suppose
e E
IleII = 1 , i s n o t a c o m p l e x
E,
extreme p o i n t o f t h e u n i t b a l l . Ile+zxll 5 1 f o r a l l z some l z o ]
E C ,
1 we h a v e
Choose
IzI 5 1 .
If
IleII 5 y1( I l e + z o x I I
since t h i s is impossible
X E
+
Ile-zoxI1)
Ile+zxII = 1 f o r a l l
The f u n c t i o n f ( z ) = e + z x , z
E
D,
E such t h a t
Ile+zoxII < 1 f o r
Z E
i s non-constant
< 1 and
E,
121
but
5 1.
Ilf(z)
11
i s a c o n s t a n t f u n c t i o n o f z r e s t r i c t e d t o D and hence w e
have proved t h e theorem i n one d i r e c t i o n . Now s u p p o s e E i s a s t r i c t l y c - c o n v e x B a n a c h s p a c e .
-
L e t U b e a c o n n e c t e d o p e n s u b s e t o f E. suppose t h e mapping z
E
U
Let
f
E
H(U;E)
and
Ilf(z)II i s a c o n s t a n t m a p p i n g .
By u s i n g t r a n s l a t i o n s i f n e c e s s a r y w e may s u p p o s e D C U a n d s u p I l l f ( z ) \ \ ; z E D ) 5 1.
By lemma 4 . 2
and t h e Hahn-Banach
165
Holomorphic functions on Banach spaces theorem
(If(o) + A ( f ( z ) - f ( o ) ) ( l i l i f
Z E 1x1 I#, 1- z
D\(o).
S i n c e t h e u n i t b a l l o f E c o n t a i n s no complex extreme p o i n t s it follows t h a t
for a l l z near zero.
f(z) = f(o)
This
completes the proof.
F o r a r b i t r a r y f u n c t i o n s we a l s o h a v e t h e f o l l o w i n g method f o r r e c o g n i s i n g n o n - c o n s t a n t which a r e of
holomorphic functions
constant modulus.
L e t E be a c o m p l e x Banach s p a c e and l e t f ( z ) non c o n s t a n t hoZomorphic mapping from D
=
1"
a n z n be a
n=o
{ Z E( c ; l z l
=
11
<
.
.. . I i f and o n l y o n E and an o p e n s u b s e t U o f D s u c h t h a t ( ( l f ( z ) ( ( Iis c o n s t a n t on U. {
Then a.
into E.
C l o s e d Span { a l , a Z ,
i f t h e r e e z i s t s a n e q u i v a Z e n t norm111
1))
We now l o o k a t t h e C a u c h y - H a d a m a r d
formula.
formula i n one dimension s t a t e s t h e f o l l o w i n g ;
This m
if
i s a sequence o f complex numbers and r = ( l i m s u p l a ll'n)-l n - - - + m n d n a z then the series c o n v e r g e s u n i f o r m l y on I z ; l z l 5 r o ) E=o n r. f o r any r 0
T h e s i t u a t i o n i n i n f i n i t e l y many v a r i a b l e s
is quite
different. Lemma 4 . 5
then
1"
n=o
I f
$nn
-
i f and onZy i f $ , ( x )
for e v e r y x i n E ( i . e . $ n Proof
If
1" e:
n =1
E
HIE)
then
f o r e v e r y x i n E a n d z i n C.
-
o in t h e
1"
n=l
n - m
+n ( x ) 1" en H G ( E ) .
Conversely if
n =1
IT
E
0
W*
o a s n-
m
topology).
(Gn(x))"zn
converges
By t h e C a u c h y - H a d a m a r d
-
i n one v a r i a b l e l i m sup I @ , ( X ) ~ ~
then f =
-
E i s a Banach s p a c e and $ n E' ~ for a l l n
H(E)
E
as n
=
n-l- "imlm ~5
Since t h e nth
$n ( x )
I
formula
= 0.
f o r every x i n E derivative o f f a t 0
Chapter 4
166
i s $" a n d t h i s i s c o n t i n u o u s we may a p p l y t h e o r e m 2 . 2 8 t o n complete t h e proof.
Let E be a s e p a r a b l e H i l b e r t s p a c e w i t h
Example 4 . 6 orthonormal
cm z n e n
E.
lm
-
$n w h e r e $n i s e v a l u a t i o n n n=l S i n c e $no as n m a t the nth coordinate of E.
all
n=l
E
i n (E',o(E',E))
n
Hence f =
lemma 4 . 5 i m p l i e s t h a t f
= 1 and l i m s u p
n - -
II
d"fj0) n.
p
E
H(E).
However
= 1.
Example 4 . 6 shows t h a t i n i n f i n i t e d i m e n s i o n s we h a v e t o d i s t i n g u i s h between t h e " r a d i u s o f p o i n t w i s e convergence" and t h e " r a d i u s o f u n i f o r m convergence".
A f u r t h e r concept
i s t h e r a d i u s o f b o u n d e d n e s s w h i c h e n t e r s i n a n a t u r a l way and p l a y s an i m p o r t a n t r o l e i n l a t e r developments.
Let U
b e an o p e n s u b s e t o f a l o c a l l y c o n v e x s p a c e E a n d l e t B b e a balanced closed subset of E .
We l e t If E is
d B ( 5 , U ) = S U ~ I ~ A I ; A E E , ~ + A B C Uf o) r e v e r y 5 i n U .
a normed l i n e a r s p a c e a n d B i s t h e u n i t b a l l o f E t h e n d B ( 5 , U ) i s t h e u s u a l d i s t a n c e o f 5 t o t h e complement o f U i n E.
Now l e t F b e a Banach s p a c e a n d l e t f € H ( U ; F ) .
B r a d i u s of b o u n d e d n e s s o f f a t 5 , r f ( t , B ) ,
s u p C ~ A ) ; A E ~ : , ~ + X B C IUl f 1,1 5 + A B The B r a d i u s of defined a s sup
<
The
i s defined as
a).
uniform convergence of f a t 5 , Rf(c,B),
I1X(
is
; X ~ C , c + h B c U a n dt h e T a y l o r s e r i e s o f f a t
5 c o n v e r g e s t o f u n i f o r m l y on c + A B ) . Proposition 4 . 7
L e t U be an o p e n s u b s e t of a l o c a l l y c o n v e x
s p a c e E, l e t F b e a Banach s p a c e and s u p p o s e f
5
EU,
B
E
H(U;F).
is a c l o s e d baZanced s u b s e t of E and r f ( E , B )
If
o then
167
Holomoiphic functions on Banach spaces Proof
We f i r s t n o t e t h a t i f E = U t h e n d B ( S J U ) = +
a n d t h e a b o v e may r e d u c e t o
m
=
m
=
m
T h i s however s a y s
m.
t h a t f i s bounded and t h e T a y l o r s e r i e s c o n v e r g e s u n i f o r m l y on 5 + XB f o r e v e r y A E C i f a n d o n l y i f l i m s u p I ( i " f ( 5 ) n - m n! If o <
la1
< rf(S,B)
then
(by t h e Cauchy i n e q u a l i t i e s ) .
Since rf(c,B)
5 d B ( c , U ) we h a v e shown t h a t
The a b o v e a l s o shows o n t a k i n g la1 <
l a q \ <
rf(S,B),
that
(Note t h a t s i n c e r f ( S J B ) 7 every n).
0
we h a v e
Hence i f 5 + (1-E)BBCU
, 2l ft (dSl l B <
for
B
=O;
168
Since
Chapter 4
E
was a r b i t r a r y i t f o l l o w s t h a t
Now s u p p o s e y < R f ( < , B ) .
Hence f o r any
2,
o < E
B).
< Corollary 4.8
Since
4
Then
y,
2
we h a v e
and y were a r b i t r a r y t h i s i m p l i e s
r f ( < , B ) and c o m p l e t e s the p r o o f .
If E i s a L o c a l l y c o n v e x s p a c e , F is a
Banach s p a c e and K i s a compact b a l a n c e d s u b s e t of E t h e n wl$";;o) Proof
~l:/~=
o for e v e r y f s H ( E ; F ) .
Since a holomorphic function i s continuous it i s
bounded on e a c h compact s u b s e t o f E and t h e r e s u l t f o l l o w s from p r o p o s i t i o n 4 . 7 . I f E i s a f i n i t e dimensional space then rf(<,B)=dB(S,U) f o r any bounded s u b s e t B o f E , any f E H ( U ) .
any open s u b s e t U o f E and
F o r t h i s r e a s o n t h e c o n c e p t of r a d i u s of
boundedness i s n o t i n t e r e s t i n g i n f i n i t e dimensions. The r e m a i n d e r o f t h i s s e c t i o n i s d e v o t e d t o v a r i o u s p r o p e r t i e s o f t h e r a d i u s of boundedness.
These r e s u l t s
w e r e a l l m o t i v a t e d b y t o p o l o g i c a l c o n s i d e r a t i o n s , w h i c h we discuss i n the next section, but are also of independent interest.
We r e s t r i c t o u r s e l v e s t o e n t i r e f u n c t i o n s o n a
169
Holornophic functions on Banach spaces Banach s p a c e E w i t h c l o s e d u n i t b a l l B .
I n t h i s c a s e we
write r (5) i n place of rf(C,B). N o t e t h a t r (C) i s a n f f i s o m e t r i c p r o p e r t y o f t h e Banach s p a c e E and w i l l change i f t h e Banach s p a c e i s r e n o r m e d e v e n w i t h a n e q u i v a l e n t norm. If f i s t h e f u n c t i o n c o n s i d e r e d i n example 4 . 6 t h e n
p r o p o s i t i o n 4 . 7 shows t h a t r f ( o ) f o r a l l E E E)
1 + ~ ,E >
0 ,
= 1 (and hence rf(c)
m
a n d s o f i s unbounded on e v e r y b a l l o f r a d i u s
centered a t the origin.
T h i s a l s o shows t h a t
t h e Taylor series expansion a t zero converges at a l l points
o f E b u t d o e s n o t c o n v e r g e u n i f o r m l y on a n y b a l l o f r a d i u s greater than 1 centered at zero. Our n e x t r e s u l t on t h e r a d i u s o f b o u n d e d n e s s s a y s t h a t e v e r y i n f i n i t e d i m e n s i o n a l Banach s p a c e s u p p o r t s an e n t i r e function with non-trivial
o f boundedness.
(i.e.
n o t i d e n t i c a l l y + -)
radius
This i s a consequence o f t h e following
deep r e s u l t . Proposition 4.9
E i s a n i n f i n i t e d i m e n s i o n a 2 Banach
If
space t h e n t h e r e e x i s t s a sequence i n E ' ,
II$nll
= 1 f o r a22
n and $ , ( x )
---+
o as n
x i n E.
-
( $ n ) n J such t h a t
f o r every
C o r o l l a r y 4.10 I f E i s a n i n f i n i t e d i m e n s i o n a 2 Banach s p a c e t h e n t h e r e e x i s t s a C-valued e n t i r e f u n c t i o n o n E , f ,
-
suah t h a t r f ( c ) < Proof
Let f
=
1"
for e v e r y 5 i n E .
n=o
9:
where
given by p r o p o s i t i o n 4 . 9 . e x a m p l e 4.6, r f ( o )
= 1.
($n)n is t h e sequence i n E '
By lemma 4 . 5 f e H ( E ) a n d , a s i n Hence r f ( c )
<
00
for all S E E.
This completes t h e proof. O u r n e x t r e s u l t s t a t e s t h a t r f may t a k e a r b i t r a r i l y
small v a l u e s even o v e r a bounded s e t .
Chapter 4
170
Proposition 4.11 I f E i s an i n f i n i t e d i m e n s i o n a Z Banach s p a c e w i t h u n i t baZZ B t h e n t h e r e e x i s t s an e n t i r e f u n c t i o n f on E s u c h t h a t inf {rf(x);
xeB1 = o
However rf does have regularity properties as the following proposition shows. I f
Proposition 4.12 rf(o)
<
m,
where
then
(a) Irf(x) (b)
E i s a Banach s p a c e and fEH(E),
- rf(y)
I
5
IIx-rll f o r
aZZ X,Y
E
E,
- log rf i s a p Z u r i s u b h a r r n o n i c f u n c t i o n on E. I n v i e w of (a), ( b ) s a y s t h a t 1 - log If(') 5 J2"( - l o g rf (x+eif3y))de
f o r e v e r y x,y i n E
The above proposition is not difficult to prove (indeed (a) i s obvious) but we shall omit the proof since it would first require a discussion o f plurisubharmonic functions. It is rather remarkable that conditions (a) and (b) of proposition 4.12 characterize radii of boundedness of holomorphic functions o n certain spaces. A proof of this type of result involves, as condition ( b ) might suggest, methods used in solving the Levi problem. We,therefore, only state some o f the important results in this area.
It is
also worth noting that this result have no analogue in the theory o f several complex variables. Theorem 4.13
Banach s p a c e and l e t g:E conditions :
-
L e t E b e an i n f i n i t e dirnensionaZ s e p a r a b l e R+ s a t i s f y t h e folZowing
(a) Ig(x) - g(y)l 2 Ib-yll f o r X,Y i n E , (b) - log g i s a p Z u r i s u b h a r m o n i c f u n c t i o n on E .
171
Holomolphic functions on Banach spaces
Then ( 1 ) if E = 2
1
t h e r e e x i s t s f s H ( E ) such t h a t g = r f ,
( 2 ) if E has a Schauder b a s i s t h e n t h e r e e z i s t s f
such t h a t
qn 5
E
H(E)
r f 5 g.
E x a m p l e s h a v e b e e n c o n s t r u c t e d w h i c h show t h a t we c a n n o t r e p l a c e l l by lp, 1 < p <
m,
Whether
i n (1) above.
o r n o t t h e o r e m 4 . 1 3 ( 2 ) h o l d s f o r a r b i t r a r y s e p a r a b l e Banach s p a c e s i s s t i l l an open p r o b l e m . Condition
(a) of p r o p o s i t i o n 4.12 says t h a t rf i s a
Lipschitz function with Lipschitz constant l e s s than o r e q u a l t o 1.
Example 4 . 1 4 shows t h a t t h i s c o n s t a n t i s t h e
b e s t p o s s i b l e i n g e n e r a l b u t f o r f u n c t i o n s which s a t i s f y s p e c i a l growth c o n d i t i o n s one can f i n d a s m a l l e r c o n s t a n t . On t h e o t h e r h a n d i n B a n a c h s p a c e s w i t h a s u i t a b l e g e o m e t r y we may r e p l a c e t h e i n e q u a l i t y o f p r o p o s i t i o n 4 . 1 2 b y a s t r i c t inequality. Example 4 . 1 4
If f
:zl
-
t i s given by
T h i s c a n n o t h a p p e n on u n i f o r m l y c o n v e x Banach spaces. A Banach s p a c e E is s a i d t o be uniformly Definition 4.15. c o n v e x if f o r e v e r y E > o t h e r e e x i s t s a 6 > o s u c h t h a t for X,Y E
E,
I(x(I =
IIYII
= 1,
IIx+YII
'> 2 - 6
we have
Ilx-yII
i
E
172
Chapter 4
i s a u n i f o r m l y convex s p a c e f o r l < p < m b u t Z1 i s
1,
P
n o t uniformly convex. Proposition 4.16
If E i s a u n i f o r r n z y c o n v e x i n f i n i t e
dimensionaZ Banach s p a c e , f F H @ ) and r f ( o ) <
-
lrf(x)
rf(y)
1
IIx-rll f o r aZZ x , y
<
then
i n E.
Suppose t h e c o n c l u s i o n i s f a l s e .
Proof
m,
Then t h e r e e x i s t
a n e n t i r e f u n c t i o n f on E a n d a u n i t v e c t o r x i n E s u c h t h a t = 1 and r
rf(o)
f
(Ax) = 1 - A
f o r each i n t e g e r n If(xn) 1-
1 n
I
> n,
1
I
llXnII
I 1+
- A
2 we c a n c h o o s e x
1
-
1.
n
T h e n Ilynll
xn = ( l - X ) y n + Ax. @ nE E '
<
-
X
Hence
< 1.
such t h a t
n
1 n
llxn-Xxll 5 1 - X +
1 L e t yn = = ( x n - X x ) .
theorem,
for some A , o
and
1 as n
--
and
F o r e a c h n c h o o s e , by t h e H a h n - B a n a c h
such t h a t
Il@nII
= 1 and @ n ( x n ) =
llxnII.
For
a l l n we h a v e @,(Xn) =
-
=
llXnII
-
(l-Al$n(Yn)
S i n c e l $ n ( y n ) l 5 IlynII $,(x)
Hence n
1.
+
m
1 as n
xnII
L
-
and l i m n - +
$n(x
-. +
+
1 as n
xn)
=
-
A q X ) .
@,(XI
+
m
it follows t h a t
IIxnII
IIx+xnII = 2 .
By u n i f o r m c o n v e x i t y i t f o l l o w s t h a t This contradicts t h e f a c t t h a t
If(xn)
I(x-xn(I
I
-
2 as
o as n
> n f o r a l l n and
completes t h e proof. 14.2
B O U N D I N G SUBSETS O F A B A N A C H SPACE
I n t h e p r e v i o u s s e c t i o n we c o n s i d e r e d s e t s a n d r e g i o n s w h e r e a s i n g l e f u n c t i o n was b o u n d e d .
We now l o o k a t s e t s o n
which e v e r y holomorphic f u n c t i o n i s bounded.
-.
173
Holomorphic functions on Banach spaces Definition 4.17
space E .
L e t U b e a n o p e n s u b s e t o f a ZocaZZy c o n v e x
A s u b s e t .4 o f U i s s a i d t o b e b o u n d i n g f o r U i f
We s h a l l u s e t h e t e r m b o u n d i n g s e t when t h e d o m a i n
s p a c e U i s e a s i l y u n d e r s t a n d a b l e from t h e c o n t e x t .
Bounding
s e t s arise n a t u r a l l y i n problems o f a n a l y t i c continuation, c o n s t r u c t i o n o f t h e envelope o f holomorphy and i n problems c o n c e r n i n g t o p o l o g i e s on H ( U )
.
We b e g i n b y c o l l e c t i r g s o m e s i m p l e p r o p e r t i e s o f b o u n d i n g s e t s .
L e t U b e a n o p e n s u b s e t of a l o c a l l y c o n v e x
Lemma 4 . 1 8
s p a c e E and Z e t F b e a Z o c a Z l y c o n v e x s p a c e .
Then
( a ) e v e r y compact s u b s e t o f U i s b o u n d i n g , ( b ) t h e cZosure o f a bounding s e t i s bounding, (c) i f f cH(U;F)
and A i s a b o u n d i n g s u b s e t o f U t h e n f ( A )
i s a b o u n d i n g s u b s e t of P
\\fllA
( d ) i f A i s a bounding s u b s e t o f U t h e n
i s a
T*
c o n t i n u o u s semi-norm on H ( U ) .
Proof
( a ) , (b) and ( c ) a r e obvious.
Let V = ( f
E
H(U)
T*
closed.
fl
XEA
{f EH(U);
Since (H(U),r6)
neighbourhood of z e r o i n (H(U),T*)
is a
T*
XEA
)f(x)
I
We p r o v e ( d ) .
; Ilf{lA 5 1 1 . V i s c o n v e x b a l a n c e d a n d
a b s o r b i n g and s i n c e V = also
= sup
I f ( x ) l 5 1) i t i s
is barrelled V is a and h e n c e
11 1,
continuous semi-norm.
Corollary 4.19
L e t U b e a balanced open s u b s e t o f a ZocalZy
convex space E t h e n (a) A C U
i s b o u n d i n g i f and onZy i f
(b) A C E
i s b o u n d i n g i f and onZy i f
Chapter 4
174
Proof
The c o n d i t i o n s g i v e n i n ( a ) and ( b ) a r e o b v i o u s l y If A i s
s u f f i c i e n t a n d w e now s h o w t h a t t h e y a r e n e c e s s a r y . b o u n d i n g t h e n p ( f ) = IlfIIA i s a
on H(U). -r6
s e m i - n o r m o n H(U)
irn ll*]lA
1
<
=o
m
If U = E then
number B and e v e r y *n
1"
gn
Em
=0
1"
l l ~ l l A<
q
) is also a
E
H(U).
(a).
m
f o r e v e r y complex
E
m
f
n!
n=o
i=o B n dnfo n! H(E) l m 4H ( E ) . n=o anf n.
p
(proposition 3.17).
f o r every
This completes t h e proof of
11 = 0
c o n t i n u o u s semi-norm
Since U i s balanced y ( f ) =
continuous
Hence
T~
Hence, by ( a ) ,
0
f o r e v e r y p o s i t i v e number 8 and s o
If U is a baZanced o p e n s u b s e t o f a
Corollary 4.20
l o c a l l y c o n v e x s p a c e E t h e n t h e b a l a n c e d h u Z l of e v e r y b o u n d i n g s u b s e t o f U is b o u n d i n g . I t i s n o t known i f t h e c o n v e x h u l l o f a b o u n d i n g s u b s e t of E i s s t i l l bounding.
T h i s would amount t o showing t h a t
t h e bounding s e t s form a bornology. Corollary 4.21
t h e locally c o n v e x s p a c e E c o n t a i n s a non-precornpact b o u n d i n g s e t t h e n T # T~ o n H ( E ) . Proof
I f
Suppose A i s a non-precompact bounding s e t .
p r o v e o u r r e s u l t we show t h a t t h e s e m i - n o r m
on H(E) i s n o t
T
w
continuous.
Suppose
pA were T
and p o r t e d by t h e compact s u b s e t K o f E . (IfllA 5 C ( V )
IlfIIV
5
IlfnIIA
w
To
I/fllA
continuous
For every
neighbourhood V o f K t h e r e would e x i s t C(V) Hence
pA(f) =
> o such t h a t
for every f i n H(E). C(V) IlfnI/V f o r a l l f i n H ( E ) a n d e v e r y p o s i t i v e
i n t e g e r n and t h u s
-
Letting n
m
K we h a v e
V-
175
Holomolphic functions on Banach spaces we s e e t h a t
Ilfl(A
Ilflh
I l f I I v.
5
On l e t t i n g
IIfIIK f o r e v e r y f i n H ( E ) .
By t h e Hahn-Banach t h e o r e m i t f o l l o w s t h a t A l i e s i n t h e S i n c e K i s compact i t s c l o s e d
c l o s e d convex h u l l o f K .
convex h u l l i s precompact and t h i s c o n t r a d i c t i o n completes the proof. Our n e x t p r o p o s i t i o n , and i t s c o r o l l a r i e s ,
show t h a t
b o u n d i n g s e t s b e h a v e more l i k e compact t h a n bounded s e t s . I f E i s a l o c a l l y convex s p a c e f E H ( E ) fx : E
C be d e f i n e d by f x ( y )
I
that fx EH(E)
and
X E
f(x+y).
=
E let I t i s immediate
f o r every x i n E.
Proposition 4 . 2 2
If E
is a m e t r i z a b l e l o c a l l y c o n v e x s p a c e and A i s a b o u n d i n g s u b s e t o f E t h e n ( f x ) x A i s a T
0
bounded s u b s e t of H ( E )
Proof
for e a c h f i n H ( E ) .
F i r s t l e t K be a f i x e d compact s u b s e t o f E .
I f K1
i s any o t h e r compact s u b s e t o f E t h e n , f o r a given f i n H(E)
3
s i n c e K+K1
i s a compact s u b s e t o f E .
Hence ( f x ) x E K i s a metrizable H(E)
T
and
T
T&
Since E is
bounded s u b s e t o f E . define
t h e same b o u n d e d s u b s e t s o f
and hence
YEA
S i n c e K was a r b i t r a r y ( f )
Y
YEA
is a
a n d we h a v e c o m p l e t e d t h e p r o o f .
T
0
bounded s u b s e t o f H ( E )
Corollary 4 . 2 3
If E i s a m e t r i z a b z e l o c a l l y c o n v e x s p a c e , f E H ( E ) and A i s a b o u n d i n g s u b s e t o f E t h e n t h e r e e x i s t s a neighbourhood V o f z e r o i n E such thatIlfIIA+V < . Proof
By p r o p o s i t i o n 4 . 2 2 ( f x ) X E A i s a
T
bounded s u b s e t
Chapter 4
176 of H(E).
S i n c e E i s m e t r i z a b l e i t i s a l s o a l o c a l l y bounded and hence t h e r e e x i s t s a neighbourhood V o f
subset of H(E)
zero such t h a t sup XEA
llfxIIV
<
Hence s u p If(x+Y)l XEA
m .
=IlfllA+V
<
YEV
Corollary 4.24
I f E is a m e t r i z a b l e l o c a l l y c o n v e x s p a c e
t h e n t h e v e c t o r sum of f i n i t e l y many b o u n d i n g s u b s e t s of E
i s a l s o a bounding s u b s e t o f E . Proof
L e t A1
and A2 b e b o u n d i n g s u b s e t s o f E and l e t
f EH(E).
By p r o p o s i t i o n 4 . 1 8 t h e s e m i - n o r m g~ H ( E )
_ _ _ f
c o n t i n u o u s a n d p r o p o s i t i o n 4 . 2 2 imp i e s t h a t bounded s u b s e t o f H ( E ) .
~-
T
6
(fx)xEA2 i s a
T~
Hence +
and t h u s A1+A2
llgl~lis
A2
<
m
i s a bounding s u b s e t of H ( E ) .
The e x t e n s i o n
t o f i n i t e l y many b o u n d i n g s e t s i s c a r r i e d o u t i n a n o b v i o u s manner and t h i s completes t h e p r o o f . We now i n v e s t i g a t e t h e s i z e o f b o u n d i n g s e t s .
Since
a n y t w o l o c a l l y c o n v e x t o p o l o g i e s w h i c h d e f i n e t h e same c o n t i n u o u s d u a l h a v e t h e same b o u n d e d s e t s , b o u n d i n g s e t s a r e bounded.
On t h e o t h e r h a n d s i n c e e v e r y h o l o m o r p h i c
f u n c t i o n i s c o n t i n u o u s it f o l l o w s t h a t r e l a t i v e l y compact s e t s are always bounding. o f t h e above,
We h a v e a l r e a d y g i v e n a r e f i n e m e n t
c o r o l l a r y 4.10,
w h i c h we may r e s t a t e a s f o l l o w s .
Each b o u n d i n g s u b s e t of a n i n f i n i t e Proposition 4.25 d i m e n s i o n a l l o c a l l y c o n v e x s p a c e E i s a nowhere d e n s e s u b s e t of
E.
Our n e x t t w o r e s u l t s show t h a t f o r a l a r g e c l a s s o f s p a c e s t h e bounding s e t s c o i n c i d e w i t h r e l a t i v e l y compact s e t s . I f t h i s were t r u e o f a l l l o c a l l y c o n v e x s p a c e s t h e n
bounding s e t s would n o t b e v e r y i n t e r e s t i n g .
We p r o v e s o m e
o f o u r r e s u l t s f o r b o u n d i n g s u b s e t s o f Banach s p a c e s .
These
177
Holomorphic functions on Banach spaces can be extended t o a r b i t r a r y l o c a l l y convex spaces q u i t e e a s i l y a n d t o a r b i t r a r y o p e n s e t s i f we r e p l a c e r e l a t i v e l y compact sets by precompact s e t s .
The d i s t i n c t i o n
between r e l a t i v e l y compact and precompact s e t s i n v o l v e s p r o b l e m s o f a n a l y t i c c o n t i n u a t i o n w h i c h we b r i e f l y d i s c u s s
later. Proposition 4.26
Every bounding s u b s e t o f a separabZe
ZocaZZy e o n v e x s p a c e is r e t a t i v e l y c o m p a c t .
quasi-compZete
Irn b e a d e n s e s e q u e n c e i n t h e l o c a l l y n n=l convex space E and l e t A b e a bounding s u b s e t o f E . Let Let cx
Proof
p
E
Let
cs(E) and
vm
=
f J n=1
E
> o be arbitrary.
{xn+x ; p(x)
<
m
( v ~ ) ~ i s =a n~ i n c r e a s i n g
€1. Since
c o u n t a b l e open c o v e r o f E .
11 1,
i s a rC6c o n t i n u o u s
s e m i - n o r m on H(E) i t f o l l o w s t h a t t h e r e e x i s t C > o a n d N a p o s i t i v e i n t e g e r such t h a t H(E).
n
Ilf(/A
5
IlfllVN
C
On r e p l a c i n g f b y f n , t a k i n g n t h
+m
Ilfla 5
it follows t h a t
By t h e H a h n - B a n a c h
roots,
and l e t t i n g
f o r every f i n H(E).
theorem A l i e s i n t h e closed convex
S i n c e p and
h u l l o f VN,
IlfllVN
for every f i n
E
were a r b i t r a r y it f o l l o w s t h a t
A i s a precompact subset o f E .
S i n c e E is quasi-complete
A i s i n f a c t r e l a t i v e l y compact and t h i s completes
the
proof. Using t h i s r e s u l t w e o b t a i n a f u r t h e r class o f spaces f o r w h i c h t h e same r e s u l t h o l d s . Theorem 4 . 2 7
L e t E be a Banach s p a c e w h i c h is i s o m o r p h i c
t o a s u b s p a c e of C(T),
T a s e q u e n t i a l l y compact H a u s d o r f f
s p a c e , where C ( T ) i s endowed w i t h t h e s u p norm t o p o l o g y . Then t h e b o u n d i n g s u b s e t s of E a r e r e l a t i v e l y c o m p a c t . Proof
Let
CxnIn b e a bounded s e q u e n c e i n subsequence. Then t h e r e e x i s t
E which h a s no c o n v e r g e n t E
> o and t
n,m
E
T with
I X , ( ~ , , ~ )- xm(t - xE m ~ ~ n,m ) ~ = ~ ~ x n 2
178
Chapter 4 CO
f o r a l l n,m.
Now c h o o s e a s u b s e q u e n c e o f ( t n , m ) , ( t k ) k , l ,
which i s d i s c r e t e and c o n v e r g e s t o t e T and a s u b s e q u e n c e m
m
E if of (XnIn=l> (YnIn=lI such t h a t l y n ( t k ) - y m ( t k ) l m S i s a compact s u b s e t o f n,m>k. Let S = I t k l k = l u { t ] .
T a n d (y,
m
i s n o t a r e l a t i v e l y compact s u b s e t o f C(S).
1s) k = l
S i n c e C(S) i s s e p a r a b l e t h e r e e x i s t s by p r o p o s i t i o n 4 . 2 6
f EH(C(S)) such t h a t sup If(yk)
k
I
=
m.
I f RS d e n o t e s t h e
r e s t r i c t i o n m a p p i n g f r o m C(T) t o C(S) t h e n g = f0RSlE
E
H(E)
pro0 f .
and s u p l g ( x n ) l = n
-.
This completes t h e
I f E i s a Banach s p a c e a n d t h e c l o s e d u n i t b a l l B of E'
i s weak* s e q u e n t i a l l y c o m p a c t t h e n we may embed E
i n C(B) a n d h e n c e t h e b o u n d i n g s u b s e t s o f E a r e r e l a t i v e l y compact.
Thus t h e b o u n d i n g s e t s a r e r e l a t i v e l y compact i n
E whenever E '
h a s t h e Radon-Nikodym p r o p e r t y a n d i n
p a r t i c u l a r whenever E i s r e f l e x i v e o r weakly compactly generated.
A complete geometric o r l i n e a r c h a r a c t e r i z a t i o n
o f Banach s p a c e s i n w h i c h a l l b o u n d i n g s e t s a r e r e l a t i v e l y c o m p a c t i s s t i l l n o t known. We now c o n s t r u c t a n e x a m p l e o f a n o n - r e l a t i v e l y c o m p a c t b o u n d i n g s u b s e t o f a Banach s p a c e .
Here,
once
a g a i n , t h e g e o m e t r y o f t h e Banach s p a c e p l a y s an i m p o r t a n t role.
S i n c e i t c a n e a s i l y b e shown t h a t a n o n - r e l a t i v e l y
c o m p a c t b o u n d i n g s u b s e t o f a Banach s p a c e , i f s u c h e x i s t s , c a n b e c o n t i n u o u s l y mapped o n t o a n o n - r e l a t i v e l y c o m p a c t b o u n d i n g s u b s e t o f 1- i t i s n a t u r a l t h a t we i n v e s t i g a t e bounding s u b s e t s o f 1-.
B o u n d i n g s u b s e t s o f 1, h a v e ,
i n f a c t , b e e n c o m p l e t e l y c h a r a c t e r i z e d and we now q u o t e , without proof, t h i s characterization. Theorem 4 . 2 8
( a ) If A i s a bounded s u b s e t o f Z,
following conditions are equivalent; ( i ) A is a bounding s u b s e t o f I,.
then the
179
Holomorphic functions on Banach spaces
(ii)
every sequence ($,),cZL
which converges pointwise
t o z e r o c o n v e r g e s u n i f o r m l y t o z e r o on A, ( i i i ) t h e r e is no s e q u e n c e (a
)
i n A w h i c h is e q u i v a t e n t
n n t o t h e u n i t v e c t o r b a s i s in l 1
(iv)
-
t h e r e is no c o n t i n u o u s l i n e a r mapping T: w i t h c o n t i n u o u s i n v e r s e , T - ~ : T ( Z 1)
\
2
+
zl,
m
such
t h a t T ( B ) C C o n v e x H u l l o f A uhere B is t h e u n i t b a l t
o f 11’ (v)
A i s weakly c o n d i t i o n a l t y compact
(i.e.
each sequence
i n A c o n t a i n s a weak Cauchy s u b s e q u e n c e ) . (b)
The c o n v e x h u l l of a b o u n d i n g s u b s e t of 2 , is b o u n d i n g .
(c)
E v e r y bounded s u b s e t of c o is a b o u n d i n g s u b s e t of 1 ,
We now s h o w t h a t bounding sets.
z m
contains
T h i s , of c o u r s e ,
c l o s e d non-compact
is a l s o a c o r o l l a r y
of
theorem 4.28. We f i r s t n e e d s o m e p r e l i m i n a r y r e s u l t s n I f p(z) = 1 + aiZ1 + z is a p o l y n o m i a l
i:il
Lemma 4 . 2 9
of one c o m p t e x v a r i a b l e t h e n s u
IlzTi=1
By P a r s e v a l ‘ s f o r m u l a
Proof
2 + 1 = 1 ‘/2 n
Ip(z)(
1
C(xnIncl
E
zm;
defined on 2 ,
Lemma 4.30
xn = o i f n let
Let p
L
S)
1
and
(S)
2
=
and if f i s a f u n c t i o n
IIfIIS = s u p { f ( x )
B(nzm)
Ip(eie) I2d0
-71 TI
If S i s a s u b s e t of N l e t 2 , m
fi
E
I; 7
x
E
Z,(S),
IIxII 5 1 1 .
o be a r b i t r a r y .
Then
t h e r e e x i s t s an i n f i n i t e s u b s e t S of N s u c h t h a t IIplls 5
E.
180
Chapter 4 Let P =
Proof
where A
E
Ls(("Im).
T h e n we c a n c h o o s e a s e q u e n c e o f m u t u a l l y d i s j o i n t
i s false.
(Sn)n, such t h a t
infinite subsets of N,
By h o m o g e n e i t y w e c a n f i n d x1
Iblll(
and P ( x l ) I(x211 ~ , 1
1,
B Y lemma 4 . 2 9
sup Ixl.1
we h a v e s h o w n IIpII S i m i l a r l y IIp
IPII
L
since
1 1s g u
IF 11p Ilf' ( 1
Suppose t h e c o n c l u s i o n
>
0 si i=l <
-
Now l e t u
=
P(x2) =
s 1 u s2 s4
->
n
=
( 0 ,
. . .
1, o nt'1
.o,
7
=u
. . . .
with
C then
s l n s2
=
n=l
Theorem 4 . 3 1
a b o u n d i n g s u b s e t of
A i s
Suppose A i s n o t bounding.
e x i s t s an e n t i r e f u n c t i o n f on 2,
f o r every x i n
I,,
)
f o r each
pos it i o n A is a closed
fun}.
s u b s e t o f 1-
00
Since
E
(S2)
and h e n c e w e h a v e completed t h e p r o o f
bounded non-compact
-
2,
This is impossible f o r a l l n
m
as n
E
GE a n d a p p l y i n g t h e s a m e m e t h o d
p o s i t i v e i n t e g e r n and l e t A
Proof
E
> ATE. -
(JT)".
E
for all n.
>
If A
E.
JTE.
I P ( x ~ + xX I ) /
Sn
and x2
Zm(S1)
E
IlPII
2.,
By c o r o l l a r y 4 . 1 9 t h e r e
such t h a t
we c a n c h o o s e ( i f n e c e s s a r y
181
Holomorphic functions on Banach spaces
z l m ]a n
b y r e s t r i c t i n g f t o z,(S) Q1
positive integers,
(nj)j=l,
( n j
An. J
u
E
j
zm)
An,
where
I
S1 i n f i n i t e s u c h t h a t k
c IxTz1 o < r L n l
(,1) n
su
for all j .
1"
) = 1 for all j .
Ls ( " j
0
belongs t o H(Z-1 'n'f(o) j=1 P f ( o ) (uj)
The f u n c t i o n g =
h
such t h a t
-> 25 >
l-(uj)\l'nj n.! I
increasing sequence of
1
4
For each i n t e g e r j
- dnJn(o) n ' j' S1 a n d
.
Cet k l
( X U ~ ) ~ ~ -5 ' ;l!. 1~ ~
]]An
and
let = 1.
Choose
This i s possible
s1
1
b y lemma 4 . 3 0 . Now s u p p o s e k i km
E
and Si h a v e b e e n c h o s e n f o r
Sm-1 a n d l e t C,
4
s u c h t h a t k,
S,
=(kl,
. .
.,km}.
1ziLm-1.
Choose
Choose S m ~ S m - l
, Sm i n f i n i t e a n d
S i n c e C m i s a f i n i t e s e t t h e c l o s e d u n i t b a l l o f loo ( C m )
i s c o m p a c t a n d w e a r e t a k i n g t h e supremum o v e r a c o m p a c t s e t o f a f i n i t e sum o f c o n t i n u o u s f u n c t i o n s e a c h o f w h i c h c a n b e made a r b i t r a r i l y s m a l l b y a n a p p r o p r i a t e c h o i c e o f S m (lemma 4 . 3 0 ) . By i n d u c t i o n we o b t a i n a n i n c r e a s i n g s e q u e n c e o f p o s i t i v e integers
m
(km)m,l.
By r e s t r i c t i n g e v e r y t h i n g t o l , ( S )
where
m
S = ul{km}
l e t CA
For each m
we may s u p p o s e k m = m f o r a l l m .
= S\Cm
= (n;n>m}.
Each z
i n a u n i q u e manner a s x+y where x
E
E
Z,(S)
lm(C,,,),
can be w r i t t e n y
E
lm(Clm)
Chapter 4
182
II.II
and = sup (nxiiSiiyii). U s i n g t h e a b o v e n o t a t i o n we d e f i n e f o r e a c h p o s i t i v e i n t e g e r
m , T,
@ ( n k m Z,(S)),
E
by
fink
Tm(z) =
nkm
1,
t h i s implies that
m= 1
IAj I
i s a polynomial
(x) km*
m
Tm(x) converges f o r each x i n t,(S)
and hence, by theorem 2.28,
been chosen,
n
(x) = Ank
1, and
1"
m= 1
Trnc H ( l w ( S ) ) .
i Amurn)! 2 1 f o r i = 1 ITi(l m= 1
of degree s t r i c t l y l e s s than n
( i n A)
By C a u c h y ' s i n e q u a l i t i e s t h e r e e x i s t s A . J+1
. .
Let a = ( A 1 J A 2 , .we
.,AnJ
have ITj(a)l
=
lTj(A1,
By c o n s t r u c t i o n
. .
. .
.)
.,Aj,o,
E
z,(S).
. .
E
CC,
kj+l'
IAj+ll
. ) [ = I T j ( cj
m= 1
Amurn)[ 2 1 .
2 1 and t h i s c o n t r a d i c t s t h e
1"
and completes t h e p r o o f .
fact that
d
m
j=l Tj
J
E
H(Z,(S))
5 1,
For each i n t e g e r j
Hence l i m s u p IT.(a)ll'kj j
,..., j .
183
Holomoiphic functions on Banach spaces
We d o n o t know o f a n y B a n a c h s p a c e E f o r w h i c h T
#
'c6
o n H(E) a n d i n w h i c h t h e c l o s e d b o u n d i n g s e t s
a r e compact. § 4 . 3 H O L O M O R P H I C FUNCTIONS O N B A N A C H SPACES WITH A N
BASIS
UNCONDITIONAL -
I n t h i s s e c t i o n we l e a v e c o u n t e r e x a m p l e s a s i d e a n d o n H(U) i f U i s a b a l a n c e d o p e n s u b s e t 6 o f a Banach s p a c e w i t h a n u n c o n d i t i o n a l b a s i s . The p r o o f show t h a t
T~
involves,
as i n t h e previous section, geometric properties
= T
o f Banach s p a c e s .
A l s o we s e e t h e u s e f u l n e s s o f a b a s i s
o r a coordinate system
-
used h e r e re-appear
i n studying holomorphic functions
-
and v a r i a t i o n s o f t h e t e c h n i q u e s
on f u l l y n u c l e a r s p a c e s i n t h e n e x t c h a p t e r . We o r d e r t h e f i n i t e s u b s e t s o f N ,
t h e n a t u r a l numbers,
by s e t i n c l u s i o n . Definition 4.33
i n a Banach s p a c e E i s
(en);=1
A basis
c a l l e d an u n c o n d i t i o n a l b a s i s i f f o r any x =
1"
n =1
xn en
E
E
-
l i m J+
JCN,J
( i . e . given
E>O
IIx
-
1
i E J
finite
x
j
'
e.11 = o
t h e r e e x i s t s a f i n i t e s u b s e t J E of N such
t h a t for any f i n i t e s u b s e t J o f N w h i c h c o n t a i n s J c we have
IIx -
1
j EJ
xjej
11
5
E)
.
l p , 1 5 p < -, a n d c o a l l h a v e u n c o n d i t i o n a l b a s e s and t h e f i n i t e p r o d u c t o f s p a c e s w i t h an u n c o n d i t i o n a l b a s i s a l s o h a s an u n c o n d i t i o n a l b a s i s .
The s p a c e o f a l l
c o n v e r g e n t s e r i e s i s an example o f a Banach s p a c e ( w i t h a b a s i s ) which h a s n o t a n u n c o n d i t i o n a l b a s i s . The f o l l o w i n g r e s u l t i s w e l l known a n d c o n s e q u e n t l y we d o not include a proof.
Chapter 4
184
Lemma 4 . 3 4
g i v e n by
-
I f E is a Banach s p a c e with an u n c o n d i t i o n a 2 m
-
( e n ) n = l b t h e n t h e b i l i n e a r mapping from l m x E
basis,
1"
((Bn)n=19
il=l
xnen)
1"
n=l
BnXnen
E
is w e l l d e f i n e d and c o n t i n u o u s . The a b o v e p r o p e r t y i n f a c t c h a r a c t e r i s e s B a n a c h s p a c e s with an unconditional b a s i s . Lemma 4.34
a l l o w s us t o renorm E w i t h an e q u i v a l e n t
b u t more u s e f u l norm.
L e t ( E , \ ] 11)
Lemma 4 . 3 5
b e a Banach s p a c e with an
CO
( e n ) n = l , t h e n t h e norm
unconditional basis,
i s e q u i v a l e n t to t h e o r i g i n a l norm o n E . H e n c e f o r t h we s h a l l a s s u m e t h a t t h e g i v e n n o r m o n E satisfies
lm x e n=l n n
E
IIC" n =1
111
xnen[[ = sup XnXn enII f o r a l l ~cl\r,J finite ~ E J 1 E and i n t h i s c a s e t h e b i l i n e a r mapping o f
bj15
lemma 4 . 3 4 h a s n o r m 1 . We now i n t r o d u c e s o m e n o t a t i o n f o r t h e B a n a c h s p a c e E
with unconditional basis If o 5
m i n 5 "
generated by e
j'
n =
(e )" n n=l'
w i l l
E:
5 n.
rn < j
.
denote t h e closed subspace of E I f m=o we w r i t e E
n
and i f
m
m we write E Note t h a t E o = E . We l e t B d e n o t e m t h e u n i t b a l l o f E a n d l e t B m d e n o t e t h e u n i t b a l l o f 2,.
n Let n m ,
0
5 m 5 n 5
n o n t o Em w h e r e
IT
n
and
denote the natural projection of E
m, 71
m
are given t h e i r obvious meanings.
'The f o l l o w i n g s i m p l e f a c t s a r e e a s i l y v e r i f i e d ,
185
Holomorphic functions on Banach spaces (a)
T;(B)
(b)
(B,
x
Now l e t
= B
n
E*m
B)n
E:
=
m
(Bn)n=l
E
B f l E.:
lm
Now suppose S1,
then
. .,
.
Sm-l
is a finite increasing
s e q u e n c e o f p o s i t i v e i n t e g e r s a n d B1,
. . .,
B,
are non-
We d e f i n e t h e s e q u e n c e ( a n ) n a s
n e g a t i v e r e a l numbers. f o 11o w s
B1' a
=
n
n 5 'i-1
s1 < n i S
i'
25izm-1
i f n > Sm-l
'm a n d we l e t
lAil
5 1,
z.
E
E
'i
si-1
w h e r e S o = o a n d Sm =
Lemma 4 . 3 6
basis,
L e t E be a Banach s p a c e w i t h an u n c o n d i t i o n a Z (en)n=l and norm s a t i s f y i n g t h e c o n d i t i o n o f m
Zemrna 4 . 3 5 . (a) i f
Then m
(pn)n=l
E
co,
(Bn)n
x B
i s a r e l a t i v e l y compact
subset of E , (b) i f
m}
(Bi)y=l i s any f i n i t e sequence
o f p o s i t i v e reaZ
186
B
Chapfer 4
. .,
*
S1'
B1'
3
6,
* ,
*
f
Sm-1
yB f o r any f i n i t e i n c r e a s i n g s e q u e n c e
. . .,
of p o s i t i v e i n t e g e r s S1, Proof
(a)
Sm-l.
(Bn)n=l x B i s a bounded s u b s e t of E .
By lemma 4 . 3 4
Hence i t s u f f i c e s t o n o t e t h a t s u p
1-
;
(b)
x e n n
n=l
If z
E
yB t h e n
E
z
i=l
1' Bi 5
implies z
E
1 we h a v e
B
=
m z
B~
S1'
*
*
.,
B,,
*
*
.,Bm
E
Hence z =
si
E ~ i - l
Sm-1
o as j
z i where z i
-.
i
Bnxnenll
n=j
T
y-lm
=
a l l i where So = 1 and S since
lBnl
B} 5 s u p nlj
{Ill"
.
-
and
03
i '
B n ES
1m
i=l
for
i-1
Bi(;,zi)
and
1
f o r a l l i and t h i s
This completes t h e proof.
We now d e c o m p o s e h o m o g e n e o u s p o l y n o m i a l s b y u s i n g t h e d e c o m p o s i t i o n g i v e n by t h e b a s i s . Let P
E
@("E),
A e &S(nE) k
i n t e g e r then E = E +Ek
A
and A = P .
and h e n c e each x
i n a u n i q u e f a s h i o n a s x + x 2 where x1 1
= P(x1+x2)
P(x)
continuous k
by A j .
-
= A(x1+x21n
The mapping x
E
If k is a positive
E
=
In
j =o
(y)
E
E
Ek
E can be w r i t t e n
and x 2
A(xl)j(x2)
E
Ek.
Hence
n- j
~ ( x ~ ) j ( nx- j~ ) d e f i n e s
a
n-homogeneous poZynomiaZ w h i c h we shaZZ d e n o t e
We t h e n h a v e P
=
1"
j=o
k
A.. J
W i t h t h e a b o v e n o t a t i o n we o b t a i n t h e f o l l o w i n g lemma.
187
Holornorphic functions on Banach spaces
If K i s a c o m p a c t s u b s e t of E ' m - 1
Lemma 4 . 3 7
f o r a22 j , Proof
5 j 'n
(a) Let
x =
Irn
xm
Es m-1
i=l
Y E
K and
(y
=
1 2ri
. +
Hence,
l:i
Bixi)
B be arbitrary.
Then
i E s i - l, 2 5 i Z r n - 1 , x 1 € E S 1
and
by t h e Cauchy i n t e g r a l f o r m u l a ,
=
(7)A(y+Irn-' i =1
P(y
lAl=
X E
S
x i ' where xi
A 'm-1
j
0
then
+
i=l
Pixi)'
Pixi
+
n-j (Bmxm)
A Bmxm)
dX
A j+ 1
1
I t now s u f f i c e s t o n o t e t h a t
Im-' Bixi i=l
+ A Bmxm
E:
and t o take t h e supremum o f b o t h s i d e s ( b ) U s i n g t h e a b o v e n o t a t i o n we h a v e
Chapter 4
188 Hence y +
1"
$ i ~ i + 'mxm+ 1 = y i=l
+
iy = - l l! 3 i x i
B m ( x m + mx t l 1
+
and t h i s p r o v e s t h e d e s i r e d e q u a l i t y In order t o avoid unnecessary subscripts i n t h e f o l l o w i n g p r o p o s i t i o n we s h a l l a d o p t t h e f o l l o w i n g c o n v e n t i o n ; when w e s a y I t b y t a k i n g s u b s e q u e n c e s i f n e c e s s a r y " t h e n we s h a l l assume w i t h o u t l o s s o f g e n e r a l i t y t h a t t h e o r i g i n a l sequence has t h e d e s i r e d p r o p e r t y .
If U i s a baZanced o p e n s u b s e t of a Banach
Theorem 4 . 3 8
on H(U).
s p a c e E w i t h an u n c o n d i t i o n a Z b a s i s t h e n -ccw = -c6
is the
S i n c e E i s a n o r m e d l i n e a r s p a c e -c6
Proof
b o r n o l o g i c a l t o p o l o g y a s s o c i a t e d w i t h -ccw a n d h e n c e i t s u f f i c e s t o show t h a t a n y Banach v a l u e d
bounded
T~
h e n c e T6 b o u n d e d ) l i n e a r f u n c t i o n o n H(U)
is
Tcw
(and
continuous.
The i d e a o f t h e p r o o f i s t o f i r s t
Let T b e s u c h a f u n c t i o n .
show, u s i n g t h e f a c t t h a t T i s
T~
continuous,
i s s u p p o r t e d b y a c e r t a i n open s e t .
that T
Then u s i n g i n d u c t i o n
we c h i p a w a y a t t h i s o p e n s e t a n d s h o w t h a t T i s s u p p o r t e d by a s e q u e n c e o f open s e t s which t e n d t o a compact s e t . Let B b e t h e open u n i t b a l l o f E . (nB);,l
i s an open c o v e r i n g o f E and h e n c e ,
confinuous, such t h a t U
If U = E,
there exist a positive integer n
llT(f)
11
C
IlfIln
then
since T is 0
T~
and C > o
f o r e v e r y f i n H(E)
(*).
If
0
# E l e t k d e n o t e t h e s e t o f a l l compact b a l a n c e d s u b s e t s
o f U which l i e i n En each K
u
E
3?
f o r some p o s i t i v e i n t e g e r n . 1
For
l e t V K = K + Td(K,,&U)B w h e r e d(K,,&U) i s t h e
VK i s open and d i s t a n c e from K t o t h e complement o f U . VK = u . S i n c e E i s s e p a r a b l e we c a n c h o o s e f r o m t h i s K C k
189
Holomophic functions on Banach spaces covering a countable subcover o f U,
is
m
(VK ) i = l . i
Since T
r and
continuous there exist a positive integer
T~
C ' > o such t h a t
llT(f)
11
L e t ai
5
sup
C' 1
=
4
. ,r
i=l,.
d(Ki,AU)
llfllv
f o r e v e r y f i n H(U)
(**).
Ki f o r i = 1 , . . . , r and suppose
f o r i = l , , . . rif U # E .
S
K i C E
I f U=E l e t r = l , K = { o ) and al=n 1
1
:
Then ( * ) a n d ( * * ) i m p l y t h a t t h e r e e x i s t s C > o s u c h t h a t 1
f o r a l l Pn
E
L?(nE) 2 n d f o r a i l n
Now s u p p o s e we a r e g i v e n m + l p o s i t i v e n u m b e r s C , Y @ 1 J ' * . J 8 , Y a s t r i c t l y i n c r e a s i n g sequence o f p o s i t i v e i n t e g e r s S
3
and y > 1 such t h a t
* * * 'Sm-l
for all ?ll+l
P
n
E
@(nE) and a l l n .
> o a n d y1
T h e n we c l a i m t h a t g i v e n
> y t h e r e e x i s t Cm+l
> o and sm > s
~
-
such t h a t
for all
pn
E
(nE) and f o r a l l n .
Suppose o t h e r w i s e . Then f o r e v e r y p o s i t i v e i n t e g e r n t h e r e a homogeneous p o l y n o m i a l o f d e g r e e k n , s u c h t h a t e x i s t s P,,
~
Chapter 4
we
f i r s t show sup kn =
n
-.
Otherwise, by t a k i n g a subsequence
i f n e c e s s a r y , w e may s u p p o s e k n = M f o r a l l n . By lemma 4 . 3 6 ( b )
and hence t h e sequence
m
'n SUP i = lJ
. .. , T
(Y 1a i l k n IIPnlb K i + ~sB1 a. 1""
I .
1
- - J sm - l ' s m - l
+n
9Bm+l
Y
i s a l o c a l l y bounded s u b s e t o f @ ( - E ) and a subset of H(U).
T
6
n=l
bounded
f o r a l l n t h i s i s i m p o s s i b l e a n d h e n c e s ~ kpn =
m.
t a k i n g a s u b s e q u e n c e i f n e c e s s a r y we may a s s u m e t h a t
By (knIn
i s s t r i c t l y i n c r e a s i n g sequence o f p o s i t i v e i n t e g e r s . Asm-l+n a n d h e n c e , b y (4.4), t h e r e e x i s t s pn = l k n J j =o
Now
f o r each i n t e g e r n ,
jn,
0
2 jn I kn, such t h a t
,
Now s u p p o s e l i m s u p n - m
kn
-
jn
kn
i f n e c e s s a r y we may s u p p o s e
=
E
k -j
>
0 .
By t a k i n g a s u b s e q u e n c e
n n +E 7 n
as n
-
3
2
K
F a II
w
(0
v
K
n
3
.4
c,
F
cd a,
m 0 0
u
c x k
cd
k c, .ti
P
a,
3
0
z
W
vl
x II
z
-4
a,
.4
-
8 w.4 -
cd
w
F
w
c
W
I
-
F
.ri
F
m
A
E
I
d
+
.ri
x
.4
a,
.4
w.4 V
x
a,
K
'n Y
K
+ 3 I
F
v
cl
n
K
F
E d m II w -4
Y
V
II
-
n
x
V
>
F
-n
4 -
I
3
w
.4
E m
0
c4
111
.4
a,
F:
c,
+F
.rl
k a, M a,
z
7 m
V
a
A
0
P
<+I k cd
Y
rd
F
.rd w .4
W
a,
II
d
. t i
x
a, eII
w
x
II
w .4
8
w K el
c, a,
F
rl
V
c cd a, k
0 Cr,
K
a,
n .4
K
.4
x
+ E
,-I I
VI A W.4
II N
x
a,
F
K
\o
M
d
E E
d
: a, v)
0
a
c4 3 ln
3 0
z
a,
3:
K a V,
P
x
m
E
I
+3
A
z
+
a
x
d
.4
a,
.4
F
F
II
cd
I
3
+
E m
w.4 II rl
x
3
a, k a, F V
K
+
E
I
d
m
E
I
E
-
m
d r l I +
m
.
. . . .
'
3
3
%
. . *
m
4
v ) m
+
3
II W.4
E
v
u
+
+ x x
V
x
n
F
m
4
.r
EE
3 I
+F
Y
d '
a
n
M
r .
V
a
d
E
192
Chapter 4
S i n c e 6 > o was a r b i t r a r y a n d lirn s u p i s z e r o .
s
s e q u e n c e a n d A m-l jn
i s p o s i t i v e it follows t h a t t h e
E
Hence,
since k
+n
k
E
p(
s
i s a s t r i c t l y increasing
n
the series
nE),
+n
A.m-l
Jn
1"
n=o
- , sm..l'sm-l+" 1' ~ * i + ( a i Y 1 k n ~ ~ p n B~ s ~ i = l , . ,r B1' * * *,Bm+l
SUP
.
d e f i n e s an e n t i r e f u n c t i o n on E More o v e r s
A m-1
+n
1
jn
l i m sup
k
sl"
UP i=l,
. . ,r
S
> -
-
m
However i f f = is a
$,,...
l i m sup
n
T
> -
n
1"
(and hence a
consequently l i m n - +
(1
m - 1 ' sm - 1 + n
2 s
Bm+1
1.
E
n.
n=o
* .
H(E)
) bounded
then {gn
~
T(
dnf(o) 'In n! 111
m
g i v e s a c o n t r a d i c t i o n and s o l i m n-+
dnfo n! 'n=o
s u b s e t o f H(E) a n d
T
-
=
k
n
-j
kn
-
Thus t h e above
0 .
n
o as n
__f
m .
We now c o n s i d e r t h i s c a s e . S i n c e r i s f i n i t e and f i x e d and t h e sequence i n f i n i t e we may s u p p o s e ,
(kn)nmZ1 i s
from ( 4 . 2 ) and t a k i n g a subsequence
i f n e c e s s a r y , t h a t t h e r e e x i s t s i,l2izr, s u c h t h a t
for all n.
'
I
kn
193
Holomoiphic functions on Banach spaces
On t h e o t h e r h a n d
(lemma 4 . 3 7 ( a ) )
2
Bm+l
n Y 1 (m (-1 n
II
k
k -j
n
8,
s
n
1
-+n
'/k,
( b y lemma 4 . 3 7 ( b ) ) .
'
II
194
Chapter 4
T h i s i s i m p o s s i b l e s i n c e y 1 > y a n d t h u s we h a v e p r o v e d t h e r e q u i r e d s t e p i n o u r induction argument. Aside
A s i m p l i f i e d v e r s i o n o f t h e above goes a s f o l l o w s ;
i f t h e i n d u c t i o n s t e p d i d n o t w o r k t h e n we c o u l d f i n d jn kn - j n , where is evaluation a t the nth fn= c o o r d i n a t e , such t h a t t h e sequence ( f n ) n did n o t s a t i s f y
4,
If
(4.4).
nth
4,
hn
-
k -j
n n ___ kn
E
> o then the rapid decrease of the
c o o r d i n a t e overcomes t h e g e o m e t r i c growth
first coordinate so that
kn-jn/k
n
--+
1,
fn
E
H(E).
of the
Otherwise
o so t h a t t h e e f f e c t of t h e nth coordinate i s
n e g l i g i b l e and
1n
fn behaves
like
1, 4,j n .
In both cases
we s a w t h a t t h i s l e d t o a c o n t r a d i c t i o n .
We now c o m p l e t e t h e p r o o f o f t h e t h e o r e m . Let
(yn)n d e n o t e a s e q u e n c e o f r e a l numbers,yn
such t h a t
T:=~
Now u s i n g ( 4 . 1 )
'n
= y
< 2'
> 1,
and i f U # E such t h a t
a s t h e f i r s t s t e p i n t h e i n d u c t i o n and
s i n c e ( 4 . 2 ) --J ( 4 . 3 ) we c a n f i n d a s t r i c t l y i n c r e a s i n g sequence of p o s i t i v e integers,
m
( s ~ ) ~ a=n d~ ,(Cn):=l
a
195
Holomorphic functions on Banach spaces s e q u e n c e o f p o s i t i v e numbers s u c h t h a t llTIPn)
II
f o r a l l Pn Let K =
n
F cIT,+l h l . .-Ym) SUP
i=l,
...
@ ( n E ) and a l l n .
E
n i Ki+B , r a i l I p n l I"i
sl,. .,s m + l l J I J * * J
1 m'
where % = ( % , ) , and 1 i f n < s2
< n < s i i+l Since
E
c0 3 K i s a compact s u b s e t o f E .
F o r e a c h i , liicr, l e t L i
= yKi+ya
i
K.
Li
is a
compact s u b s e t o f E and moreover f o r e a c h i
Hence L =
cfj i=l
Li
i s a compact s u b s e t o f U . Moreover i f V i s
any open s e t which c o n t a i n s L t h e r e e x i s t s a p o s i t i v e i n t e g e r nv such t h a t
S i n c e U i s b a l a n c e d we c a n c h o o s e A > 1 s u c h t h a t XL i s a g a i n a compact s u b s e t o f U .
I f W i s any open s u b s e t o f U
w h i c h c o n t a i n s XL t h e n t h e r e e x i s t s a n e i g h b o u r h o o d V o f K s u c h t h a t XLCXVCW. Hence, f o r any f = proposition 3.16
1" dllfo n!
n=o
E
H(U) we h a v e , b y
196
Chapter 4
H e n c e T i s p o r t e d b y t h e c o m p a c t s u b s e t XL o f U . This completes the proof. By m o d i f y i n g t h e a b o v e p r o o f p l a c e of t h e b a s i s )
o n e c a n s h o w t h a t T~ =
whenever E i s a s u b s p a c e o f sense of Shilov.
( u s i n g C e s r . r o sums i n T~
on H(E)
1
L [ o , ~ I Th]o m o g e n e o u s i n t h e
The p r o o f however i s j u s t as d i f f i c u l t
a s t h e a b o v e a n d we d o n o t i n c l u d e i t . §
4.4
F U R T H E R RESULTS A N D EXAMPLES CONCERNING HOLOMORPHIC
FUNCTIONS O N B A N A C H SPACES
Iie c o m m e n c e t h i s s e c t i o n b y e x h i b i t i n g a g e n e r a t i n g f a m i l y o f semi-norms
f o r (H(U),ru),
s u b s e t o f a Banach s p a c e .
U a b a l a n c e d open
We t h e n g i v e a n u m b e r o f e x a m p l e s
a l l o f which i n v o l v e bounding s e t s . Proposition 4.39
L e t U be a baZanced open s u b s e t o f a
Banach s p a c e E .
The
T~
topoZogy on H ( U )
i s generated
b y t h e semi-norms
where
B i s t h e u n i t baZZ o f
C J ( a n ) ~ z Or a n g e s o v e r c
K r a n g e s o v e r t h e compact s u b s e t s o f U .
and
197
Holomorphic functions on Banach spaces Let K b e a c o m p a c t b a l a n c e d s u b s e t o f U a n d l e t
Proof (IJ
E C ~ .
I f V i s any balanced neighbourhood o f K
t h e n t h e r e e x i s t A < 1 and n o , a p o s i t i v e i n t e g e r ,
such t h a t
AV i s a neighboushood o f K and K + a n B C V f o r a l l n 2 n 0 .
By u s i n g t h e C a u c h y i n e q u a l i t i e s w e c a n f i n d c l > o s u c h t h a t
Thus p
is a
K,
T
W
c o n t i n u o u s semi-norm on H(U).
Conversely suppose p is a
Tw
c o n t i n u o u s s e m i - n o r m o n H(U)
p o r t e d by t h e compact b a l a n c e d s u b s e t K o f U .
If X > 1 is
such t h a t A K C U then proposition 3.24 implies t h a t t h e s e m i -norm $(f) =
1” n=o
A”p(-) d n f ( 0 n.
is
c o n t i n u o u s and p o r t e d by
T W
AK.
Moreover p
5 $.
c ( n > > o such t h a t
Fo? e a c h p o s i t i v e - i n t e g e r n t h e r e e x i s t s mf 0 2 c(n) d m f o
IlnK+3
$*(I
f o r e v e r y f i n H(U) a n d a l l m .
f o r e v e r y f i n H(U) a n d a l l m . For each i n t e g e r n choose a p o s i t i v e i n t e g e r jn such t h a t
‘(”)/,j
2
1 for all j
2 jn.
We may a s s u m e , w i t h o u t l o s s o f
Chapter 4
198
m
generality,
t h a t t h e sequence (jn)n=l
increasing.
1'
is s t r i c t l y
1 for j A j2
Now l e t
a. =
-
J The s e q u e n c e
(aj)Yzo
for j
n
< j c j n + l , nL2
lies i n c
and,
for e a c h f i n H ( U ) ,
we h a v e
Hence any
T~
c o n t i n u o u s s e m i - n o r m on H ( U )
i s dominated by
a semi-norm o f t h e r e q u i r e d t y p e and t h i s c o m p l e t e s t h e proof. We now l o o k a t h o l o m o r p h i c f u n c t i o n s o n a c o u n t a b l e d i r e c t sum o f B a n a c h s p a c e s .
P r o p e r t i e s of bounding sets
enable us t o s e t t l e t h e "completion
problem" f o r such
spaces and t h e techniques developed prove useful i n d i s c u s s i n g h o l o m o r p h i c f u n c t i o n s on s p a c e s o f d i s t r i b u t i o n s (chapter 5). E i where e a c h E i is a Let E = 1=1 On H ( E ) , T and ~ T~ d e f i n e t h e same bounded
Proposition 4.40
Banach s p a c e . sets. Proof let
L e t Fn =
p be a
T~
In
i = l Ei
f o r e a c h p o s i t i v e i n t e g e r n and
c o n t i n u o u s s e m i - n o r m on H ( E ) .
without loss of generality t h a t
We may a s s u m e
199
Holomoiphic functions on Banach spaces
p(f)
1-
=
pc-1
n=o
n
f o r every f =
1
n=o
dnf(o) 7
We f i r s t c l a i m t h a t t h e r e e x i s t s a p o s i t i v e
i n H(E). integer n
such t h a t i f f E H ( E )
P
Suppose o t h e r w i s e . can choose P
n’
and f I F n
= o then p ( f )
=
0.
P
T h e n f o r e v e r y p o s i t i v e i n t e g e r n we
a homogeneous p o l y n o m i a l ,
s u c h t h a t pn
= 0
I Fn
Vie now s h o w t h a t t h e s e q u e n c e (Pn):=l is and p(Pn) # 0. l o c a l l y bounded. For each n l e t B denote t h e u n i t b a l l n of En. I f x E E t h e n X E Fn f o r s o me i n t e g e r n . Hence t h e r e
. . .n ,
e x i s t A.>o,
i=l,
such t h a t
By u s i n g t h e b i n o m i a l e x p a n s i o n we c a n f i n d X n + l > o such t h a t
’ [I
l n * l XiBif
IIP.
M +
X+iZl
1.
2n+1
f o r i=l,.
..,
n+l
and by p r o c e e d i n g i n t h i s manner, s i n c e e a c h s t e p o n l y i n v o l v e s a f i n i t e n u m b e r o f p o l y n o m i a l s , we c a n f i n d a sequence o f p o s i t i v e numbers, I I p j IIx+Cm n=l
n Bn
<
m
Hence {Pn,)n,l polynomials.
M+l
m
(Xn)n=1’ such t h a t
for all j.
i s a l o c a l l y bounded f a m i l y o f
S i n c e we o n l y u s e d t h e p r o p e r t y P m
= o
n
f o r each n it follows t h a t {a p 1 is also a locally n n n=l bounded f a m i l y o f p o l y n o m i a l s f o r any s e q u e n c e o f s c a l a r s co
(an)n=l
*
n L e t a =-p ( p n )
f o r each n .
T h e n {anPnl:=l
i s a l o c a l l y bounded and h e n c e a r 6 bounded s e q u e n c e of
Chapter 4
200
S i n c e p ( a P ) 2 n f o r a l l n and p n n continuous t h i s i s impossible and e s t a b l i s h e s o u r c l a i m .
holomorphic f u n c t i o n s .
is
T~
Let
( f a ) a E A b e an a r b i t r a r y
l e t p be a
T
b o u n d e d s u b s e t o f HIE) a n d
0
c o n t i n u o u s s e m i - n o r m on H ( E ) .
T&
To c o m p l e t e
t h e p r o o f we m u s t s h o w s u p p ( f ) m . By t h e a b o v e we may a = 0 choose a p o s i t i v e i n t e g e r N such t h a t i f f E H ( E ) and f FN
I
then p(f) = ?,(x+y)
'L
F o r e a c h C(EA l e t f a
0 .
f a ( x ) where x E F
=
and
N
YE
H(E) b e d e f i n e d by
E
lm
n=N+1
Since
En.
lac* i s a l o c a l l y bounded f a m i l y i n H(FN)
'fa[
follows
it
FN %
( f a ) a ~ Ai s a l o c a l l y b o u n d e d a n d h e n c e a
that
subset of H(E).
S i n c e fa
'L
I
2.
we h a v e p ( f a ) = p ( f , )
= fa, I
FN
f o r a l l a a n d s o s ~ pp( f a )
pN 'L
= s;p
bounded
T~
p(fa)
m.
This completes
the proof. Proposition 4.41
T~
if a n d only if each E i
Proof
1-
i=l
i =1
Ei
where each E i
Then ( H ( E ) , r ) is complete f o r
space. OT
1"
Let E =
E i s
(H(c"))
I f each Ei (C")
,TI
T
= T ~ , T
~
,
~
,
~
,
is a finite dimensional space.
a n d we h a v e a l r e a d y s e e n ( e x a m p l e 2 . 4 7 )
i s complete.
ow
s u p p o s e a t l e a s t o n e E~
an i n f i n i t e d i m e n s i o n a l Banach s p a c e .
ll$nII
= 1.
an e n t i r e f u n c t i o n f on El
that is
Without l o s s o f
is i n f i n i t e dimensional.
.rn d e n o t e t h e n a t u r a l p r o j e c t i o n f r o m E o n t o E n .
nE A~ w i t h
T
is a f i n i t e dimensional space then
g e n e r a l i t y we may s u p p o s e E l
n let @
is a Banach
Let
For each
By c o r o l l a r y 4 . 1 0 t h e r e e x i s t s with r f ( o ) = 1.
T
~
,
~
20 1
Holomorphic functions on Banach spaces
1"
En and e a c h compact s u b s e t o f E i s c o n t a i n e d a n d n=l c o m p a c t i n some Fm i t f o l l o w s t h a t t h e p a r t i a l sums o f g
Fm =
form a Cauchy s e q u e n c e i n ( H ( E ) , T ~ ) . g
I
We now s h o w t h a t
H(E).
Suppose o t h e r w i s e .
Then t h e r e would e x i s t a convex b a l a n c e d Ilgllv = M <
neighbourhood V o f zero i n E such t h a t
For
03.
e a c h p o s i t i v e i n t e g e r n l e t Bn b e t h e u n i t b a l l o f E n . e x i s t s f o r e a c h n a p o s i t i v e number 6
There
such t h a t &,B,CV.
n
4
Choose n a p o s i t i v e i n t e g e r such t h a t n >
and choose
7
O1
6n x E-B f o r which @ (x) 2 n n
#
- -
e x i s t s a sequence i n El,
I f ( y m )I
m
(ym m = l '
as m
m
By o u r c o n s t r u c t i o n t h e r e
0 .
IlymII 5 2 ,
m
1
T(2X) =
as m
--
4
t h i s shows t h a t g
not complete.
By e x a m p l e
such t h a t
1.24
Ym
7
X E
V for a l l m.
H e n c e ( H ( E ) , T ~ )i s
H(E).
( p (nE)
t
, T ~ )and
( 6( n E ) , 8 )
a r e complete 1,ocally convex spaces f o r each p o s i t i v e i n t e g e r n.
Hence H(E) i s n o t T . S . T o
H(E) i s
n o t T.S.T
0
complete.
complete and hence
,b
a complete l o c a l l y convex s p a c e . TO
,b
=
T~
T
=
To,
T
~
,
~
,
T
~
(H(E),T
0
,b
)
is not
By p r o p o s i t i o n 4 . 4 0
a n d t h u s we h a v e s h o w n t h a t
complete f o r
By c o r o l l a r y 3 . 3 4
,
oTr
(H(E),T) i s n o t
~T 6, . ~
This completes
the proof. Note t h a t t h e above a l s o shows t h a t t h e r e e x i s t
'c6
Chapter 4
202
b o u n d e d s u b s e t s o f H(Cm En) w h i c h a r e n o t l o c a l l y b o u n d e d n =1 whenever a t l e a s t one E space.
n
i s an i n f i n i t e d i m e n s i o n a l Banach
We now b r i e f l y c o n s i d e r a n e x t e n s i o n p r o b l e m w h i c h a r i s e s o n l y in i n f i n i t e d i m e n s i o n a l a n a l y s i s .
If
F is a
s u b s p a c e o f a l o c a l l y c o n v e x s p a c e E when c a n e v e r y h o l o m o r p h i c f u n c t i o n on F b e e x t e n d e d t o a h o l o m o r p h i c f u n c t i o n on E ?
Two i n t e r e s t i n g d i s t i n c t c a s e s o f t h i s
p r o b l e m a r i s e when ( a ) F i s a c l o s e d s u b s p a c e o f E a n d ( b ) when F i s a d e n s e s u b s p a c e o f E .
Problem ( a ) c o n c e r n s
a n a t t e m p t t o f i n d a h o l o m o r p h i c Hahn-Banach
t h e o r e m and
w i l l r e a p p e a r i n o u r d i s c u s s i o n on h o l o m o r p h i c f u n c t i o n s
Example 4 . 4 2 , w h i c h u s e s p r o p e r t i e s
on n u c l e a r s p a c e s .
o f b o u n d i n g s e t s , shows t h a t i n g e n e r a l we d o n o t o b t a i n a Problem (b) i s t h e
p o s i t i v e s o l u t i o n t o t h i s problem.
holomorphic analogue o f f i n d i n g t h e completion of a E x e r c i s e s 1 . 8 9 and 2 . 9 4 a r e r e l a t e d
l o c a l l y convex s p a c e .
t o problems ( a ) and (b) r e s p e c t i v e l y . This example i s devoted t o showing t h a t
Example 4 . 4 2
n o t e v e r y holomorphic f u n c t i o n on c h o l o m o r p h i c f u n c t i o n on l m . m
Let A = (u ) where u = (o,.. n n=l n positive integer n. c
0
. , 1,o
can be extended t o a
. . .)
f o r each
T nth place
A i s a c l o s e d non-compact
subset of
By p r o p o s i t i o n 4 . 2 6 A i s n o t a b o u n d i n g
and o f lm.
s u b s e t o f c o and by t h e o r e m 4 . 3 1 A i s a b o u n d i n g s u b s e t o f NOW suppose each holomorphic function o f co has a
1m .
h o l o m o r p h i c e x t e n s i o n t o lm. f E H ( c ~ )s u c h t h a t
IlfIIA =
a.
By t h e a b o v e t h e r e e x i s t s %
I f f c H( 2,)
%
and f
=
f then
IC 0
11?11,
=
llflk =
a
and t h i s c o n t r a d i c t s t h e f a c t t h a t A i s
a bounding s u b s e t of
lm.
H e n c e we h a v e shown t h a t t h e r e
e x i s t h o l o m o r p h i c f u n c t i o n s on c o w h i c h c a n n o t b e e x t e n d e d h o l o m o r p h i c a l l y t o tm.
203
Holomoiphic functions on Banach spaces
I f E i s a l o c a l l y convex space w i t h completion E t h e n
t h e r e e x i s t s a subspace of E , E u J
which i s c h a r a c t e r i z e d by
the following properties (1) E (2)
r.
C E , C
E,
each holomorphic f u n c t i o n on E can b e extended t o a h o l o m o r p h i c f u n c t i o n on Eo,
(3)
fi
A
F C E , F a s u b s p a c e of E , and e a c h h o l o m o r p h i c f u n c t i o n on E c a n b e e x t e n d e d t o a h o l o m o r p h i c f u n c t i o n on F t h e n F C E L o .
if E C
E & i s c a l l e d t h e h o l o m o r p h i c c o m p l e t . i o n of E .
Proposition
4.43
space t h e n E6
=
u
If E
i s a metrizable l o c a l l y convex
A
i s t h e c l o s u r e of
where
ACE, A bounding
fi
in E
A
Let E B =
Proof
(cn)nC E
Ux
-
I f 5 E ECp t h e n t h e r e e x i s t s
ACE A bounding
s u c h t h a t 5,
-
.5 a s n
m.
Since 5
EB.
E
Ls)
n
n
and 5
E
and h e n c e s u p [ f ( S n ) l <
l i m f ( s ) e x i s t s f o r e v e r y f i n H(E)
- - + n
f o r every f i n H(E).
E
i s a bounding subset o f E
Thus {En),
On t h e o t h e r h a n d i f
T h i s shows t h a t E @ C E B .
A i s a bounding s u b s e t o f E and f
E
H(E) t h e n b y c o r o l l a r y 4 . 2 3
t h e r e e x i s t s a convex balanced neighbourhood V of z e r o i n
E such t h a t about
Ilf[(A+V <
a.
By u s i n g T a y l o r s e r i e s e x p a n s i o n s
p o i n t s o f A w e f i n d t h a t t h e r e e x i s t s a holomorphic
function
1
%
o n A+W s u c h t h a t f
IA + V A
i n t e r i o r o f t h e closure of V i n E. t h a t rCEOa n d h e n c e Eu
= EB.
-
= f
'
A+V
where
W is
the
S i n c e A + W 3 K t h i s shows
This completes t h e proof.
204
Chapter 4 P r o p o s i t i o n 4.43 and t h e s o l u t i o n t o t h e Levi problem
may b e u s e d t o p r o v e t h e f o l l o w i n g r e s u l t . Proposition 4.44
If E i s an i n f i n i t e d i m e n s i o n a Z
m e t r i z a b l e ZocaZZy c o n v e x s p a c e of c o u n t a b l e a l g e b r a i c d i m e n s i o n t h e n E i s hoZomorphicaZZy c o m p l e t e if and o n l y
if E a d m i t s a c o n t i n u o u s norm. I n p a r t i c u l a r p r o p o s i t i o n 4 . 4 4 s a y s t h a t an i n f i n i t e countable a l g e b r a i c dimension
d i m e n s i o n a l normed s p a c e o f
i s holomorphically complete. 1 4 . 5 EXERCISES
Let X b e an H a u s d o r f f
4.45
t o p o l o g i c a l s p a c e and l e t
b e t h e s p a c e o f bounded c o n t i n u o u s complex v a l u e d
,fb(X)
f u n c t i o n s on X w i t h t h e s u p norm t o p o l o g y .
Show t h a t f
i s a r e a l o r complex extreme p o i n t o f t h e u n i t s p h e r e o f g b ( X ) i f and o n l y i f 4.46"
( f ( x )\
= 1 f o r every x i n X.
Let E and F b e complex Banach s p a c e s w i t h o p e n u n i t
b a l l s U and V r e s p e c t i v e l y .
Show t h a t e v e r y f
E
H(U;V)
w i t h d f ( o ) = L and f ( o ) =o i s l i n e a r i f and o n l y i f L i s m
a complex e x t r e m e p o i n t o f t h e u n i t b a l l o f H (U,v)
4.47*
Let f b e a c o n t i n u o u s f u n c t i o n mapping t h e c l o s e d
u n i t d i s c o f t h e complex p l a n e i n t o a complex Banach Suppose f i s holomorphic on t h e open d i s c .
algebra B. If
1x1
( f(h)
1,
5 1
(1 la
= 1 show t h a t
4.48"
d e n o t e s t h e s p e c t r a l r a d i u s ) whenever
(f(x)l,
5 1 for a l l h,IXI
2
Let B b e a Banach a l g e b r a a n d l e t f : D = { z c
b e a n a n a l y t i c f u n c t i o n s u c h t h a t f ( o ) = o and for all z
z
E
D\{o)
4.49 -
1.
E
D.
or
Show t h a t e i t h e r I f ( z ) If(z)la
5
IzI
lo
<
c
:lzl
(f(z)
I,
5 1
( z ( for all
f o r a l l z i n D.
L e t U b e an o p e n s u b s e t o f a Banach s p a c e E which
B
205
Holomophic functions on Banach spaces c o n t a i n s t h e o r i g i n and l e t f
-
spectrum of f t h a t f-XI
as I X E C ;
,a(f),
: V
We d e f i n e t h e
H(U;E).
E
-fi
V,W
open, OEV, W C U such
W i s a b i h o l o m o r p h i c mapping}.
Show t h a t
o(f) = o(df(o)).
Let E b e a B a n a c h s p a c e w i t h o p e n u n i t b a l l B ,
4.50*
For
each x i n E l e t
-
K(x) = : B
If f
f4
E
II+II
E';
= sup
= 1)
E then the numerical range of f,W(f),
4
d e f i n e d as { d ( f ( x ) ) ; IW(f)l
= 4(x)
(1x1
; A
E K ( x ) , IIxI( = 1 1 .
is
Let
W(f)).
E
I f f I B i s h o l o m o r p h i c show t h a t
m w h e r e k = 1, k 1= e a n d k m = m / m - 1
If f ( o )
for m 2 2.
= o
a n d W(f) i s r e a l show t h a t f i s a l i n e a r m a p p i n g . 4.51*
-
L e t E and F b e Banach s p a c e s w i t h open u n i t b a l l s
U and V r e s p e c t i v e l y .
Let f
: U
V be a biholomorphic
mapping from U t o V ( i . e .
f i s holomorphic and b i j e c t i v e
and f - l
I f f ( o ) = o show t h a t f i s t h e
i s holomorphic).
restriction t o U of a linear isometry of E onto F. 4.52* -
6
h
Let B b e t h e open u n i t b a l l o f a Banach s p a c e .
o and suppose f
E
C ,
[hi
K O = 1 , K1
E
H(B;g)
5 1 and x E B . = e a n d Km
=
s a t i s f i e s I(x+Af(x) Show t h a t
mm/m-l
for m
1 d n f1 2.
( 0 )
7 (1
Let
11
2 1+6 for all 2 Km6 w h e r e
By c o n s i d e r i n g
t h e c a s e 6 = o deduce t h a t t h e i d e n t i t y mapping on E i s a r e a l e x t r e m e p o i n t o f t h e u n i t b a l l o f Hm(B;E) endowed w i t h t h e s u p norm t o p o l o g y . 4.53*
L e t L(H) b e t h e a l g e b r a o f a l l b o u n d e d l i n e a r
o p e r a t o r s from t h e H i l b e r t s p a c e H i n t o i t s e l f and l e t U b e t h e open u n i t b a l l o f L(H). t h a t t h e mapping
I f S E U and A = 1-S*S
show
206 2
E
Chapter 4 L(H)
___f
(s-z)( I - ~ * z ) - ~ A ' / ~
A*-'/2
i s a b i h o l o m o r p h i c mapping of U i n t o i t s e l f which h a s S(1+A1/q-' 4.54*
a s a unique f i x e d p o i n t .
L e t U b e a b o u n d e d o p e n s u b s e t o f a Banach s p a c e
and l e t f EH(U;U).
I f d ( f ( U ) , % U ) > o show t h a t f h a s a
unique f i x e d p o i n t . 4.55 -
If f
-
: c
C i s d e f i n e d by
1"
m
f((xn}n=l ) =
show t h a t f
n=2
f =
I f E i s a B a n a c h s p a c e , (bne
1" 0:
n=1
E
H(co)
and
-1
m
4.56
E
E l
all n,
and
H(E) show t h a t r f ( x ) i s a c o n s t a n t and
find t h i s constant
4.57*
L e t E a n d F b e Banach s p a c e s a n d l e t f e H ( E ; F ) .
that A =
EF';
f
( 0 ) )
i s a s e t of f i r s t category
I f E i s s e p a r a b l e show t h a t t h e r e e x i s t s g
i n E.
such t h a t rf = r
4.58
r + o f( 0 ) > r
Show
. g
E
H(E)
G i v e an e x a m p l e o f a n i n f i n i t e d i m e n s i o n a l Banach
s p a c e E and an f i n H(E) many c o o r d i n a t e s b u t r variables. 4.59* -
f
such t h a t f "depends"
I f T i s an i n f i n i t e d i s c r e t e s e t and f
i s such t h a t r
show t h a t
$
on i n f i n i t e l y
" d e p e n d s " o n l y on f i n i t e l y many
Ir,(x)
-
rf(y)
E
I
H(c
0
(T))
< 1lx-y
11
f o r a l l x, y i n co(T). 4.60
L e t E b e a Banach s p a c e .
f EH(E) Frgchet
f o r which r f topology
T
+
0~
Show t h a t t h e s e t o f a l l
can be g i v e n a unique
which i s f i n e r t h a n t h e compact open
207
Holomorphic functions on Banach spaces topology
.
Show t h a t
((fEH(E); rf
E
- 1 , ~ ) is a locally m
+
convex
Fr6chet algebra 4.61 -
I f e a c h c o m p a c t s u b s e t o f a Banach s p a c e E l i e s i n a
s e p a r a b l e c o m p l e m e n t e d s u b s p a c e show t h a t t h e c l o s e d
bounding s u b s e t s o f E a r e compact.Using t h i s r e s u l t g i v e an e x a m p l e o f a Banach s p a c e whose c l o s e d b o u n d i n g s e t s a r e a l l compact b u t which i s n o t a weakly c o m p a c t l y g e n e r a t e d Banach s p a c e .
4.62
By u s i n g b o u n d i n g s e t s show t h a t l m d o e s n o t c o n t a i n
any i n f i n i t e d i m e n s i o n a l s e p a r a b l e complemented s u b s p a c e s . 4.63* -
Let f
E
H
G
(U;F)
where U i s an open s u b s e t o f a l o c a l l y
c o n v e x s p a c e E a n d F i s a Banach s p a c e whose d u a l b a l l i s
weak* s e q u e n t i a l l y c o m p a c t .
Show t h a t f
E
HHY(U;F) i f
gof EH(U) f o r e v e r y g i n H(F). 4.64*
gl
I f f E H ( c o ) show t h a t t h e r e e x i s t s g e H(1,) i f and o n l y i f r = + m . f -
CO
4.65
L e t E b e a Banach s p a c e a n d l e t td,);=,
n u l l sequence i n E ' .
Let
m
b e a weak*
(kn)n=l be a s t r i c t l y i n c r e a s i n g
s e q u e n c e o f p o s i t i v e i n t e g e r s and f o r e a c h n l e t j negative i n t e g e r with o 5 j n 2 kn. jn
f
such t h a t
= f
=
kn-jn
1" 6, 4, n=l
be a non-
Show t h a t
H(E) i f a n d o n l y i f l i m i n f nm
E
n
kn-jn
kn
is positive. 4.66
If E =
1"
n=1
E
n
w h e r e e a c h E n i s a Banach s p a c e w i t h
a n u n c o n d i t i o n a l b a s i s show t h a t 4.67 E,
T
w
= T~
on H(E).
I f K i s a c o m p a c t b a l a n c e d s u b s e t o f a Banach s p a c e
F i s a Banach s p a c e a n d B i s t h e u n i t b a l l o f E show t h a t
t h e t o p o l o g y o f H(K;F)
i s g e n e r a t e d b y t h e semi-norms
208
Chapter 4
m
where
(an)n=o ranges over co
=*
If f
:
show t h a t f
E
z2
-
C i s d e f i n e d by f ( { x
Z2,
p(L(Z2)).
}m
n n=l
)=I"
2
xn,
n=l
s i n c e it i s s e p a r a b l e , can be
i d e n t i f i e d with a closed subspace of &[o,l]
(say n ( 2 ) ) . 2
Show t h a t t h e r e e x i s t s n o h o l o m o r p h i c f u n c t i o n o n , g [ o . 11 w h o s e r e s t r i c t i o n t o ~ ( 2 ~i s) e q u a l t o f . /
4.69"
Let
fi
b e a c o n t i n u o u s s u r j e c t i o n from 2,
onto co.
Show t h a t t h e i d e n t i t y m a p p i n g from c o t o c o c a n n o t b e l i f t e d t o Z1 t o Zl
c
i.e.
show t h a t no h o l o m o r p h i c m a p p i n g , %
e x i s t s s u c h t h a t n o f = I d on c
.
%
from
f,
I f U i s a b a l a n c e d open s u b s e t o f a Banach s p a c e
4.70*
show t h a t e v e r y n u l l s e q u e n c e i n ( H ( U ) , T ) sequence where T =
T
0'
T
w
i s a Mackey n u l l
or T ~ .
Let U b e an o p e n s u b s e t o f a Banach s p a c e E and l e t
4.71* -
F b e a Banach s p a c e .
Let
T~
of
b e t h e t o p o l o g y on H ( U ; F )
uniform convergence o f f u n c t i o n s and t h e i r f i r s t n d e r i v a t i v e s on t h e compact s u b s e t s o f U where n = o , l ,
. . . ,m .
I f E i s i n f i n i t e d i m e n s i o n a l show t h a t
T
~
, n
=o,l,
m , W
Show t h a t
H(U;F). 4.72*
...,
Let
a l l d e f i n e t h e same bounded s u b s e t s o f ( H ( U ; F ) , T ~ )i s c o m p l e t e f o r n = o , l ,
( P a ) @ b e a f a m i l y o f s c a l a r v a l u e d homogeneous
p o l y n o m i a l s on t h e Banach s p a c e E , degree n
.
...,-.
If
sup
I Pa(~)ll/n.
<
m
p a b e i n g homogeneous o f
f o r e v e r y x i n E show t h a t
t h e r e e x i s t s a neighbourhood V o f zero i n E such t h a t
209
Holomorphic functions on Banach spaces
*
4.73
I f E i s a Banach s p a c e show t h a t t h e f o l l o w i n g
conditions are equivalent a) E has t h e approximation property, b ) ( H ( E ) , T ~ )h a s t h e a p p r o x i m a t i o n p r o p e r t y c ) IH(E) @ E i s d e n s e i n ( H ( E ; E ) , T ~ ) .
F o r e a c h p o s i t i v e i n t e g e r n show t h a t
4.74
and hence deduce t h a t
(6' ( n l l ) , @ ) z 2,
( H ( l l ) , ~ u )has t h e approximation
property.
4.75*
I f E i s a s e p a r a b l e m e t r i z a b l e l o c a l l y convex space
show t h a t
(H(U),.ru) i s q u a s i - c o m p l e t e f o r any open s u b s e t
U of E.
4.76"
x =
L e t E b e a Banach s p a c e w i t h a monotone b a s i s
1"
n=l
x e n n
E
E.
Let B b e t h e open u n i t b a l l o f E
If
e v e r y automorphism o f B h a s t h e form
where
IT
i s a permutation of t h e positive integers,
for all n and
SLIP
i s m o r p h i c t o .c 4.77* -
Let
r
lanl
<
1 , show t h a t E i s i s o m e t r i c a l l y
.
b e an u n c o u n t a b l e d i s c r e t e s e t .
H ( ( c o ( r ) ) , T ~ )= ( H ( C ~ , ~, (T ~~ ) ). s u b s e t s o f H ( c o ( r ) ) a n d t h e -c0 H ( c ~ , ~ ( ~a r)e ) l o c a l l y b o u n d e d . shows t h a t t h e l o c a l l y bounded
=
T~
Show t h a t
Show t h a t t h e r o - b o u n d e d
bounded s e q u e n c e s i n Give an example which
bounded s u b s e t s o f H(c
0,P
(r))
are not
1
Chapter 4
210
4.78*
L e t E b e a s e p a r a b l e Banach s p a c e w i t h o p e n u n i t
b a l l B and l e t D b e t h e open u n i t d i s c i n C . t h e r e e x i s t s an f i n H ( D ; B ) 4.79
Show t h a t
-
such t h a t f(D) 3 8 .
L e t E b e a Banach s p a c e a n d s u p p o s e t h e r e e x i s t s a
sequence i n
I
m
(4n)n=1,such t h a t
E l ,
I1xII = S X P ~ + ~ ( Xf o) r
By u s i n g e x e r c i s e 4 . 7 8 show t h a t E c a n b e
every x i n E .
embedded i s o m e t r i c a l l y i n H m ( D ) , D = E Z E C ; I z \ < l ) . 4.80* -
m
Let B be t h e open u n i t b a l l o f co and l e t
b e a s e q u e n c e i n CC s u c h t h a t
( a n ( < 1 and
Im ( l - ( a n l )<
n=l
m.
Show t h a t t h e r e e x i s t s an f i n A ( B ) s u c h t h a t
-
f l B E H ( B ) a n d { z c B ; f ( z ) = o ) = {(B,),EB;
B m = a m f o r some m}.
A s u b s e t A o f a l o c a l l y convex s p a c e E i s c a l l e d
4.81*
a (holomorphic o r a n a l y t i c ) f cH(E)
determining s e t i f
and f l A = o i m p l i e s f E
0 .
Show t h a t E c o n t a i n s a
compact d e t e r m i n i n g s e t i f and o n l y i f
( E ' , T ~ )admits a
Show t h a t a m e t r i z a b l e l o c a l l y c o n v e x
c o n t i n u o u s norm.
s p a c e c o n t a i n s a compact d e t e r m i n i n g s e t i f and o n l y i f i t is separable. 4.82
I _
L e t E and F b e Banach s p a c e s and l e t K b e a compact
-
d e t e r m i n i n g s e t f o r h o l o m o r p h i c f u n c t i o n s on E .
f : K
F i s a mapping s u c h t h a t f o r each $ i n F '
e x i s t an open neighbourhood V that
+
o f K and an f
+
E
+
there
H(V ) s u c h
= + o f show t h a t t h e r e e x i s t s an o p e n n e i g h b o u r h o o d
'1K V o f K and a n
14.6
If
2
E
H(V)
%
such t h a t f l K = f.
NOTES A N D R E M A R K S The i m p o r t a n c e o f c o m p l e x e x t r e m e p o i n t s
( d e f i n i t i o n 4.1)
i n t h e t h e o r y o f v e c t o r v a l u e d maximum m o d u l u s t h e o r e m s w a s f i r s t n o t e d by E . prove theorem 4 . 4 .
Thorp and R .
I " h i t 1 e y i n [685] w h e r e t h e y
They a l s o g i v e e x a m p l e s
-
e.g L(o,l)
-
Of
21 1
Holomoiphic functions on Banach spaces s p a c e s which a r e s t r i c t l y c-convex b u t n o t r o t u n d .
The
proof o f theorem 4 . 4 given h e r e i s due t o L . Harris
[305].
Many o f t h e o t h e r known i n f i n i t e d i m e n s i o n a l maximum modulus theorems a l s o i n v o l v e complex extreme p o i n t s .
L.
H a r r i s [304] shows t h a t t h e complex e x t r e m e p o i n t s o f a
convex b a l a n c e d compact s e t K form a boundary f o r t h e a l g e b r a o f a l l c o n t i n u o u s complex v a l u e d f u n c t i o n s on K which a r e "holomorphic" on t h e a n a l y t i c d i s c s i n K .
This r e s u l t
may b e r e g a r d e d a s a h o l o m o r p h i c K r e i n - M i l m a n t h e o r e m . J.
Globevnik [257] proves t h e r e s u l t quoted a f t e r theorem 4.4
and a p p l i e s it i n s t u d y i n g t h e r e s o l v e n t f u n c t i o n o f an operator.
Further p e r t i n e n t r e s u l t s a r e t o be found i n For i n s t a n c e J . Globevnik
[32,258,260,276,631,693,694].
c h a r a c t e r i s e s h o l o m o r p h i c f u n c t i o n s o f c o n s t a n t norm i n [260],
J . Globevnik and 1.Vidav d i s c u s s o p e r a t o r v a l u e d
holomorphic f u n c t i o n s i n [276] and g e n e r a l i s e e a r l i e r r e s u l t s
of A.
Brown a n d R . G .
Douglas
[lo61 and i n [258] 3. Globevnik
p r o v e s a maximum m o d u l u s t h e o r e m f o r B a n a c h a l g e b r a s i n w h i c h t h e s u p norm i s r e p l a c e d b y t h e s p e c t r a l r a d i u s . Schwarz's v a r i a b l e s by C .
l e m m a was e x t e n d e d t o f u n c t i o n s o f t w o C a r a t h e o d o r y [lll],
dimensional algebras by C . L .
t o some f i n i t e
S i e g e 1 [658] and t o c e r t a i n
i n f i n i t e d i m e n s i o n a l o p e r a t o r a l g e b r a s by R . S. [568]. t o L. H.
Phillips
i s due
The g e n e r a l r e s u l t g i v e n h e r e , t h e o r e m 4 . 3 , Harris
[304,305].
An a l t e r n a t i v e p r o o f u s i n g
Cartan's uniqueness r e s u l t
[113]
(see I .
Shimoda [656]
f o r a normed l i n e a r s p a c e v e r s i o n o f t h e u n i q u e n e s s theorem) papers
i s giv'en i n L .
[306,307,308]
Harris
In a s e r i e s o f
L . H a r r i s p r o v e s a number o f o t h e r
g e n e r a l i z a t i o n s o f Schwarz's t o J*algebras,
[304,313].
l e m m a and g i v e s a p p l i c a t i o n s
Mzbius t r a n s f o r m a t i o n s ,
t h e numerical
range and t h e c h a r a c t e r i s a t i o n o f extreme p o i n t s , [327, c o r o l l a r y 6.81, A.
Renaud
S.J.
[604] and J . P .
G r e e n f i e l d and N . R .
M.Herv6
Wallach
[282],
Vigug [696, p r o p o s i t i o n 1 . 2 . 1 1
a l s o obtain i n f i n i t e dimensional generalizations o f t h i s
lemma.
212
Chapter 4 A.E.
Taylor [678, p.474-4751
was t h e f i r s t t o n o t e
t h a t e n t i r e f u n c t i o n s o n c e r t a i n B a n a c h s p a c e s may h a v e f i n i t e r a d i i of uniform convergence.
Subsequently L.
Nachbin
[SO91
g a v e t h e d e f i n i t i o n o f r a d i u s of b o u n d e d n e s s a n d p r o v e d Around t h e same t i m e H .
proposition 4.7.
Alexander
[5]
w r o t e h i s d i s s e r t a t i o n on a n a l y t i c c o n t i n u a t i o n i n Banach spaces.
H i s i n v e s t i g a t i o n s l e d t o t h e concept of bounding
s e t ( h e w i s h e d t o know i f
( H ( E ) , T ~ ) was b a r r e l l e d )
a n d showed
t h a t t h e bounding s u b s e t s o f a H i l b e r t s p a c e were precompact. T h i s r e s u l t was p r o v e d i n d e p e n d e n t l y b y S . who was n o t i v a t e d b y L .
Dineen
[177]
Nachbin's proof of proposition 4.7.
'The n e x t d e v e l o p m e n t was t h e o r e m 4 . 2 7 w h i c h was p r o v e d i n d e p e n d e n t l y by S . Dineen [336,339].
[181] and A .
The p r o o f o f t h e o r e m 4 . 2 7
preliminary r e s u l t , proposition 4.26) t o M.
S c h o t t e n l o h e r [531,638]
t h e o r e m may b e f o u n d i n K .
Hirschowitz
(including the g i v e n h e r e i s due
a n d f u r t h e r comments on t h i s
Rusek
[616] and M.Bianchini
[64].
6
Complete l i n e a r c h a r a c t e r i z a t i o n s o f t h e Banach s p a c e s
i n which a l l bounding s e t s a r e r e l a t i v e l y compact and t h e
class
o f Banach s p a c e s w i t h weak* s e q u e n t i a l l y compact
d u a l b a l l s a r e s t i l l unknown. M.
By t h e o r e m 4 . 2 7
6c a.
S c h o t t e n l o h e r [642] n o t e s t h a t e v e r y we5kly compactly
generated
(WCG) s p a c e ( s e e J . D i e s t e l
[171])
lies in
and
t h a t & i s closed under t h e operation of taking closed subspaces.
S i n c e t h e r e e x i s t c l o s e d s u b s p a c e s o f WCG
s p a c e s w h i c h a r e n o t W C G we s e e t h a t W C G # and F .
J.
llagler
S u l l i v a n [ 3 0 1 ] p r o v e t h a t smooth Banach s p a c e s l i e
i n @I a n d
C.Stegal1
Asplund s p a c e . i n J.
&.
F u r t h e r r e s u l t s on t h e c l a s s
H a g l e r and W . B .
[300] and R .
[ 6 7 1 ] s h o w s t h a t 03 c o n t a i n s e v e r y w e a k
Haydon
Johnson
63
are given
[299], J . Hagler and E.Odel1
[320].
A s u b s e t A o f a Banach s p a c e E i s l i m i t e d i f e v e r y
weak* n u l l s e q u e n c e i n E '
t e n d s t o z e r o u n i f o r m l y on A .
L i m i t e d s e t s were i n t r o d u c e d b y J . D i e u d o n n 6 [ 1 7 3 ] lemma 4 . 2 s h o w s t h a t b o u n d i n g s e t s a r e l i m i t e d .
and
Limited
s e t s may b e r e g a r d e d a s t h e l i n e a r a n a l o g u e o f b o u n d i n g s e t s .
213
Holornorphic functions on Banach spaces We d o n o t know o f a n y l i m i t e d s e t w h i c h i s n o t b o u n d i n g . J.
Bourgain, J . D i e s t e l and D .
IVeintraub
[98] summarise
m o s t o f t h e known r e s u l t s o n l i m i t e d s e t s a n d p r o v e s o m e new r e s u l t s ,
They show,
s u b s e t s o f L 1 (X,F,v) e x i s t s a n L1(X,F,v) e x a m p l e s show t h a t
for instance, that the limited
a r e r e l a t i v e l y compact and s i n c e t h e r e which does n o t belong t o
a#
63.
Bayoumi
A.
this
[55] s t u d i e s
bounding s e t s i n complete separable metrizable topological vector spaces.
He proves f o r c e r t a i n non-locally
convex
s p a c e s t h a t t h e bounding s e t s are r e l a t i v e l y compact i f t h e open b a l l s d e f i n e d by t h e m e t r i c are p o l y n o m i a l l y convex and deduces t h a t t h e bounding s u b s e t s o f I p , o < p < l , a r e r e l a t i v e l y compact. The p r o o f o f t h e o r e m 4 . 2 7 m o t i v a t e d a number o f t h e
In t h i s
o t h e r r e s u l t s p r e s e n t e d i n s e c t i o n 4 . 1 and 4 . 2 . c a t e g o r y we may i n c l u d e lemma 4 . 5 , 4.18,
c o r o l l a r i e s 4.19.4.20
and c o r o l l a r i e s 4 . 2 3 and 4 . 2 4 . A.
Hirschowitz
corollary 4.8,
and 4 . 2 1 ,
lemma
proposition 4.22
T h e s e may b e f o u n d i n
[339] and S . Dineen
[181,184].
N e x t came t h e p r o o f o f t h e e x i s t e n c e o f a c l o s e d non-compact bounding s u b s e t o f Zm,
theorem 4.31,
and t h i s
i s due t o S . Dineen [178].
This motivated t h e deep
a n a l y s i s o f t h e geometry o f
Lm
undertaken by B.
i n [361] and l e d t o theorem 4 . 2 8 .
Josefson
Josefson only gives the
first f o u r conditions o f theorem 4.28(a) but we have included condition
( v ) which i s e q u i v a l e n t t o ( i i i ) by
t h e celebrated' theorem o f H . P .
Rosenthal
Tzafriri
[447]).
[611] ( s e e
J.
L i n d e n s t r a u s s and L .
Recently
R.
Haydon [ 3 2 0 , 3 2 1 ] o b t a i n e d e x a m p l e s o f B a n a c h s p a c e s
w h i c h d o n o t c o n t a i n lm b u t c o n t a i n c l o s e d non-compact bounding s e t s D.
( s e e a l s o J . Bourgain, J . D i e s t e l and
Weintraub [ 9 8 ] ) . With t h e o r e m 4 . 2 7 as m o t i v a t i o n B .
Josefson
[359]
proved p r o p o s i t i o n 4 . 9 ( a l s o due i n d e p e n d e n t l y and w i t h a
214
Chapter 4
different proof t o A.
Nissenweig [529]) and a p p l i e d it t o
o b t a i n c o r o l l a r y 4.10 and p r o p o s i t i o n 4.25.
In c o n s t r u c t i n g
a counterexample t o t h e Levi problem B . J o s e f s o n [358] showed t h e e x i s t e n c e o f a c l o s e d p r e c o m p a c t non-compact b o u n d i n g s u b s e t o f a p o l y n o m i a l l y convex domain i n c o ( r ) , r uncountable. P r o p o s i t i o n s 4 . 1 1 and 4.12 a r e due t o R. and P . C.O.
Aron
[20]
Lelong [425] r e s p e c t i v e l y and t h e s e m o t i v a t e d
Kiselman [385,386,387]
t o undertake a detailed
i n v e s t i g a t i o n o f t h e r a d i u s of convergence.
Kiselman
f i r s t posed t h e problem of p r e s c r i b i n g t h e r a d i u s of uniform c o n v e r g e n c e a n d p r o v e d t h e o r e m 4 . 1 3 when E=co o r
l s <
P’ and t h e r a d i u s o f c o n v e r g e n c e depended o n l y on a f i n i t e
number o f v a r i a b l e s .
m,
Subsequently t h e f i n i t e n e s s
r e q u i r e m e n t was r e m o v e d a n d t h e o r e m 4 . 1 3 p r o v e d b y G . C o e u r 6 A more a c c e s s i b l e p r o o f o f t h i s t h e o r e m h a s b e e n
[136].
given by M .
Schottenloher i n [641].
N.
Cherfaoui [122]
s t u d i e s t h e r a d i u s of boundedness of plurisubharmonic f u n c t i o n s on O r l i c z s p a c e s . are due t o C . O .
Example 4 . 1 4 and p r o p o s i t i o n 4 . 1 6
Kiselman [385].
Many f u r t h e r e x a m p l e s
a n d r e s u l t s o n t h e r a d i u s o f b o u n d e d n e s s Lor t h e r a d i u s o f uniform convergence) are contained i n t h e paper of Kiselman quoted above.
M. S c h o t t e n l o h e r [ 6 4 2 ] h a s w r i t t e n a v e r y r e a d a b l e s u r v e y a r t i c l e on b o u n d i n g s e t s and t h e r a d i u s of boundedness. If
T~
=
T&
o n H(U)
and U i s a holomorphically
convex open s u b s e t of a l o c a l l y convex s p a c e t h e n U i s a domain o f holomorphy.
This provided the motivation f o r
theorem 4.38 proved i n S . Dineen [180].
G. Coeur6 [130] 1 g r o v e d t h e same r e s u l t f o r s u b s p a c e s o f L [ o , 2 1 ~ 1 w h i c h a r e
homogeneous i n t h e s e n s e o f S h i l o v . An i n i t i a l d i f f i c u l t y i s d e a l i n g w i t h t h e
T~
topology
i s t h a t a g e n e r a t i n g s e t o f semi-norms i s n o t p r e s c r i b e d . T h i s d i f f i c u l t y i s overcome by p r o p o s i t i o n 4.39
215
Holomoiphic functions on Banach spaces w h i c h was f i r s t p r o v e d f o r e n t i r e f u n c t i o n s b y S . D i n e e n [177] and a f t e r w a r d s extended t o holomorphic f u n c t i o n s
on b a l a n c e d o p e n s e t s b y R .
Aron [ 1 7 ] .
Propositions 4.40
and 4 . 4 1 a r e due t o S . Dineen [185] and show, i n a s i m p l e fashion,
t y p i c a l phenomena which a r i s e i n t h e s t u d y o f
h o l o m o r p h i c f u n c t i o n s on n o n - m e t r i z a b l e l o c a l l y c o n v e x spaces.
E x a m p l e 4 . 4 2 may b e f o u n d i n S . D i n e e n [ 1 8 4 ] a n d
a g e n e r a l i n v e s t i g a t i o n o f Hahn-Banach t y p e e x t e n s i o n
t h e o r e m s f o r h o l o m o r p h i c f u n c t i o n s on Banach s p a c e s i s undertaken i n R.
Aron a n d P .
Berner [26] and R.
Aron [ 2 5 ] .
They show i n [ 2 6 ] t h a t f & H ( c O ) h a s a h o l o m o r p h i c e x t e n s i o n
t o Zm i f and o n l y i f rf
=
Hahn-Banach e x t e n s i o n
+m.
t h e o r e m s f o r h o l o m o r p h i c f u n c t i o n s on f . u l l y n u c l e a r s p a c e s a r e discussed i n 15.4. The c o n c e p t o f h o l o m o r p h i c c o m p l e t i o n i s d u e t o Proposition 4.43 i s due t o
A.
Hirschowitz
[339].
A.
Hirschowitz
[339] f o r normed l i n e a r s p a c e s and t o
S.
Dineen [184]
f o r metrizable l o c a l l y convex spaces.
Proposition 4.44
is given i n
[184].
F u r t h e r r e s u l t s on
h o l o m o r p h i c c o m p l e t i o n may b e f o u n d i n A . H i r s c h o w i t z [337,339,343], R.R.
G.
Baldino [43],
Ph. N o v e r r a z
CoeurB [ 1 3 5 ] , Ph. Noverraz
[206] and M .
S.
Dineen [ 1 9 0 ] ,
[542,547],
S . Dineen and
Schottenloher [633,642,645].
For
example G . Coeur6 [135] shows t h a t a h o l o m o r p h i c a l l y complete p r o p e r s u b s p a c e o f a Banach s p a c e i s a p o l a r s e t M.
S c h o t t e n l o h e r [633] shows t h a t
(ExF)"
= E m x FG
f o r any
m e t r i z a b l e l o c a l l y convex s p a c e s E and F and S . Dineen and Ph. N o v e r r a z [206] show, u s i n g G a u s s i a n m e a s u r e s ,
that
" a l m o s t a l l " a l g e b r a i c h y p e r p l a n e s i n a Banach s p a c e a r e not holomorphically complete.
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Chapter 5 HOLOMORPHIC FUNCTIONS ON NUCLEAR SPACES WITH A BASIS
We h a v e a l r e a d y s e e n t h a t n u c l e a r h o l o m o r p h i c f u n c t i o n s a n d n u c l e a r s p a c e s e n t e r i n t o t h e g e n e r a l t h e o r y o f i n f i n i t e dimens i o n a l holomorphy i n a n a t u r a l way.
For i n s t a n c e ,
i n 53.3,
we
saw t h a t t h e d u a l o f t h e s p a c e o f h o l o m o r p h i c f u n c t i o n s o n a Frgchet space
E
with t h e approximation property,
endowed w i t h
t h e c o m p a c t o p e n t o p o l o g y , may b e i d e n t i f i e d v i a t h e B o r e 1 t r a n s f o r m w i t h t h e s p a c e o f n u c l e a r germs a t t h e o r i g i n i n Moreover,
E
if
i s nuclear,
isomorphism between
(H(E)
we o b t a i n a l i n e a r t o p o l o g i c a l
, T ~ )
and
E' B'
H(OEl). B
I n t h i s c h a p t e r , w e i n v e s t i g a t e holomorphic f u n c t i o n s on f u l l y nuclear spaces basis.
( d e f i n i t i o n 1.49) which have a Schauder
T h e b a s i s p r o v i d e s u s w i t h a coordinate s y s t e m a n d t h u s
i t i s n o t s u r p r i s i n g t h a t we u s e r e f i n e m e n t s o f a n u m b e r o f t h e
t e c h n i q u e s d e v e l o p e d i n t h e l a s t two c h a p t e r s .
The f i n a l
r e s u l t s we o b t a i n c o n c e r n i n g t h e d i f f e r e n t t o p o l o g i e s d o n o t i n t r i n s i c a l l y depend on t h e e x i s t e n c e o f a b a s i s and h a v e i n s p i r e d a number o f a n a l o g o u s r e s u l t s o n s p a c e s n o t n e c e s s a r i l y having a basis, section 5.4,
w h i c h we p r e s e n t e d i n c h a p t e r s 1 a n d 3 .
c e r t a i n cases.
The e x i s t e n c e o f a b a s i s a l s o a l l o w s u s t o
d e f i n e i n f i n i t e d i m e n s i o n a l a n a l o g u e s o f Reinhardt domains
poZydiscs.
In
w e a l s o s h o w how t o r e m o v e t h e b a s i s h y p o t h e s i s i n
and
When we d o n o t h a v e a b a s i s , we a r e o b l i g e d t o
confine ourselves t o entire functions. The f i r s t s e c t i o n o f
t h i s chapter i s devoted t o
l i n e a r and
geometric properties of c e r t a i n c l a s s e s of nuclear spaces with a basis.
We o n l y p r e s e n t r e s u l t s n o t a l r e a d y c o n t a i n e d i n t h e
s t a n d a r d t e x t s on n u c l e a r spaces.
217
I n 5 5 . 2 w e show t h a t
218
Chapter 5
(H(U),-ro)
i s an open poly-
U
h a s a n a b s o l u t e b a s i s whenever
disc i n a f u l l y nuclear space with a basis.
T h i s b a s i s problem
a n d t h e f o r m o f i t s s o l u t i o n m o t i v a t e d much o f t h e r e s e a r c h reported i n t h i s chapter. S o l u t i o n s t o similar b a s i s problems l e d t o a c h a r a c t e r ization of
set of
(H(U),ro)'
as a s p a c e o f germs on a compact sub-
This duality theory clarifies the relationship
Ei.
between t n e d i f f e r e n t t o p o l o g i e s on
H(U)
and p l a c e s i n p e r -
s p e c t i v e a number o f t h e c o u n t e r e x a m p l e s from p r e v i o u s c h a p t e r s
I n 9 5 . 3 , we s h o w t h a t a nuclear
DN
chapter 6 . t h o s e of
=
T~
T
space with a basis.
6
on
H(E)
A converse
whenever
E
is
i s given i n
The methods u s e d h e r e w e r e p a r t i a l l y m o t i v a t e d b y
94.3
w h e r e we d i s c u s s e d h o l o m o r p h i c f u n c t i o n s o n
Banach s p a c e s w i t h a n u n c o n d i t i o n a l b a s i s .
In 55.4,
we d i s c u s s
holomorphic f u n c t i o n s on c e r t a i n n u c l e a r s p a c e s which do n o t n e c e s s a r i l y have a b a s i s and d e s c r i b e a r e l a t i o n s h i p between e x t e n s i o n theorems f o r holomorphic f u n c t i o n s and p r o p e r t i e s o f t h e compact open t o p o l o g y . The r e s u l t s o f t h i s c h a p t e r a r e proved i n d e p e n d e n t l y o f o u r p r e v i o u s r e s u l t s on n u c l e a r holomorphic f u n c t i o n s and h o l o morphic f u n c t i o n s on n u c l e a r s p a c e s .
55.1
NUCLEAR S P A C E S WITH A B A S I S We f i r s t d i s c u s s a b s o l u t e b a s e s i n l o c a l l y c o n v e x s p a c e s .
This i s a p a r t i c u l a r c a s e of a b s o l u t e Schauder decompositions previously discussed i n §3.1
-
one t a k e s each o f t h e sub-
spaces t o b e one dimensional
-
and a number o f t h e t e c h n i q u e s
used t h e r e are a l s o a p p l i e d h e r e .
Let space
E.
p E cs(E)
m
(en)n=l
b e a S c h a u d e r b a s i s i n t h e LocaZZy c o n v e z
T h e b a s i s i s s a i d t o b e a b s o l u t e i f for a n y
there exists
q
E
cs(E)
such t h a t
219
Holomorphic functions on nuclear spaces with a basis
p
E
This is obviously equivalent to saying that for any cs(E) the mapping
defines a continuous semi-norm on
E
We now give two elementary but useful results concerning the uniqueness o f topologies on locally convex spaces with an absolute basis. Let
Lemma 5 . 1
and
T~
T~
o g i e s on t h e v e c t o r space
b e two ZocaZZy c o n v e x t o p o Z and suppose
E
absoZute b a s i s f o r both topoZogies.
if
i s an
(en);,l
Then
T~
if a n d onZy
= T~
( E , T ~ )= ~ ( E , . r 2 ) l .
Proof
If
=
T~
then, trivially,
T~
Suppose conversely that continupus semi-norm on generality, that
Let
$:
Then
+
E
E
-+
C
(E,T~)!
( E , T ~ ) ! = (E,.r2)!.
E.
(E,r2)!.
=
Let
p
be a
T~
We may assume, without l o s s o f
be given by
( E , T ~ ) !=
( E , T ~ ) !
continuous semi-norm every x in E .
q
x
=
Hence for all
on
m
and hence there exists a E
such that
xnen
in
E
I$(x)l
we have
d
T1
q(x)
for
220
Chapter 5 ?d
By the definition of absolute basis q is a T~ continuous semi-norm on E. Hence p is T~ continuous and T~ T ~ . Interchanging T~ and T~ we find that This completes the proof. T1 - T 2 .
T~
and hence
T~
Our next result is similar to proposition 3.12 Let T~ and T~ Lemma 5.2 topoZogies o n t h e v e c t o r s p a c e
basis for both
( E , T ~ ) and
b e two l o c a l l y c o n v e x b a r r e l l e d
E.
If
(en)i=lis
( E , T ~ ) then
T~
=
T
~
Proof Let p be a T~ continuous semi-norm on may suppose, without loss of generality, that
W
Since
,
absolute
CIY! .
E.
We
is an absolute and hence a Schauder basis f o r
( E , T ~ ) the semi-norms
are
T~
continuous for each positive integer
m.
Let
V is a T~ closed convex balanced absorbing subset of E and hence, since ( E , T ~ ) is barrelled, a T~ neighbourhood of continuous semi-norm and T 2 < Ti. zero. Thus p is a T~ and T~ in the above, we see that T~ 6 T * Interchanging T~ and hence T~ = T ~ . This completes the proof.
22 1
Holornotphic functions on nuclear spaces with a basis L o c a l l y convex spaces w i t h an a b s o l u t e b a s i s can b e ident i f i e d with dense subspaces of sequence spaces. Definition 5.3
Let
b e a c o l l e c t i o n o f s e q u e n c e s o f non-
P
r>O
n e g a t i v e r e a l numbers s u c h t h a t f o r e a c h p o s i t i v e i n t e g e r there exists space
m
in
P.
P
E
with
ar
The sequence
> 0.
i s t h e s e t of all s e q u e n c e s o f c o m p l e x n u m b e r s ,
A(P)
(Xn) n = 1'
m
(an)n=l
=
01
m
ln=l/xnlan <
s u c h that We endow
pa,
semi-norms
w i t h t h e t o p o l o g y g e n e r a t e d by t h e
A(P) Q
The e l e m e n t s o f
=
m
a = (an)n=l
f o r every
m
m
("n)n=l
P
where
P,
E
are called weights.
If
m
is
(4n)n=1
any s e q u e n c e o f n o n - n e g a t i v e numbers and
i s a neighbourhood of zero then t h e sequence c a l l e d a c o n t i n u o u s weight on
Hausdorff
The c o n d i t i o n
P.
Q
l o c a l l y convex space.
forms an a b s o l u t e b a s i s f o r
Now l e t (en);=1
9
p
=
l i n e a r mapping from in
E
is a
A(P)
i s a complete l o c a l l y
m
)
n(P).
+
We r e f e r t o t h i s b a s i s a s
h(P).
b e a l o c a l l y convex space w i t h a b s o l u t e b a s i s
E
Let
A(P)
( u ~ ) ~ =u ~ =, ( 0 , . . . 0 , 1 , 0 , . . . ) nth position
convex s p a c e and t h e s e q u e n c e
the unit vector basis of
(Bn)n=l
a d m i t s a c o n t i n u o u s norm
i f and o n l y i f t h e r e e x i s t s a c o n t i n u o u s w e i g h t on
A(P)
m
that
ensures t h a t
> 0
A(P)
c o n s i s t i n g o f p o s i t i v e numbers.
is
We may t h e n a s s u m e , w h e n -
A(P).
e v e r n e c e s s a r y and w i t h o u t l o s s o f g e n e r a l i t y , belongs t o
m
(Bn)n=l
{ (P(en)
E
m
I n = 1 3 pEcs ( E ) .
into
A(P)
There is a natural
which t a k e s t h e b a s i s
onto the u n i t vector basis of
(en =: 1 mapping i s g i v e n by
A(P).
This
Chapter 5
222
and
E
is then linearly isomorphic with its image in
E
is isomorphic to h(P) locally convex space.
A(P).
if and only if it is a complete
When we identify a locally convex space containing an absolute basis with a subspace of a sequence space, we shall always assume that the above identification is used. Nuclear sequence spaces have a particularly nice and practical characterization as the following fundamental result shows.
This is the Grothendieck-Pietsch criterion.
Proposition 5 . 4 The s e q u e n c e s p a c e A(P) i s n u c Z e a r if m m and onZy i f f o r e a c h E P there e x i s t ( u ~ ) ~E = II'~ m
and
E
such t h a t
P
an 6 u
0.1
n n
f o r all
This criterion can be rephrased as follows: n u c Z e a r if and onZy i f f o r e a c h s e q u e n c e of n o n - n e g a t i v e
I:=,
n
reaZ numbers, m
and
<
m
(6nan)n=1
E
1
n. A(P)
i s
there exists a
P m
(6n)n,l,
such t h a t
i s a c o n t i n u o u s w e i g h t on
A(P).
Since a locally convex space is nuclear if and only if its completion i s nuclear, this criterion can also be applied to locally convex spaces with an absolute basis.
We obtain the
following: i f is an absolute basis for the locally convex space E then E is nuclear if and only if for each q 3 p, such that p E cs E) there exists q E cs(E),
U S ng proposition 5 . 4 it is possible to obtain a further representation of nuclear sequence spaces. If A(P) is nuclear, then
A(p)
=
m
{(XnInZl;
SUP
n
223
Holomorphic functions on nuclear spaces with a basis
Furthermore,
the topology of
a l l semi-norms
o f t h e form
where
m
a =
ranges over
i s a l s o g e n e r a t e d by
A(P)
The neighbourhood s y s t e m s
P.
generated by t h e d i f f e r e n t systems o f semi-norms d e s c r i b e d above have d i f f e r e n t geometric p r o p e r t i e s - one being o f t h e Q1
t y p e and t h e o t h e r being o f t h e
system has i t s advantages.
-
type
Q~
and each
Since they are equivalent systems
we may u s e w h i c h e v e r i s m o r e s u i t a b l e . D e f i n i t i o n 5.5 IJ
B,
B
T h e m o d u l a r h u l l of a s u b s e t
A(P)
of
,
i s defined as
The modular h u l l i s a l s o c a l l e d t h e s o l i d h u l l . Lemma 5 . 6 A(P)
I f
E v e r y bounded s u b s e t
of
B
t h e nuclear space
E
i s a n u c l e a r s p a c e w i t h an a b s o l u t e b a s i s , h
E
bounded s u b s e t of
Proof i
i
Let
B = (x ) .
m
f o r each
x i - (Xn)n=l
E)
( t h e c o m p l e t i o n of
t h e c l o s u r e of a bounded s u b s e t of
1 E
I
i.
then every
i s contained i n
E.
b e a bounded s u b s e t o f A(P), i Let xn = s y p l x n I f o r each
T o c o m p l e t e t h e p r o o f o f t h e f i r s t h a l f o f t h e lemma,
i c e s t o show
m
( x ~ ) E~ A(P1. = ~
Let
m
(an)n=l
Grothendieck-Pietsch c r i t e r i o n , there e x i s t m
(aA)n=1 bounded and
A(P).
i s c o n t a i n e d i n t h e m o d u l a r h u l l of a n e l e m e n t of
E
P.
E
where n.
it s u f f -
By t h e m
( u ~ ) ~ = ~ and E II;
such t h a t a n < u a' for all n. n n i s u p suplaAxnl = M < m. Hence Ia;x,I i n
Since
P
$
M
B
is
for all
n
m
Thus
(Xn)n=l
point
and
A(P)
E
B
l i e s i n t h e modular h u l l o f t h e
(Xn);=l.
Now s u p p o s e m
,(en)n=l.
i s a nuclear space with absolute basis
E
Identify If
E S A(P).
with a dense subspace of
E
i s a bounded s u b s e t of
B
Then
A(P).
then
h(P),
c o n t a i n e d i n t h e modular h u l l of
( x ~ ) T =E ~A ( P ) .
i s o b v i o u s l y a bounded s u b s e t o f
E
is
B
The s e t
h
and i t s c l o s u r e i n
E
can b e i d e n t i f i e d w i t h t h e modular h u l l of an element of
A(P).
This completes t h e proof. Corollary 5.7
basis,
then
i s a n u c l e a r space w i t h an a b s o l u t e
A
(E);.
=
E
is a n u c l e a r s p a c e w i t h a n a b s o l u t e A is a n i n f r a b a r r e l l e d s p a c e if and o n l y if E
Corollary 5.8
basis, then
E
If
EA
E
I f
is b a r r e Z l e d . We now l o o k a t t h e s t r o n g d u a l o f a n u c l e a r s p a c e w i t h a n
We m a y ,
absolute basis. E
i f we w i s h
by t h e above,
assume t h a t
i s complete.
Proposition 5.9
ute basis
m
o v e r , if (e;)n=l Proof
If m
ordinate then m
Let m
(en)n=l. (e;)n,l
(E,.r) e;
be a n u c l e a r space w i t h a b s o l -
is e v a l u a t i o n a t t h e
is a n a b s o l u t e b a s i s f o r
is a l s o a S c h a u d e r b a s i s f o r
i s a n a b s o l u t e b a s i s for Let
T
nth E;I.
coMore-
(E,.rb)
(E,T~);I.
b e a l i n e a r f u n c t i o n a l on
E
which i s
then
Holornorphic functions on nuclear spaces with a basis bounded on t h e
bounded s u b s e t s o f
b e a bounded s u b s e t of
B
r.
in
[B]
=
E,
R;
en
x n en '. J C N f i n i t e ,
i s a bounded s u b s e t of
M = sup
m
x* =
i o n B,
E.
by t h e Grothendieck-Pietsch E
B = (x B )
and l e t
en
x:
BEr
f o r each
'
We c l a i m t h a t
{I n d e
(u,,):=~
where
E
225
such t h a t
q ( x Bn e n ) .
If
p
E
E
[O,ZII],
p(en) 6
E
r
all
n}.
then w e can choose,
cs(E)
q
criterion,
B,
E
unq(en)
cs(E)
for all
and
n.
Let
Then
Rsr,nEN
Hence
Hence i f basis for
T
[B]
is
i s a bounded s u b s e t o f
T
continuous o r
( E , T ~ ) then,
E
m
(en)n=l
and
-
B C [B].
i s a Schauder
f o r any p o s i t i v e i n t e g e r
m,
we h a v e
226 and
Chapter 5 m
i s a Schauder b a s i s f o r t h e dual space i n both
(e;)n=l
cases.
Moreover,
m
and hence
i s an absolute b a s i s .
(eA)n,l
This completes t h e
proof. On c o m b i n i n g lemma 5 . 6 a n d p r o p o s i t i o n 5 . 9 ,
we i m m e d i a t e l y
obtain the following r e s u l t . C o r o l l a r y 5.10
basis.
E C A(P)
IS
E
Let
h
be a n u c l e a r space w i t h an a b s o l u t e E' B
then
A(P')
where
C o r o l l a r y 5 . 1 0 a n d t h e G r o t h e n d i e c k - P i e t s c h c r i t e r i o n may b e u s e d t o d e c i d e when a g i v e n n u c l e a r s p a c e w i t h a n a b s o l u t e b a s i s i s a dual nuclear space.
We p r e f e r , h o w e v e r ,
t o r e l y on
t h e f o l l o w i n g more p r a c t i c a l c r i t e r i o n which c o v e r s most n o t a l l ) cases i n which
is nuclear,
E
(if
d u a l n u c l e a r and h a s
an a b s o l u t e b a s i s . Definition 5.11
A
l o c a l l y convex space
n u c l e a r s p a c e is i t h a s a n a b s o l u t e b a s i s e x i s t s a s e q u e n c e of p o s i t i v e r e a l numbers
1;=1
6 n
<
m
E
A-
i s an m
(en)n=l
and t h e r e
(Sn)ZZ1
such t h a t
and t h e s e m i - n o r m
is continuous w h e n e v e r
p
E
cs(E).
T h i s d e f i n i t i o n i s e a s i l y compared w i t h t h e G r o t h e n d i e c k Pietsch criterion for nuclearity - for sequence
m
(6n)n=1
shows t h a t e v e r y
A - nuclearity the
i s i n d e p e n d e n t o f t h e semi-norm A-nuclear
p
-
and
space i s also a nuclear space.
Holomotphic functions on nuclear spaces with a basis T h e s t r o n g d u a l of
Proposition 5.12 A-nuclear
an
Proof
a n A-nuclear
s p a c e is
space.
Let
03
b e a n A-nuclear
E
We i d e n t i f y
(en)n=l.
221
t h e u s u a l manner.
space with absolute b a s i s
w i t h a dense subspace of
E
A(P)
By t h e d e f i n i t i o n o f A - n u c l e a r i t y
in
the
mapping m
6 : (Xn)n=l
E
A(P)
__f
(Gnxn)n=l
i s a l i n e a r t o p o l o g i c a l isomorphism from m
Hence
(xnln=l,
Since
E'
P'
6 -
P A(Pl)
' . ( I x n / ) n" = l 1
=
i f and o n l y i f
A(P)
E
(E);
w e h a v e shown t h a t
m
El
B
A(P)
E
onto
(Gnxn)n=l
E
A(P).
A(€').
( c o r o l l a r i e s 5 . 7 and 5.10)
is an A-nuclear space.
The c o m p l e t i o n o f a n A-nuclear space.
A(P)
where
(Xn)n=l
the proof.
A(p)
E
This completes
space i s an A-nuclear
C o u n t a b l e p r o d u c t s a n d c o u n t a b l e d i r e c t sums o f A -
n u c l e a r spaces a r e A-nuclear
spaces.
space with a b a s i s i s an A-nuclear
Every Frcchet n u c l e a r
space and hence everyB3'LL
space w i t h a b a s i s i s A-nuclear. Let CD
absolute basis
(en)n=l.
b a s i s for t h e A - n u c l e a r Proof 5.11
Let
for
E.
m
( q n = 1 Since
(E,T)
be an A-nuclear space with m
Then space
(en)n=l
is also an absolute
(E,.rb).
be t h e sequence s a t i s f y i n g d e f i n i t i o n
I;=, 6n
<
we may
w
assume, by
reordering t h e b a s i s i f necessary, t h a t t h e sequence i s i n c r e a s i n g and t h a t
i t y implies that
>1.
If
p
E
cs(E)
m
(6n)n=1
then A-nuclear-
Chapter 5
228
d e f i n e s a c o n t i n u o u s semi-norm on
m- 1 In=1 ~
t h e sequence ISm(x -
of
E.
Since
6-
E.
Let
rb
as
e
n+m
~
i) s }a
{6;x;
1
Er
~r b - = b o u~n d e d
c o n t i n u o u s semi-norm on
, n = l *, ,
C = s u p q(6:xg)
subset
it f o l l o w s t h a t
i s a bounded s u b s e t of
x:en}Bsr implies that of
+ m
is a
q
Now, i f
+
m
~
E.
...
E
E
and
then A-nuclearity
i s a l s o a bounded s u b s e t
and l e t
n,B
it follows t h a t
Hence since
m
(en)n=l
is a
rb
c o n t i n u o u s semi-norm on
i s an a b s o l u t e b a s i s f o r
(E,rb).
E.
Moreover,
229
Holomorphic functions on nuclear spaces with a basis
(E,rb)
i s an A-nuclear
space.
This completes t h e proof
An i n f r a b a r r e l l e d A - n u c l e a r s p a c e
Corollary 5.14
(E,T)
is b o r n o l o g i c a l . Proof
p
Let
be a
p
proposition 5.13, norm
Let
q
c o n t i n u o u s s e m i - n o r m on
T~
i s dominated by a
By
which h a s t h e form
U = {xeE;q(x)$l}.
is ?-closed
U
bounded s e t s . S i n c e ( E , r ) bourhood o f z e r o i n norm on
E.
c o n t i n u o u s semi-
T~
E.
Since
and h e n c e
E
q>p
c o n t i n u o u s and h e n c e
and a b s o r b s a l l
is infrabarreled, q
is a
T
p
t h i s shows t h a t
U
T-
i s a neighcontinuous semi-
is also
T
This completes t h e proof.
T = T ~ .
We r e c a l l f r o m c h a p t e r 1 t h a t a l o c a l l y c o n v e x s p a c e i s a f u l l y nuclear space i f nuclear spaces. Schauder b a s i s
and
E'
a
E
are both reflexive
i s a f u l l y n u c l e a r s p a c e and h a s a
E
If
E
m
(en)n=l
( h e n c e f o r t h we u s e t h e t e r m f u l l y
nuclear space with a b a s i s ) then t h e b a s i s equicontinuous b a s i s s i n c e an a b s o l u t e b a s i s s i n c e
E
E
co
(en)n=l
i s an
i s b a r r e l l e d and h e n c e it i s
is nuclear.
By p r o p o s i t i o n 5 . 9
t h e strong dual of a f u l l y nuclear space with a b a s i s i s a l s o a f u l l y nuclear space with a b a s i s .
Every r e f l e x i v e A-nuclear
s p a c e i s a f u l l y n u c l e a r s p a c e w i t h a b a s i s a n d we d o n o t know o f any f u l l y n u c l e a r s p a c e w i t h a b a s i s which i s n o t an Anuclear space.
C o u n t a b l e p r o d u c t s a n d c o u n t a b l e d i r e c t sums
of f u l l y nuclear spaces with a b a s i s a r e a l s o f u l l y nuclear
spaces with a b a s i s .
We i n t r o d u c e f u r t h e r c l a s s e s o f n u c l e a r
s p a c e s i n l a t e r s e c t i o n s o f t h i s c h a p t e r a n d a l s o g i v e a number of examples.
Most o f t h e c l a s s i c a l n u c l e a r s p a c e s e n c o u n t e r e d
i n a n a l y s i s a r e r e f l e x i v e A-nuclear nuclear space with a b a s i s ,
m
spaces.
(en)n=l,
If
E
is a fully
we f i x o n c e a n d f o r a l l
Chapter 5
230
a representation of
and
E
EA
as sequence spaces
such t h a t t h e cadonical d u a l i t y between
A(P')
n a t u r a l l y t r a n s f e r r e d t o t h e d u a l i t y between
and
A(P)
and
E
and
A(P)
is
E'
A(P')
We t h u s h a v e
m
Cn=1 W n Z n
=
Z E E
where
and
W E E ' .
Definition 5.15
Ic)
A
be a nucZear sequence
is said t o b e R e i n h a r d t if w h e n e v e r
A
z
A(P)
A C A ( P ) .
s p a c e and l e t
(a)
E
Let
=
m
(ZnIn=1
A
(en)n
and
is a p o l y d i s c if
A
R~
then
h a s e i t h e r of
ie (e
n
m
ZnIn=1
E
A.
the
foZZowing forms
where
if
B,
a >0
E
f o r a22
[O,+m]
and
0
.
(+-)
n,
a
.
(+a)
+m
= 0.
The R e i n h a r d t h u l l o f a r b i t r a r y s u b s e t s of d e f i n e d i n a n o b v i o u s way.
=
E
is
A p o l y d i s c which h a s t h e form
i s open i f and o n l y i f ( R ~ E ) P ~ and a p o l y d i s c which has t h e form (**) is always closed.
(*)
Holomorphic functions on nuclear spaces with a basis
23 I
The origin is a compact polydisc and the whole space is a n open polydisc.
By the Grothendieck-Pietsch criterion for
nuclearity every nuclear space with an absolute basis has a fundamental neighbourhood system consisting o f open polydiscs. It is also immediate that every polydisc is modularly decreasing and every modularly decreasing set is Reinhardt. In studying holomorphic functions on fully nuclear spaces with a basis, we find that the multiplicative polar, which we now introduce, is more useful than the usual linear polar o f functional analysis.
~f E 2 A ( P ) is a fuzzy nuclear space A C E w e define AM (the muZtiplicatiue
Definition 5.16
with a basis and poZar of
A)
as
It i s immediate that subset o f
EE,
AM
is a closed modularly decreasing
and
is a closed subset o f
E
which contains
A.
The following two simple, but technical, results play a crucial role in the sequel.
Let
U
space with a basis
E
Lemma 5.17 4
E ' = A(PI). B
b e a n open polydisc in a fuZZy nucZear A(P).
Then
UM
is a compact poZydisc
U contains a fundamental system of compact sets consisting of compact poZydiscs and UM has a fundamentaz neighbourhood system consisting of open polydiscs. K Interior(K M ) establishes a one-to-one The mapping correspondence between compact poZydiscs in U and open polyM disc neighbourhoods of U . in
Furthermore
-
232
Chapter 5 u)
Proof m (nn)n= 1
E
Let U = { ( z ~ ) ~ =E ~A(P); P and let
v
m
{(zn)n=l
=
E
U M = V o = {(w,):,~
Then
*(PI; E
suplznan/ < 1) n
m
where
Cn=l/zn"n/h 11.
A(P');
sup1 w n n
,<
11.
is a compact Since E is complete and dual nuclear V o subset of E; and hence UM is a compact polydisc in A(P'). K
Let
be a compact subset of AK
such that
X
m
m
(Bn)n=l
E'
U.
for all
(Bn)n=l
Hence
E
U
this shows that
then for all
n
and
and
Since the modular hull of any element of U,
n
By lemma 5 . 6 ,
m
= Modular Hull o f
polydisc in
U.
hBnan 6 1 99
I
There exists
is also a compact subset o f
x
if B n = suP{lznl; (ZnIn=l E K } m ('n)n= 1 E A(P). Since A K C U, hence
U.
U
U
is a compact
contains a fundamental
system of compact sets consisting o f compact polydiscs. Now let W be a neighbourhood o f U M in We can choose
Ei.
a,
(aA)n=l
where
Since
E
Bn
P'
such that
1
0
=
m
(Bn)n=l
if
a' = 0
n
I/(an+l/a
E
P'
'1
if
a,:
# 0.
n
this implies that
Ui4
has a fundamental
neighbourhood system consisting o f open polydiscs.
Holomorphic functions on nuclear spaces with a basis i s any open p o l y d i s c neighbourhood of
V
If
may c h o o s e
a,
O<5<1,
such t h a t
B
and,
since
vM c
Ei
(UM)M =
rr
E
K
KM
i s a n e i g h b o u r h o o d of z e r o i n
and hence
i s a neighbourhood of
KM
m
M (K )
interior
and h e n c e i n
E
(B)" A m
(6n)n,l
Lemma 5.18
sup/ n
<1 W
such
:: 11
and
UM
c o r r e s p o n d e n c e i s now e v i d e n t i f o n e n o t e s t h a t
for a n y c l o s e d p o l y d i s c B,
A(P');
U.
Ei.
in
UM
is a
and hence
KM
KOC
i s a n open p o l y d i s c neighbourhood of
The one-to-one
6 =
then
U,
VM
E' Choose X > 1 B' i s a l s o a compact s u b s e t of U . Then
XK
If
OF
which l i e s i n
i s a compact p o l y d i s c i n
If
disc
Hence
UM.
is also f u l l y nuclear with a basis,
compact p o l y d i s c i n
that
t h e n we
UM
i s a l s o an open poly-
BV
d i s c n e i g h b o u r h o o d of t h e compact s e t 1 (BLqM = -
233
=
A,
(AM)M = A ,
and for a n y o p e n p o l y -
This completes t h e proof.
BM.
i s a subset of a sequence space
A(P)
and
i s a s e q u e n c e of s c a l a r s , w e l e t
Let
space w i t h a b a s i s ,
b e an open p o l y d i s c i n a f u l l y n u c l e a r
U
E
A(P),
and l e t
K
be a compact
234
Chapter 5 U.
poZydisc i n m
T h e r e e x i s t a s e q u e n c e o f reaZ numbers
6n > 1
6 = (6n)n=1,
hood o f z e r o i n U
s u b s e t of
Since
Proof
m
<
such t h a t
A(P)
and
In=,
and
m,
6K
and
.= A ( P t )
E;
i s a f u l l y n u c l e a r space, w e can
m
( B ~ ) ~i n = P~ '
such t h a t
, i s a r e l a t i v e l y compact s u b s e t o f E
2"
Bn
if
1 +-
Then
U.
= 0
n
and l e t E~
E
=
> 1
( E
a neighbour-
G(K+V)CU.
choose a sequence
Let
V
is a r e Z a t i v e Z y compact
if
)"
n n=l'
for all
n
and
B,
f 0
235
Holomorphic functions on nuclear spaces with a basis Moreover,
if
(
then
z ~ ) L = K~ E
and h e n c e
EK
Bn(l
6
IEnznl
+
Bn # 0
and
B:, - ) Bn
=
B,
(otherwise +
c K1.
Now c h o o s e a n e i g h b o u r h o o d o f z e r o , EK + V' C U .
Let
Next c h o o s e
(an);=1
an d u l a ' n n
Let
{ ( z ~ ) ; = ~ A(P);
=
hood o f z e r o i n
Let
for a l l
A(P).
if
2"
Moreover,
cV ' .
= inf(En,sA)
We h a v e
6,,
>
3
1
6n
all all
W
i s a neighbour-
E '
=
n n,
and moreover,
This completes t h e proof.
W
( E : ) ~ = ~ .
an#O
( z ~ ) E~ W= ~ then
6n
U
< 1).
n,
all
< 1
and l e t
m
if
E'W
E~
suplzna;l n
u,!,
an=0
Now l e t
Since
n.
=
E:
if
hence
,ti,
W
(u;),,=~
P,
(a;),"=1
such t h a t
V',
be chosen so t h a t
P
E
such t h a t W
zn = 0)
for all and 6K
Izn~;anI
n
I",l q 1
s
Izna;I
and l e t
6
I",l
6 =
1
En
+
6
and
1
m
(6n)n=1.
I;=,
l' <
-
'n
m.
i s a r e l a t i v e l y compact s u b s e t o f
236
Chapter 5
If
E
happens to be a reflexive A-nuclear space, then the
proof o f the above results f o r the particular case much simpler. 00
(6nzn)n=l
m
A(P),
E
U=E
are
m
( z ~ ) ~E =A ( P~ ) +
This is because the mapping
the sequence defining A-nuclear-
ity.is a linear topological isomorphism. T h i s observation also leads to the following useful lemma. A(P)
Let
Lemma 5.19 let
m
6 = (6nn)n=1
b e a r e f Z e x i v e A - n u c l e a r s p a c e and
be t h e s e q u e n c e g i v e n by t h e A - n u c Z e a r i t y
of
E.
(a) If
f o r m s a f u n d a m e n t a l s y s t e m of c o m p a c t
(K)KE'k
s u b s e t s of
(6K)KEk
then
A(P)
i s also a
f u n d a m e n t a l s y s t e m of c o m p a c t s e t s for
ibi
A(P).
If
9
i s a f u n d a m e n t a l s y s t e m of n e i g h b o u r h o o d s
of
0
in
A(P)
(K+6V)VEv
then
i s a fundamental
s y s t e m of n e i g h b o u r h o o d s o f t h e c o m p a c t s u b s e t
ic)
If
K
A(P).
Of
V
i s a SundamentaZ
s y s t e m of i n c r e a s i n g
c o u n t a b l e o p e n c o v e r s of
R(P)
then so also
i s {
§5.2
m
(6vn)n=l;
m
(VnIn=l
E P1.
HOLOMORPHIC FUNCTIONS ON FULLY NUCLEAR SPACES WITH A BASIS
In this section, we show that the monomials form a basis for certain spaces o f holomorphic functions and analytic functionals on fully nuclear spaces. Definition 5 . 2 0 integer for a l l large).
I f
zm
m
n E
=
N
Let
N")
=
and
mn
= 0
and
m mn r! n = l 'n .
z
=
m
{(mn)n=l;
mn
m
(zn)n=l
a non-negatioe
n sufficiently
for a l l E
A(P)
we l e t
237
Holomotphic functions on nuclear spaces with a basis
The function
z
m
N").
each
in
A monomial
Iml
m
A(P)
E
-
m
z
E
i s c a l l e d a monomial f o r
CC
i s a Im(-homogeneous p o l y n o m i a l on
A(P)
where
mn.
=
Theorem 5 . 2 1
Let U
b a s i s and l e t
E
A(P)
--J
be a f u l l y n u c l e a r s p a c e w i t h a
be an open p o l y d i s c i n
E.
The m o n o m i a l s
form a n a b s o l u t e b a s i s f o r t h e c o m p l e t e n u c l e a r s p a c e (HHy(U) ,
Moreouer,
T ~ ) .
i f
E
is A - n u c l e a r
(H(E)
then
, T ~ )
i s a l s o A-nuclear. Proof
Let
be a compact s u b s e t o f
K
By lemma 5 . 1 8 ,
U.
t h e r e e x i s t s a sequence of p o s i t i v e r e a l numbers, m 1 6 n ' 1 and 6 < m , such t h a t 6K i s a ('n)n=l> m r e l a t i v e l y compact s u b s e t o f n U . Let 5 = (En)n=l E U .
I;=,
each p o s i t i v e i n t e g e r
r
Izn( c ( E n [
and
nsr
let
[tlr
= { ( z ~ ) E~ E=; ~
f o r n>r}. n f i n i t e d i m e n s i o n a l compact p o l y d i s c i n U . If and
m
E
5
=
for
z
For
m
= 0
[Elr f
E
is a HHy(U)
let
N
where
and
Support
m
(5n)n=1
6, # 0
i s chosen s o t h a t
C S u p p o r t (m) 1 .
ts)
if
# 0, ( i . e .
mn
The Cauchy i n t e g r a l f o r m u l a i n s e v e r a l v a r i a b l e s i m p l i e s that
am
f(z) all
d o e s n o t depend on t h e c h o i c e o f
1
=
a zm
for all
z
mENr
E
[t],,
6 all
and
r
E
N
and
6 i n U.
Now l e t
6
E
K.
If
m
E
N
and
Support (5)
S u p p o r t (m)
238 then
Chapter 5
Em
and
= 0
S u p p o r t (m) (z { 1 ,
lamEml 6
. . . ,r 1
In b o t h cases, w e have
m I/amz
/IK
i
I/f
11 .
and
l a m (m
I
i
I I f / ( K and hence
I / f l I K . Applying t h i s r e s u l t t o
Hence
Since
m
Otherwise, S u p p o r t ( c ) C
1 6
n
<
-
t h i s means t h a t
6K
we o b t a i n
239
Holornorphic functions on nuclear spaces with a basis Hence U,
f(z) =
I
4
and s i n c e
1
(N)
mcN f and
a z m , u
m
d e f i n e s a h y p o a n a l y t i c f u n c t i o n on a r e b o t h h y p o c o n t i n u o u s on
f
a g r e e on a s e t whose s e q u e n t i a l c l o s u r e i s f(z) f(z) = N
f o r every
z
in
U.
N")
Let E>O be a r b i t r a r y . Choose J 1 such t h a t f 6K 6 mEN(N)\ J 6 m
1 1 11
subset of
N")
1
which c o n t a i n s
J
and
it follows t h a t a ,mcN (N) ,
U,
The c o e f f i c i e n t s
are obviously u n i q u e l y determined by
U
f.
m
a f i n i t e subset of E .
If
J'
i s any f i n i t e
then
and hence
Since
and
6K
i s a r e l a t i v e l y compact s u b s e t o f
U
w e h a v e shown
t h a t t h e monomials form a n a b s o l u t e b a s i s f o r t h e s p a c e o f h y p o a n a l y t i c f u n c t i o n s on As t h e s p a c e A(Q)
U
w i t h t h e compact open t o p o l o g y .
( H H y ( U ) , ~ O ) i s c o m p l e t e , we may i d e n t i f y i t w i t h
where
and
&(U)
i s t h e s e t o f a l l compact s u b s e t s o f
Since
the Grothendieck-Pietsch c r i t e r i o n implies t h a t
U.
240
Chapter 5
(HHy(U),~o) If
U
is a nuclear space. E
=
E
and
is a reflexive A-nuclear space with
as the sequence occurring in t h e definition o f A (6 n);=1 nuclearity, then 6 K is a compact subset of E whenever K E (lemma 5 . 1 9 ) .
is a compact subset o f T
0
continuous semi-norm o n
1m e N (N)
m
a z m
H(E),
E
is also
(H(E),.ro)
H(E)
-io
n u c l e a r space
continuous and hence
is an A-nuclear space.
(H(U),T~)
space,
Corollary 5.23 =
is any
then the semi-norm
whenever
U
a f u l l y n u c l e a r space w i t h a b a s i s .
(U)
p
The m o n o m i a l s form a n a b s o Z u t e b a s i s f o r t h e
Corollary 5.22
A-nuclear
Hence if
then Let
HHy(U)
(H(E),.rO) E
I f
is a n o p e n p o l y d i s c i n E i s a reflexive
i s a l s o an A - n u c l e a r
space.
be a f u z z y n u c l e a r s p a c e with a b a s i s
f o r any open s u b s e t
= (HHy(U),~o) and any open p o l y d i s c U i n E.
(H(U),T~);
U
of
E.
Moreover,
9 (HHy(U),.ro)~ f o r
F o r corollary 5.22, it suffices to notice that each monom-
ial is continuous, that a subspace o f a nuclear space is nuclear and a subspace o f an A-nuclear space which contains the absolute basis is also A-nuclear.
T h e first part o f corollary
5.23-follows from the fact that we only used the boundedness o f hypoanalytic functions on compact sets in proving theorem 5.21. The second equality is a consequence o f the continuity of !nonomials and the completeness o f the space (HHy(U),~o). Theorem 5.21 shows that any bounded subset of (HHy(U),.ro)
is
contained in the closure of a bounded subset o f (H(U),ro). An application o f lemma 5.6 now completes the proof o f corollary 5.23. If
U
is a connected Reinhardt domain containing the
241
Holornoiphic functions on nuclear spaces with a basis o r i g i n , t h e n , by u s i n g f i n i t e d i m e n s i o n a l r e s u l t s on a n a l y t i c c o n t i n u a t i o n , o n e c a n show t h a t e a c h
f
E
H H Y W )2
(resp. H(U)
1
h a s a u n i q u e e x t e n s i o n as a h y p o a n a l y t i c ( r e s p . holomorphic) f u n c t i o n t o t h e modular h u l l o f
The method o f theorem 5 . 2 1
U.
c a n a l s o b e e a s i l y e x t e n d e d t o show t h a t t h e m o n o m i a l s f o r m a n absolute basis for
( H H y ( U ) , ~ o ) whenever
is a modularly
U
d e c r e a s i n g open s u b s e t o f a f u l l y n u c l e a r s p a c e w i t h a b a s i s . We n o w p r o v e a n a n a l o g o u s r e s u l t f o r t h e
H(U).
To s h o w , h o w e v e r ,
t o p o l o g y on
T~
( H ( U ) , T ~ ) i s n u c l e a r we m u s t
is an A-nuclear space.
E
assume t h a t
that
Proposition 5.24
Let U be a n o p e n polydisc i n a fully nucZear space with a basis E. Then the m o n o m i a Z s f o r m a n absolute basis for ( H ( U ) , T ~ ) . If E is also a n A-nuclear space, t h e n ( H ( U ) , r W ) a n A-nuclear space. Proof
Let
p
be a
is a nuclear space and
c o n t i n u o u s semi-norm on
T~
p o r t e d by t h e compact p o l y d i s c
let
b e a n open p o l y d i s c ,
V
By lemma 5 . 1 8 ,
in
K
Let
U.
t h e r e e x i s t an open p o l y d i s c
W
In=, 6
lamzml
where
a
sion of
m
Let
J
1 m - Ilamz
s
grn
f
‘K+W
H(U) H(U)
E
IIflIV
E
in >
1
z
E
z
m
and <
m.
and a for all. K+W
and
i n t h e monomial e x p a n -
(H(U),T~).
E
is
“6
is the coeff cient of
CK+W
P(g) If
n
then
f
such t h a t
K C V C U ,
s e q u e n c e -o f r e1 a l n u m b e r s 6 = (6 6,, n n = l where n and < such t h a t 6(K+W) C V . If mEN
(H(E),ru)
b e a p o s i t i v e r e a l number s u c h t h a t IlglI K + W
for all
i s any f i n i t e s u b s e t o f
N“)
g
in then
H(U).
242
Chapter 5
1
Since
i s f i n i t e , i t f o l l o w s t h a t t h e monomials 6" m c N "1 form an u n c o n d i t i o n a l b a s i s f o r (H(U),-cU). Moreover, s i n c e
t h e semi-norm
is
c o n t i n u o u s and h e n c e t h e monomials form an a b s o l u t e
-cU
basis for s'pace.
N o w suppose
(H(U),rU).
Let
m
(6n)n=1
be t h e sequence occurring i n t h e d e f i n -
i t i o n of A-nuclearity. m
numbers,
In=, 6'< m
1
n
m,
and
i s a l s o an A - n u c l e a r
E
Choose a s e q u e n c e o f p o s i t i v e r e a l
such t h a t 6'6'K
1 < 6' 6 6
n
n
for all
n,
i s a r e l a t i v e l y compact s u b s e t of
T h i s i s p o s s i b l e b y lemma 5 . 1 8 .
U.
243
Holornophic functions on nuclear spaces with a basis
f = C mcN "1 U
a z m
m
which contains
neighbourhood
Hence
W
in
6'6'K.
C
mcN
and let
b e an open subset o f
V
By lemma 5.18, there exists a 6'6'K
o f zero such that
q ,is ported by
Since
H(U)
6'6'K
pO
,q ( zm) # O q ( zm)
and
6 ' 6 ' W C V.
(N)
<
and all
p
If
U=E
then letting
6n
in the above proof, we see that
A-nuclear space.
m
meN
the Grothendieck-Pietsch criterion implies that a nuclear space.
Hence
continuous.
T
1
+
= 6;
(H(U),.rw)
is
for all
(H(E),-cw)
n
is an
This completes the proof.
In proving proposition 5.24, we have also shown the follm owing: if f E H(u), f(z) amz , where U is an open mEN polydisc in a fully nuclear space with a basis, and K is a
=I
compact subset of zero such that
U,
then there exists a neighbourhood
V
We now investigate when the monomials form a basis for H(U)
endowed with the barrelled and bornological topologies
of
244
Chapter 5
associated with
and
T~
T
~
.
We f i r s t s h o w t h a t t h e y a l w a y s
form an u n c o n d i t i o n a l equicontinuous b a s i s f o r t h e s e t o p o l o g i e s b u t t o s h o w t h a t t h e b a s i s i s a b s o l u t e we n e e d e x t r a h y p o t h e s e s on
E.
and
U
Proposition 5.25
If
E
nuclear space
is a n o p e n p o l y d i s c in a fully
U
with a basis,
t h e n t h e monomials form a n
unconditional equicontinuous basis for T
=
T
o"u'
Proof
or
b ' T w ,b
' 0 ,
Y
(H(U),T),
6 .
We h a v e a l r e a d y p r o v e d t h i s r e s u l t f o r
s i n c e an a b s o l u t e b a s i s i s a n u n c o n d i t i o n a l , equicontinuous,
basis.
If
and
T
T
and h e n c e a n
is an A-nuclear space, then we
E
may a p p l y p r o p o s i t i o n 5 . 1 3 t o s h o w t h a t t h e m o n o m i a l s f o r m a n absolute basis for
(H(U),T),
ever, does not cover
or
T h i s , howw,b' t h e g e n e r a l c a s e w h i c h we now p r o v e . We T
=T
o,b
Y
f i r s t show t h a t t h e m o n o m i a l s f o r m a n u n c o n d i t i o n a l b a s i s f o r ( H ( U ) , T & ) . Let every Vn
z =
in
Iz
E
J
of
and l e t
f
E
H(U).
f(z) =
1
mEN
For each p o s i t i v e integer
U.
/ I m E aJm z m /
U;
Then
s n
(N)
n
a zm m
for
let
f o r every f i n i t e subset
N")} be the interior of
Wn
Vn.
By o u r o b s e r v a t i o n co
preceding t h i s proposition, it follows t h a t U. If p i n c r e a s i n g c o u n t a b l e open cover o f c o n t i n u o u s semi-norm on
(Wn)n=l is a
H(U), t h e n t h e r e e x i s t
is an
C > 0
and
a p o s i t i v e i n t e g e r such t h a t p(g)
s c
IlglI
wN
f o r every
g
in
H(U).
J finite
and
{
I ~ E a Jm z m
JCN"),
J finite
is
T&
bounded.
N
245
Holomorphic functions on nuclear spaces with a basis
By lemma
lmsJ,J finite am zm
3*28,
+
f
as
J + W
in
;;Tlfo
as J + m a zm + 'mEJ,J finite m n ! lml=n in ( ? ("E),T~) for each non-negative integer n . Since the monomials form an absolute basis for (H(U),TW) this shows (H(U),T~)
if and only if
that they also form an unconditional basis for
( H ( U ) ,T6).
Since T~ t T this also shows that the monomials form ~ b T,o,b~ an unconditional basis for (H(U) ,Tw,b) and (H(U) ,TO,b).
(H(U),-r6)
fa
then
+
f
in
(H(U),-ro)
as
and hence,
since the monomials form a Schauder basis for
(H(U),T~)
aa + am as a + m for each m in N") and the monomials m form a Schauder basis for (H(U),T,),(H(U),T~,~) and
(H(U) 9To,b) .
This completes the proof.
If U is a n open polydisc in a fully Corollary 5 . 2 6 is the barrelled topology nuclear shace with a basis, then T~ assoeiated with T ~ .
Proof T
0
If
Let
on
p
Since
H(U).
is a
T
p'(f)
=
be the barrelled topology associated with
T
(H(U) ,-r6)
is barrelled,
cont n u o u s semi-norm o n
6
H(U)
T I T S T 0 6' then
sup
JCN")
J fini e is also a T~ finite subset
is
continuous semi-norm and p ' of N") the semi-norm
continuous and hence the set
T~
5 p
F o r each
J
V = {fEH(U);p(f)
$
l}
is T O (and hence T ) closed, convex, balanced and absorbing. Thus V is a T-neighbourhood o f zero and p' is continuous. Hence
T
3
T
6
and this completes the proof.
246
Chapter 5
space T
=
E
U
If
Corollary 5.27
with a basis,
T o>TwJ T
o , bJ?w, b
O r
i s a n open p o l y d i s c i n a f u z z y nucZear t h e n t h e foZZowing a r e e q u i v a Z e n t for T6;
(a)
(H(U),T)
is c o m p l e t e ,
(b)
(H(U)
, T )
i s quasicompzete,
(el
(H(U)
,T)
i s s e q u e n t i a l l y complete,
if
id)
is a s e t of c o m p l e x numbers
{arn} mE N “1
and
m amz 1
{ImEJ
i s a r-Cauchy
If (el
T
=
T
if and
or
0
I
net,
mEN
is a s e t of c o m p l e x numbers
“1
(N)
then
t h e n t h e above a r e equivaZent t o
w
{am}
, J finite
JCN“)
p(amz
m
) <
m
f o r every
mEN
c o n t i n u o u s semi-norm on
H(U)
then
We omit the proof since it is similar to the analogous result proved f o r holomorphic functions on balanced open sets in chzipter 3 (proposition 3 . 3 6 ) .
E i s a r e f Z e x i v e A-nuclear space, t h e n i s a n A-nucZear s p a c e f o r T = T ~ , ~T ’ ~ o , b ’ ~ w , obr
Proposition 5.28 (H(E),r)
If
Tc6’
E
Proof
If
shown
(H(E),.ro)
is a reflexive A-nuclear space, then we have is A-nuclear (corollary 5.22) and
is A-nuclear (proposition 5.24).
By proposition 5 . 1 3 ,
(H(E),T~)
the
241
Holomorphic functions on nuclear spaces with a basis associated bornological topologies, A - n u c l e a r t o p o l o g i e s on
H(E).
and
o,b
‘ w,b’
are a l s o
T h e r e r e m a i n s o n l y t h e case = ‘I w h i c h we now d i s c u s s . 6 Our p r o o f f o r t h i s c a s e c a n a l s o b e e a s i l y a d a p t e d t o g i v e 02
d i r e c t proofs for the other topologies. Let 6 = (6n)n=1 t h e sequence occurring i n t h e d e f i n i t i o n of A-nuclearity. Since
E
is a complete A-nuclear space,
i s a l i n e a r i s o m o r p h i s m when
H(E)
t h e mapping:
i s endowed w i t h t h e
topology ( i t i s a l s o an isomorphism f o r any o f t h e o t h e r topologies). then H(E).
p
If
XEB.mEN
is a
T~
‘6
i s a T~ bounded s u b s e t o f H(E) is also a bounded s u b s e t of
{ f
If
{ (62)matzm}
be
“1
c o n t i n u o u s semi-norm on
H(E)
and
then
f o r all
1
m amz
E
H(E).
mEN
The a b o v e shows t h a t
p1
i s bounded on t h e
T~
bounded sub-
sets of H(E) and h e n c e i s T~ continuous since (H(E) , T ~ ) is a bornological space. As t h e monomials form a n u n c o n d i t i o n al basis for
(H(E),-c6),
p
6
pl.
2 48
Chapter 5
t h i s shows t h a t
( H ( E ) , T ~ ) i s an A-nuclear space and completes
the proof. U
If
is a n o p e n s u b s e t of a l o c a l l y c o n v e x s p a c e arid H(U)
is a Z o c a l l y c o n v e x topology o n (H(U),r)'
a r e c a l l e d a n a l y t i c f u n c t i o n a l s (or t o b e more
precise,
analytic functionals on
space form,
(H(U],T)?
U).
can be interpreted,
T
t h e n t h e e l e m e n t s of We s h o w t h a t t h e
v i a t h e "Borel"
as a s p a c e o f holomorphic germs on
trans-
U
whenever
UM
is
a n open p o l y d i s c i n a f u l l y n u c l e a r space with a b a s i s . C o n t i n u i t y p r o p e r t i e s o f t h e germs w i l l depend on o u r c h o i c e o f T
-
f o r example
t i n u o u s germs and a n a l y t i c germs.
T~
a n a l y t i c f u n c t i o n a l s g i v e r i s e t o con-
T
a n a l y t i c f u n c t i o n a l s g i v e r i s e t o hypo-
T h i s i n t e r p r e t a t i o n a l l o w s u s t o s e t up a
correspondence between p r o p e r t i e s o f
H(U)
t r a n s l a t e s e e m i n g l y d i f f i c u l t problems on t r a c t a b l e problems on find that
(H(U),ro)
and H(U)
and conversely.
H(UM)
H(UM)
and t o
i n t o more
For
instance, we
i s i n f r a b a r r e l l e d i f and o n l y i f
H(U')
is a regular inductive l i m i t . The key t o o u r i n v e s t i g a t i o n s i s t h e e x i s t e n c e o f an absolute basis i n
(H(U),T)
and,
indeed,
certain results
a b o u t a n a l y t i c f u n c t i o n a l s a l r e a d y f o l l o w f r o m t h e r e s u l t s we have j u s t proved. that
(H(E),r);
P r o p o s i t i o n s 5 . 1 2 a n d 5 . 2 8 t o g e t h e r show i s an A - n u c l e a r s p a c e (and h e n c e h a s an
a b s o l u t e b a s i s and i s n u c l e a r ) s p a c e and
T
E
i s a r e f l e x i v e A-nuclear
i s any one o f o u r u s u a l t o p o l o g i e s on
Propositions 5.9,5.21 ( H ( U ) 9 T o , b) '
i f
and 5 . 2 5 imply t h a t
H(E).
( H ( U ) , T ~ ) ~a n d
b o t h have a n a b s o l u t e b a s i s whenever
U
i s an
open p o l y d i s c i n a f u l l y n u c l e a r space with a b a s i s while propositions 5.9,5.24 (H(U),Tw,b)i
and 5.25 imply t h a t
( H ( U ) , T ~ ) ; and
b o t h h a v e a n a b s o l u t e b a s i s when
U
i s a n open
polydisc i n a r e f l e x i v e A-nuclear space. T h e B o r e 1 t r a n s f o r m d e f i n e d h e r e I S N O T T H E SAME a s t h a t g i v e n i n c h a p t e r s 1 a n d 3 , b u t i t i s u s e d f o r more o r l e s s t h e same p u r p o s e s .
F o r t h i s r e a s o n , we a d o p t t h e s a m e t e r m i n o l o g y
b u t u s e a d i f f e r e n t n o t a t i o n t o d i s t i n g u i s h between t h e
249
Holomorphic functions on nuclear spaces with a basis d i f f e r e n t Borel transforms Let basis
b e an open p o l y d i s c i n a f u l l y n u c l e a r space w i t h a
U
E Z A(P).
all
m
for
(H(U),r6),
If
N").
E
T
E
(H(U),T~)' we l e t
S i n c e t h e monomials form a n u n c o n d i t i o n a l b a s i s
T
i s f u l l y determined by T,%T,
The Bore2 t r a n s f o r m of E; S A(P')
N
B1'
w h e r e t h e r i g h t hand s i d e of
and i s G-holomorphic. mapping. E
N
"1.
is g i v e n b y t h e s e t of p o i n t s converges).
tf
neighbourhood o f
%
One e a s i l y s e e s t h a t
0
in
E'
is an injective
The r e l a t i o n s h i p b e t w e e n o u r two B o r e l t r a n s f o r m s , cc,
and
B T
mE
i s defined on a subset of
(*)
i s t h u s d e f i n e d on a
BT
{bm)
by t h e formula
( t h e d o m a i n of d e f i n i t i o n of
N
bm = T ( Z ~ ) f o r
i s contained i n t h e following formula.
B,
(H(U),T&)'
and
bm = T ( z m )
for all
mcN
If then
and
where (
' .1
m
1
Iml!
=
ml!
...
m
n
!
i f
m = (ml,.
..,mn,O,. ..).
We now l o o k m o r e c l o s e l y a t t h e B o r e l t r a n s f o r m o f analytic functionals. Theorem 5 . 2 9
Let
space w i t h a b a s i s .
U
T~
be an open p o l y d i s c i n a f u l l y n u c l e a r
The B o r e l t r a n s f o r m ,
t o p o Z o g i c a Z i s o m o r p h i s m from
(H(U)
isomorphism e s t a b l i s h e s a one-to-one
,T~);
N
B,
onto
i s a l i n e a r and H(UM).
This
correspondence between
Chapter 5
250
equicontinuous subsets of ( H ( U ) , T ~ ) I and sets o f germs which a r e defined and uniformZy bounded o n a neighbourhood of U M . Let
Proof in
T
K
polydisc
U
s m
There e x i s t
(T(f)/
such t h a t
6,>l
i s a neighbourhood of
s
all
f
n,
i s a r e l a t i v e l y compact s u b s e t of
SK
UM
N"),
in
and a compact
C>O
C / ( f ( ( K f o r every
we may c h o o s e a s e q u e n c e o f p o s i t i v e
m
6 = (6n)n=1
< m and In=1 n lemma 5.17,(6K)*
any
, T ~ I) .
such t h a t
By l e m m a 5 . 1 8 ,
H(U).
real numbers, m
(H(U)
E
in
in
E' B
By
U.
and,
for
-C gm
where
and (6K)
bm
N
M
T(zm).
d e f i n e s a holomorphic f u n c t i o n on t h e i n t e r i o r o f
BT
.
=
We h a v e t h u s s h o w n t h a t t h e B o r e 1 t r a n s f o r m m a p s
(H(U),T~)' into depends o n l y on
H(UM) C
and
and moreover, K
s i n c e o u r bound on
d
BT
w e h a v e a l s o shown t h a t t h e i m a g e
o f an equicontinuous subset o f
( H ( U ) , T ~ ) ' i s a s e t of germs
which i s d e f i n e d and u n i f o r m l y bounded on a neighbourhood o f M U . We now p r o c e e d i n t h e o t h e r d i r e c t i o n . Let
g
E
H(UM).
d i s c neighbourhood of
UM
is
g,
?(w)
By l e m m a 5 . 1 7 , UM,
V,
and
t h e r e e x i s t an open polyE
H(V),
such t h a t =
I
mEN
(N) bmwm
for all
w E V
and
whose germ on
25 1
Holomotphic functions on nuclear spaces with a basis
By lemma 5.18, V and C ' can be chosen uniformly for any family o f germs which are defined and uniformly bounded on a fixed neighbourhood o f
1
for each
If
mE N (N)
mEN")
lambml
and
a z m
m
IIwmllV
m C 1 / l a m z 11 M . V
$
UM.
=
Let
in
H(U).
+a,
then
bm = 0
and hence
Otherwise
and
Hence
and
Tg
E
(H(U),T~)'.
Since
d
BTg(w)
=
$(w)
for all
WCV
this implies that the Bore1 transform is a linear isomorphism from
(H(U),T~)'
ute basis for
is
T~
onto
(H(U)
H(UM).
, T ~ )
A s the monomials form a n absol-
the semi-norm
continuous and thus we have established the required
result about equicontinuous subsets o f Finally, we show that On
N
B
(H(U),T~)'.
is a topological isomorphism.
we have two locally convex topologies - the natural inductive limit topology, T ~ , and the topology transferred by. H(U')
252
%
Chapter 5 from
that
( H ( U ) , T ~ ) ~ T, =
T .
T
c o m p l e t e t h e p r o o f , we m u s t s h o w M We u s e lemma 5 . 2 . (H(U ) , T ~ ) i s a b a r r e l l e d
B.
.To
~
l o c a l l y c o n v e x s p a c e s i n c e it i s a n i n d u c t i v e l i m i t o f Banach spaces. By c o r o l l a r y 5 . 2 3 , ( H ( U ) , . r o ) i = (HHy ('1 9 T o ) and hence, s i n c e ( H H y ( U ) , ~ o ) i s a complete n u c l e a r and t h u s a semi-reflexive
space,
(H(UM) ,
To c o m p l e t e t h e p r o o f ,
i s also a barrelled space.
T ~ )
u s i n g lemma 5 . 2 ,
w e m u s t show t h a t t h e M monomials form an a b s o l u t e b a s i s f o r b o t h (H(U ) , T ~ ) a n d ( H ( U M ) p B ) . By p r o p o s i t i o n 5 . 9 , basis for (H(U') ,T B ) . If
g
H(UM)
E
containing g
on
g
=
1
g
=
1
N
mc N
mE
N
t h e n t h e r e e x i s t s an open p o l y d i s c
and
UM
H(UM).
7
E
(N) b m w m
(N)
1 mEN
Since
H(V)
such t h a t
V
d e f i n e s t h e germ
By p r o p o s i t i o n 5 . 2 4 ,
in
b wm m
and hence
M
(H(U ) , T ~ ] . I f
p
is a
T .
H ( u ~ ) , let
bmwm
"1
(H(V),T~)
in
c o n t i n u o u s semi-norm on
for e v e r y
t h e monomials form a n a b s o l u t e
in
H(UM)
I
i s c o n t i n u o u s f o r every open p o l y d i s c P I (H(V) Jw)
containing
UM
V
and t h e monomials form an a b s o l u t e b a s i s f o r
( H ( V ) , T ~ ) it f o l l o w s t h a t e v e r y open p o l y d i s c
'1
(H(V)
containing
V
,Tu)
UM
is
and h e n c e
continuous f o r 4
p
is
T.
continuous. Hence t h e monomials form a n a b s o l u t e b a s i s f o r M (H(U ) , T ~ ) a n d t h i s c o m p l e t e s t h e p r o o f . C o r o l l a r y 5.30
Let
U
b e an o p e n p o l y d i s c i n a fully
25 3
Holomorphic functions on nuclear spaces with a basis
with a basis.
E
nuclear space erties:
Consider t h e f o l l o w i n g prop-
(a)
(H(U),.ro)
i s a bornological space,
(b)
(H(U)
i s an infrabarreZled space,
(c)
H ( u ~ ) =
, T ~ )
11 \ l V )
l i m
(H~(v),
regular
'Ls a
----f
v 2 UM
V open i n E '
inductive l i m i t , id)
H(UM)
B
i s complete,
(fj
bounded l i n e a r f u n c t i o n a l s o n continuous, H ~ U M ) is q u a s i - c o m p l e t e ,
(g)
H ( u ~ ) i s sequentially complete,
(el
H(U)
T~
are
T
( a ) < = >( b ) < = > ( c ) = > ( d ) < =(>e ) < = > ( f ) < = > ( g ) .
then
E
Furthermore, i f
i s A - n u c l e a r a l l of t h e a b o v e p r o p e r t i e s U = E.
a r e e q u i v a l e n t when Proof
I n any l o c a l l y convex space
(d)=>(f)=>(g).
Since
e a s i l y show t h a t
H(UM)
(g)=>(d).
( a ) = > ( b ) , and
Now s u p p o s e
b e a semi-norm on
H(U)
subsets of
By p r o p o s i t i o n 5 . 2 5 ,
H(U).
(a)=>(e)=>
has an a b s o l u t e b a s i s , (b)
which i s bounded on
one can
holds.
Let
p
bounded
T~
w e may s u p p o s e
J finite
for every
Let
absorbs every t h i s shows t h a t continuous.
Then
V
bounded s u b s e t o f
T
e a s i l y seen t o b e T~
c 11.
V = {fEH(U);p(f) T
V
0
c l o s e d and
i s convex, balanced and H(U).
(H(U),.ro)
Since
V
is
is infrabarrelled
i s a neighbourhood of z e r o and h e n c e
Thus
(b)=>(a).
(b)
and
(c)
p
is
are equivalent
25 4
Chapter 5
by theorem 5 . 2 9 ,
s i n c e a l o c a l l y convex s p a c e
is infra-
F
b a r r e l l e d i f and o n l y i f t h e e q u i c o n t i n u o u s s u b s e t s and t h e
F'
s t r o n g l y bounded s u b s e t s o f NOW
H ( u ~ ) i s complete.
suppose
and 5.25,
coincide.
6
BY p r o p o s i t i o n s 5 . 9 , 5 . 2 1 ,
t h e monomials form an a b s o l u t e b a s i s f o r b o t h ( H ( U ) , T ~ , ~. ) ~I f
( H ( U ) , T ~ ) ~a n d
T
( H ( U ) , T ~ , ~ ) t' h e n t h e
E
p a r t i a l sums i n t h e monomial e x p a n s i o n o f in
(H(U),ro)l
and hence
t h i s completes t h e proof Now s u p p o s e (H(E),.ro) only i f
and
(H(U),ro)l.
E
i s an A-nuclear
E
p r o p o s i t i o n 5.28,
T
for arbitrary
form a Cauchy n e t
T
Thus
(d)=>(e)
and
U.
s p a c e and
U = E.
By
t h e monomials form an a b s o l u t e b a s i s f o r b o t h
( H ( E ) , T ~ , ~ ) .By lemma 5 . 1 ,
T~
( H ( E ) , T ~ ) ' = ( H ( E ) , T ~ , ~ ) 'a n d h e n c e
= T
o,b (e)=>(a).
i f and This
completes t h e proof. Corollary 5.31
basis. on
H(U)
Proof
Then
E
Let T~
= T
be a f u Z l y nucZear space w i t h a on
0,b
H(E)
i f and o n l y i f
f o r e v e r y open p o l y d i s c By c o r o l l a r y 5 . 3 0 ,
T
i n E.
U =
T
0,b is a regular inductive l i m i t . 0
on
H(E)
T~
= T
0,b
i f and o n l y
if H(OEl) Since t h e space of B germs a b o u t any compact p o l y d i s c i s r e g u l a r i f and o n l y i f t h e
s p a c e o f germs a t t h e o r i g i n i s a l s o r e g u l a r ,
a further applic-
a t i o n o f c o r o l l a r y 5.30 completes t h e p r o o f . Corollary 5.32
U
i s an o p e n p o Z y d i s c i n a F r g e h e t n u c l e a r s p a c e w i t h a b a s i s , t h e n T~ = T~ o n H(U) i f and onZy i f
H(UM)
Example 5 . 3 3
I f
i s a regular inductive l i m i t . (a)
If
admit a c o n t i n u o u s norm, 2.52).
then
i s a Frgchet s p a c e which does n o t T
Hence, by c o r o l l a r y 5.31,
has a basis then p a r t i c u l ar ,
EN
E
H(OEl)
H(Oc(N)) 'is
#
on
y6
if
E
H(E),
(example
i s a l s o n u c l e a r and
is not a regular inductive l i m i t . not a regular inductive l i m i t since
d o e s n o t a d m i t a c o n t i n u o u s norm.
We h a v e a l r e a d y p r o v e d
In
255
Holomotphic functions on nuclear spaces with a basis t h i s d i r e c t l y i n example 3 . 4 7 . that a
83n
s p a c e w i t h a b a s i s and
norm. (b) of
E
More g e n e r a l l y ,
t h e a b o v e shows
i s n o t a c o m p l e t e i n d u c t i v e l i m i t whenever
H(OE)
does not admit a continuous
Ei
i s a F r g c h e t s p a c e and
E
If
then H(K)
(H"(V),
l i m
=
---t
is
E
II
i s a compact s u b s e t
K
llv)
V 3K V open
is a regular inductive l i m i t c o r o l l a r y 5.30, whenever
U
i s a k-space,
E
since
( p r o p o s i t i o n 2 . 5 5 ) and h e n c e , by T
i s an open p o l y d i s c i n a & J k
0
on
= T~
H(U)
space with a b a s i s .
This i s a p a r t i c u l a r case of t h e r e s u l t proved d i r e c t l y i n example 2.47. We now c h a r a c t e r i z e t h e B o r e 1 t r a n s f o r m o f functionals.
T~
analytic
T h i s c h a r a c t e r i z a t i o n was o r i g i n a l l y u s e d t o
prove t h e t o p o l o g i c a l isomorphism o f theorem 5.29,
and l e a d s t o
a s i m p l e c r i t e r i o n f o r comparing
H(U),
U
T
and
0
on
T~
when
i s an open p o l y d i s c i n a f u l l y n u c l e a r s p a c e w i t h a b a s i s .
Proposition 5.34
U
Let
be a n o p e n p o 2 y d i s c i n a f u 2 2 y &
B, is a n u c l e a r space w i t h a b a s i s . The Bore2 t r a n s f o r m , M v e c t o r s p a c e i s o m o r p h i s m from (H(U) , T ~ ') o n t o HHy(U ) . V
Moreover, a s u b s e t
( H ( U ) , T ~ ) ' i s e q u i c o n t i n u o u s i f and
of &
o n 2 y if t h e germs i n B(V) a r e d e f i n e d and uniformly bounded M o n t h e compact s u b s e t s o f some n e i g h b o u r h o o d of U . Proof K
in
have
Let
T
E
( H ( U ) , T ~ ) ' . T h e r e e x i s t s a compact p o l y d i s c
s u c h t h a t f o r e v e r y open p o l y d i s c
U
IT(f)l
G
C(V)
d e p e n d s o n l y on
T
IlflIV and
for all V.
f
Moreover,
in
V,
K C V C U ,
H(U)
where
the set of all
we c(V)
T
which
s a t i s f i e s t h e above i n e q u a l i t i e s forms an equicontinuous subset of (H(U),T~)'. bourhood V of m
6 = (6n)n=13
By lemma 5 . 1 8 , K
6n >1
we c a n c h o o s e f o r e a c h n e i g h -
a s e q u e n c e o f p o s i t i v e r e a l numbers m 1 m, and a n f o r a l l n and In=, dn
256
Chapter 5
open p o l y d i s c
and
subset of
U
in
If
r
N").
= {mcN(N);
in
W
6( K + W ) C V.
11 zmllV
) ) z r n ) l V<
By lemma 5 . 1 7 ,
such t h a t
E
BT
E
then
=
.
m}
i s a r e l a t i v e l y compact
6K
Let
IIwmll
bm = T ( z m ) VM
m
f o r each Let
= 0.
Then
Y
HHY(U').
By t h e u n i f o r m i t y o f o u r b o u n d s
we h a v e a l s o s h o w n t h a t t h e B o r e 1 t r a n s f o r m m a p s e q u i c o n t i n u o u s subsets of
(H(U),rw)'
onto subsets of
H(UM)
which are
d e f i n e d and u n i f o r m l y bounded on t h e compact s u b s e t s o f neighbourhood of
some
U". rJ
i s a s u r j e c t i v e mapping. Let V b e an open p o l y d i s c neighbourhood of U M a n d l e t .& = ( C K ) K E ~ . B
We now s h o w t h a t
b e a s e t o f p o s i t i v e r e a l numbers indexed by t h e f a m i l y , of compact p o l y d i s c s i n
Ha
Let If
g
m IIbmw If
E
H a
I/K
<
f = X
then CK
"1 mEN
V .
{gEHHy(V); I l g ) l K < C K
=
g =
1
b wm m N (N) f o r each K in k mE
a z m
m
E
k ,
( H ( U ) , T ~ ) and
f o r each
in
in
(HHy(V),To)
and g
K
in
m E
HA
N").
we let
k and
1.
251
Holornorphic functions on nuclear spaces with a basis
For any open polydisc
W
in
containing
VM we can find a 6 n > 1 all n and sequence o f real numbers, 6 = (6n)n,l, 1 m, and W1 a polydisc neighbourhood o f zero in E n such that 6VM is a relatively compact subset o f U and (vM+wl) is a compact polydisc 6(v"+w1) c W. B Y lemma 5 . 1 7 , U
m
in
V.
C = C
Let
C m -b m z
lambml
6m
I1
(VM+W1 1
i
I/zrnll
is finite and
*
Vb'+W1
such
IIzml/
'I
M . C
6
IIw
.
[If
IIwmI/
M
=
v +wl
m
Otherwise
M
=
1.
then /IzmlI
v +wl
For all
( V +W1)
m
i
i
for every Since
f
in VM
H(U)
is a compact subset o f
U
and
W
was N
T g E ( H ( U ) , T ~ ) I . A s BTg = g this proves that B is surjective, and since our bounds a r e uniform over g in H A , we have a l s o shown that (Tg )g E H is a n equicontinuous subset o f (H(U),rU)l. This completes the proof. arbitrary, we have shown that
25 8
Chapter 5
Corollary 5.35
E
Let
be a f u l l y nuclear space w i t h a
The f o l l o w i n g a r e e q u i v a l e n t :
basis. (a)
T
0
H(U)
on
= T
for e v e r y o p e n p o l y d i s c
U
E;
in
( H ( U ) , T ~ ) ; = ( H ( U ) , T ~ ) ; I for e v e r y o p e n p o l y d i s c
ibi
(H(U),ro)'
(c)
(H(U),T~)
=
U
for e v e r y o p e n p o l y d i s c
E,
i n
for e v e r y o p e n s u b s e t V of El 8' i s c o m p l e t e for e v e r y o p e n s u b s e t V
id)
HHy(V) = H(V)
(el
(H(V),.ro)
ifi
o f Eb, t h e b o u n d e d s u b s e t s of ( H ( V ) , - r o ) a r e loea2Z.y b o u n d e d for e v e r y o p e n s u b s e t V o f E L . I t is clear that
Proof
U
E,
in
and
(d)=>(e).
Since
t h e monomials form a n a b s o l u t e b a s i s f o r b o t h
(a)=>(b)=>(c)
(H(U),-ro)
and
( H ( U ) , T ~ ) , lemma 5 . 1 s h o w s t h a t (d) are e q u i v a l e n t .
( c ) and
(c)=>(a).
Since
(
H
m
By p r o p o s i t i o n 5 . 3 4 ) = HHy(V)
open p o l y d i s c i n a f u l l y n u c l e a r space w i t h a ' b a s i s
(f)
If where
i s s a t i s f i e d and
ImEJ is a
T
g =
i s an open polydisc i n
V
-bounded
amzm
JCN"),J
subset of
sup (g(z)1
Hence
<
m
and
"1 mEN
a z m
m
E
HHY(V)
9
then
finite
H(V)
K C V C U ,
V,
1
EA,
and hence i s l o c a l l y bounded K
Hence f o r e a c h compact p o l y d i s c open p o l y d i s c
(corollary
(e) are equivalent.
(d) and
5.23),
f o r any
M <
and
g
in
E
H(V).
V
t h e r e e x i s t s an
such t h a t
T h i s shows t h a t
ZEV
( f ) = >( d )
.
Finally,
if
(a) i s s a t i s f i e d , then t h e equicontinuous
259
Holomorphic functions on nuclear spaces with a basis
subsets of ( H ( U ) , T ~ ) and ( H ( U ) , T ~ ') coincide. By theorem 5 . 2 9 , and proposition 5 . 3 4 , this means that (a)=>(f) and completes the proof. Example 5 . 3 6 (a) If U is an o p e n p o l y d i s c in a Frcchet nuclear space with a basis or in a 8 3 n space with a basis, then
=
T~
T
on
w
H(U).
This result follows from corollary
5 . 3 5 since condition (d) is easily seen to be satisfied.
We
have already proved this result for arbitrary open subsets o f
a3'Lz
spaces (example 2 . 4 7 )
and for entire functions o n
Frzchet nuclear spaces (corollary 3 . 5 4 ) . we showed that T~ # T~ on H ( C N x (E")).
I n example 1 . 3 9 , This is a particular
(b)
case of the following result which is an immediate consequence o f corollary 5 . 3 5 and example 1 . 2 3 .
If
is a n i n f i n i t e d i m e n s i o n a l f u l l y n u c l e a r s p a c e w i t h T~ # T~ on H(E x E l ) .
E
a basis,
then
B
We also obtain a topological characterization of (H(U),rw); in certain situations. This is illustrated by the following proposition. Proposition 5 . 3 7
Let
U
b e a n o p e n p o l y d i s c in a f u l l y E.
n u c l e a r space w i t h a b a s i s (a) (b)
(H(U),T~);
EHy(U
M
1
= lim +
(HHy(V),ro),
V3UM V open
(H(U),-cw);l
has t h e monomials a s an a b s o l u t e b a s i s (H(U) , T ~ ) i s s e m i - r e f l e x i v e ,
and (c)
=
The f o l l o w i n g a r e e q u i v a l e n t :
t h e T~ bounded s u b s e t s o f bounded.
Moreover,
if
E
i s an A-nuclear space,
equivalent t o the following: (d)
(H(U)
, T ~ )
(e)
(H(U)
, T ~ )
H(U)
is s e m i - r e f l e x i v e , is q u a s i - c o m p l e t e .
are l o c a l l y
t h e n t h e above a r e
Chapter 5
2 60
Proof The m o n o m i a l s f o r m a n a b s o l u t e b a s i s f o r H H Y by theorem 5.21. By t h e o r e m 5 . 2 9 , lemma 5 . 1 7 a n d c o r o l l a r y 5 . 2 3 , H H y ( U M ) ’ may b e i d e n t i f i e d , v i a t h e B o r e 1 t r a n s f o r m , w i t h H(U). (b)
An a p p l i c a t i o n o f lemma 5 . 1 now shows t h a t are equivalent.
be a K
T
in
Now s u p p o s e
bounded s u b s e t o f
w
H(U).
and Let
For each compact p o l y d i s c
t h e r e e x i s t s an open p o l y d i s c
U
(a)
is satisfied.
(c)
W
such t h a t
6,>1 6 = ( 6 n ) mn = 1 , an open p o l y d i s c i n E
Choose a s e q u e n c e o f p o s i t i v e r e a l numbers for all such t h a t
n
and
m
Cn=l
1
6(K+V) C K + W .
<
m,
and
V
Hence
and
i s a l o c a l l y bounded and h e n c e a Since
T
bounded s u b s e t o f
H
26 1
Holomoiphic Jirnctions on nuclear spaces with a basis t h i s proves t h a t N (N 1 (H(U),T~)~ h a s t h e monomials a s an a b s o l u t e b a s i s .
f o r any s e t o f s c a l a r s
{bm}
mE
then t h e r e e x i s t s a that
x
aml
{I
B =
where
1
lies in
i s s e m i - r e f l e x i v e and
Suppose be a
T
EN'^)
(b)
.
is satisfied.
bounded s u b s e t o f
w
am = suplakl
x
m
@ E
H(U).
X E r
( c ) = >( b )
B
m
f o r every
in
Let
H(U).
3A E r
amz
N (N)
suplamlz ,
a n d we h a v e shown
H(U)
bounded s u b s e t
w
/(wmII
A m
mE
I t now f o l l o w s t h a t
T
I am[ 6
( @ ( w m ) /=
If
1
and h e n c e
C =
mE
For each
N
m
m
mE N (H(U),T@)
Hence
{I
( N ) amz
(N)
in
A m
amz
1
A E r
let
N")
and l e t
A
1
f o r every
rnE
N
(N)
bmwm
in
(H(U)
( H ( U ) , T ~ ) ~ +,,
m i a l s form an a b s o l u t e b a s i s f o r c o n t i n u o u s form on
(H(U),T@)A.
i d e n t i f i e d with an element of
Hence,
H(U),
f o r e v e r y compact p o l y d i s c
By n u c l e a r i t y ,
i t now f o l l o w s t h a t
subset of
If
H(U)
E
and h e n c e
in
B
(b)=>(c)
and h e n c e
b a s i s by p r o p o s i t i o n 5 . 9 .
Hence
U,
such t h a t
.
i s an A-nuclear space then
by p r o p o s i t i o n 5 . 2 4 ,
T
+,
w
may b e
that is,
K
W
K
is a
By s e m i - r e f l e x i v i t y
open p o l y d i s c
containing
,T~)'.
t h e r e e x i s t s an A zm s y p llam < m.
I(w
i s a l o c a l l y bounded
(H(U),T~) is nuclear
( H ( U ) , T ~ ) h~a s a n a b s o l u t e (b)
and
(d)
a r e equivalent.
Chapter 5
262
In g e n e r a l , it i s e a s i l y seen t h a t satisfied then
(c)=>(e).
If
(e)
is
( H ( U ) , T ~ ) i s a q u a s i c o m p l e t e n u c l e a r s p a c e and
hence it i s semi-reflexive.
Thus
( e ) + (d)
and t h i s completes
the proof. The Bore1 t r a n s f o r m o f
T~
analytic functionals is
t r e a t e d i n e x e r c i s e 5.81,
H O L O M O R P H I C FUNCTIONS ON
55.3
DN
SPACES W I T H A BASIS
Using t h e r e s u l t s o f t h e preceding i o n s o f t h e t e c h n i q u e s u s e d t o show
T
s e c t i o n and m o d i f i c a t =
on
T~
i s a Banach s p a c e w i t h a n u n c o n d i t i o n a l b a s i s
show t h a t nuclear
=
T
DN
T~
on
H(U)
when
H(E)
when
E
(section 4.3),
we
i s an open p o l y d i s c i n a
U
space with a basis.
We b e g i n b y r e c a l l i n g s o m e f u n d a m e n t a l f a c t s a b o u t
DN
spaces. S,
t h e space of rapidly decreasing sequences,
is the
Frgchet nuclear space with a b a s i s consisting of a l l sequences, m
(Zn)n=l
o f complex numbers such t h a t
is finite for all positive integers m
g e n e r a t e d b y t h e norms
( P , ) ~ = ~ .s
m.
The t o p o l o g y o f
is a universal generator
f o r t h e c o l l e c t i o n o f n u c l e a r l o c a l l y convex s p a c e s , i . e . ,
l o c a l l y convex s p a c e
E
is
s a
i s n u c l e a r i f and o n l y i f i t i s
isomorphic t o a s u b s p a c e o f
sA
f o r some i n d e x i n g s e t
A.
iz
depends on t h e c a r d i n a l i t y o f a fundamental neighbourhood system a t t h e o r i g i n i n
E.
In p a r t i c u l a r ,
s p a c e i s i s o m o r p h i c t o a c l o s e d s u b s p a c e of D e f i n i t i o n 5.38
Let
E
n.
E
i s a
DN
s
N
.
b e a m e t r i z a b l e ZocaZZy c o n v e x
s p a c e w i t h g e n e r a t i n g f a m i l y o f semi-norms
for a 2 2
any F r g c h e t n u c l e a r
m
(pn)n=l,
pn 6 pn+l
(dominated norm) space i f t h e r e i s a
263
Holomorphic functions on nuclear spaces with a basis
P
c o n t i n u o u s norm k
there e x i s t
E
on
s u c h t h a t for a n y p o s i t i v e i n t e g e r
a positive integer
n
and
such t h a t
C>O
The f u n d a m e n t a l r e s u l t c o n c e r n i n g n u c l e a r
spaces i s
DN
the following proposition. Proposition 5.39 DN
i s a
A metr2izable n u c l e a r l o c a l l y convex space
s p a c e i f and o n l y i f i t i s i s o m o r p h i c t o a s u b s p a c e
s.
of
Now l e t
be a Frgchet nuclear space with a basis.
E
is isomorphic t o
w
m
=
and
m
m
m,
for all
(Wm,n ) n = 1
for all
w h e r e w e may s u p p o s e
A(P)
w
~
+ 2 ~w
E
m
P = ( w ~ ) ~ = ~
, ~ for all m,n
m
and
n
(by t h e Grothendieck-Pietsch c r i t e r i o n f o r n u c l e a r -
ity). The c o l l e c t i o n where t h e is the
row i s t h e
mth
n th
may b e d i s p l a y e d a s a n i n f i n i t e m a t r i x
P
For e a c h p o s i t i v e i n t e g e r
vm and l e t
07
= {(zn)n=l
m
(ni)i=l
column
=
07
{(zn)n=l
m
let
E
o f a1
continuous weights on
E.
i s a s t r i c t l y increasing sequence
of positive integers with (n,)
n th
s u p z nwm,n n
E;
E
denote t h e se
[PI
Now s u p p o s e
LJ
weight and t h e
mth
coordinate.
n E;
1
=
SUP
n
.
Let
Chapter 5
264
nm s n
for
m
It is immediate that V
if
U(ni)i,l
k
E,
The sequence
I/(
C =
1
n
E
then
1
-
m
) n = l ( (1 .
a
C
K
then there exists a
V 3 U(nl,
K
...,n k ) ,
K is is contained in a compact If
)n=l lies in E . Let n Now choose a strictly increasing sequence
a = (
II ( O , O , ... ,o>- 1 N
U(ni)i,l
and
say
m
o f positive integers,
Then
m
such that
a compact subset of polydisc in
E
is a compact subset of
is a neighbourhood of
positive integer
m=1,2, . . . I .
nm+l'
m
C CU(ni)i,l
"i
(ni)i=l, n l = l ,
such that
1 , ... )I/
c
for all
and consequently t h e s e t s
CU(ni)i,l
,
a
n .1+1
i
r a n g e s o v e r a l l p o s i t i v e r e a l numbers and
i.
m
m
(ni)i=l
as
ranges
o v e r a l l s t r i c t f y i n c r e a s i n g s e q u e n c e s of p o s i t i v e i n t e g e r s with
n l = 1,
f o r m a f u n d a m e n t a l s y s t e m of c o m p a c t s u b s e t s o f
E. We now give a characterization of nuclear
DN
spaces
with a basis. Proposition 5.40 L e t E be a Frgchet nuclear space w i t h a basis. The f o l l o w i n g a r e e q u i v a l e n t :
265
Holomorphic functions on nuclear spaces with a basis
(a)
E
i s i s o m o r p h i c t o a s u b s p a c e of
(b)
E
i s a
(c)
E
i s isomorphic t o
a
m+l,n
( W m + ,l n )
Id)
E
2
w
m,n
,
<
wm,n
W
m
( w m , n 1 2 2‘
and
cw l , n
11.
m
where P = ( w ~ ) ~ = ~ , m and f o r e a c h
such t h a t for all n,
k,n
i s isomorphic t o A ( P ) w h e r e P = (w m ) mm = l ’ m for a l l m and t h e f o l l o w i n g w m = (Wm, n ) n = l hold: E
( i ) wm,.,
-
Bm,n
then
m
for a l l
> 0
(ii) i f
B,,,
W
-
2
m+l ,n W
m,n 1 all m
(Wm,n(Bm,nP)n=l
for a l l m
m
and n
and
n
and
[PI f o r any m and p.
E
positive integers Proof
and
there e x i s t a positive
C>O W
m
A(P)
m
positive integer k
for a l l
for all
(Wm,n ) n = 1
integer
m
m+2,n
i s isomorphic t o
wm =
(e)
A(P)
where for a l l m, wm = (Wm,n ) n = 1 for a l l m and n, and
P = (w m ) mm = l ’ W
s,
space,
DN
( a ) and ( b ) a r e e q u i v a l e n t b y p r o p o s i t i o n 5 . 3 9 .
do n o t p r o v e t h e e q u i v a l e n c e o f
(b),
( c ) and
(d) h e r e .
We
See
t h e n o t e s a n d r e m a r k s a t t h e end o f t h i s c h a p t e r f o r a r e f e r ence.
and
(c)=>(e)
n
.
we h a v e W
m + l ,n W
W
Since
W
4
-
m,n
6
m+2,n W
m,n
m+l,n W
(Wm+1,nI2
W
m+2,n
for a l l
m
and h e n c e
m+l,n
m+j+l,n W
W
Wm,n
m+j , n
for all positive integers
m,
n
and
j .
266
Chapter 5
fir'
W
Hence W
wm , n ( R m , n ) P and
m+p,n
m + l ,n w m , n ( ____ W m,n
=
j=O
(WIJ
m
A m n W
m+j , n
)P)n=l
E
m,
assuming ( e ) , t h a t
p
for all positive integers
[PI
and
m.
1, n
The case
p
m=l,
arbitrary is t r i v i a l .
is true for the positive integer
m
induction hypothesis there exist
C1>O
j
w
--
(c)+(e).
( e ) = > ( d ) . We f i r s t p r o v e b y i n d u c t i o n o n W
Wm+j+l,n
Now s u p p o s e t h e a b o v e
and f o r a l l
p.
By o u r
and a p o s i t i v e i n t e g e r
such t h a t for all
By c o n d i t i o n
(e) there e x i s t
C2> 0
n.
and a p o s i t i v e i n t e g e r
such t h a t for all
n.
Hence
c1 . c*
<
C = JC,. C2
where Thus
w.j , n w k , n
W
W
1,n
If we let
(
~
and
m+l,n )P< 1, n
p=2
I,
c
w
' c2
wt,n
= j+k.
&,n
for all
w e o b t a i n ( d ) and hence
completes t h e proof.
n
and
(e)=>(d).
This
k
267
Holornotphic functions on nuclear spaces with a basis Condition
(e) of p r o p o s i t i o n 5.40 a r o s e i n our study of
h o l o m o r p h i c f u n c t i o n s on F r g c h e t n u c l e a r s p a c e s w i t h a b a s i s a n d i s t h e o n l y o n e o f t h e a b o v e e q u i v a l e n t c o n d i t i o n s t h a t we s h a l l u s e f r o m now o n .
I n t h e o r i g i n a l p a p e r s on h o l o m o r p h i c
f u n c t i o n s on n u c l e a r s p a c e s , a F r g c h e t n u c l e a r s p a c e w h i c h s a t i s f i e d c o n d i t i o n ( e ) was known a s a B - n u c l e a r r e l a t i o n s h i p between B-nuclear
s p a c e s and
DN
space.
The
spaces with a
b a s i s was n o t i c e d a f t e r w a r d s . W
be a s t r i c t l y i n c r e a s (an)nz 1 m qCLn< m i n g sequence of p o s i t i v e r e a l numbers s u c h t h a t
Example 5 . 4 1
(a)
Let
a =
In=,
q, O
f o r some and l e t
E = A(P)
Let
where
f o r every positive integer
w
m,n
P =
=
1 m c1 )
(
m't
for all
n
( w ~ ) ~a n = d ~ wm =
m.
Since
m
and
n
m
(Wm,n ) n = 1
wm+l,n
is finite for a l l basis.
m
for all
and
E
m,
Furthermore,
and
i s a Frgchet nuclear space with a
n,
is a nuclear
If w e l e t
E
since
an=n
DN
space with a b a s i s
for all
n
we o b t a i n
c o m p a c t o p e n t o p o l o g y a n d i f we l e t we o b t a i n
s.
H(C)
with the
an = log(n+l) for a l l
n
268
Chapter 5 The s p a c e
i s c a l l e d a power s e r i e s s p a c e of i n f i n -
h(P)
i t e type and w e u s e t h e n o t a t i o n m
Let
(b)
~
~
(
af o )r t h i s s p a c e .
be a s t r i c t l y increasing sequence. "n q is f i n i t e for
a = (a,In=1
of p o s i t i v e r e a l numbers and suppose q, O
all
and l e t
P
for all
p
'
an f o r a l l n = p P.n (Wp,n);=l'O
Let
=
and
w
p',
.
O
( a ) and c a l l Po A power s e r i e s s p a c e
i t a power s e r i e s s p a c e of f i n i t e t y p e .
of f i n i t e type is not a E,
and
space.
DN
t h e n we o b t a i n
po=l
O
i s a FrGchet n u c l e a r
A(€')
0'
p,
We d e n o t e t h i s s p a c e b y
space with a basis.
n
and
I f we l e t
H(D),
an=n
for all
t h e open u n i t d i s c i n
D
w i t h t h e compact open t o p o l o g y . I t i s p o s s i b l e t o c o n s t r u c t f u r t h e r examples of
(c)
DN
s p a c e s w i t h a b a s i s by c h o o s i n g w e i g h t s o f even more r a p i d
For example,
growth. ers
m
and
all
m.
Let
Since
for all Also,
n.
w
for all positive integ-
= 2~~
m, n P = (w ) " m m=l
wm =
where
( W m ,n ) n = l
for
is a Frcchet nuclear space with a basis.
A(P)
m,
let
since
w
W
m,n
(---
m+l ,n W
m,n
I
m
P
2n
=
(2"
f o r any p o s i t i v e i n t e g e r s large,
h(P)
is a
DN
m+l
IP
.(2n
m
space.
m
6
2"
m+2
IP
and
p
and a l l
n
sufficiently
I t c a n b e shown t h a t
n o t i s o m o r p h i c t o a power s e r i e s s p a c e .
A(P)
is
269
Holomorphic functions on nuclear spaces with a basis
I n p r o v i n g o u r m a i n r e s u l t , w e s h a l l make f r e q u e n t u s e o f t h e f o l l o w i n g e q u a l i t y c o n c e r n i n g t h e supremum o f a m o n o m i a l o n a polydisc:
co
V = { ( z , , ) ~ = ~ s; u p ( z n a n l
if
Theorem 5 . 4 2
Proof
By c o r o l l a r y 5 . 3 1 ,
H(E)
(nl,:..,nk)
i n t e g e r s with
is
continuous.
T
m
I1
in
a positive integer
H(U).
on
T~
T~
on
= T~
that
T
0
Let
integer
T~
N"),
= T
0'
bounded
T
be such a
k
there exist
6
and
c(k) > 0
c(k)
II zmll
such t h a t
6U(nl'
* .
T h e n we c l a i m ,
N").
j
and
c(k+l)>1
*
,nk) if
6' > 6,
there exist
such t h a t
I f n o t , w e can choose f o r each p o s i t i v e i n t e g e r
N"). E
"1
N
a s t r i c t l y i n c r e a s i n g sequence of p o s i t i v e
nl = 1
IIT(zm)
I
=
i t s u f f i c e s t o show
S u p p o s e f o r some n o n - n e g a t i v e
m.
T~
which i s bounded on t h e
H(E),
subsets of H(E), linear functional.
f o r every
m E
i t s u f f i c e s t o show t h a t a n y B a n a c h v a l u e d l i n e a r
f u n c t i o n a l on
in
then
E,
S i n c e we h a v e a l s o s h o w n , e x a m p l e 5 . 3 6 ( a ) ,
H(E).
6>0,
and
is an o p e n p o Z y d i s c i n a c o m p l e t e
U
s p a c e w i t h a basis,
DN
nuclear
on
If
11
1.
m = (ml,m 2 , . . . , m n , 0 , . . .
and
$
j,
such t h a t
We f i r s t s h o w t h a t
O t h e r w i s e we c a n c h o o s e s u p / m j j = m. m I and a p o s i t i v e i n t e g a s t r i c t l y i n c r e a s i n g sequence (jn)n=l, er
J
such t h a t
Imj
n
I
= J
for all
jn.
Since
270
Chapter 5
U (nl,.
.. ,nk,
j+n,) 3 V k c l
for all
j
t h e sequence
i s u n i f o r m l y bounded on a f i x e d n e i g h b o u r h o o d o f z e r o i n and hence i s a
bounded s u b s e t of
T
and
6"('E)
E
H(E).
This contradicts the fact that
By t a k i n g a s u b s e q u e n c e i f n e c e s s a r y we may t h u s
for all
n.
suppose
Imj/
-f
+m
as
j
-f
m.
For each j l e t m . = ( r . , s . ) where r . are t h e f i r s t 1 1 1 I j + n -1 c o o r d i n a t e s a n d s. are t h e remaining coordinates of k 3 m.. We i d e n t i f y r . a n d s . w i t h e l e m e n t s o f N") in the 3 1 I usual fashion. For a l l j and a l l z m. r. s . z J = z J z J .
C o n s i d e r t h e f o l l o w i n g two p o s s i b i l i t i e s :
5.
where point and
(**)
S .
i s t h e v a l u e o f t h e monomial
(@k,n)_ (@k,n)n=l.
Since
B
~ 2 , 1 ~f o r a l l
cover a l l p o s s i b i l i t i e s .
z
k
and
at the
n
(*)
Holomorphic functions on nuclear spaces with a basis We f i r s t s u p p o s e t h a t
Hence
(*)
is satisfied.
27 1
Chapter 5
212
m
for all
N"),
E
we m u s t h a v e
This is a contradiction,
since
and h e n c e
6' > 6 ,
(*)
cannot
hold. We now s u p p o s e
(**)
holds.
By t a k i n g a s u b s e q u e n c e i f
n e c e s s a r y we may s u p p o s e
and
i s a s t r i c t l y increasing sequence.
lmjl Let
m. f(z)
lj=1 m
=
m IIZ
S i n c e e a c h monomial
Z J
'11
U(nl,
i s c o n t i n u o u s and
imp,lies t h a t
theorem 2 . 2 8
. . . ,n k , j + n k )
f
n
let
Q
;;In
m
(an)n,l
= anun
E.
be an a r b i t r a r y element o f
R
where
E.
For each
m
( u ~ ) ~i s= t ~h e u n i t v e c t o r b a s i s o f
a p o s i t i v e i n t e g e r such t h a t 1 2 l e t m j , m j , . . . , m! all nbR. For each j J coordinate of m . E N ") and l e t 3 E.
Choose
=
i s a Frgchet space,
i s an e n t i r e function i f t h e
above s e r i e s converges a t a l l p o i n t s of Let
E
d
an
E
Vk+l
for
be t h e f i r s t
27 3
Holornorphic functions on nuclear spaces with a basis
We h a v e 1
m a j
6
m.
'(/u(nl,. ..
11'
2
m . m. clJ c 2 J .
. .
II m.
ceJ ~ ( a , m ~ )
, n k , j+nk)
where
for all
j
such t h a t
j>a
( t h e t e r m s between
II
and
j+nk
a r e a l s o l e s s t h a n o n e b u t we n e e d a s h a r p e r e s t i m a t e ) . NOW
given any p o s i t i v e i n t e g e r
and h e n c e
nk+j
have
>
II1;
and t h u s
l a w k , n ( ~ k , n ) P I6 1
p,
for all
in particular for all
j
E )[ P ( w ~ , ~ ( R ~ , ~ ) P ; I= ~
n
2
k1
> II.
Hence i f
s u f f i c i e n t l y l a r g e , we
Chapter 5
214 where
for Since all
As
1s
i h 1.
i m. 5 / m j l 1 i . Hence
for all
is greater than
w
zero and h e n c e
f
E
1
and
i
and
P
j
we h a v e
0 < r . <1
for
is arbitrary, the l i m i t is
H(E).
Hence N
f(z)
=
1"
m.
1
j = 1 (6') lmj
Z
I
i s a l s o an e n t i r e f u n c t i o n on
is a that
T
bounded s u b s e t o f
J
~~zmjl~U(n . .l ,,n. k , j + n k )
E.
H(E).
The s e q u e n c e
This contradicts the f a c t
275
Holornotphic functions on nuclear spaces with a basis
and e s t a b l i s h e s o u r c l a i m . m
c o n t i n u o u s and (nVl)n=l i s an i n c r e a s i n g T6 c o u n t a b l e open c o v e r o f E there exist c ( l ) > O and 61 a Since
is
T
p o s i t i v e i n t e g e r such t h a t
f
f o r every
in
H(E).
In p a r t i c u l a r
for all
m
E
m
("In=2
Let
nl
where
N
= 1
be a sequence o f p o s i t i v e r e a l numbers, m
6, = 6 i s f i n i t e . By t h e n=l a b o v e , we c a n c h o o s e i n d u c t i v e l y a s t r i c t l y i n c r e a s i n g s e q u e n c e all
6,>l
n,
such t h a t
m
(nk)k=l, n l = 1 ,
of p o s i t i v e integers, p o s i t i v e numbers
and a s e q u e n c e o f
( ~ ( k ) ) ; , ~ such t h a t
. . ., n k )
Let
m
K = 6U (ni)i=l.
If
V
lemma 5 . 1 8 , E > 1 all n
K
i s a compact p o l y d s c i n
i s any neighbourhood o f
K
a sequence o f r e a l numbers, 1 n and < m, and W
L=l
n
E.
t h e n we c a n c h o o s e , b y E
=
m
( E ~ ) ~w i=t h ~
a neighbourhood o f
276
Chapter 5 E(K+W)C
zero such that K
V.
Since
there exists a positive integer S
...
SkU(nl,.
Hence, €or any
€
E
. . ,nk)C
K+IV
k
is a neighbourhood of such that
6 U(nl,.
. . ,nk) C K + W
H(E),
(proposition 5.25)
V was arbitrary, this shows that compact subset K o f E. Hence T is this completes the proof. Since
T
is ported by the
T
continuous and
w
Theorem 5.24 immediately leads to a strengthening of some
of our earlier results. (a)
If
E
particular if
The following are now easily verified.
is a nuclear E
=
s or
H((C))
DN
space with a basis (in then
(H(E),ro)
is a reflex-
ive A-nuclear space. (b)
If
U
is an open polydisc in a nuclear
with a basis, then basis.
(H(U),.ro)
DN
space
is a fully nuclear space with a
(c) If E i s the strong dual of a nuclear D N space with a basis then H(OE) = lim (Hm(V), I/ is a complete +
V30,V open
]Iv)
271
Holomorphic functions on nuclear spaces with a basis regular inductive l i m i t . Thus w e h a v e examples o f non-rnetrizable
l o c a l l y convex
s p a c e s i n which t h e s p a c e o f germs a b o u t t h e o r i g i n i s c o m p l e t e and r e g u l a r . I n c h a p t e r 6 , w e prove, u s i n g t e n s o r p r o d u c t s and a r e s u l t of Grothendieck,
I f
on
55.4
E
H(E)
t h e following converse t o theorem 5.42.
i s a F r g c h e t n u c l e a r s p a c e w i t h a b a s i s and then
E
is a
DN
T
~
=
space.
TOPOLOGICAL PROPERTIES INHERITED B Y STRICT INDUCTIVE LIMITS A N D SUBSPACES The r e s u l t s a n d m e t h o d s d e v e l o p e d i n t h e f i r s t t h r e e
s e c t i o n s o f t h i s c h a p t e r h a v e a number o f i n t e r e s t i n g c o n s e quences such as a k e r n e l s theorem f o r a n a l y t i c f u n c t i o n a l s , regularity r e s u l t concerning
H(K),
where
compact s e t i n c e r t a i n n o n - m e t r i z a b l e and r e p r e s e n t a t i o n theorems f o r spaces of i n f i n i t e type.
K
a
is an a r b i t r a r y
l o c a l l y convex s p a c e s ,
H(Am(a)i)
as power s e r i e s
Some o f t h e s e w i l l b e d i s c u s s e d i n
t h e next chapter. We c o n f i n e o u r s e l v e s i n t h i s s e c t i o n t o a p p l i c a t i o n s w h i c h
y i e l d new e x a m p l e s c o n c e r n i n g t h e r e l a t i o n s h i p b e t w e e n t h e topologies
TO , ~ W
and
T&.
As w e have a l r e a d y had
s u c c e s s w i t h F r z c h e t n u c l e a r and
a4'n
some
spaces, it is n a t u r a l
t h a t w e i n v e s t i g a t e holomorphic f u n c t i o n s on s u b s p a c e s and i n d u c t i v e and p r o j e c t i v e l i m i t s of t h e s e s p a c e s . The p r o j e c t i v e l i m i t c a s e i s n o t v e r y s a t i s f a c t o r y ( e . g . consider
(EN x
C"))
and any p o s i t i v e r e s u l t s w e o b t a i n i n t h i s
d i r e c t i o n are g i v e n i n t h e s e c t i o n on s u r j e c t i v e l i m i t s i n chapter 6.
Since arbitrary inductive l i m i t s are too general
we c o n f i n e o u r s e l v e s t o
(countable)
s t r i c t i n d u c t i v e l i m i t s of-
T
~
Chapter 5
218
Frgchet n u c l e a r spaces ( t h e s t r i c t i n d u c t i v e l i m i t of & J 4 L spaces i s a
8JQ
This class,
a s we s h a l l s e e , i s r a t h e r r e s t r i c t i v e b u t d o e s
s p a c e a n d s o l e a d s t o n o new e x a m p l e s ) .
y i e l d new n o n - t r i v i a l e x a m p l e s .
The t e c h n i q u e s u s e d ,
from t h o s e d e v e l o p e d i n t h i s c h a p t e r , those used f o r
a
3 q
apart
a r e somewhat s i m i l a r t o
s p a c e s a n d d i r e c t sums o f Banach s p a c e s
Our i n v e s t i g a t i o n s i n t o h o l o m o r p h i c
( c h a p t e r s 1 , 2 a n d 4).
f u n c t i o n s on s u b s p a c e s l e a d t o a correspondence between h o l o m o r p h i c e x t e n s i o n t h e o r e m s ( s e e c h a p t e r 4 f o r t h e Banach s p a c e c a s e ) and t h e c o m p l e t e n e s s o f q u o t i e n t s p a c e s o f h o l o m o r p h i c functions. We b e g i n b y d i s c u s s i n g h o l o m o r p h i c f u n c t i o n s o n f u l l y n u c l e a r s p a c e s w i t h a b a s i s which can b e r e p r e s e n t e d a s a s t r i c t inductive l i m i t of Frgchet nuclear spaces.
A typical
e x a m p l e o f s u c h a s p a c e i s t h e c o u n t a b l e d i r e c t sum o f F r g c h e t nuclear spaces with a b a s i s .
Our n e x t r e s u l t shows t h a t t h i s
i s , i n f a c t , t h e o n l y p o s s i b l e example. Lemma 5 . 4 3
E = l i m En
Let
be a s t r i c t i n d u c t i v e l i m i t o f E
F r g c h e t n u c l e a r spaces a n 2 s u p p o s e Then
OD
E
t h e n each Proof
En
DN
is a
space
space w i t h a b a s i s .
m
Let
i s a Frgehet n u c l e a r space
M o r e o v e r , if e a c h
DN
is a
Fn
Fn
where each
l n = 1 Fn
w i t h a Schauder b a s i s .
has a Schauder b a s i s .
(en)n=l
be a Schauder b a s i s f o r
Since
E.
E
m
i s an a b s o l u t e b a s i s f o r i s a f u l l y nuclear space, (en)n=l E. For each p o s i t i v e i n t e g e r n let Fn be t h e c l o s e d sub-
space of Since
E
g e n e r a t e d by
FnCEn
follows that
all
a l s o i s each
e
m .d E j
j
for
is a s t r i c t inductive l i m i t ,
E
and i f each
n
Let
Fn.
F =
En
is a
En
m
Fn.
DN
space then so
Using t h e n a t u r a l
i n j e c t i o n s from
Fn
inductive l i m i t ,
we o b t a i n a c o n t i n u o u s i n j e c t i o n II
into
E.
If
it
i s a Frgchet n u c l e a r space with a Schauder
Fn
b a s i s f o r each
I e m ;e m € E n ,
and
n
x =
into m
i s a bounded s e q u e n c e i n l i m i t is regular,
and t h e c o n s t r u c t i o n of an
xnen€E E,
then t h e sequence
from
F
{xnen}t=l
and s i n c e e v e r y s t r i c t i n d u c t i v e
there exists a positive integer
m
such
279
Holomoiphic functions on nuclear spaces with a basis that
Em
xnen'
n.
for all
Hence
{en;xn
# 01 C E m
as
and,
the basis is absolute
m
i s a Cauchy s e q u e n c e i n s u r j e c t i v e mapping.
and
Fn
X E
Thus
F.
By t h e o p e n m a p p i n g t h e o r e m ( b e t w e e n
c o u n t a b l e i n d u c t i v e l i m i t s o f B a i r e s p a c e s ) TI phism and hence
W
E =
Proposition 5.44 DN
Proof
By lemma 5 . 4 3 ,
nucZear s p a c e s .
is a
En
b e a fuZly n u c l e a r s p a c e w i t h a
can be r e p r e s e n t e d a s a s t r i c t i n d u c t i v e
Z i r n i t of
each
E
Then
=
T~
we may s u p p o s e
-r6 E =
space with a basis.
DN
i s a homeomor-
This completes t h e proof.
Fn.
Let E
b a s i s and s u p p o s e
is a
IT
H(E).
on
ln=lE n W
Hence
where
is a
E
r e f l e x i v e A - n u c l e a r s p a c e a n d s o , b y lemma 5 . 1 a n d p r o p o s i t i o n 5.28,
Let
i t s u f f i c e s t o show
T
E
(H(E),T~)I.
For each
n
choose a neighbourhood o f zero i n
s u c h t h a t e a c h monomial on possible since each then
n
W
z = (Zn)n=l
j =
IW
(jn,m n,m=l h(j)
E
z n c En
where
Hence e a c h monomial on
This i s
Vn.
a d m i t s a c o n t i n u o u s norm.
En
sufficiently large.
i s bounded on
En
En,Vn,
For each
E
( ~ ~ 1 " ) = sup{ncN;3
all n,
n zn
i s indexed by
and
zn
If = 0
pl
= (Zn,m)m=l
z
E
E
for all
and so
(N2)").
If
we l e t
m
such t h a t
We c l a i m t h e r e e x i s t s a p o s i t i v e i n t e g e r
Z n
0).
such t h a t
280 if
Chapter 5 jE(N2)")
and
h(j) >no,
thenm
b y r e s t r i c t i n g o u r s e l v e s t o some
can f i n d a s e q u e n c e . i n h(jn) = n
T(zJn) # 0
and
s e q u e n c e of complex numbers. l
m
subset of Let
i f n e c e s s a r y , we
k (jn)n=l,
n.
for all
such t h a t m
Let
be any
We f i r s t s h o w t h a t
i s a l o c a l l y bounded,
{ a n z n}n,l
l k = lE n m
(N2)"),
I f not, then
T(zj) =O.
and hence a
bounded,
T &
H(E). be a r b i t r a r y ,
say
. . . ,L
I/anzJn((
where
L'>R
and
ZEE
M = 1 + sup n=l,
Now s u p p o s e
z
E
R
En.
Let
V = z + V1 x v 2
x...x
VQ
c ~ + ~ , . . . , c a~ r~e p o s i t i v e r e a l
numbers s u c h t h a t
If
c > 0
then
is positive since
where
e(L')
Hence,
by choosing
c
h[jQ,+l) = R'+l.
s u f f i c i e n t l y s m a l l a n d p o s i t i v e , we
have
Since
h(jn)
=
n
t h e same e s t i m a t e a l s o h o l d s f o r a l l
jn anz , n i Q 1 . T h u s we c a n c h o o s e a s e q u e n c e o f p o s i t i v e r e a l m numbers ( c ~ ) & =s ~ uch t h a t
Holomorphic functions on nuclear spaces with a basis T h i s shows t h a t t h e sequence i s a very strongly
Jn
)mnZ1
is locally
w e also see t h a t t h e sequence
in fact,
and,
bounded
(an z
28 1
convergent sequence i n
T&
{zjnlm
n=1
H(E).
n
contini?ous. Hence t h e r e e x i s t s a p o s i t i v e i n t e g e r Jn "0 such t h a t T(z ) = 0 if h(jn) > no. Let F = n = lE n ' is
F
T~
1
is a
space with a basis.
DN
0
If
t h e n , by t h e above,
Since
on
i s a complemented s u b s p a c e o f
F
H(F)
T(flF) = T(f)
As t h e b a s i s i n
F
1 a m T ( z m )I N
meN
for every
By t h e o r e m 5 . 4 2 , c>O
N
and
K
is a
T
N
T
T
is
is a
T
c /(f/IK f o r every
,<
H(E).
N (N)
E
E
a z m
m
we h a v e E
H(F).
c o n t i n u o u s and h e n c e t h e r e
f
F
in
such t h a t H(F).
and so f o r any
c o n t i n u o u s l i n e a r f u n c t i o n a l on
T
Since
c o n t i n u o u s semi-norm on
T~
a compact s u b s e t o f
i s a l s o a compact s u b s e t o f
T
1
mE
exist
Hence
in
f o r every
H(F).
l'?(f)l
f
extends t o a basis i n
<
(H(E),T~) is barrelled,
K
N
by t h e formula
Lu
1
w e may d e f i n e
E
f
in
H(E)
and t h i s c o m p l e t e s t h e p r o o f . Example 5 . 4 5
If
n
i s any open s u b s e t o f
Rn
then
H(E)
282
3
Chapter 5
,$ (Q)
=
(the space of
f u n c t i o n s on
& ,m
w i t h compact
s u p p o r t endowed w i t h t h e s t r i c t i n d u c t i v e l i m i t t o p o l o g y o f
&
the spaces Hence =
To
= T
T~ T
0
on
,b
[-n,+n], n
on
6
H(E).
H(U)
Example 5 . 4 6 each T
on
w
Then
=
T~
on
W
and hence
.
in 8
U
i s a .83Q En i f and o n l y i f
where each T
=
H(E;)
In p a r t i c u l a r
H(2').
each
If
m
E = n=l En
Let
s")
1",1S we a l s o h a v e
By c o r o l l a r y 5 . 3 1 ,
i s a f i n i t e dimensional space.
En
#
i s isomorphic t o
)
f o r every open p o l y d i s c
space with a basis. T~
Z)
E
is a f i n i t e dimensional space then
En =
T~
may s u p p o s e
T
on
W
H(E)
by example 5 . 3 6 .
s cN
O t h e r w i s e , we
i s an i n f i n i t e dimensional space.
El
EB
By
c o r o l l a r i e s 5 . 2 3 a n d 5 . 3 5 , w e may c o m p l e t e t h e p r o o f by s h o w i n g t h e e x i s t e n c e o f a non-convergent ,T~),
Let
(Vn);=l
the origin in I J J ~ E
f
=
Cauchy s e q u e n c e i n
.
(H(E;)
be a fundamental neighbourhood system a t
(El);.
En
such t h a t
m
n
For each
Ilonl\
=
'n
n
choose
x:=l(En)E;
f o r some i n t e g e r
I:=1
n I I $ n + n IIK
f o r e a c h compact s u b s e t
of
E
El
and
for all
n.
Let
Cn=l+nQn* S i n c e each compact s u b s e t o f
in
9,
$n # 0
and
m
f
i s c o n t a i n e d a n d comp-act
we h a v e
N(K) n Zn=1 ll$n$nllK
=
of
K
k,
E' B
E'.
form a Cauchy s e q u e n c e i n
B
m
(Wn)n=l,
zero i n
such t h a t
f o r each
n,
m
H e n c e t h e p a r t i a l summands If
HIE;).
t h e r e would e x i s t a s e q u e n c e (En);
<
Wn
t h i s i s i m p o s s i b l e and completes t h e p r o o f .
f E
H(Eb)
then
a neighbourhood of
28 3
Holornorphic functions on nuclear spaces w i t h a basis
Ne now c o n s i d e r h o l o m o r p h i c f u n c t i o n s o n c l o s e d s u b s p a c e s o f We f i r s t p r o v e t h e f o l l o w i n g lemma
f u l l y nuclear spaces.
c o n c e r n i n g l i n e a r f u n c t i o n a l s on a s u b s p a c e o f a f u l l y n u c l e a r space. Lemma 5 . 4 7
Let
n u c l e a r space
E
Proof
Since
space,
b e a c l o s e d s u b s p a c e of t h e f u l l y
F
t h e n , if
FL
= I $ E E ' ; $ I ~= 0 } ,
is a closed subspace of a f u l l y nuclear
F
it i s a complete n u c l e a r space and hence i s s e m i - r e f l e x -
Thus F' i s a b a r r e l l e d space and (F')'d F. Also E' B B B i s a c o m p l e t e r e f l e x i v e s p a c e and hence EA/FI is a barrelled
ive.
l o c a l l y convex s p a c e . spaces.
Thus
B
FA
and F' a r e b o t h Mackey B s u f f i c e s t o show
T O
n
E
E'
To c o m p l e t e t h e p r o o f , / i t )'
If
T
=" F .
( E i / F ~ ) '
E
then
c a n o n i c a l p r o j e c t i o n from reflexive,
E l
t h e r e e x i s t s an
B
x
in
$ ( x ) # 0.
Since
$
E
F
,
n($)
This contradicts the fact that
By t h e H a h n - B a n a c h
F.
such t h a t $(F) = 0 and t h u s
= 0
$(x) # 0
T O
and s o
This completes t h e proof.
If
E
?IF
H(E);flF = g}.
and I f we l e t
= 01
HE(F) r:
r e s t r i c t i o n m a p p i n g f r o m H(E) into E E Kernel ( r F ) and HE(F) = r a n g e ( r F ) Theorem 5 . 4 8
n u c l e a r space
Let E.
F
.
=
n($)
and = 0.
F ' = Ei/FI. B
i s a subspace o f a l o c a l l y convex s p a c e
F
H(F)I = If such t h a t
is the
Il
Since E is A/F I' such t h a t T o n ( $ ) = $(x)
E
L
where
B
onto
f o r every $ in E' Suppose x B' in E' theorem t h e r e e x i s t s a $ I
E'
E
we l e t
{ g E H ( F ) ; 3 Z E H(E) denote t h e natural
H(F)
then
H(Ff
b e a c l o s e d s u b s p a c e of a f u z z y
Then (HE(F),ro)
(by
=
284
Chapter 5
Proof
E rF : H(E)
The mapping
s u r j e c t i o n and h a s k e r n e l
-t
i s a continuous
HE(F)
H(F)L.
We c o m p l e t e t h e p r o o f b y
showing t h a t t h e i n v e r s e mapping from
HE(F)
i t s u f f i c e s t o show
By t h e o r e m 1 . 2 7 a n d p r o p o s i t i o n 1 . 4 1 , t h a t f o r e a c h compact s u b s e t subset
of
M
f o r every If that
and
F
f
in
c > 0
~ ~ ( $= 1 i n f { l ( $ I / K : T E E ' ,
subset
If
all in
of
L
ji
E
suppose i
F,
n
6
P
lT=l
and
=
$1,
( n ~ ) . If n
))IJJ~/\
liZl
of zero i n
W
\]Ti\\n
<
m
-
and
Y n (jii) = P
$EF',
is a positive integer,
<
m
E
yiiF
in
is a
then
W
pIF = li=l(jii)n
where
f o r some n e i g h b o u r h o o d
t h e n , b y t h e Hahn-Banach theorem,
bourhood
l i m m-
and
@lF
t h e n lemma 5 . 4 7 i m p l i e s
E
nl
and h e n c e t h e r e e x i s t s a compact
F' B such t h a t
F
E'
t h e r e e x i s t a compact
H(E).
i s a compact s u b s e t o f
K
E
of
such t h a t
c o n t i n u o u s semi-norm on
NOW
K
onto
and = jii
Ti
€
all
( H ( E ) , T ~ ) and
all
i.
Since P
V
E
of
F' 0
there e x i s t s a neigh-
E',
N
jii
lF
=
i,
PiF
such t h a t
w e have
285
Holomoiphic functions on nuclear spaces with a basis
Hence
= nL(p)
f o r every
8 ("E).
in
P
If f = C ___ n=O n!
then
9
Hence t h e r e e x i s t s in
H(E).
such t h a t
c>O
q ( f ) ,< c n L ( f )
for every
An a p p l i c a t i o n o f p r o p o s i t i o n 1 . 4 1 i m p l i e s t h e
e x i s t e n c e o f a compact s u b s e t q ( f ) 6 cllf/(M f o r every
f
M
in
of
such t h a t
F
H(E).
This completes t h e
proof.
We now a p p l y t h e o r e m 5 . 4 8 t o t h e f o l l o w i n g p r o b l e m s :
(a)
When i s
HE(F) = H(F)?
f u n c t i o n on
i.e.
when c a n e v e r y h o l o m o r p h i c
b e e x t e n d e d t o a holomorphic f u n c t i o n on
F
E? (b)
H(E)
What t o p o l o g i c a l p r o p e r t i e s o f
are inherited by
H(F)? S i n c e e v e r y c o n t i n u o u s l i n e a r form on Hahn-Banach
theorem,
extends,by the
t o a continuous l i n e a r form on
t h e polynomials o f f i n i t e t y p e on it f o l l o w s t h a t
F
HE(F)
F
are
T~
i s a dense subspace o f
E
and
dense i n H(F)
, T ~ ) .
H(F) W e
immediately o b t a i n t h e following c o r o l l a r i e s t o theorem 5.48: Corollary 5.49
Let
F
b e a c l o s e d s u b s p a c e of t h e f u l l y
f
286
Chapter 5 E.
n u c l e a r space
i s complete then every hoZomorphic f u n c t i o n on
(bl
F
( H ( E ) , T ~ ) is c o m p Z e t e ,
If
F
f u n c t i o n on
Corollary 5.50 E
space
t h e n e v e r y hoZornorphic
e x z e n d s t o a hoZornorphic f u n c t i o n o n
if and o n l y i f
E
( H ( E ) . T ~ ) / ~ ( ~is) ~c o m p Z e t e .
F
If
F
We h a v e a l r e a d y s e e n ( c o r o l l a r y 3 . 5 6 )
i s a Frgchet nuclear space. (H(E)
Since
H(F)l
it f o l l o w s t h a t
, T ~ )
extends t o a
E.
h o l o m o r p h i c function o n
space of
83g
is a c Z o s e d s u b s p a c e o f a
t h e n e v e r y h o Z o m o r p h i c function o n
Proof
E.
e x t e n d s t o a hoZornorphic function o n
a l s o a Frgchet nuclear space.
that
(H(E),T~)
i s a closed sub-
( H ( E ) , T ~ ) / . ~ ( ~ i) s ~
An a p p l i c a t i o n o f c o r o l l a r y
5 . 4 9 ( a ) now c o m p l e t e s t h e p r o o f . Corollary 5.51
If E
nucZear space
is a c Z o s e d s u b s p a c e of a F r g c h e t
F
t h e n e v e r y h o Z o m o r p h i c function o n
F
extend
E i f and onZy if is a c o m p Z e t e ZocaZZy c o n v e x s p a c e .
e x t e n d s t o a hoZomorphic f u n c t i o n on (H(E)
, T ~ )
/H(F)l We now t u r n t o problem ( b ) . If Proposition 5.52 T~
= T
6 subspace
on F
Proof p
If
: H(E)
then
p +
be a R
T~
on
H(F)
for a n y c Z o s e d
c o n t i n u o u s semi-norm on
T&
b e d e f i n e d by t h e formula
$(f)
H(F). = p(flF).
i s an i n c r e a s i n g c o u n t a b l e o p e n c o v e r o f
E
Wnzl
F.
C
C
\\f\\v,
consequently p
=
i s an i n c r e a s i n g countable open cover of
(Vn n
N
i s a fuZZy n u c Z e a r s p a c e and
(vn);=l
Hence t h e r e e x i s t s
T(f)
E T~
E.
of
Let d
Let
H(E)
c > 0
for all T~
f
and
N
E
H(E).
c o n t i n u o u s on
i n d u c e s a c o n t i n u o u s semi-norm
then
a p o s i t i v e i n t e g e r such t h a t Hence H(E). q
on
4. /
p
Since
is @ p
-r6
and
= o
JH(F)L (H(E),T~) / H ( F ) ~ .
By
287
Holornotphic functions on nuclear spaces with a basis t h e r e e x i s t a p o s i t i v e number
theorem 5.48,
K
subset
f o r every p(f)
F
of f
in
H(E).
y(f)
Hence,
In particular
6 clIflIK.
@(nE).
such t h a t
= p(fIF)
= q(f+H(F)
f o r every p(p)
f
c(~PI(~
6
and a compact
c in
L
) 6 c(lf(lK
HE(F),
f o r every
P
in
Hence
n
m
.
d n f ( o ) E H(E) By t h e o r e m 3 . 1 9 , t h e h o m o g e n In=, n! eous polynomials form an a b s o l u t e decomposition f o r (H(E),ro) f o r every and t h u s
is
p
T
continuous.
0
This completes t h e proof.
A similar r e s u l t holds f o r the
Proposition 5.53
n u c l e a r space
F
If
E
and
p
be a
is a c l o s e d s u b s p a c e o f a f u l l y = T
T~
topology.
T~
H(E)
on
then
= T
T~
on
H(F).
Let
Proof Suppose
p
i s p o r t e d by t h e compact s u b s e t
N
define
on
p
H(E)
&
p(f)
If
c o n t i n u o u s semi-norm on
T~
bourhood o f
= p(fIF)
in
K
We
by t h e formula f o r every
i s a neighbourhood of
W
F.
of
K
H(F).
F.
K
f
in
in E
H(E).
then
Hence t h e r e e x i s t s
Wn F
i s a neigh-
c(Wn F ) > O
such
that
Thus N
P
1
N
p
H(Ff
semi-norm
c o n t i n u o u s semi-norm on
is a = 0
q
and
T~
= T
H(E),
.
on
e x i s t a p o s i t i v e number
0
on
c
N
p
H(E).
Since
defines a continuous
By t h e o r e m 5 . 4 8 ,
and a compact s u b s e t
L
of
there
F
288
Chapter 5
f o r every
f
in
H(E).
i s now c o m p l e t e d a s i n t h e p r e c e d i n g p r o p o s i t i o n .
The p r o o f
A s c o r o l l a r i e s t o t h e above p r o p o s i t i o n s ,
we may d r o p t h e
b a s i s r e q u i r e m e n t i n some o f o u r p r e v i o u s r e s u l t s . Corollary 5.54
then
T~
=
E
I f
on
T~
is a c o m p l e t e n u c l e a r
E
i s a complete n u c l e a r
s p a c e i f and o n l y i f i t i s a c l o s e d s u b s p a c e o f 5.42,
T~
=
on
T &
space
H(E).
By p r o p o s i t i o n 5 . 3 9 ,
Proof
,DN
H(s)
s.
DN
By t h e o r e m
and an a p p l i c a t i o n o f p r o p o s i t i o n
5 . 5 2 now c o m p l e t e s t h e p r o o f . A similar proof using proposition 5.44, =
T
T &
on
H(F)
whenever
F
shows t h a t
i s a c l o s e d sub'space o f
2
.
We a l s o o b t a i n a new p r o o f o f c o r o l l a r y 3 . 5 4 Corollary 5.55 T
0
=
on
T
Proof
E
If
E
i s a Fre'chet n u c l e a r s p a c e t h e n
H(E). i s a closed subspace of
nuclear space with a basis, To
=
15.5
on
T
N
.
sN
i s a Frgchet
and h e n c e by example 5 . 3 6 ( a ) ,
~ ( s ~ )B Y. p r o p o s i t i o n 5 . 5 3 ,
To
=
T
0
on
H(E).
EXERCISES
5 . 5_ 6* _ A-nuclear 5.57* -
s
Show t h a t a F r g c h e t n u c l e a r s p a c e w i t h a b a s i s i s a n space. Let
(E,T)
be an A-nuclear space.
t h e t o p o l o g y o f u n i f o r m c o n v e r g e n c e on
the
convex modularly d e c r e a s i n g s u b s e t s o f
E'.
Show t h a t u(Ei,E)
T
com~act
is
289
Hobmorphic functions on nuclear spaces with a basis 5.58 -
Show t h a t a n y B a n a c h s p a c e w i t h a n a b s o l u t e b a s i s i s
isomorphic t o
R
If
5.59* -
~
.
i s a s t r i c t inductive l i m i t of Frgchet
E = l i m En --f
s p a c e s and E
n has an unconditional b a s i s ,
E
where each
Fn
show t h a t
i s a Frgchet space w i t h an uncon-
Fn
ditional basis. 5.60 -
If
A
~
i n f i n i t e type,
exists
h m ( ~ ) a r e p o w e r s e r i e s s p a c e of
show t h a t
A
If
c an
log(n+l) a n
5.62 -
<
m )
6
an > O
n
all
5.63 -
when
g
If
_ 5 . 6_ 4
B,
6 Can
for all
n.
(xn)”,=,
H(C), g (n)
g(z) =
+ 0
as
AR(a)
(resp.
0
=
R i m ) .
lies in
l:=Oan~ n +
n
s
i f and o n l y
,
where
m .
i s a n open p o l y d i s c i n a f u l l y n u c l e a r space
U
If
in
such t h a t
with a basis,
~ ~ ( 5i f) a n d o n l y i f t h e r e
log(n+l) an (resp.
l i m n+R <
Show t h a t a s e q u e n c e
if there exists a
c
i s a p o w e r s e r i e s . s p a c e , show t h a t
AR(a)
i s n u c l e a r i f and o n l y i f sup n
(a)
1-
such t h a t
C > 0
5.61
~ an ) d
(
-
show t h a t
U = U
MM .
i s a l o c a l l y convex s p a c e w i t h t h e approx-
E
imation property
(i.e.
t h e i d e n t i t y mapping on
E
can be
u n i f o r m l y a p p r o x i m a t e d on compact s e t s by f i n i t e r a n k o p e r a t o r s ) show t h a t t h e a l g e b r a g e n e r a t e d b y (H(U),To) _ 5 , 6_ 5*
If
n
, T ~ )=
5.66* basis,
and
E’
i s dense i n
U
of
E.
i s a quasi-complete d u a l n u c l e a r space and
E
i s an open s u b s e t o f (HtU)
1
f o r any b a l a n c e d open s u b s e t
E,
show t h a t
Hn(U)
= HHy(U)
U
and
HHy(u).
Let
be a r e f l e x i v e n u c l e a r space with a Schauder
E
b e a modularly d e c r e a s i n g open s u b s e t o f E l B’ Show t h a t t h e m o n o m i a l s f o r m a n a b s o l u t e b a s i s f o r (H(U),To) and l e t
U
290
Chapter 5
5.67" -
Let
b e a r e f l e x i v e n u c l e a r space with Schauder
E
b a s i s and l e t
Ei.
subset of
N
Show t h a t e a c h
and t h a t t h e mapping
f
E
from
(HHy(U) , T o )
Y
f
HHy(U)
f
-f
(resp.
H(U))
in
-4
N
f
can
HHy(U)
(resp.
i s a l i n e a r isomorphism
-4
onto
Show
a modularly d e c r e a s i n g open
U,U,is
b e e x t e n d e d i n a u n i q u e way t o a f u n c t i o n
H(G))
Ei.
b e a c o n n e c t e d R e i n h a r d t domain i n
U
t h a t t h e modular h u l l o f
(HHy(U) , T o )
and from
( H ( U ) ,To)
onto
( H ( G ) , T ~ ) . Hence deduce t h a t t h e monomials f 0 r . m a n a b s o l u t e basis for
5.68*
(H(U) Let
, T ~ )
and
(HHy(U) ,
T ~ ) .
be a n open p o l y d i s c i n a h e r e d i t a r y Lindelgf
U
f u l l y nuclear space with a basis.
If
(H(U),T&)
show t h a t t h e monomials f o r m a n a b s o l u t e b a s i s f o r
5.69*
Let
Show t h a t t h e s e t o f a l l
l i n e a r f u n c t i o n a l s on with
(H(U),T~).
be an open p o l y d i s c i n a f u l l y n u c l e a r space
U
with a basis.
i s complete
H(U),
spec(H(U)
T
6
,T&),
multiplicative . may b e i d e n t i f i e d
U.
5.70*
Let
that
H(K)
be a f u l l y nuclear space with a b a s i s .
E
l i m ( H ( V ) , ~ ~ )f o r a n y c o m p a c t p o l y d i s c
=
4
Show in
K
V 3 K
E
if and o n l y if
disc
in
V
__ 5.71*
Let
Let K
( H ( V ) , T ~ )= ( H ( V ) , r ( , ) )
f o r every open poly-
E.
b e a l o c a l l y convex space w i t h completion
E
b e a compact s u b s e t o f
l o o k e d upon as a s u b s e t o f
A
E.
E
and l e t
Show t h a t
A
K
denote A
H(K) = H(K)
h
E.
K
alge-
b r a i c a l l y and t o p o l o g i c a l l y .
5.72 which t h e
G i v e an e x a m p l e o f a r e f l e x i v e A - n u c l e a r bounded s u b s e t s o f
T
b u t i n which t h e 5.73* -
Let
U
T~
H(E)
space
E
in
a r e l o c a l l y bounded
bounded s u b s e t s a r e n o t .
b e an open p o l y d i s c i n a r e f l e x i v e A-nuclear
29 1
Holomoiphic functions on nuclear spaces with a basis space
and l e t
E
w = (Wn)n=l 00
be t h e mapping which t a k e s
f
M R ~ : H ~ ~ +( ~u
~
o n t o i t s germ a t t h e p o i n t
w
uM.
E
Let
~
and l e t tRw d e n o t e t h e t r a n s p o s e o f t h i s mapping. Show t h a t t R w ( H ( E ) ) C H(U) a n d t R w ( H H y ( E ) ) C H H y ( U ) . Hence d e d u c e t h a t
Rw : (H(uM)
,
o(H(UM),H(U)))
i s c o n t i n u o u s and t h a t Mackey s p a c e i f
(H(U),y0)
(H(E),.ro)
-
(H(OE,),o(H(OE,),H(E)))
(resp.
(resp.
4
6
(H(U),T~)) is a
( H ( E ) , T ~ ) ) i s a Mackey
space. If
5.74* -
E
and
limit,
i s a q u o t i e n t s p a c e o f t h e l o c a l l y convex s p a c e
F
H(OE) = l i m (Hm(V),II ---+ O E V , V open show t h a t
Ilv)
H(OF) = lirn
(Hm(V),I(
--t
OEV,V
regular inductive l i m i t .
is a regular inductive (Iv)
is also a
open
Use t h i s r e s u l t t o g i v e a n a l t e r -
n a t i v e proof of c o r o l l a r y 5.55.
Why c a n n o t t h i s r e s u l t b e u s e d
t o give a f u r t h e r proof of proposition 5.52? Let
5.75* -
b e a Banach s p a c e .
E
A sequence
( x 00~ ) ~i n= ~E
i s s a i d t o be rapidly decreasing i f f o r every i n t e g e r
n p I/xn
I\+ 0
as
n+-.
Let
ES
denote t h e vector space
p, E
endowed w i t h t h e t o p o l o g y o f u n i f o r m c o n v e r g e n c e o n t h e r a p i d l y
E’.
decreasing sequences i n
and
Show t h a t
ES
HM(Es) = H H y ( E s ) ’
5.76*
E = CN
If
x
C(N)
show t h a t
( p ( n E ) , ~ o ) i s a borno-
l o g i c a l s p a c e which i s n o t b a r r e l l e d f o r any 5.77*
If
5 . 7_ 8* _
and
If
limit,
.
)i
( ( H ( U ) , T ~ ) ~ f (H(U),.rg).
K
i s a compact s u b s e t o f a f u l l y n u c l e a r s p a c e
H(K) = l i m
4
V3K,V
space.
n z 2.
i s a b a l a n c e d open s u b s e t o f a F r e c h e t n u c l e a r
U
s p a c e show t h a t
E
i s a nuclear space
show t h a t
(Hm(V),II
11,)
is a regular inductive
open
H(K)
i s a quasi-complete
l o c a l l y convex
(
0
292
Chapter 5
5.79" -
Show t h a t a f u l l y n u c l e a r s p a c e w i t h a b a s i s i s
5.80 -
Let
ultrabornological. be an open p o l y d i s c i n a f u l l y n u c l e a r space
U
with a basis.
i s a s e t of complex numbers,
{bm} meN "1 show t h a t t h e r e e x i s t s a T * " such t h a t
1
mEN
If
= bm
T(zm)
( N ) la,b,I
analytic functional
m
for all
5.81* -
Let
numbers,
show t h a t t h e r e e x i s t s a
with
E
infinite subset
5.82*
Let
(m,n)'
and f o r
N")
(mama
j
1 b'
in
=
n
,
i s not
5.83"
T
i f and o n l y i f e a c h J'
. i s an i n t e g e r and
an
ma,,,
m
and
...
0,
t
n
a
n+l
Let
> a +n
n
let
)
an p o s i t i o n
m
let
N")
lo
T(zj) 0
a n a l y t i c f u n c t i o n a l on
contains an i n f i n i t e subset
. . .,
0,
otherwise.
Show t h a t t h e r e e x i s t s a such t h a t
space. and suppose
N
For any p o s i t i v e i n t e g e r s
=
H(u).
i s a s e t o f complex
T~
E
E
b e a n u c l e a r power s e r i e s o f t y p e 1.
A,(a)
Suppose each
(an)n=l* for all n.
m
"1
mEN
m
for all
of
J
{bm}
C m E J , b m w mE H ( O E , ) B
m
a =
If
= bm
T(zm)
such t h a t
a zm
be a r e f l e x i v e A-nuclear
E
(H(E),T~) i s complete.
U
i f and o n l y i f
N
c m c N "1
for every
<
E
on
T
= b.
1
T~
a n a l y t i c f u n c t i o n a l on
f o r every
in
j
N").
A,(a)
Show t h a t
T
continuous.
For each non-negative i n t e g e r
n
let
LA(nE)
denote the vector space of continuous alternating forms on t h e l o c a l l y convex space
E.
Let
n
linear
293
Holomorphic functions on nuclear spaces with a basis
HAW)
aA ( n E ) , C n , O / I P n l ( K"
a,
=
m
{(Pn)n=O;Pn~
m
= P ~ ( { P ~ } ~ < = m ~j )
a n d endow
HA(E)
w i t h t h e t o p o l o g y g e n e r a t e d b y t h e semi-
norms
as
r a n g e s o v e r t h e compact s u b s e t s o f
E
pK
K
is a f u l l y nuclear space with a basis,
a c l o s e d complemented s u b s p a c e o f Hence deduce t h a t E
is a nuclear
5.84* -
If
HA(E)
n=1,2,.
..
.
is
~ ) .
is a reflexive nuclear space i f
n
show
form a Schauder b a s i s f o r
If, i n addition,
E
has t h e
p r o p e r t y and i t s b a s i s i s s h r i n k i n g ,
( p("E)
t h e y a l s o form a b a s i s f o r
55.6
HA(E)
i s a Banach s p a c e w i t h a Schauder b a s i s ,
t h a t t h e monomials o f d e g r e e
Dunford-Pettis
( H ( E ) ,T
If
space with a basis.
DN
E
(6( n E ) , ~ o ) ,
show t h a t
E.
show t h a t
,B).
NOTES A N D REMARKS Although D .
Hilbert
[ 3 3 2 ] f i r s t s u g g e s t e d , i n 1909, a
monomial e x p a n s i o n a p p r o a c h t o h o l o m o r p h i c f u n c t i o n s i n i n f i n i t e l y many v a r i a b l e s ,
t h i s i d e a was n o t d e v e l o p e d u n t i l
r e c e n t l y and most o f t h e r e s u l t s p r e s e n t e d i n t h i s c h a p t e r were discovered w i t h i n t h e last f o u r y e a r s .
I n d e e d , many o f
the original research a r t i c l e s are still only available i n p r e p r i n t form. N u c l e a r mappings and n u c l e a r s p a c e s , a s well a s most o f t h e i r fundamental p r o p e r t i e s ,
are due t o A.
Grothendieck
[287].
Further accounts of the l i n e a r theory are given i n t h e very r e a d a b l e book o f A . i n the notes of Y.C.
Pietsch Wong
[570],
[717].
in P.
t h i s c h a p t e r i s given i n S. Dineen [198], from a s l i g h t l y d i f f e r e n t p o i n t o f view, S . Dineen
Kr6e [ 4 0 3 , 4 0 4 ] ,
and
A survey of t h e r e s u l t s of
and a f u l l a c c o u n t , can b e found i n
[197].
Lemma 5 . 1 i s due t o P . J . A p a r t from i t s u s e s i n [ 9 1 ] ,
Boland and S . Dineen
[91].
we h a v e u s e d i t i n t h i s c h a p t e r
t o s h o r t e n a number o f t h e o r i g i n a l p r o o f s .
Proposition 5.4
294
Chapter 5
i s t h e c l a s s i c a l characterization of nuclear sequence spaces due t o A.
Lemma 5 . 6 ,
[570].
i s given i n A.
G r o t h e n d i e c k [287] and a p r o o f
c o r o l l a r i e s 5 . 7 and 5 . 8 ,
Pietsch
p r o p o s i t i o n 5 . 9 and
c o r o l l a r y 5 . 1 0 a r e a l l p r o b a b l y known t o r e s e a r c h w o r k e r s i n n u c l e a r space t h e o r y , b u t w e have been unable t o l o c a t e an exact reference.
A-nuclear
s p a c e s were i n t r o d u c e d by S . Dineen
[2023 as a t o o l i n studying holomorphic f u n c t i o n s
P r o p o s i t i o n 5 . 1 2 i s d u e t o S . Dineen
of independent i n t e r e s t .
w h i l e p r o p o s t i o n 5 . 1 3 and c o r o l l a r y 5 . 1 4 a r e g i v e n i n
[202 P.J.
b u t may b e
Boland and S . Dineen 1913. F u l l y n u c l e a r s p a c e s and f u l l y n u c l e a r s p a c e s w i t h a b a s i s
were i n t r o d u c e d by P . J .
Boland and S . Dineen
(H(E),ro)
s o l v e t h e b a s i s problem f o r space with a basis.
This a r t i c l e ,
[go] i n o r d e r t o
when
[go],
is a nuclear
E
contained t h e motiva-
t i o n f o r many o f t h e r e s u l t s i n t h i s c h a p t e r ,
introduced t h e
concept of multiplicative polar
( d e f i n i t i o n 5.16)
t h e key r e s u l t s ,
Lemma 5 . 1 9
5 . 1 7 and 5 . 1 8 .
and c o n t a i n s
i s due t o S. Dineen
[202]. Theorem 5 . 2 1 ,
a r e due t o P . J .
corollaries 5.22,5.23
Boland and S . Dineen
and p r o p o s i t i o n 5.24
[go],
and S. Dineen
t h e l a t t e r c o n t a i n i n g t h e r e s u l t s on A - n u c l e a r i t y . using a proof
similar t o t h a t of theorem 5.21, A.
h a s shown t h a t
(H(E),-Io)
decomposition ( i . e . whenever
E
5.29 i s due t o P . J .
dim(En)
<
f o r each
n
i n d e f i n i t i o n 3.7)
Propositions 5.25,5.28
a r e g i v e n i n S . Dineen
Boland and S . Dineen
proof of t h i s proposition,
[go].
[202],
Boland and S. Dineen
and
Theorem
The o r i g i n a l
Boland and S . Dineen
w h i l e p r o p o s i t i o n 5.34 [go].
i s due t o
C o r o l l a r y 5 . 3 5 and example
b o t h o f which i n v o l v e a n a p p l i c a t i o n o f l e m m a 5.1,
due t o P . J .
Boland and S . Dineen
has
Corollaries 5.30,5.31,
5 . 3 2 a n d e x a m p l e 5 . 3 3 may b e f o u n d i n P . J . and S . Dineen
[202].
which used p r o p o s i t i o n 5 . 3 4 ,
b e e n s h o r t e n e d h e r e b y u s i n g lemma 5 . 1 .
5.36,
[56]
i s a F r g c h e t - S c h w a r t z s p a c e w i t h a f i n i t e dimen-
c o r o l l a r i e s 5.26,5.27
P.J.
Benndorf
admits a f i n i t e dimensional Schauder
s i o n a l Schauder decomposition.
[91],
[202],
Recently,
1911.
are
Proposition 5.37 i s a
s l i g h t l y improved v e r s i o n o f a r e s u l t proved i n P . J .
Boland and
295
Holomorphic functions on nuclear spaces with a basis S . Dineen
[91]
I n h i s i n v e s t i g a t i o n s on t h e mathematical f o u n d a t i o n s o f q u a n t u m f i e l d t h e o r y w i t h i n f i n i t e l y many d e g r e e s o f P.
freedom,
Kr6e [ 4 0 7 , 4 0 8 , 4 0 9 , 4 1 0 ] u s e d t h e n u c l e a r i t y r e s u l t o f P . J .
Boland
[86] and L .
Waelbroeck
[713!
l o c a l l y convex space.
This,
and s u b s e -
was a b o r n o l o g i c a l
together with the r e s u l t s of
f u r t h e r q u e s t i o n s o f Krge c o n c e r n i n g
s e c t i o n s 5 . 1 and 5 . 2 ,
H(E)
(theorem 3.64)
(H(s),.ro)
q u e n t l y h e w i s h e d t o know i f
and t h e p o s s i b i l i t y of a k e r n e l s theorem f o r a n a l y t i c
f u n c t i o n a l s i n i n f i n i t e l y many v a r i a b l e s ( s e e c h a p t e r 6 ) m o t i v a t e d much o f t h e r e s e a r c h d e s c r i b e d i n s e c t i o n s 5 . 3 a n d 5.4. To s o l v e t h e s e p r o b l e m s , t h e concept of B-nuclear Subsequently, B-nuclear
D.
Vogt,
Dineen [195,202],
S.
introduced
s p a c e and proved theorem 5 . 4 2 .
i n a p r i v a t e c o m m u n i c a t i o n , showed t h a t
s p a c e s and n u c l e a r DN-spaces w i t h a b a s i s c o i n c i d e d
DN
(proposition 5.40).
spaces a r e due t o D.
Vogt and h a v e
played an important r o l e i n t h e development of s t r u c t u r e We r e f e r t o D .
theorems f o r Frgchet nuclear spaces.
Vogt
[703]
f o r an e x c e l l e n t survey a r t i c l e on n u c l e a r DN s p a c e s and t o E.
Dubinsky
[212] f o r a comprehensive account o f r e c e n t d e v e l -
opments i n nuclear Frgchet space theory. ences f o r p r o p o s i t i o n 5.39 are D. [211].
Vogt
Proposition 5.40 i s due t o D .
Lemma 5 . 4 3 i s d u e t o S . D i n e e n case of proposition P.J.
[198,199].
is given i n P . J .
[704] and M.
Vogt
[199].
[702,703]. The p a r t i c u l a r
Valdivia
while the general r e s u l t An e a r l i e r p a r t i a l
Boland and S . Dineen [ 9 1 ] . i.e.
Vogt
B ( n ) & s “1 .
Example 5.46 f o l l o w s from r e s u l t s proved i n P . J . Dineen
result for D.
[690] have proved, i n d e p e n d e n t l y , t h e
deep r e s u l t quoted i n example 5.45, S.
Dubinsky
5.44 d e s c r i b e d i n example 5.45 i s due t o
Boland and S . Dineen [ 9 2 ] ,
a p p e a r s i n S . Dineen H ( a )
The o r i g i n a l r e f e r -
[702] and E .
Boland and
[91].
The r e m a i n i n g r e s u l t s i n s e c t i o n 5 . 4 a r e t o b e f o u n d i n
296
Chapter 5
S . Dineen
[199], where t h e connection between e x t e n s i o n
theorems f o r holomorphic f u n c t i o n s on s u b s p a c e s and t o p o l o g i c a l H(E)
properties of
is established.
Corollary 5.50 (the
h o l o m o r p h i c H a h n - B a n a c h t h e o r e m ) was f i r s t p r o v e d b y P . J . In [161],
J.F.
Boland
[83] using a d i r e c t approach.
Colombeau
and B .
P e r r o t g e n e r a l i s e t h i s c o r o l l a r y by showing t h a t a
F r g c h e t - S c h w a r t z v a l u e d e n t i r e f u n c t i o n on a c l o s e d n u c l e a r
a38
subspace of a to
E.
space,
E,
can be extended holomorphically
An e x t e n s i o n t h e o r e m f o r n u c l e a r h o l o m o r p h i c f u n c t i o n s
on Banach s p a c e s i s g i v e n i n R .
Aron and P .
Berner [26] and
u s i n g t h i s theorem and a u n i f o r m f a c t o r i z a t i o n theorem f o r holomorphic f u n c t i o n s on J.F.
of
23q
(see exercise 2.105),
spaces
Colombeau and J . M u j i c a [ 1 5 6 ] g i v e a n a l t e r n a t i v e p r o o f
corollary 5.50.
of theorem 5.48, 6.54),
A further proof
of t h i s c o r o l l a r y and a l s o
using t h e symmetric t e n s o r a l g e b r a ( d e f i n i t i o n
has r e c e n t l y been o b t a i n e d by R .
I n [488],
R.
Meise and D .
Meise and D .
Vogt
[487].
Vogt show t h a t t h e h o l o m o r p h i c Hahn-
Banach theorem i s n o t v a l i d f o r c e r t a i n n u c l e a r F r g c h e t s p a c e s . A different kind of
e x t e n s i o n r e s u l t f o r e n t i r e f u n c t i o n s on a
n u c l e a r subspace o f a l o c a l l y convex space i s g i v e n i n A . Martineau
[453].
In chapter 6,
s e c t i o n 4 , w e p r o v e a number o f s t r u c t u r e
theorems f o r holomorphic f u n c t i o n s on i n f i n i t e t y p e power series spaces. The r o l e o f n u c l e a r i t y i n i n f i n i t e d i m e n s i o n a l holomorphy i s much m o r e e x t e n s i v e t h a n t h a t o u t l i n e d i n t h i s b o o k . i s mainly due t o o u r choice of t o p i c s .
This
I n Appendix I , w e s e e
t h a t i t a p p e a r s i n t h e s t u d y o f c o n v o l u t i o n o p e r a t o r s on s p a c e s of holomorphic f u n c t i o n s ,
5
i n s o l v i n g t h e Levi problem and t h e
p r o b l e m a n d i n i n f i n i t e d i m e n s i o n a l h o l o m o r p h i c s h e a f theory.
A p a r t f r o m t h e s e t o p i c s we a l s o f i n d n u c l e a r i t y a p p e a r i n g i n A.
M a r t i n e a u ' s s t u d y [453] o f t h e s u p p o r t s o f a n a l y t i c functiorr
als i n several variables,
c o n t i n u a t i o n [712]
~
i n L.
i n B.
W a e l b r o e c k ' s r e s u l t on a n a l y t i c
Kramm's
[398,399,400]
/
interesting
c l a s s i f i c a t i o n theorems f o r Frechet n u c l e a r a l g e b r a s , a n a l y t i c s p a c e s and S t e i n a l g e b r a s and i n M .
a &?
S c h o t t e n l o h e r [&61
Chapter 6 GERMS, SURJECTIVE LIMITS, E -PRODUCTS AND POWER SERIES SPACES
The l a s t two c h a p t e r s d e a l t w i t h s c a l a r - v a l u e d holomorphic f u n c t i o n s d e f i n e d on s p e c i a l domains i n s p e c i a l s p a c e s . return i n t h i s chapter t o the general theory, f u r t h e r methods - t h e
We
and p r e s e n t t h r e e
t o p o l o g y , s u r j e c t i v e l i m i t s , and
T~
&-products - f o r studying t h e r e l a t i o n s h i p between t h e t o p o l ogies
T
0'
T
and
w
T
~
.
The
T~
t o p o l o g y aims a t r e m o v i n g
g e o m e t r i c r e s t r i c t i o n s on t h e domain,
surjective l i m i t s are
used t o g e n e r a t e spaces o f holomorphic i n t e r e s t and u s i n g
6 - p r o d u c t s we s t u d y v e c t o r - v a l u e d f u n c t i o n s . A p a r t f r o m t h e p r o b l e m o f t h e d i f f e r e n t t o p o l o g i e s we a l s o d i s c u s s i n t h i s chapter o t h e r problems of general i n t e r e s t such as t h e r e p r e s e n t a t i o n o f a n a l y t i c f u n c t i o n a l s and t h e comple'teness of
H(K).
The f i n a l
section of t h i s chapter i s devoted t o
holomorphic f u n c t i o n s on t h e s t r o n g d u a l s o f c e r t a i n power In t h i s case, t h e r e s u l t s o f c h a p t e r f i v e are
s e r i e s spaces.
c o m b i n e d w i t h some i n t e r e s t i n g e s t i m a t e s t o o b t a i n a number o f representation theorems.
§6.1
HOLOMORPHIC G E R M S O N C O M P A C T SETS I n c h a p t e r s 2 , 3 a n d 5 we o b t a i n e d v a r i o u s r e s u l t s
c o n c e r n i n g t h e r e g u l a r i t y and c o m p l e t e n e s s o f compact s u b s e t o f a l o c a l l y convex s p a c e .
H(K),
K
a
The p o s i t i v e a n d
n e g a t i v e r e s u l t s we o b t a i n e d s h o w t h a t t h e s e a r e i n d e e d c o m p l e x q u e s t i o n s and n o t u n r e l a t e d t o one a n o t h e r . assumed t h a t condition
-
K e.g.
I n m o s t c a s e s , we
s a t i s f i e d a rather r e s t r i c t i v e geometric that
K
consisted of a single point o r w a s a
298
Chapter 6
balanced set o r a polydisc. internal structure of
T h i s e f f e c t i v e l y meant t h a t t h e
p l a y e d no p a r t i n o u r i n v e s t i g a t i o n s
K
a n d t h a t e s s e n t i a l l y we w e r e s t u d y i n g l o c a l p h e n o m e n a .
H e r e we
l o o k a t t h e g l o b a l t h e o r y b y c o n s i d e r i n g a r b i t r a r y compact s e t s . We d o n o t c o n c e r n o u r s e l v e s w i t h t h e r e g u l a r i t y a n d c o m p l e t e ness of
-
H(0)
as w e have considered t h e s e q u e s t i o n s i n chap-
t e r s 3 and 5 and s h a l l g i v e a f u r t h e r example i n t h e n e x t
-
section
b u t look a t
H(K)
where
K
i s a compact s u b s e t of a
Frgchet space o r of a f u l l y nuclear space with a basis.
The
m e t h o d o f p r o o f u s e d f o r m e t r i z a b l e s p a c e s was m o t i v a t e d b y t h e proof o f p r o p o s i t i o n 3.40 and t h i s ,
i n turn, provided t h e
m o t i v a t i o n f o r the f u l l y n u c l e a r space case.
We h a v e a l r e a d y s e e n i n p r o p o s i t i o n 2 . 5 5 , r e g u l a r when
K
that
H(K)
is
i s a compact s u b s e t o f a F r g c h e t s p a c e and
K
t h a t i t i s c o m p l e t e when
is balanced.
We now e x t e n d t h e
l a t t e r r e s u l t t o a r b i t r a r y compact s e t s . Theorem 6 . 1
then
If K i s a compact s u b s e t o f a F r g c h e t s p a c e i s cornpZete.
H(K)
Proof
Since
is a
H(K)
t h a t it i s quasicomplete. t o p o l o g y on H(K)
H(K).
Let
Let T
space,
DF
-rl
E
i t s u f f i c e s t o show
denote the inductive l i m i t
b e t h e l o c a l l y convex topology on
g e n e r a t e d by a l l semi-norms
which have e i t h e r o f t h e
following forms:
is a
where
p
where
(xnInzl,
m
T~
c o n t i n u o u s semi-norm on
m
(XA)n=1
H(O),
a r e two s e q u e n c e s i n
K,
m
(yn)n=l,
are n u l l sequences i n E, x n + y n = x n’ + yn’ f o r a l l and (Y;l);=l n and (kn)m i s a s t r i c t l y i n c r e a s i n g sequence o f p o s i t i v e n=l
Germs, surjective limits,
E
299
-products and p o w e r series spaces
integers
By p r o p o s i t i o n 2 . 5 6 , subsets of
H(K)
and
We now s h o w t h a t (gB)BEB be a
n
f
T
T
-rl
and
T ~
d e f i n e t h e same b o u n d e d
.
(H(K),T)
is quasi-complete.
bounded Cauchy n e t i n
T
H(K).
If
Let X E
and
K
is a positive integer,
o f t h e form
( g(
net in
then using T c o n t i n u o u s semi-norms fin we s e e t h a t (d g B ( x ) / n ! ) B E B i s a Cauchy
(*),
n ~ ) , ~ u ) s. i n c e
(corollary 3.42),
(0'
( n ~ ) , ~ u )i s c o m p l e t e
t h e r e e x i s t s an element of
@("E),
Pn,x,
such t h a t
( g B ) B E B i s bounded t h e r e e x i s t a neighbourhood
As
zero i n €3
B E B,
for all
and
Y
and
y,y'
in
arbitrary.
4n
exists
qnl
E
X E
gB
E
H(K+4W)
and a l l non-negative
K
for all
w W
satisfy
and
6
In=, m
x+y = x ' + y '
x.
n.
Bo
E
T
B
For any
Pn , x ( y ) .
and l e t
c o n t i n u o u s semi-norm on such t h a t
in
This implies
Let 6'
0
x
in
x,x' be
C h o o s e nl a p o s i t i v e i n t e g e r such t h a t 6 E . F o r a n y f E H(K) let
is a
of
W
B
Hence
for all n 4" let f(x)(y) =
llPn,xll,v
Since
such t h a t
M,O
s u P l l g B l l K + 4 w= M < m . BEB
and
that
and
E
H(K)
there
K E
K
300
Chapter 6
Hence
and
Thus, there exists an
f
and
for all
f(x)(y)
f(x+y)
=
in
Hm(K+W) x
in
such that K and y
Due to the form o f the semi-norms immediate that (H(K),T)
gf3
+
f
as
is quasi-complete.
ogy associated with
T
barrelled space and
T
hence
+
(H(K),T~)
on 4
and
Let H(K).
in T~
(**)
it is
(H(K),T). Hence be the barrelled topol-
Since
(H(K),T~)
it follows that
T~
(H(K),T~)
This shows that (H(K),.r2) over, by proposition 3 . 6 , h e n c e complete.
m
and
(*)
/ I f l / K + w6 M in W .
is a and
T~
have the same bounded sets.
is a barrelled DF space. More(H(K),.r2) is quasi-complete and
We complete the proof by showing that
T
~ T~=
T h e situation now is rather similar to that o f proposition and an examination o f the proof o f that proposition
3.40,
shows that we only need find a fundamental system o f bounded
m k subsets o f H(K), (Bn)n=l, such that closed for any finite sequence o f numbers to complete the proof.
AnBn
is
T~
in order
(~~)i=~
Let be a decreasing fundamental neighbourhood system at the origin in E consisting o f convex balanced open sets and f o r each H" (K+v,) . Let
r
h
E
n
l:=lAnBn
Bn
let
+
h
E
be the closed unit ball o f
H(K)
By corollary 3.39, Bn (H(K+V,),T~). F o r each y in r
y
h
E
Y>n
+
E
m .
Bn.
in t h e
T
topology as
is a compact subset o f
let
h, =
l n = 1An h Y,n where By using subnets, if necessary, we see that there
301
Germs, surjective limits, e -products and power series spaces exists a in
hn
E
(H(K+V,),T~)
rv
hy + h
in
Bn, n = l , . . . , k , for all n.
This implies t h a t n.
all
as
(H(K+Vk),~o)
%(x) n!
Iy
Hence
h = h
closed subset of
y
and
is a
l!=l~nBn
B
B
of
H(K)
H(K) V
some n e i g h b o u r h o o d o f
for every
f
in
B
in
K
for
K
and -r2,
an a r b i t -
i s c o n t a i n e d and bounded i n of
K
(i.e.
Hm(V),
i f and only i f t h e elements
K,
Taylor series development o f elements of W of
+ w
and hence
T,
s a t i s f y u n i f o r m Cauchy e s t i m a t e s o v e r
neighbourhood
y
of a l o c a l l y convex s p a c e .
f o r some n e i g h b o u r h o o d of
x
for all
n!
We now l o o k a t t h e r e g u l a r i t y o f
A subset
as
hn
+
This completes t h e proof.
H(K).
r a r y compact s u b s e t
hy,n
m .
-f
'nh(x)
-
~
such t h a t Hence
B
and t h e l o c a l
K
is coherent i n
i f and o n l y i f t h e r e e x i s t s a
zero such t h a t
whenever
x,x'
E
y,y'
K,
E
and
W
x+y = x'+y'). We h a v e p r e v i o u s l y u s e d t h i s r e d u c t i o n i n o u r a n a l y s i s , a s f o r example i n p r o p o s i t i o n 2 . 5 6 ,
where t h e semi-norms
were u s e d t o o b t a i n C a u c h y e s t i m a t e s a n d t h e s e m i - n o r m s were u s e d t o p r o v e c o h e r e n c e .
If
H(0)
(*)
(**)
is regular, then we
h a v e Cauchy estimates and i t i s p o s s i b l e t h a t t h i s a l s o i m p l i e s coherence.
We a r e n o t ,
however,
able t o prove t h i s .
To p r o v e c o h e r e n c e , w e n e e d e x t r a h y p o t h e s e s a n d t h e s e
can t a k e various forms.
O n e may p l a c e c o n d i t i o n s o n
a s l o c a l c o n n e c t e d n e s s , or c o n d i t i o n s o n i l i t y o r a c o m b i n a t i o n o f c o n d i t i o n s on
E K
K,
such
such as metrizaband
E.
We s h a l l
302
Chapter 6
assume t h a t
i s m e t r i z a b l e and t h a t
K
E
satisfies a certain
t e c h n i c a l c o n d i t i o n which appears t o b e s a t i s f i e d by most, not all,
s p a c e s f o r which
examples o f non-metrizable H(K)
is regular.
H(0)
if
This gives us
l o c a l l y convex s p a c e s i n which Our m e t h o d s a r e
K.
i s r e g u l a r f o r e v e r y compact s e t
e a s i l y seen t o be i n f l u e n c e d by t h e p r o o f s o f p r o p o s i t i o n 2.56 and theorem 6 . 1 . Proposition 6.2
a)
K
b)
H(0)
c)
if
b e a c o m p a c t s u b s e t of a ZocaZZy
K
Let
E
convex space
and s u p p o s e
i s metrizable, is r e g u Z a r , i s a c o n v e x b a l a n c e d o p e n s u b s e t of
V
(fn)n c H ( V ) ,
and
fn f 0
fn(xn) # 0
such t h a t H(K)
then Proof
n,
for e a c h
t h e r e e x i s t s a bounded s e q u e n c e i n
E
then (xn)nmZl,
V,
n
for a22
i s a regular inductive l i m i t . Let
b e a bounded
B
subset of
H(K).
Since each
semi-norm o f t h e form
where on
p
H(K)
i s a c o n t i n u o u s semi-norm on
and
H(0)
a neighbourhood
I/ f o r every
in
Now s u p p o s e
K,
zero i n
E,
such t h a t
i s continuous
of
V
0
in
and
E
MzO
such t h a t
A
x
nets in
H(O),
is regular it follows that there exists
K,
and
f
in
is not coherent.
B
( x a )a E
nzO
r
(ya) acr y,,yA E V
and and all
(x:)
acr *
B.
Then t h e r e e x i s t s two two n e t s c o n v e r g i n g t o and
(fa)aEr a net i n B (Y;)aEr’ a, xa+ya = x ’ + y ’ f o r a l l a and a a
303
Germs, surjective limits,E -products and power series spaces
Since set
is metrizable
K
{Y,-Y;IaEr W
and
(Xn)n=l
i s a l s o m e t r i z a b l e and hence t h e
K-K
(X;)n=l
be t h e corresponding sequences i n
For each p o s i t i v e i n t e g e r
3 f ,
(x,)
n
(XI
j!
Each
hn
n
I;=,
-
and each
such t h a t
x
A .
in
# 0
let
V
(X+YA-Y,)
j!
and
V
for all
n.
# 0.
hn(yn)
m
t h e r e e x i s t s a bounded s e q u e n c e
hn(zn)
K.
d'fan(xr;)
i s a holomorphic f u n c t i o n on
By c o n d i t i o n ( c ) , V
m
( ~ ~ - y ; ) ~ =L e~t .
contains a n u l l sequence
( z ~ ) ~i n= ~
By t h e i d e n t i t y t h e o r e m
f o r h o l o m o r p h i c f u n c t i o n s o f o n e c o m p l e x v a r i a b l e we c a n c h o o s e
a n u l l sequence i n n.
for all
2~~ > 0
m
C,
loss of,generality,
Hence
such t h a t
Anzn+O
w e may s u p p o s e
Xnzn+yA-yn~V f o r a l l
n.
Xnzn
E
all
> n
n
V
and,
=
without
and
Now choose i n d u c t i v e l y a s t r i c t l y m
(kn)n=l
increasing sequence of p o s i t i v e integers, 2 k n B,
Ihn(Anzn)l
n-tm
as
such t h a t
and
Let q(f)
2.kn
=
f o r every If
f
f c H(K)
in
H(K).
then t h e r e e x i s t a neighbourhood
and a p o s i t i v e i n t e g e r
IIf(IK+4W
6
M,
Xnzn
E
no W
and
such t h a t
W
of
0,
M>O
f e Hm(K+4W),
Xnzn+yA-yn
E
W
all
n
2
n
.
304
Chapter 6
Hence
Since
H(K)
Since
q(fa?
is barrelled k n Bn > n L 2
q
is a continuous semi-norm. m and (fa)n=l is bounded n B is a coherent family
for all n this leads to a contradiction. Hence and
H(K)
is a regular inductive limit.
This completes the
proof. If
E
is a fully nuclear space with a basis, then every
E
compact subset o f
is metrizable.
Condition (c) o f prop-
osition 6 . 2 is satisfied by E if E;i admits a continuous norm. Hence we have the following corollary to proposition 6 . 2 and this applies, in particular, to strong duals o f n u c l e a r
DN
spaces.
Eb
then
H(K)
subset
If
K
E
E
If
Corollary 6 . 3 such t h a t
i s a fuzzy n u c Z e a r s p a c e w i t h a b a s i s
a d m i t s a c o n t i n u o u s norm and
H(OE) i s regular i s a r e g u l a r i n d u c t i v e Z i m i t f o r e v e r y compact of E . is a nuclear locally convex space, E
compact subset o f
and
V
K
is a
is a neighbourhood o f zero in
E
then, using Cauchy estimates, one can show there exists a neighbourhood W o f zero such that H ( K ) and Hm(K+W) induce the same uniform structure and hence the same topology o n t h e unit ball o f H m ( K + W ) . Since H m ( K + W ) is a Banach space, corollary 6 . 3 yields the following result. Corollary 6.4
If
E
i s a fully nuclear s p a c e w i t h a b a s i s ,
305
Germs, surjective limits, E -products and power series spaces Eb
a d m i t s a c o n t i n u o u s norm a n d
H(OE)
i s regular,
n chapter 2, on
H(U)
we d e f i n e d ( d e f i n i t i o n 2 . 5 9 )
for
then
of
K
is q u a s i c o m p l e t e f o r a n y c o m p a c t s u b s e t
E.
the
T
71
top-
an open s u b s e t o f a l o c a l l y convex
U
T h i s t o p o l o g y h a s good l o c a l p r o p e r t i e s and c o i n c i d e s i n certain cases,
W
and
71
i n d e e d i t has been c o n j e c t u r e d
always c o i n c i d e .
w
We now e x a m i n e t h i s t o p -
and b e g i n by showing t h a t i t i s i n d e e d well d e f i n e d . Lemma 6.5
space
E.
=
lirn
+ KE
&(U)
Proof
t h e l o c a l l y convex
Then a l g e b r a i c a l l y
H(U)
where
be a n o p e n s u b s e t of
U
Let
H(K)
k(U,
i s t h e s e t of a l l c o m p a c t s u b s e t s of
U.
Under t h e n a t u r a l r e s t r i c t i o n mappings
i s c l e a r l y a projective system.
{ H ( K )' K E A ( U ) mapping A :
H(U)-
lirn
c-
The c a n o n i c a l
H(K),
K E .k(U 1
where
and
[fIK
i s t h e h o l o m o r p h i c germ on
K
induced by
l i n e a r and i n j e c t i v e .
I t r e m a i n s t o show t h a t
ive.
E
Let
(EK)KEt(u)
l i m
H(K)
f,
is
is surject-
A
We d e f i n e a
be given.
t-
function
f
on
U
by
f E H(U) and compact s u b s e t o f U,
claim
KEP<(U) f(x) = fIxl(x)
for all
A(f) = (fK)KEk(U). If then since
K
x
in
i s any
U.
We
306
Chapter 6
n!
n!
f o r any compact s u b s e t negative integer Hence i f
x E U
of zero such t h a t in
V
of
K
containing
U
and
i s a convex balanced neighbourhood
V
x+VCU
and
fix)
H"(x+V)
E
f{x+y}(X+Y)
=
f [X,X+Yl
=
n!
t h e n f o r any
y
U,
x
E
f K
n!
E
H(U).
Moreover,
and
n
is arbitrary,
n!
if
K
[fIK
=
n! Hence
fK.
i s a compact sub-
then
n!
and c o n s e q u e n t l y
(X+Y)
{x+Ay; 0 i X i 11
[x,x+y] =
T h i s shows t h a t
set of
and any non-
we have f(x+y)
where
x
n.
A(f)
=
( f K l K E& ( u ) .
This
completes t h e proof. Remark 6.6 U an a r b i t r a r y - M!e h a v e T ~ S 7TS1T w o n I l ( U ) , Our n e x t p r o p o s i t i o n open s u b s e t of a l o c a l l y convex s p a c e . d e s c r i b e s a s i t u a t i o n i n which
T~
and
T~
coincide.
Proposition 6.7
If U is a b a Z a n c e d o p e n s u b s e t of a ZocaZZy c o n v e x s p a c e , t h e n T~ = T on H(U). Proof Suppose
Let p
p
be a
T
c o n t i n u o u s semi-norm on
i s p o r t e d by t h e compact b a l a n c e d s u b s e t
H(U). K
of
U.
307
Germs, surjective limits,E -products and power series spaces By t h e o r e m 3 . 2 2 , we may s u y p o s e , w i t h o u t l o s s o f g e n e r a l i t y , that
Let
If
V
i s a neighbourhood of
such t h a t
then there ex i s t s
K
p ( f ) 6 ~ ( V ) l ( f / (f o~ r e v e r y
f
in
c(V)> 0
H(U).
Hence f o r
every
N
is and t h u s p i s a c o n t i n u o u s semi-norm on H(K). If 'U,K t h e c a n o n i c a l mapping o f H(U) i n t o H(K), then p = OpU,K
and h e n c e than
T~
5
is
T~
continuous.
Since
T~
i s always f i n e r
t h i s completes t h e proof.
We now show t h a t Frschet-Schwarz
T~
spaces.
i s a s h e a f o r l o c a l t o p o l o g y on
To p r o v e t h i s , we n e e d some p r e l i m -
The f i r s t , a n open mapping theorem,
inary results.
we g i v e
without proof. Proposition 6.8
t h e semi-Monte2 s p a c e i s bounded i n o p e n mapping.
T
Let E
E
whenever
b e a c o n t i n u o u s l i n e a r mapping from
into the B
DF
F.
space
i s bounded i n
F
If then
T-I(B)
T
i s an
Chapter 6
308 Proposition 6.9
If
Schwartz space E, Proof
Since
is a c o m p a c t s u b s e t of a F r g c h e t -
K
is a ,X13d'space.
H(K)
then
i s a complete
H(K)
show t h a t f o r e v e r y n e i g h b o u r h o o d e x i s t s a neighbourhood
B
If
=
E
Hm(K+W).
Let
p
it suffices to
space,
of zero i n
V
c 11
there
E
of zero contained i n
W
Ilf 11 K + V
H"(K+V) ;
DF
such t h a t
V
i s a precompact s u b s e t o f
b e t h e Minkowski f u n c t i o n a l o f
E
Since
V.
i s a S c h w a r t z s p a c e , t h e r e e x i s t s a c o n t i n u o u s semi-norm on
q,q 32p,
s u c h t h a t t h e c a n o n i c a l mapping from
E
into E = (E/ ,p) i s precompact. By q-l(O) P P-l(O) L i o u v i l l e ' s t h e o r e m we may s u p p o s e , w i t h o u t l o s s o f g e n e r a l i t y ,
E
(E/
=
9
that
p
K+W
,q)
q
and
a r e norms.
i s a compact s u b s e t o f
t h e c l o s u r e oE
K+V
compact s u b s e t o f
in
A
E
Let
.
W
P (H(K+V),r,)
{ x ~ E ; q ( x ) <} .
=
= K + Interior
V1
(v)
Hence
is
is a
B
By p r o p o s i t i o n 3 . 3 7 ,
and hence e v e r y n e t i n
c o n t a i n s a s u b n e t which i s c o n v e r g e n t i n
v
where B
This
H"(K+W).
completes t h e proof. Theorem 6.10
The
lT topology
is a sheaf o r local topology on Fre'chet-
Schwartz spaces. Proof
Let
let
be an open subset of the Frgchet-Schwartz
U
be an open covering of
(Ui)iEI
denote t h e r e s t r i c t i o n mapping from
H(U)
theorem we must show t h a t t h e mapping l i m (H(Ui),rT) c
F o r each
U.
into
J:f
E
iE1,
H(Ui).
space
let
E
and
P
U,Ui To prove our
( H ( U ) , T ~ )+ ( f l u . ).
E
1 1 E I
i s a l i n e a r isomorphism.
i
i s e a s i l y seen t o be a l i n e a r b i j e c t i o n .
J
mappings from
(H(U),rv)
compact subset hence
J
Let
of q
T~
on
K
of
U
into
H(K),
p
i s continuous. p
U,Ui
be a continuous semi-norm on
t h e r e e x i s t a compact subset H(K)
a r e continuous f o r every
IIK,
t h e mappings
such t h a t
p s q o IIK.
K As
Since t h e canonical
a r e a l l continuous and
( H ( U ) , T ~ ) . By t h e d e f i n i t i o n of K
U
and a continuous semi-norm
i s compact and
E
i s metrizable
Germs, surjective limits, we may w r i t e
OK.
as
K
f o r some j i n m H(K) + II H ( K ; ) j=1 t h e definition of
E
309
-products and power series spaces
i s a compact s u b s e t o f U . J m I . The n a t u r a l i n j e c t i v e l i n e a r mapping A = TI A;; j=1 i s c o n t i n u o u s . By r e g u l a r i t y o f t h e i n d u c t i v e l i m i t s i n J=1
where each
K. J
J
J
.
H(K.), j = 1 , . . , m , one s e e s t h a t A-l(B) i s bounded J m f o r e a c h bounded s u b s e t B o f n H(Kj). F u r t h e r m o r e , j =1 i s a DF s p a c e and H(K) i s a Monte1 s p a c e by p r o p o s i t i o n
i n H(K) m n H(K.) j=1 3 6.9. An a p p l i c a t i o n o f p r o p o s i t i o n 6.8 now i m p l i e s t h a t Hence t h e r e e x i s t s c o n t i n u o u s semi-norms
on
qj
H(K.) J
such t h a t q(g) d sup j=1,. Since
A . o II J
q . (A.(g)) 1 3
f o r all
. .,m
g
in
i s open.
A
for
j=l,
..., m
H(K)
-
K -*Kj
p ( f ) 6 q(nK(f)) 6 sup qj(nK j = 1 , . ,m j
~ ~ , ~ . ( ff o) r) e v e r y
..
f
in
H(u).
1
q . o I I K , i s a c o n t i n u o u s semi-norm on ( H ( U j ) , ~ . r r ) f o r each ' 3 tliis i m p l i e s t h a t J-' i.s c o n t i n u o u s and c o m p l e t e s t h e p r o o f .
Since
j
A s a c o r o l l a r y we improve a r e s u l t o f c h a p t e r f i v e . C o r o l l a r y 6.11 E
space
By c o r o l l a r y 5.23,
T~
Since
0
and
i s a baZanced open subset of a Frgehet nucZear T~ = T
c
-c0
whenever U T
U
with a b a s i s , then
Proof E.
If
T~
T
7
=
on H(U). whenever
T
c1 T 0 t h i s i m p l i e s t h a t
i s a n open p o l y d i s c i n
E.
T~
U
i s a n open p o l y d i s c i n
and
c o i n c i d e on
T~
By lemma 2 . 3 9 , and theorem 6 . 1 0 ,
a r e l o c a l t o p o l o g i e s and hence, s i n c e open p o l y d i s c s form a
fundamental neighbourhood b a s i s f o r t h e neighbourhoods o f z e r o i n
E
since
=
and
T
T
are both t r a n s l a t i o n i n v a r i a n t topologies,
H(U) f o r any open s u b s e t whenever
U
H(U)
U
of
E.
By p r o p o s i t i o n 6 . 7 ,
i s b a l a n c e d and open and hence
T~
= T
.
T
=
T
T
0
7
on
and T
1
on
H(U)
T h i s completes t h e
proof.
We now g i v e a few o t h e r r e s u l t s which show f u r t h e r p o s s i b i l i t i e s o f
Chapter 6
3 10
applying germs to obtain global results about the
topology
T~
Let U be an open subset of a locally convex space E
and let K be
a compact subset of U. We let
If U is an open subset o f a ZocaZly convex space,
Proposition 6.12 then (H(U),rw)
lim
=
kcU
HU(K)
K compact
Proof
=
lim
kcU
-
HU(K)
.
K compact
The algebraic identification follows at once, as in lemma 6.5,
from the fact that holomorphic functions are locally defined.
If K
is a
compact subset of U we let IIK denote the canonical mapping from H(U) into HU(K). If pK is a continuous semi-norm on HU(K) then we claim that p K o n K is a continuous semi-norm on (H(U),T~). If V is open and K C V C U then there exists c(V) > 0 such that pK(f) 6 c(V) I/f/lV for every f in HU(K). Hence
and this proves our claim. Thus the canonical linear bijection from (H(U),.cU) onto lim HU(K) is continuous. KCU K compact
Conversely, let p
be a
T
0
continuous semi-norm on H(U)
which is
ported by the compact subset K of U. If f,g E HU(K) and nK(f) = n,(g) then f and g coincide on a neighbourhood V
of
K.
31 1
Germs, surjective limits, E -products and power series spaces
Hence p(f) = p(g) since p(f-g) 5 c(V)(If-g/IV = 0. Thus, we may write p = pK o n K where pK is a well defined semi-norm on HU(K). Since p is ported by
K the restriction of pK to each Hm(V)nH(U)
and hence pK
is a continuous semi-norm on HU(K). (H(U),T~) = lim HU(K).
is continuous
This shows that
f-
KCU,K compgct The canonical injection from HU(K) into H (K) U
is clearly continuous
and hence it suffices to show that any continuous semi-norm p extends to a continuous semi-norm on Ff,(K). of p
to H"(V)n
H"(V)
n H(U)
p(f)
6
H(U)
Let pv be the restriction
for any open set V,
is dense in the Banach space
-
KCVCU.
(H"(V)
c(V) ((fl/Vfor every f in Hm(V)f7 H ( U ) ,
n
-
and
TIHu(K) = p.
If H"(V)n H(U) = H"(V) for all V neighbourhood system of K then H(K) =
Since
H(U),
pv
sion to a continuous semi-norm pv on (H"(V) n H(Uj, d rr/ z = pv for every V. p p on H"(K) by 'ir continuous semi-norm on HU(K)
on HU(K)
11
Ilv)
and
has a unique exten-
11 ( I v ) .
We define is a well defined
This completes the proof.
belonging to some fundamental = HU(K) for
gu(K) and if H(K)
a Eundamental system of compact subsets of U
then proposition 6.12 implies that T = T on H(U). This is the case if U is balanced and also 0 7 1 occurs in the next example. Example 6.13
Let
R
be a holomorphically convex open subset of (En.
. Any compact subset of nxcCN is contained in a compact set of the form KxL where K is a holomorphically convex compact subset of n and L is
a balanced convex compact subset of EN. If f E H(KxL) then f depends only on a finite number of coordinates and hence, by a reduction to finite dimensions and an application of the finite dimensional Oka-Weil approximation theorem (see Appendix I), f can be uniformly approximated on some N neighbourhood of KxL by holomorphic functions on G x C . Hence (H(RX~$,T,? T
0
= T
0
=
(H(QxE N 1
on H(U)
, ~ ~ )We. shall use this result in 56.3 to show
for any open subset
3
of
CN.
We now use proposition 6.12 to show that (H(U),T~) is complete whenever U is an open subset of a quasi-normable metrizable locally convex space.
Chapter 6
312 Definition 6.14
The i n d u c t i v e limit
( E , T ) = l i m ( E L YT ,L Y )
7
i s b o u n d e d l y r e t r a c t i v e i f f o r e a c h bounded s u b s e t t h e r e e x i s t s an
B C ( E a , ~ L Y ) and
such t h a t
a
T
B ~
E
of = B T
~
~
B
Boundedly r e t r a c t i v e i n d u c t i v e l i m i t s a r e r e g u l a r and i f each
i s quasi-complete then
ELY
Proposition 6.15
(E,T)
Let
is a l s o quasi-complete.
E
( E n , ~ n ) be a c o u n t a b Z e
= l i m
* n
boundedZy r e t r a c t i v e i n d u c t i v e Z i m i t o f Banach s p a c e s , and l e t F
b e a s u b s p a c e of
E.
If
-+ n nl
(F,T')
then
is b o u n d e d l y r e t r a c t i v e and c o m p l e t e .
For e a c h p o s i t i v e i n t e g e r
Proof
c l o s e d u n i t b a l l of
( 7 , ~ ' ) .S i n c e
integer
n
Now
Bnfl F
and
LY
xB
xB
E
x
+
x
c
is a
DF
B
n
let
Bn
be the
-cn
b e a bounded s u b s e t o f
space there e x i s t
a positive
such t h a t
h > O
-T
and t h e r e e x i s t
BnnF
a positive integer
m
such t h a t
> O
If then
(p,~')
and
T I
and l e t
En
E
h(BnO F) in
Em
X(BnCI F )
+
as
x
E
B+m
A(Bnn F ) and
TmC A(Bmfi F ) T m .
T'
as
B+-
in
(F,T)
.
Germs, sutjective limits, Hence
B C A (Bm r \ F )
shows t h a t
E
313
-products and p o w e r series spaces
Tm
and T and T a g r e e on B. This m i s a boundedly r e t r a c t i v e i n d u c t i v e l i m i t
(F,T')
and completes t h e p r o o f .
We a l s o n e e d t h e f o l l o w i n g r e s u l t . Proposition 6.16
( E , T ) = l i m ( E n , ~ n ) be a c o u n t a b l e
Let
-2
i n d u c t i v e l i m i t o f Banach s p a c e s .
l i m ( E n , ~ n ) i s bound-
+ n
Then
co
e d l y r e t r a c t i v e if and onZy i f f o r e a c h nu22 s e q u e n c e in
E
there e x i s t s a positive integer m
co
i s a (x,,,)~=~
( x , ) , = , C ( E ~ ~ T ~ )and Definition 6.17
T
n
such t h a t
n
n u l l sequence. E
A l o c a l l y convex space W
e x i s t s a z e r o neighbourhood B
i s quasi-
of z e r o t h e r e
V
n o r m a b l e i f f o r any g i v e n n e i g h b o u r h o o d c a n f i n d a bounded s u b s e t
such t h a t f o r every E
of
( x ~ ) ~ = ~
we
a>O
WCB+aV.
with
Every normed l i n e a r s p a c e i s q u a s i - n o r m a b l e and a l o c a l l y convex s p a c e i s a Schwartz s p a c e i f and o n l y i f it i s q u a s i normable and i t s bounded sets a r e precompact.
Thus a F r g c h e t -
Monte1 s p a c e i s q u a s i - n o r m a b l e i f and o n l y i f it i s a F r c c h e t Schwartz space, Proposition 6.18
I f
is a c o m p a c t s u b s e t of a q u a s i -
K
normabZe r n e t r i z a b Z e s p a c e
E
then
H(K)
(Hm(V),
= l i m
+
VDK,
v
I(
I(v)
open
i s a boundedZy r e t r a c t i v e i n d u c t i v e l i m i t . We a p p l y p r o p o s i t i o n 6 . 1 6 .
Proof
sequence i n
H(K).
Since
H(K)
Let
m
be a null
(fn)n=l
is a regular inductive l i m i t
( p r o p o s i t i o n 2 . 5 5 ) , t h e r e e x i s t s a convex balanced neighbourhood
V
of zero such t h a t
IlfnlI K + V = M <
m.
Since
(fn)E=,CHm(K+V) E
n a convex balanced neighbourhood t h a t f o r every
a>O
and
is quasi-normable t h e r e e x i s t s W
of
zero,
ZWCV,
we c a n f i n d a b o u n d e d s u b s e t
B
such of
E
3 14
Chapter 6
with
We c o m p l e t e t h e p r o o f b y s h o w i n g t h a t
WCB+clV.
i s a null
sequence i n
Since
fn(x+Y)
=
(H~(K+w),
/I /I K+W).
fim d fJX)
Cm=o
( f n ) nm = l
(y)
for every
x
in
K
m! and
y
in
for all
x
in
K
1 O < 6 c T
Given
For a n y
where with
and
W
n,
,
xeK,
dmfn(x)
/\m
m! fn(X)
.
and a l l
choose ylcB,
n
B y2
i t s u f f i c e s t o show
bounded i n E
V
and
is t h e symmetric
Since
E
y1+6y2
n
E
w
y1 = y 1 + 6 y z - 6 y Z €W + 6 V C T1V + S V C V
m!
m!
WCB+6V.
l i n e a r form a s s o c i a t e d
that
+
such t h a t
V
we s e e
315
Germs, surjective limits, E -products and power series spaces A
Since
p(f)
semi-norm on
for all
m
dnf (x)
sup
=
f
XEK
is a continuous
and
H(K)
and
H(K),
this implies
n
and this completes the proof
is a n o p e n s u b s e t of a quasi-normabZe
If U
Corollary 6.19
E
metrizabZe ZocaZZy c o n v e x s p a c e compZete. By proposition 6 . 1 8 ,
Proof
for any compact subset
K
of
then
H(K)
U.
(H(u),T~)
is
is boundedly retractive
By proposition 6.15,
UNJK V open is also boundedly retractive and hence complete. Since = lim (lim H"(V)nH(U)) (proposition 6.12) (H(U),.rU) c-
--f
KCU U 3V3K K compact and a projective limit o f complete spaces is complete, this shows that
(H(U),.rw)
is complete.
This completes the proof.
A weak converse to proposition 6 . 1 8 is also true as one
can easily prove the following: if E H(K)
=
is a distinguished Frgchet space and lim (Hm(V), I / I l v ) is boundedly d
V3K V open retractive for some non-empty compact subset
316
Chapter 6 K
of
E
then
E
In particular, E
is quasi-normable.
H(OE)
i s n o t b o u n d e d l y r e t r a c t i v e when
i s a F r g c h e t Monte1 s p a c e which i s n o t a F r g c h e t Schwartz
space.
16.2
SURJECTIVE LIMITS O F L O C A L L Y C O N V E X SPACES We now d e s c r i b e a m e t h o d o f d e c o m p o s i n g s p a c e s o f h o l o m o r -
p h i c f u n c t i o n s i n t o a u n i o n o f more a d a p t a b l e s u b s p a c e s . Alternatively,
t h i s m e t h o d may b e d e s c r i b e d a s a way o f g e n e r -
a t i n g l o c a l l y convex s p a c e s w i t h u s e f u l holomorphic p r o p e r t i e s . Our m e t h o d , t h e u s e o f s u r j e c t i v e l i m i t s a n d L i o u v i l l e ’ s i s b a s e d o n t h e f a c t o r i z a t i o n r e s u l t s o f c h a p t e r two
theorem,
a n d a r i s e s n a t u r a l l y i n many p r o b l e m s o f i n f i n i t e d i m e n s i o n a l
I t s range o f u s e f u l n e s s f o r problems of topologies
holomorphy. on
i s n o t a s g r e a t a s i n some o t h e r a r e a s a s f o r
H(U)
i n s t a n c e i n s o l v i n g t h e Levi problem. Definition 6.20
A c o l Z e c t i o n of l o c a l l y c o n v e x s p a c e s and
(Ei,ni)iEA
l i n e a r mappings
i s called a s u r j e c t i v e represen-
t a t i o n of t h e l o c a l l y e o n v e x s p a c e t i n u o u s l i n e a r mapping f r o m
E
E
ni
i f each Ei
onto
and
(ni
-1
i s a con(Vi))iEA
forms a b a s e ( a n d n o t a s u b b a s e ) f o r t h e f i l t e r of n e i g h b o u r 0 i n E as Vi ranges o v e r t h e neighbourhoods o f h o o d s of 0 i n Ei and i r a n g e s o v e r A . E i s called the surjective l i m i t of ( E i , ~ i ) i c A and we w r i t e E = l i m (Ei,ni). f-
iEA
I f each
ni
i s a n open mapping,
we call
l i m (Ei,iIi)
ci EA
an open s u r j e c t i v e l i m i t and i f f o r each subset
K
such t h a t
of
Ei
ni(Ki)
i E A
and e a c h compact
t h e r e e x i s t s a compact s u b s e t = K
then we say
l i m (Ei,ni)
f-
Ki
of
E
i s a compact
i EA
surjective l i m i t . E v e r y l o c a l l y c o n v e x s p a c e i s a s u r j e c t i v e l i m i t o f normed
Germs, surjective limits,
E
317
-products and power series spaces
nuclear spaces are s u r j e c t i v e limits of separ-
linear spaces,
s p a c e s and a l o c a l l y convex s p a c e which h a s
able inner product
t h e weak t o p o l o g y i s a s u r j e c t i v e l i m i t o f f i n i t e d i m e n s i o n a l spaces.
i s a s u r j e c t i v e l i m i t o f TI. E. 'iEA E i iaAl 1 ranges over all the f i n i t e subsets of A. This
Example 6 . 2 1 where
A1
s u r j e c t i v e l i m i t i s e a s i l y seen t o b e open and compact. Example 6 . 2 2 and
& ,
If
is a completely regular Hausdorff space
X
i s t h e s p a c e of a l l c o n t i n u o u s complex v a l u e d
(X)
f u n c t i o n s on
endowed w i t h t h e t o p o l o g y o f u n i f o r m c o n v e r -
X
g e n c e on t h e compact s u b s e t s o f .Q,cx)
=
l i m ( &(K), f-
11
then
X,
llK)
KCX where
K
r a n g e s o v e r t h e compact s u b s e t s of
and
X
&(K)
i s t h e Banach s p a c e o f complex v a l u e d c o n t i n u o u s f u n c t i o n s on K
endowed w i t h t h e s u p norm t o p o l o g y .
X
Since
i s a complet-
e l y r e g u l a r space, t h e T i e t z e extension theorem implies t h a t
I(
l i m (,&(K),
t-
I/K)
i s a compact s u r j e c t i v e l i m i t and t h e open
KCX
mapping theorem f o r Banach s p a c e s i m p l i e s t h a t i t i s a n open surjective l i m i t . The s t r o n g d u a l o f a s t r i c t i n d u c t i v e l i m i t
Example 6 . 2 3
o f F r c c h e t Monte1 s p a c e s i s a n open and compact s u r j e c t i v e l i m i t of
33W
Proof
Let
.spaces. E
=
l i m ( E n , ~ n ) be a strict inductive l i m i t ----f
n
o f Frgchet-Monte1 spaces.
Since
E
induces on
En
its
o r i g i n a l t o p o l o g y , w e s e e , by t h e Hahn-Banach theorem,
that
t h e transpose of the canonical injection of
E
s u r j e c t i v e mapping from
on
E'
Eb
onto
(En);.
En
into
is a
The s t r o n g t o p o l o g y
i s t h e t o p o l o g y o f u n i f o r m c o n v e r g e n c e on t h e bounded
subsets of
E
and,
s i n c e e a c h bounded s u b s e t o f
E
318
Chapter 6
i s c o n t a i n e d a n d c o m p a c t i n some
t h e topology on
E' is B t h e weakest t o p o l o g y f o r which a l l t h e t r a n s p o s e mappings a r e En,
E' is a surjective l i m i t of B An a p p l i c a t i o n o f t h e o p e n m a p p i n g t h e o r e m s h o w s t h a t i t i s
continuous.
Hence
an open s u r j e c t i v e l i m i t . (En);.
Now l e t
b e a compact s u b s e t o f
Kn
There e x i s t s a convex balanced neighbourhood
zero in
whose p o l a r i n
En
(En);,
contains
Vo,
of
V
Since
Kn.
E
is a strict inductive l i m i t ,
t h e r e e x i s t s a neighbourhood
W
of
Wn
in
0
s
I$(WnEn)I a
$EE'
such t h a t
E
Monte1 s p a c e
Hence i f
En.
lv(W)
I
4
and
1
$IEn = $.
i s a compact s u b s e t
Wo
of
In particular, we note t h a t
CNx
C")
=
there exists
E'.
B
is a
E
As
completes t h e proof.
then
+EVO
a n d , by t h e Hahn-Banach theorem,
1
such t h a t
N
Vy)
This
is a
l i m C c-
n
compact and open s u r j e c t i v e l i m i t o f a 3 R s p a c e s w i t h a b a s i s . The u s e f u l n e s s o f s u r j e c t i v e l i m i t s stems from t h e f o l l o w i n g r e s u l t which i s e a s i l y proved u s i n g t h e method o f proposition 2.24.
lemma 6 . 2 4
E = l i m
Let
(E.,n.) 1
f-
u
iEA
H(U) i
if
=
in
iEA
H(ni(U))
and
A
iEA
c H(U)
f
1
f-
X
N
H(ni(U))
if
f
E
H(U)
such t h a t
such t h a t
bounded s u b s e t of
E.
Then
then there e x i s t s an N
f = foni).
Moreover,
is a l s o a c o m p a c t s u r j e c t i v e l i m i t and
1
f a e t o r s uniforrnzy through
X CH(ni(U))
subset o f
E
(i.e.
(E.,n.)
E = l i m
d
be a c o n v e x b a l a n c e d o p e n s u b s e t of
U
l i m i t and Zet
b e an o p e n s u r j e c t i v e
1
H(U)
ieA
fi.e.
X = nifi))
then
i f and o n l y
if
X
, 4
X
there e x i s t s
is a
is
G
T~ T~
bounded
H[ni(u)).
To o b t a i n p o s i t i v e r e s u l t s i t i s t h u s n e c e s s a r y t o f i n d c o n d i t i o n s u n d e r which a f a m i l y o f f u n c t i o n s on a s u r j e c t i v e
319
Germs, surjective limits, E -products and power series spaces
limit factors uniformly through some Ei. Frequently such conditions involve the structure of the indexing set A . For example, an analysis of
(X)
of
If T
in the surjective representation
leads to the following result. i s a c o m p l e t e l y r e g u l a r Hausdorff space t h e n t h e
X
H(&,(X))
bounded s u b s e t s o f
0
A
consequently
= -c6
T
on
a r e 1ocaZZ.y b o u n d e d and
H(& (X)))
if and o n l y i f ,&(X)
o,b i s an i n f r a b a r r e l l e d l o c a l l y convex space.
This result can be proved using methods similar to those employed in the proof of proposition 6.29.
We will not under-
take here a detailed study of the inde.xing set but instead confine ourselves to a few representative examples. Let
Proposition 6.25
l i m i t o f Frgchet-Montel U
t i n u o u s norm.
I f
HHy(U)
and t h e
= H(U)
l o c a l l y bounded.
E = lim E be a s t r i c t i n d u c t i v e - - b n n s p a c e s and s u p p o s e E a d m i t s a c o n -
i s a n o p e n s u b s e t of EL T~ bounded s u b s e t s o f
Hence
(H(U),-co)
then H(U)
i s c o m p l e t e and
H(U).
on
Proof
are T
0,b
=
T
We may suppose without loss of generality, that U is an E i = lim ((En)b,nn) cn
is convex and balanced and since
open surjective limit, u = ~,'(]L~(U)) for some positive integer m. Let V be the unit ball of a continuous norm on E . Since E and E t ; are complete Monte1 spaces, Vo is a compact subset of We first claim that V o is a deter-
Ei.
mining set for hypoanalytic functions o n f
E
HHy(U)
and
fIVon
=
0,
then
f
U;
i.e. if
0.
By using a Taylor series expansion, we see that if
6
320
Chapter 6
An d f(O) n! that
L o n
VO
u
for all
0
=
n
and hence it suffices to show
is a determining set for hypoanalytic homogeneous
polynomials.
n=l
If
this is clear since hypocontinuous
homogeneous polynomials o f degree continuous linear forms o n
E'
1
are nothing more than
greater than
1
E,
i.e. elements o f
B'
is the unit ball o f a continuous norm o n
N o w let
E.
and suppose we have shown that
VO
and V n be is deter-
mining for all hypocontinuous homogeneous polynomials o f degree strictly less than n . Let P be a hypocontinuous n-homogeneous polynomial which vanishes o n symmetric
n
Vo
and let
P.
linear form corresponding to
ization formula (theorem 1.5) we see that and vanishes on
Vo
x
Vo.
..
L
L
be the
By the polaris hypocontinuous
since
xV0
i=l,. . , n Fix
x
in
V O .
Then
Lx:E'
B
+ C
defined by
L(x,z, . . . , z) is an n - 1 homogeneous hypocontinuous polynomial which vanishes o n Vo. By our induction hypothesis Lx(z)
=
Lx : 0 . Now let y be arbitrary and let Ly:'L' + C be B B defined by Ly(z) = L(z,y, ...,y). Then Ly is a hypocontinuous linear form o n E' which vanishes o n V o , and hence by B
I n particular, we have Ly(y) = P(y) = 0 induction, o n E ' 6' for any y in E A . Hence V o is a determining set for hypoanalytic functions on U. Now let f E Hrjy(U). Since
E' B
=
lim ((En);,nn) n
is an open and compact surjective limit,
4-
it suffices to show that
f
factors through some (E;lB.If
were not true, then for each integer zn and zn+yn E U such that
F o r each
n
the function
z
-+
n,
f(z+yn)-f(z)
msn,
this
there exist
defines a non-zero
hypoanalytic function o n some convex balanced neighbourhood o f
32 1
Germs, surjective limits, E -products and power series spaces EB
zero i n
1
and hence t h e r e e x i s t s
x n ~ V o n 7 U such t h a t
f(xn+yn) # f ( x n ) . For a l l
n t m
Hence
gn w h i c h m a p s
i s a non-constant
f(xn+Xyn)-f(xn) gn(0) # gn(l),
the function
I
> n
it f o l l o w s t h a t
U.
Ixn+hnyn};=m
xn
to
C since
such t h a t
E (c
m
Since
v e r y s t r o n g l y convergent sequence and of
An
n 5 m.
for all
E
entire function,
a n d h e n c e we c a n c h o o s e
If(xn+hnyn)
h
(yn),=,
1
is a
for all
E V O ~ Y U
n
i s a r e l a t i v e l y compact s u b s e t
This contradicts the fact that
i s unbounded on
f
and hence f f a c t o r s t h r o u g h s o m e (E ) ' a n d {Xn+hnYn};=m nt3 f E H ( U ) . Now s u p p o s e (fa)aEr is a T bounded s u b s e t o f
We c l a i m t h a t
H(U).
(fa)aEr
f a c t o r s u n i f o r m l y t h r o u g h some
(E ) ' . S i n c e E; i s a compact s u r j e c t i v e l i m i t and (En)A is a n B space f o r each n t h i s would c o m p l e t e t h e p r o o f ( s e e
Jjgq
r e l a t i v e l y compact i n (fn):=1
t h e n w e c o u l d f i n d , as i n
I f t h i s were n o t s o ,
example 2 . 4 7 ) .
the f i r s t part of the proof,
m
a sequence
( X ~ + X ~ Y , ) , =w ~hich i s
and a sequence o f f u n c t i o n s
U
~ ( f a ) a E r such t h a t
Ifn (xn+anyn)
This contradicts the fact t h a t
I
is
(fa)aEr
n
>
for all
n.
bounded and
T
completes t h e proof. Corollary 6.26 H(U)'
' 0
'U
a
Proof
s p a c e s and
T
#
o,b
=
p(f)
open subset
= IIfllR
of
U T~
;I
T~
3'.
w,b
=
T
t h e n on
6'
i s a c o n t i n u o u s norm on T
0,b Since
; T ~ , ~ .
The r e q u i r e m e n t
6.25
T
is a s t r i c t inductive l i m i t of Frgchet nuclear
proposition 6.25 implies 5 . 4 6 show
i s an open s u b s e t o f
U
If <
of
=
T
w,b
on
T~
$nn=l
S'
a.
H(U)
Hence f o r any
examples 2.52
and
This completes t h e proof.
a c o n t i n u o u s norm on
i s n o t s u p e r f l u o u s as t h e example
corollary 5.35
=
and example 5 . 3 6 ( b ) ) .
i n proposition
E
shows ( s e e
C"XC(~)
N e v e r t h e l e s s i f w e do
n o t h a v e a c o n t i n u o u s n o r m w e may r e p l a c e
T~
by
T~
in
322
Chapter 6
proposition 6.25 and we obtain a similar result. Let
Proposition 6.27
E = lim (En,nn)
b e a n o p e n and
c-
n
c o m p a c t s u r j e c t i v e Zimit o f $ F W 2 o p e n s u b s e t of
The
E.
locaZZy b o u n d e d and
T
w,b
T~
= T
s p a c e s and l e t
bounded s u b s e t s of
on
6
U
be an are
H(U)
H(U).
As in the previous proposition we may suppose where U is a convex balanced open subset o f E. Let B be a T bounded subset o f H(U). .It suffices to show that B factors uniformly through some En. If not, Proof
U
=
n m- 1 (nm(U))
then w e can find a sequence in
u,
and
(Xn);=l
=
0
(Xn+Yn);=l’ for all
n.
B,
(fn)mnYl,
such that
two sequences in
fn(xn+yn)
# fn(xn)
By considering, for positive
and nn(yn) integers k and r , the T~ continuous seminorm Pk,r(f) = sup Id k f(0)(xn)r(yn)n-rl we s e e that Cd k f(0)JfEB factors n for each k . Hence we can find a uniformly through some E nk m sequence o f scalars, two sequences o f increasing positive integers,
for all
(R~):=~
and
(kn);=l
sucii t h at
n.
T h e semi-norm
m
on H(O), O E E, is continuous since the sequence (yL )n=l n is very strongly convergent and H(0) is a barrelled locally convex space. Since the canonical mapping (H(U),r,J H(0) is continuous it follows that p continuous semi-norm on H ( U ) . Hence sup p(f) w fE B +
T
is also a <
m.
323
Germs, surjective limits,E -products and power series spaces This contradicts the f a c t t h a t B
f a c t o r s t h r o u g h some
En
T
bounded s u b s e t s of
H(V),
s p a c e , a r e l o c a l l y bounded
p(f,
n
) > n
and i s a l s o
n.
Hence
bounded.
Since
for all T
an open s u b s e t o f a
V
(example 2.47),
23q
t h i s completes t h e
proof. I t i s worth n o t i n g t h a t even though p r o p o s i t i o n s 6.25 and 6 . 2 7 a r e s i m i l a r i n s t a t e m e n t ( b o t h h y p o t h e s i s and c o n c l u s i o n ) , q u i t e d i f f e r e n t methods are used t o g e t uniform f a c t o r izations. C o m b i n i n g p r o p o s i t i o n s 6 . 2 5 a n d 6 . 2 7 we o b t a i n t h e following r e s u l t . Proposition 6.28
-
E = l i m E
Let
n
n
be a s t r i c t i n d u c t i v e
l i m i t of F r z c h e t Montel ( r e s p . Frgchet Schwartz, Frgchet nuc l e a r ) s p a c e s . Then (H(E;I,T~)
-
(H(E;),T~,~= ) lim(H(En);) 9 ~ o ) n
=
i s a s t r i c t i n d u c t i v e Z i m i t of F r g c h e t - M o n t e 2
( r e s p . Frgchet
Schwartz,
F r g c h e t n u c l e a r ) s p a c e s and t h e
T
of
a r e ZocaZZy b o u n d e d .
(H(E;),.r6)
H(E)
E
c o m p l e t e and i f on
Moreover,
w
bounded s u b s e t s
a d m i t s a c o n t i n u o u s norm, t h e n
T~
H(EA).
Proof
I n example 2.47,
If
Frgchet space.
En
we s h o w e d t h a t
i s Montel
(H(En)b) , T ~ ) i s Montel
then
c o r o l l a r y 3.38 ion 6.9,
=
T
0,b
is a
Schwartz, nuclear),
Schwartz, nuclear), by
(resp. a modification o f the proof of proposit-
corollary 3.65).
Since limit,
(resp.
(resp.
(H(En);),~o)
is
E l
B
=
l i m (E f-
n
it f o l l o w s t h a t
) I
i s a n open and compact s u r j e c t i v e
l i m
(H(En)A),~o) i s a s t r i c t inductive
n B --f
n
324
Chapter 6 /
I i - m i t o f F r e c h e t s p a c e s and c o n s e q u e n t l y it i s c o m p l e t e .
As
T and t h e i n d u c t i v e l i m i t t o p o l o g y on H ( E i ) are both 6 b o r n o l o g i c a l a n d h a v e t h e same b o u n d e d s e t s , b y p r o p o s i t i o n
they are equal.
6.27,
This completes t h e proof.
Our f i n a l a p p l i c a t i o n o f s u r j e c t i v e l i m i t s i s t o h o l o m o r p h i c germs and i n t h i s example, w e do n o t assume t h a t t h e indexing set is countable. P r o p o s i t i o n 6.29
space a n d suppose
uous f u n c t i o n s on
d e n o t e t h e s e t o f a l l bounded c o n t i n By t h e T i e t z e e x t e n s i o n t h e o r e m
X.
i s a dense subspace
balanced open s u b s e t o f
w e can choose n.
be a completely regular Hausdorff is an infrabarrelzed 'ZocaZly convex
is a regular inductive limit of Banach is a compact metrizable subset of ,&(X).
,eb(X)
Let
,gb(X)
(X)
L.&
space. Then H ( K ) spaces whenever K Proof
X
Let
xn
E
of a ( X ) . and
&(X)
VnAb(X)
If
V
(fn);=l
i s a convex
CH(
(X))
fn(xn) # 0
such t h a t
then
for all
By t h e i d e n t i t y t h e o r e m f o r h o l o m o r p h i c f u n c t i o n s o f o n e m
complex v a r i a b l e , choose a sequence o f s c a l a r s ( A ~ ) ~s u = ch ~ xnIIX 6 1 a n d fn(h,xn) # 0 f o r a l l n . Hence that IX,I.ll h,x,
as
+ 0
n+m
in
&(X)
and
o s i t i o n 6 . 2 i t s u f f i c e s t o show
fn(Xnxn)
#
0.
By p r o p -
i s r e g u l a r t o complete
H(0)
the proof. Let
B
b e a bounded s u b s e t o f
H(0)
and l e t
I t o b v i o u s l y s u f f i c e s t o show t h e r e e x i s t s a n e i g h b o u r h o o d o f z e r o i n &(X) let
Wp
=
for all
{xEX;
f,g
such t h a t
E
Vx
supIIFII,, F Eg open i n
h(X1 S u p p o r t ( g ) C V x
P(f+g) # P(f)}.
<
X,
and
a.
xcVx
For each
P
there exists
in
V N
B
325
Germs, surjective limits, e -products and power series spaces
We c l a i m
W
-
o f open s u b s e t s o f n
AnOK
and i f
#
Wpnn A n # + .
(gn)n=19
such t h a t
Pn€ B
&(X),
Pn(fn+gn) # Pn(fn)
h(X)
is dense i n
n.
for all
and
we may s u p p o s e
By t h e i d e n t i t y t h e o r e m f o r h o l o m o r -
p h i c f u n c t i o n s and s i n c e e a c h s e q u e n c e i n ,fib(X) w e a k l y c o n v e r g e n t , we may a l s o s u p p o s e sequence i n
Since support ( g n ) C A
is very is a null
such t h a t
-
n
n
all X
Ann K
and
and a l l
n
=
+
f o r any
sufficiently large
is very strongly convergent t o zero.
(gnInzl
L i o u v i l l e ' s theorem,
of
K
m
t h e sequence 1
(fn)mnZ1
(XI.
g i v e n compact s u b s e t
( Bn =:
(fn)iXl,
n.
for all
S i n c e fS.,(X) f n ~.fi,(X)
&
there exists
such t h a t
(gn)CAn
for
then
X
Hence t h e r e e x i s t s e q u e n c e s i n
a,
support
AnnW # $
such t h a t
sufficiently large.
AnnW # $
Since
and
n
for all
$
(An):=l,
X,
i s a n y compact s u b s e t of
K
If n o t ,
X.
t h e r e e x i s t s a sequence
is infrabarrelled,
t h e n s i n c e ,&(X) all
.v
{ U W p ; P E B } i s a compact s u b s e t o f
=
By
we may c h o o s e a s e q u e n c e o f s c a l a r s I P , ( f , + ~ ~ g ~ )> l n
for all
b e t h e d e g r e e o f t h e homogeneous p o l y n o m i a l
n.
Pn.
Let
kn
If
sup kn < t h e n t h e r e e x i s t a p o s i t i v e i n t e g e r N and a n sequence of integers n such t h a t kn. = N f o r a l l j . j 1 Hence (pn. i s a bounded s u b s e t o f H(0) and consequently I a bounded s u b s e t of L
=
i s a c o m p a c t s u b s e t o f $(X)
sUpllPn.IIL < 1
( (PcNE) , T ~ ) .
J
m.
The set
(fn+Bngn);=l
and hence
This contradicts the fact t h a t
for all
j.
(nj)y=l
such t h a t
On t h e o t h e r h a n d ,
u { O }
j
\IL>
nj
s u p kn = then w e can n choose a s t r i c t l y i n c r e a s i n g sequence of p o s i t i v e i n t e g e r s kn.> I
kn
j -1
if
IIPn
for all
j .
The semi-norm
326
Chapter 6
H(0).
is a continuous semi-norm o n
Since
all j we again get a contradiction. subset o f X . We now claim that each rv
F E B subset L If
Let
X
of
f,g
E
LJ
lim & ( K )
W
>
n . for
I is a compact
factors through &(W).
B
factors through h(L) since
for some compact
is an open surjective limit.
t-
KCX
a(X)
W.
hood o f
F
then
in
F
Hence
p(P,,) I
and s u p p o s e
g
vanishes on some neighbour-
h l € &(X)
Choose
such that h l is equal t o 1 W and support ( h l ) C { x ; g ( x ) = 0 1 . If X E L\Vl then there exists a neighbourhood Vx o f x such that F(fx+gx) = T ( f x ) for any fx,gx E ,@,(X), support ( g X ) c V x . Choose h x E & ( X ) such that h x ( x ) = 1
o n a neighbourhood
and support
V1
(hx)CVx.
u XE
contains
of
WuL
L\V
Let
-
TXhr
V]
Vx
N
k(x)
such that
(y€X;hx(y)>
xl,
and hence there exists
=
2 -
x d V1
all
1
lu
...
N
W c r L C V l u V X 1 d Vx
such that
=
.
uVx
L,
Vx
1
identically z e r o o n some neighbourhood of n = k ( x ) + h l ( x ) + 1. h (x) for every 1 = 1 xi 1 for every x in X . and I k l ( x ) / 2 -
kl(x)
. . . ,xn rv
uVx
n KuL.
X E
2
Y
k = k/
Let i=l,
. . . ,n . N
hl
= hl/kl
Now A
k + hl +
and
kl
A
,
N
I:= hi,E
1
on X
and
d
hi
E
T h e set
L\V1 k
Now choose
...
r4
1 T}.
= hX
X.
and
k
(Xj
E
is
Let kl
E
(X)
321
Germs, surjective limits, €-products and power series spaces
(since
, d
is identically zero o n a
k
K u L)
neighbourhood of
(since
A
support (hl)
CIx,g(x)
=
support (h 1 )
= 0))
F(f)
=
(since
PJ
support ( h i g ) C V x
i = l , . . . ,n) . Now suppose
vanishes o n
g
W
choose
1
o n some neighbourhood of
of
V.
hV
Since
converges to
E
&(X)
W.
V
For each neighbourhood
of
Hence
I(hVII'<1, hV is identically W and h V : 0 o n the complement the net hVg converges to zero as V F(f+g)
vanishes on a neighbourhood of through
for
such that
gIK : 0 K.
W.
i
=
K.
lim F(f+hVg)
V-t K
Thus each
=
F
F(f) E
%
as
hVg factors
Since
lima(X) is a compact surjective limit and KCX each To-bounded subset ofH(.&(W)) is locally bounded, this completes the proof. t-
§6.3
-PRODUCTS
In this section, we use &-products to study holomorphic functions defined on a product o f open sets and to extend results concerning scalar valued holomorphic functions to vector valued holomorphic functions.
When t h e d o m a i n s p a c e s
328
Chapter 6
have t h e a p p r o x i m a t i o n p r o p e r t y ,
the
E.-product
i s a tensor
product. Let
and
E
t h e dual of
E i
be l o c a l l y convex spaces.
F
w i l l denote
endowed w i t h t h e t o p o l o g y o f u n i f o r m c o n v e r -
E
gence on t h e convex b a l a n c e d compact s u b s e t s o f
E.
c l o s e d convex h u l l o f each compact s u b s e t of
i s compact
( f o r example,
if
i s quasicomplete)
E
E
If the
t h i s topology coincides
w i t h t h e compact open t o p o l o g y . D e f i n i t i o n 6.30
Let
E
F
and
be l o c a l l y convex spaces.
E E. F
i s t h e s p a c e o f a l l c o n t i n u o u s l i n e a r mappings f r o m
into
F
endowed w i t h t h e t o p o l o g y o f u n i f o r m c o n v e r g e n c e o n
t h e e q u i c o n t i n u o u s s u b s e t s of E ' . by
E:
We a l s o d e n o t e t h i s s p a c e
&(E;,F). I t i s n o t d i f f i c u l t t o show
QE;;F)
Y J!JF;;E). A %E(Ei;F) i s g i v e n by
f u n d a m e n t a l s y s t e m o f semi-norms on
where
ranges over
a
If
U
cs(E)
and
i s a n open s u b s e t of
over
B
(In a n d
cs(F). F
i s a quasicom-
p l e t e l o c a l l y convex s p a c e , t h e n t h e mapping from into
H(U)E F
f*(v)
= vof,
which t a k e s
f
to
H(U;F)
f*:F;--9H(U),
i s a l i n e a r isomorphism and w e have
H(U;F)
=
H(U) E F . Our aim i s t o e x t e n d t h i s f o r m u l a t o i n f i n i t e d i m e n s i o n s u s i n g t h e same m a p p i n g .
Our f i r s t e x a m p l e shows, h o w e v e r ,
that
t h i s may n o t a l w a y s b e p o s s i b l e . Example 6 . 3 1 s p a c e and l e t
Let f
E
be an i n f i n i t e dimensional Hilbert
b e t h e i d e n t i t y mapping from
S i n c e t h e compact open t o p o l o g y on t h a n t h e weak t o p o l o g y
f p! H ( E ; ; E i ) .
E'
E;
onto
E;.
is strictly finer
We s h o w t h a t
f*
is in
329
Germs, surjective limits, E -products and power series spaces
E EA
H(E:)
ed.
when
i s g i v e n t h e compact open t o p o l o g y .
H(E,')
Since
f*(v) = vof = v a ( E ; ) ' C H ( E ' ) ,
Since
(Ed):
from
into
(E;);
Lemma 6 . 3 2
space
= (Ed):,
(H(E;)
U
Let
be a n o p e n s u b s e t of a ZocaZly c o n v e x -+
(HHy(U),~o):
6(x)(f) = f(x).
mapping g i v e n by
i s well d e f i n -
,r0).
6 : U
and l e t
E
f*
i s a l s o a c o n t i n u o u s mapping
f*
denote the evaZuatim
Then
A)'
HHy(U;(HHy(U)>ro) Proof and
The l o c a l l y c o n v e x s p a c e ((HHy(U) ,ro)A) 6(X+AY) ( f )
for fixed function and
6
x,y
and
A
n
and a l l
f
(y)
E
s u f f i c i e n t l y small, t h e
i s holomorphic a t t h e o r i g i n i n
(c
i s a G-holomorphic mapping. b e a compact s u b s e t o f
K
C {f
HHy(U); /lfIIK 6
1 } O
equicontinuous subset of on
is complete
Since
2 f ( x ) n!
&(x+Ay)(f)
A +
Let
6(K)
I",=,
=
(HHy(U),ro)
HHy(U).
=
induced by
6(K)
I
(HHy(U),ro):
Clearly
ogy on
a n d t h e weak t o p o l o g y o n
K
Since
it follows t h a t
(HHy(U),r0)'
topology.
6
U.
6(K)
is an
and hence t h e topology
i s e q u a l t o t h e weak
is continuous f o r the i n i t i a l topol-
(HHy(U),ro)'.
Thus
6IK
i s continuous and t h i s completes t h e p r o o f . Proposition 6.33
convex space
E
convex space.
g i v e n by
6*($)
Let and Z e t
u
be an o p e n s u b s e t of a ZocaZZy F
be a q u a s i - c o m p l e t e
ZocaZZy
The mapping
= $06
i s a c a n o n i c a Z i s o m o r p h i s m of ZocaZZy c o n v e x s p a c e s and h e n c e
Chapter 6
3 30
Proof
By lemma 6 . 3 2 ,
i s well d e f i n e d and i t i s
6"
o b v i o u s l y l i n e a r and i n j e c t i v e .
We now s h o w t h a t
surjective.
We d e f i n e
f
Let
by t h e formula
E
HHY(U;F).
v(f*(w))
(HHy(U),~o)'.
If
w(vof)
=
i s a compact s u b s e t o f
K
f(K),L,
Hence
f o r every
( ( v o f / I K6 \ ( v ( \ v
E
F'
Thus
B E cs(F)
then
i s well defined.
f*
If
i s a r e l a t i v e l y compact s u b s e t o f
B(f*(w)) = 1/w/Iv Moreover,
v(B*f*(x)) and s o
and
f o r any
x
f* in
w
F'
f*(w)
is
(F;)'
E
=
F
( H H y ( U ) , ~ o ) . Hence
i s a c o n t i n u o u s l i n e a r mapping. U
= v(f*(d(x)))
6*f* = f .
F.
and f o r f i x e d
F'
in
v
endowed w i t h t h e c o m p a c t o p e n t o p o l o g y . and
then the
U
i s c o n t i n u o u s when
w(vof)
-f
and w i n
VEF'
i s a compact s u b s e t o f
c l o s e d convex h u l l o f t h e mapping
for every
is
6"
and =
v
in
F'
B(x)(vof)
= vof(x)
I t r e m a i n s t o show t h a t
is a topol-
6*
o g i c a l isomorphism. Let all
in B
K
b e a compact s u b s e t o f
hEHHY(U),
and l e t If
(HHy(U),~o)'. unit ball in
f o r any
f
and
F',
f*
6
V E
U,
let
a(h) =
be t h e polar of the
cs(F)
and
W
a
/Ih
/IK
unit ball
is the polar of the
then
as d e f i n e d above.
This completes theproof.
33 1
Germs, surjective limits, e -products and power series spaces Proposition 6.34
u
Let E
ZocatZy c o n v e x s p a c e s
V
and F
and
be o p e n s u b s e t s of t h e respectively.
HHy(UxV) = HHy(U) & HHy(V) = H H y ( U ; H H y ( V ) )
topologically
Then
~ Z g e b r a i c a l Z y and
l e a c h f u n c t i o n s p a c e is g i v e n t h e c o m p a c t o p e n
top0 logy I . Proof
Since
(HHy(V),~o) i s quasi-complete,
6.33 implies that f
E
H H Y ( U xV ) .
proposition
HHy(U) g HHy(V) = H H y ( U ; H H y ( V ) ) . IJ
We d e f i n e
N
f(x)(y) = f(x,y).
f : U + HHy(V)
Now l e t
by t h e formula
By u s i n g t h e C a u c h y i n t e g r a l f o r m u l a o n e
s e e s t h a t t h e mapping
n! i s hypoanalytic f o r any f i x e d
negative integer
n.
xo
in
U,
and any non-
XEE,
Hence t h e f u n c t i o n
n! belongs t o
Pa(”~;~,,(v)).
f o r any fixed
xo
in
Since
x
U,
in
E,
s u f f i c i e n t l y small it f o l l o w s t h a t Now l e t
K
respectively. continuous. a.
for all
a I a
0
E
IK
x
I c
and a l l
h
HG (U;HHy(V)).
U
and
V
i s c o n t i n u o u s i t is u n i f o r m l y
~ L K + x
E
f o r a given
- f(X3Y)
f
and
b e compact s u b s e t s o f
L
Hence i f
such t h a t If(Xa’Y)
and Since
ycV N
r1>0
as
a-m
we h a v e
Tl
y
in
L.
Thus
then there exists
Chapter 6
332
and
r*
f
E
HHy(U;HHy(V)1
Now s u p p o s e t h e formula
K
Let ively.
Let
and u
a
b e compact s u b s e t s of -+
u
and
v BE L +v
i s continuous and
g(u)\
HHy(UxV)
HHy(UxV) with
as
g(u,)
u n i f o r m l y on t h e compact s u b s e t s o f Hence
on
UxV
V
respect-
by
= g(u)(v).
L
E K
We d e f i n e
HHy(U;HHy(V)).
z(u,v)
Then
since
g
.
+
U
and
respectively.
a , B+m
g(u)
as
V.
a n d we may a l g e b r a i c a l l y i d e n t i f y
HHy(U;HHy(V)).
t h i s i s a l s o a t o p o l o g i c a l isomorphism and c o m p l e t e s t h e p r o o f . Corollary 6.35
c o n v e x s p a c e s and let space.
If
U,V
U
Let
and
F UxV
and
V
be o p e n s u b s e t s of localZy
be a q u a s i - c o m p l e t e a r e k-spaces,
then
locally c o n v e x
333
Germs, surjective limits, E -products and power series spaces
Corollary 6 . 3 5 applies if
U
and
V
are both open subsets of
FrGchet spaces or both are open subsets of our next proof, we use the fact that E C F
U83mA = E
In
spaces.
0, F
if
E
is
a locally convex space with the approximation property (see Appendix 11). As our first application of the 6 product, we prove a converse to theorem 5 . 4 2 .
-r0 =
T
6
E
If
Theorem 6 . 3 6 and
on
i s a F r z c h e t n u c l e a r s p a c e w i t h a basis
H(E)
E
then
is a
DN
space.
m Proof Let (enIn=1 be an absolute basis for F be the closed subspace of E spanned by
E m
and let Since
F and E , C and F are Frgchet nuclear spaces, an application of Corollary 6 . 3 5 shows that E =
If
(c x
on
= -r0
eLF1
T&
H(C)
H(E)
then the closed complemented subspace
of H ( E ) is a bornological space. By proposition 15, chapter 2 of A. Grothendieck's thesis, this implies that
F
contains an increasing fundamental system o f weights m
(Wm)m=l'
w m = (wm,n)
m
SUP(lXn(W
n
such that )€
m,n
W m,n
W
l'n
is finite for every positive integer m , all E>O and all 1 (X&l in F . Letting p = and taking pth roots we see that
Chapter 6
334 for a l l positive integers
m
and
F
and p r o p o s i t i o n 5.40 i m p l i e s t h a t
and a l l
p
(x~);=~ in
F.
Hence
i s a c o n t i n u o u s weight on
F
is a
that
space.
DN
is also a
E
Since
s
= C x F C s x s
E
t h i s means
s p a c e and completes t h e p r o o f .
DN
Theorems 5.42 and 6 . 3 6 t o g e t h e r g i v e t h e f o l l o w i n g : E
I f =
T o
is a F r g c h e t n u c l e a r s p a c e w i t h a b a s i s t h e n on
T6
if and o n l y i f
H(E)
E
is a
DN
space.
F o r o u r n e x t a p p l i c a t i o n , a k e r n e l s theorem f o r a n a l y t i c
functionals on c e r t a i n f u l l y n u c l e a r spaces, w e need a f u r t h e r type of tensor product. E G F
s p a c e s we l e t E@F
E
If
F
and
are l o c a l l y convex
d e n o t e t h e c o m p l e t i o n of t h e v e c t o r s p a c e
endowed w i t h t h e t o p o z o g y o f uniform c o n v e r g e n c e o n t h e
s e p a r a t e l y e q u i c o n t i n u o u s s u b s e t s of E x F.
c o n t i n u o u s b i z i n e a r forms o n Proposition 6.37
complete nuclear
Proof
Since
nuclear
DN
(H(V),T~)
If DN
U,V
spaces, and
U
t h e s e t of a l l s e p a r a t e l y
and
V
a r e o p e n p o l y d i s c s in
spaces w i t h a b a s i s ,
and
U x V
then
a r e open polydiscs i n complete
theorem 5.42 implies t h a t
(H(UXV),T~)
(H(U)
, T ~ ) ,
are f u l l y nuclear spaces.
By
corollary 6.35,
Since 2,
(H(UXV),T~)A
of A .
i s complete,
the corollary p.91,
Grothendieck's t h e s i s implies
chapter
335
Germs, surjective limits, e-products and power series spaces
This completes the proof. For
a3rL
spaces, the situation is much simpler and the
following result is easily proved:
Using techniques similar to the above, a more detailed analysis o f the E - p r o d u c t o f 8 2 2 s p a c e s and propositions 6.9 and 6 . 1 8 one may prove the following results: Proposition 6.38 If K 1 and Frgchet Schwartz spaces, t h e n
If U
Corollary 6 . 3 9 Schwartz spaces,
As
V
a r e c o m p a c t s u b s e t s of
a r e o p e n s u b s e t s of F r z c h e t
then
a further corollary, w e improve corollary 6.11 in the
special case o f
cN.
Corollary 6.40 T
and
K2
=
T
on
If
u
is a n o p e n s u b s e t of
cN
then
H(U).
Proof -
By analytic continuation, it is possible to find a pseudo-convex domain i2 spread over (c", n a positive
integer, and a n embedding o f U in 0 X Q N such that each holomorphic function on U has a unique extension to a holomorphic function o n
Q x
CN
and moreover,
336
Chapter 6
Hence N
( H ( U ) , T u ) '=" (H(R
cf ( H ( R
x
2 (H(R)
1,
Tu)
( e x a m p l e 6.13)
Q: N ) , T ~ ) (H(C
, T ~ )
N),T~) N
( H ( R ) , T ~& ) ( H ( Q :1 2'
(H(R
2'
(H(U)
x
aN) , y o ) ,T
~
(corollary
, ~ ~( e)x a m p l e
5
(corollary 6.35)
.)
This completes t h e proof
POWER SERIES SPACES O F INFINITE TYPE
86.4
I n c h a p t e r f i v e , we showed t h a t s p a c e s o f h o l o m o r p h i c f u n c t i o n s on c e r t a i n n u c l e a r s e q u e n c e s p a c e s c o u l d t h e m s e l v e s
In t h i s s e c t i o n we
be represented a s nuclear sequence spaces.
s t u d y t h e s e q u e n c e s p a c e s t h a t a r i s e when we c o n s i d e r h o l o m o r p h i c f u n c t i o n s on t h e s t r o n g d u a l s o f power s e r i e s s p a c e s o f i n f i n i t e type.
We b e g i n b y r e c a l l i n g t h e d e f i n i t i o n o f power
s e r i e s s p a c e o f i n f i n i t e t y p e and i n t r o d u c i n g some n o t a t i o n .
An i n c r e a s i n g s e q u e n c e numbers w i t h
~f
1 7 <
W
q n
l i m an = n-tm
-
then
a
'n
integer
positive real
q,
0
or e q u i v a l e n t ~ y
log(n+l) < a
n
wiZ.1 b e c a l l e d a n u c l e a r e x p o n e n t s e q u e n c e .
sup a2n < * n
of
w i l l b e c a l l e d an exp o n en t s eq u en ce.
f o r some SUP n
W
(an)n=l
a =
k
o r equiuatentty
sup n
kn
n
t h e n t h e exponent sequence
a s t a b l e exponent sequence.
<
m
I f
f o r any p o s i t i v e
a =
The s e q u e n c e s p a c e
03
i s called
337
Germs, surjective limits, €-products and power series spaces m
co
Am(a) = { ( x ~ ) ~ = ~ ; l x n l P a
n
<
O < p < m ) endowed
m,
w i t h t h e topoZogy generated by t h e norms
i s c a l l e d a power s e r i e s s p a c e o f i n f i n i t e t y p e .
It i s a
F r z c h e t s p a c e w i t h a b a s i s and i s n u c Z e a r i f and onZy i f
a
i s
a nuclear exponent sequence. i s a s t a b l e exponent sequence i f and o n l y i f Am(.) x Am(a). If a and a are exponent sequences
a
Am(a)
=
then
A a ( a ) 'g Am(;) i f and o n l y i f t h e r e e x i s t s D b 1 such 1 for all n. I n t h i s s i t u a t i o n we s a y a n I a n s Dan
that a
and
are equivalent.
CL
Let
M = N").
m = (m.)w 1 J=1
(aim)
=
and d e n o t e by
a
1j E N a j m j an i n c easing r e a
6 = &(a)
((a J m ) ImEE.,. L e t
family
6, = ( a l d ( n ) )
i s an exponent sequence and
we l e t
M
E
If
n
f o r each
The s e q u e n c e
d:N+M
&(a)
in
ingement o f t h e
be a b i j e c t i o n such t h a t N.
i s caZled t h e a s s o c i a t e d sequence o f
a. A rephrasing of
Proposition 6.41
a basis.
where
theorem 5.21 y i e l d s t h e following r e s u l t .
Let
A(P)
b e a fuZly n u c l e a r s p a c e w i t h
The mapping
M = N"),
am(f)
monomial e x p a n s i o n o f
i s t h e c o e f f i c i e n t of f
t o p o l o g i c a l isomorphism.
and
PM
=
zm
i n the
{ ( pm ) m E M ; p ~ P Ii ,s a l i n e a r
338
Chapter 6
Corollary 6.42
Let
a s s o c i a t e d sequence
Proof
Since
HHy(Am(a)A).
Hence
&(a) = 6 .
Am(a)'
a
Then
is a
3312.
space
H(Am(a)i) =
By p r o p o s i t i o n 6 . 4 1
(H(Am(a)A),rO)
theorem 5.11,
be a n u c l e a r exponent sequence w i t h
a
A,(6)
a n d we h a v e a l r e a d y n o t e d ,
that t h i s i s a nuclear space.
in
This completes
t h e proof. A s a c o r o l l a r y , we o b t a i n a Cauchy-Hadamard
formula f o r
e n t i r e f u n c t i o n s o n power s e r i e s s p a c e s o f i n f i n i t e t y p e . Corollary 6.43
Let
of i n f i n i t e type.
Then
Our a i m i s t o d e s c r i b e a.
A,(a)
b e a n u c l e a r power s e r i e s s p a c e
&(a),
up t o equivalence, f o r v a r i o u s
T h i s t o p i c h a s been t h e o b j e c t o f r e c e n t r e s e a r c h and a
good d e a l o f i n f o r m a t i o n i s known. e s t i n g and t e c h n i c a l .
The p r o o f s a r e b o t h i n t e r -
We c o n f i n e o u r s e l v e s t o a d e t a i l e d d e s -
c r i p t i o n o f a p a r t i c u l a r c a s e which i s t y p i c a l o f t h e g e n e r a l s i t u a t i o n a n d c o v e r s many o f t h e c l a s s i c a l s p a c e s . look a t a particular situation.
We f i r s t
339
Germs, surjective limits, E -products and power series spaces J
Proposition 6.44 Proof
(H(si),.ro) = s.
Since
where
= hm(u)
s
6.41 implies t h a t
= log(n+l), proposition n is isomorphic t o a nuclear
(H(s;),.ro)
power s e r i e s s p a c e
Am(6).
By n u c l e a r i t y
sup n
<
m .
Since
c
m
~
~
I
~
=
~
c
6n
t h e r e e x i s t s a s t r i c t l y i n c r e a s i n g sequence of p o s i t i v e integers k(n) 3 n
{k(n)}",l for all
such t h a t n
= 6k(n)
'k(n) log (n+l)
,<
log(n+l)
T h i s means t h a t
and
=
-" n
a
i
(log(n+l));=l
i s a s t a b l e exponent
following r e s u l t
(see also corollary 6.26).
We d e n o t e b y
and,
w e o b t a i n as a c o r o l l a r y t h e
Y s
t h e cardinality of the set
IAl
D e f i n i t i o n 6.46
N(t)
1.
This completes t h e proof.
~8
[O,m)
Hence
a r e e q u i v a l e n t sequences and
6
Using t h e f a c t t h a t
E
n.
n
sequence and t h a t
t
for all
and
'n
hW(a) = s .
U,
Let n
A.
b e an e x p o n e n t s e q u e n c e .
CY.
If
i s a positive integer, w e l e t
=
sup Nn(t) n
N-(t)
=
ICmcN(N); (ulm) < t } \ .
Since
un+
+m
as
=
n + m
I I r n c N ( N ) ; ( a ) m ) 6 t}/ a n d
it follows t h a t
N(t)
is
{
G
~
~
340
Chapter 6
finite for all
Ni)
and it i s a l s o immediate t h a t
t
i s c o n t i n u o u s from t h e r i g h t
Lemma 6 . 4 7
sequence
Let
be a n e x p o n e n t s e q u e n c e w i t h a s s o c i a t e d
a
i f
n ,< N ( t )
fc)
i f
N-(t)
N-.
6n
then
< n
,< t ,
t c 6n
then
.
(a) f o l l o w s i m m e d i a t e l y f r o m t h e d e f i n i t i o n o f N(t) 3 n
(b) If
Since
6,,
t h e increasing rearrangement of t h e s e t that
N-(t) < n
(el If
6 n 6 t .
elements i n
with
M
definition of
We s h a l l c a l l
N
Proposition 6.48 H:R+
R+
+
n
A s i n fb), i t f o l l o w s , b y t h e
n 2 t. a
b e an exponent sequence w i t h
a
Let
be a c o n t i n u o u s s t r i c t l y i n c r e a s i n g f u n c t i o n such
that
H(t)
f o r some p o s i t i v e
Q N(t) B
,< H ( B t )
and a l l
t
sufficiently large.
Then
are equivalent sequences.
("):=1
Proof some
term i n
((alm))mEM it follows
and n u m b e r i n g f u n c t i o n N .
6
and
elements i n
n th
t h e numbering f u n c t i o n of
Let
a s s o c i a t e d sequence
6
is the
N
then t h e r e a r e l e s s than
(alm) < t .
that
6,
n
then there are a t least
(alm) 6 t .
with
M = N
n.
left) for all
The f o l l o w i n g a r e t r u e :
6.
fbi
Proof
(resp.
(resp.
Nn
-
l i m 6n = w e have f o r a l l k larger than n+m 26k-1 3 6k-1+a1 = 6 2 dk. For each p o s i t i v e
Since ko E N ,
integer
n
k(no) > ko.
let If
j k(n) = i n f { j ; 6 . = 6 1 . n 2 n
then
J
n
Choose
no
such t h a t
341
Germs, surjective limits, €-products and power series spaces
( b y lemma 6 . 4 7 ( a ) ) 6
H - ~ is increasing)
(since
2H-'(n)
On t h e o t h e r h a n d H
-1
(n) 6 H
-1
( b y lemma 6 . 4 7 ( a ) )
(N(6,))
This completes t h e proof. We now c o n s t r u c t a f u n c t i o n w h i c h s a t i s f i e s t h e c o n d i t i o n s
of p r o p o s i t i o n 6 . 4 8 and u s e it t o e v a l u a t e & ( a ) Lemma 6.49
function
Let
=
for all
t
be a n e x p o n e n t s e q u e n c e w i t h numbering
and l e t
N
F(t)
a
for certain u .
sup n al E
R'.
tn
...
arm! F ( t ) 6 N(t)
Then
t
for a l l
sufficiently
large. Proof
Let
n
lj,lajxj
M n ( t ) = {xeR;;
t}.
Now
Nn(t)
is the
number o f p o i n t s w i t h i n t e g e r c o o r d i n a t e s c o n t a i n e d i n Since any c u b e . w i t h edge l e n g t h one i n follows t h a t
Volume(Mn(t))
6
Nn(t).
that V o l u m e (Mn ( t ) )
N(t) = sup Nn(t) n
Lemma 6.50
increasing.
Let
Mn(t). h a s volume one it
An e a s y c a l c u l a t i o n s h o w s
tn
=
a1
Since
Rn
...
u n!
n
t h i s completes t h e proof. co
a = (YnnP~n,l
Then t h e r e e x i s t s
where
D>O
p>O
such t h a t
and
y
i s
342
Chapter 6
N ( t ) 5 F ( D t ) l'p
for a l l
t
sufficiently l a r g e .
A = {jl, . . . , j n}
is a f i n i t e s e t of positive
Proof
If
integers,
t h e n t h e m e t h o d o f t h e p r e v i o u s lemma s h o w s t h a t
I{mcM;
and
support(m) = A
(alm) 5 t
1I
tn
I
a. J1
...
a. n !
'n
By d e f i n i t i o n
1{ j , ,..., j n 3
1 n!
n Ik=pj 6t k (the extra
n!
1 a. '1
...
a.
I
Jn
a r i s e s from t h e permutations of
For each integer
n
let
An(t)
=
{ j l,...,jn})
a. 5 t } Jk
{ j E N n ;
and l e t
With t h i s n o t a t i o n
We may a s s u m e , i f n e c e s s a r y b y r e p l a c i n g
yn
by
y;
= ynn9,
Germs, surjective limits, e -products and power series spaces 1
2 :: P
that
Cn(t)
=
E
N.
1
n!
343
Hence
cj€An(t)
1 a.
J1
...
( s i n c e t h e sequence
a. 'n
( Y ~ )i ~ s increasing)
where B n ( t ) = {xER";
since (**)
Bn(t)
(*)
x. 3 0 1
for
lsjsn,
l nj Z ly 3. x p3
6 t},
may b e i n t e r p r e t e d a s t h e l o w e r R i e m a n n sum o f
corresponding t o t h e p a r t i t i o n given by t h e p o i n t s i n with integer coefficients. In o r d e r t o e s t i m a t e t h i s i n t e g r a l , w e first introduce t h e
coordinates
Sk = t -1 ykx;,
16kSn,
We now p r o v e b y i n d u c t i o n t h a t
and w e g e t
344
Chapter 6
n=l.
Obviously this formula holds for
J
A(1-S)
'0
=
--
Since
Jol
Bn
1 [( --2) 1
[n(-
P
--2 Sp
*
1
( I - S)
!In
(
n(- - 1 )
dS
1 - 2)! [n( 1 - l)] !
P
1 [(n+l)(--
- l)]!
P
I)]!
(by the induction hypothesis)
1
-
" - P-
2)!1
n+1
1 [(n+l)( - -
P
I)]
!
It f o l l o w s t h a t t h e f o r m u l a h o l d s f o r a l l
n.
Hence
Germs, surjective limits, €-products and power series spaces
w h e r e we l e t
We now h a v e
b
l
+
1 y n = l 2n
(since
"1 1 F (2Pct)P 1 -
=
1 + F(2Pct)P
To complete t h e p r o o f , F(ZP+'ct) for a l l Let
t
2
1 $
(1 + F(2'ct))'.
we show t h a t
1 + F(2'ct)
sufficiently large. t
be any r e a l number.
Choose
no
such t h a t
345
Chapter 6
346
F(2P+1ct)
-
=
n
(2%)
O
...
a1
n !
a "0
O
Then F(2P+1ct)
sup
=
2n ( 2 P c t y
n
...
al
n
2 O(2Pct)
::
...
al
n
0
n
o
!
2 F ( 2 P c t ) :: 1 + F ( Z P c t )
2
since F(t)
c1
n
n !
c1
for
t 5
t
2
c1
1
-
a1
This completes the proof P r o p o s i t i o n 6.51
m
a = ("n)n=l
Let
and Z e t F(t)
t
for a l l
E
sup
=
R'.
b e an e x p o n e n t s e q u e n c e
tn
al
...
a n!
n
T h e e x p o n e n t s e q u e n c e associated w i t h
i s equivaZent t o t h e sequence
( F - l (n)):=,
i f
a
a,&,
satisfies
e i t h e r of t h e f o l l o w i n g e q u i v a l e n t conditions:
a)
c(
is s t a b l e a n d t h e r e e x i s t s
(ann-P)EcN sequence;
bi
there exists a positive integer 1
Proof
( a n "-p);=l
inf n
"kn __ u
p>O
s u c h that
is e q u i v a l e n t t o a n i n c r e a s i n g
6
n
F i r s t suppose
%n sup n a n
k
sueh t h a t
m .
(a/ i s s a t i s f i e d .
By h y p o t h e s i s m
i s equivalent t o an i n c r e a s i n g sequence (yn)n=l 1 a n d we may s u p p o s e 2 $ - E h i . Hence t h e r e e x i s t s B > 0 such P
341
Germs, surjective limits,E -products and power series spaces 1 a ,< an f B a n where a = ynnP for all B n n N b e t h e numbering f u n c t i o n s o f a and a
that
and
n for a l l
t c [O,
a1
...
co
(Bn)n=l
N
a n!
n
there exists 1 -
If
Let
respectively
m).
By lemma 6 . 4 9 , N(t)
n.
5 N ( B t ) i F(BDt)'
D> 0
such t h a t
1 6 F(B2Dt)'.
i s a s t a b l e e x p o n e n t s e q u e n c e , t h e n f o r some
we have B2n = sup n Bn
c
<
C>O
00
and h e n c e
By i n d u c t i o n o n n2
k
w e have
k
2k
k+l j =1
for all p o s i t i v e i n t e g e r s
n
and
k.
s t a b l e exponent sequence, so a l s o i s
Since m
m
( u ~ ) ~i s= a~ and hence,
a p p l y i n g t h e a b o v e i n e q u a l i t y t o t h i s s e q u e n c e , we s e e t h a t there exists
C1>
0
such t h a t
3 48
Chapter 6
for all
t
R‘,
E
al
.. .
n
and
a
n2
k ( n 2k ) !
k
positive integers
Hence k
tn n
for all
t
k,
P ’
2
>
al
R+
E
...
n
n!
cil
. ..
and e v e r y p o s i t i v e i n t e g e r
and a l l
By p r o p o s i t i o n
ct
6.48,
t
c1
k.
n2
k k(n2 ) !
Thus f o r a l l
sufficiently large
m
(an)n=l
and
(F
-1
m
(n)ln=l
are equivalent
sequences. We now s h o w t h a t Suppose (a)
( a ) and
is satisfied.
(b) a r e equivalent c o n d i t i o n s .
For any p o s i t i v e i n t e g e r s
w e have
Now c h o o s e
q
such t h a t
> 1
B
and t h u s
n
and
q
3 49
Germs, surjective limits, E -products and power series spaces
and t h i s shows t h a t If
(a)
-
is satisfied, then t r i v i a l l y
(b)
sequence.
By h y p o t h e s i s ,
and
such t h a t
A > 1
Hence
a
kj
> hj.al
k j , < n , < kj",
where
p =
Let show t h a t
(b).
is a stable
there exist a positive integer
a k n ? Aa
for all
and a l l
m
n
k
for every positive integer Consequently,
j .
for all
n.
n,
we h a v e
j
a n d h = -a l. log k yn
A
= a n-'
for all
n
n.
T o c o m p l e t e t h e p r o o f , we
m
( Y ~ ) ~i s= e ~q u i v a l e n t t o a n i n c r e a s i n g s e q u e n c e .
We f i r s t s h o w t h a t i t i s a n i n c r e a s i n g s e q u e n c e a l o n g t h e arithmetic progression
co
(kn)n,l
and t h e n modify c e r t a i n i n t e r -
mediate values t o obtain an equivalent increasing sequence. Now
and hence t h e sequence
c3
=
sup akn n n
.
For
(
Y
m
~
~
nd j snk
i) s ~i n = c r e~a s i n g . w e have
Let
Chapter 6
350
For any p o s i t i v e i n t e g e r s let
and
j
n
with
we
kn 6 j dkn"
(T) j -kn
yj
.
kn
=
m
Since t h e sequence
For any -
(Ykn)n=l
n
and
-
kn - < j s k n + '
Yj
d Ykn+l
6
Ykn+l S c3ykn
Yj
-< c 3 y
=
C3Ykn
6
I f
a
-
,
and
C3yj
-
Hence
C3Yj.
y
1
inf n
<
y
are equivalent
a
kn "n
< sup n
f o r some p o s i t i v e i n t e g e r
for a l l
and
i s a n u c Z e a r e x p o n e n t s e q u e n c e and
a
kn an
= y
w e have
kn s e q u e n c e s and t h i s c o m p l e t e s t h e p r o o f .
Theorem 6.52
-
y
kn kn i s an i n c r e a s i n g sequence.
(yj)j=l
j,
-kn
i s i n c r e a s i n g and
m
-
we s e e t h a t
n
all
kn + 1
k
<
m
then
n.
Proof
By c o r o l l a r y 6 . 4 2 , p r o p o s i t i o n s 6 . 4 8 a n d 6 . 5 1
( H ( A m ( a ) ~ ) , ~ o )= n _ ( F - l ) F(t)
=
where
sup n
F-l
tn al
...
a n! n
=
(F-l(n));=l
and
35 1
Germs, surjective limits, e -products and power series spaces for all Fn
t
For e a c h p o s i t i v e i n t e g e r
[O,m).
E.
: R+ + R+
n
define
by
We h a v e
F ( t ) = sup Fn(t) for all t E R+. Since Fn(t) + 0 n a s n -t m f o r e a c h t sup F n ( t ) = F (t) f o r some n(t) n integer n(t). Since
F n + p ) - Fn(t) =
the equation
Fn(t) = Fn+l(t)
t n = ( n + l ) an + l . t 3 tn
.
since
Moreover
tn a1
. . . arm!
(
t (n+l)an+l
-1)
has e x a c t l y one r o o t
Fn+l(t)
2 Fn(t)
if and o n l y i f
is an increasing sequence of real
(tn):=l
n u m b e r s we h a v e F(t) = Fn(t) = for
nan
t L
,i
k
for all
n nn
...
a1
(n+l)an+l.
a n!
=
a,n
a1
... a n
Fn(tn-l)
=
and
By p r o p o s i t i o n 6 . 5 1 , such t h a t
Consequently
h k 5 Fn+l(tn)
a =
n
For each p o s i t i v e i n t e g e r
A(n)
. . . arm!
such t h a t
an a1
tn
sup n
n
a1
...
a
n +1
(n+l)!
let a n
nn
A*(n) = - A(n) = a1 n! a
n +1 ( n + l )n+l n+l
n
...
. n
n
an + l n !
-
Fn(tn-l).
i s s t a b l e and hence t h e r e e x i s t s
a2n - an
c.
C>O
Chapter 6
352
Hence f o r any p o s i t i v e i n t e g e r
.
Since
where
to
c
A(2m) L
Since A*(n)
and,
j=1 A(2j-l)
(C2)2
m L
(C2)2n = C
A(l) =
n n n!
A(n)
=
h
2 2m (C )
4n
2
2"-l
6
A*(n)
n.
sup{n;A*(n)
greater than o r equal
6 k}.
k
=
n
n
-
n!
A(n)
let
By d e f i n i t i o n ,
A*(n(k)) L k L A*(n(k)+l) a n d c o n s e q u e n t 1y 2" (k) - 1 L A*(n(k)) ,< k c A * ( n ( k ) + l ) 5
(eC4 ) n ( k ) + l
This implies (n(k)-l)log2 Hence,
.
2 2"-'
For each p o s i t i v e i n t e g e r n(k)
j=l
C2j
w e a l s o have
1
=
by S t i r l i n g ' s f o r m u l a ,
for a l l
fi
.
6 A(l)
i s t h e s m a l l e s t power o f
2m
n.
m
we h a v e
i s a n i n c r e a s i n g f u n c t i o n , we h a v e
A
A(n)
m,
f o r each p o s i t i v e i n t e g e r
and by i n d u c t i o n , A(2m) = A ( l )
n
s logk
there exists 1
s ( n ( k ) + l ) logeC4
q>1
such t h a t
- log(k+l) s n(k) 9
6
q log(k+l)
for all
k.
<
(eC4)"
353
Germs, surjective limits, e -products and power series spaces for all F
and
k. -1
(k)
F-l(k)
Thus f o r a l l =
a
,.. ...
a nn !
k )n
*
1 -
=
nan = n ( k ) a n ( k )
c
(a1
(a1
is stable,
...
we h a v e
1 -
(a1
...
n = n(k)
with
2
c
Since
(a1
k
a n!A*(n))
n
2
n
1 -(log (k+l))a 1 9 [ q log(k+l)l
a n ! A*(n+l))n
1 -
n
a n!
n
(n+l)
n+l
n+l
(n+l)!
1 n
"n+l a1
...
it follows t h a t t h e r e e x i s t s
a
1 n+1
C1 > 0
such
that
This completes t h e proof. 00
Example 6 . 5 3
(a)
Let
a = (nP)n,l
Since w e have
where
p
is positive.
Chapter 6
35 4 "2n i n f __ n an
a
2n s u p __ n an
=
2P*
=
By t h e o r e m 6 . 5 2 ,
In p a r t i c u l a r ,
m
f o r any p o s i t i v e i n t e g e r (b)
If
a =
m
(Pnln=l
where
pn
denotes the
nth
prime
t h e n t h e f u n d a m e n t a l t h e o r e m o n t h e d e n s i t y o f p r i m e s shows
(H(A,(~)A),T,)
for all
2
Am(&)
where
n.
We c o n c l u d e t h i s s e c t i o n w i t h a q u i t e d i f f e r e n t
d e s c r i p t i o n o f t h e s p a c e o f holomorphic f u n c t i o n s on a f u l l y nuclear space with a basis. Definition 6.54
Let
E
symmetric t e n s o r a l g e b r a o f
be a l o c a l l y convex space. E
A
i s a complete commutative
35 5
Germs, surjective limits, E -products and power series spaces S(E)
l o c a l l y m u l t i p l i c a t i v e l y convex algebra i:E
together with a continuous i n j e c t i o n
with unit
S(E)
-f
which has t h e
following universal property: f o r a n y c o n t i n u o u s l i n e a r mapping
0
E
of
into
a complete l o c a l l y m u l t i p l i c a t i v e l y convex algebra A
( o r e q u i v a l e n t l y i n t o a Banach a l g e b r a )
with
u n i t s a t i s f y i n g $ ( x ) $ ( y ) = $ ( y ) + ( x ) for a l l x,y E E t h e r e e x i s t s a unique c o n t i n u o u s aZgebra @:S(E)
homomorphism
-f
By s t a n d a r d a r g u m e n t s ,
A
wi-kh
0
= Qoi.
one e a s i l y shows t h a t a l l symmetric
t e n s o r a l g e b r a s o f a l o c a l l y convex space ( i f t h e y e x i s t ) are i s o m o r p h i c as a l g e b r a s . Theorem 6 . 5 5 (H
HY
A(P)
Let
be a f l ~ l 7 y ynuclear space.
i:*(P)
+
(HHy(JW);)Jo)
i s t h e symmetric t e n s o r algebra o f Proof
By o u r p r e c e d i n g r e m a r k ,
(HHy(A(P);),~o)
x,y that
E
into A(P).
II
s P,
A(P).
The mapping
be a
A
all
n
where
for all
(pn)Zzl m
E
P
such
is the unit vector
If
converges absolutely i n j .
Let
$(x)$(y) = $(y)$(x)
By c o n t i n u i t y , t h e r e e x i s t s
Il+(en)
basis for
i t s u f f i c e s t o show t h a t
b e a c o n t i n u o u s l i n e a r mapping from
$
such t h a t
A
A(P).
has the required properties.
Banach a l g e b r a and l e t A (P)
Then
together with the canonical i n j e c t i o n
(A(P);),y0)
@
w h e r e we l e t
A
: H H y ( A ( P ) i ) , ~ o )+ A
is e a s i l y seen t o extend
+
b. = $(e.) 1 J given by
for all
and t h i s completes t h e p r o o f .
356
Chapter 6 T h e o r e m 6 . 5 5 i n d i c a t e s how t o d e f i n e a f u n c t i o n a l c a l c u l u s m
f o r a sequence algebra,
o r i n a c o m p l e t e m u l t i p l i c a t i v e l y convex a l g e b r a .
A,
m
I t s u f f i c e s , given
space
o f commuting e l e m e n t s i n a Banach
(bn)n=l
b = (bn)n=l,
such t h a t
A(P).
A
If
mapping
t o choose a f u l l y nuclear
(I\bn\lln=l
i s commutative,
then the continuous linear
: ( H H y ( A ( P ) A ) , ~ o )+ A
@
is the desired functional
c a l c u l u s and t h e j o i n t spectrum of
i s a c o n t i n u o u s weight on
00
i s t h e n a compact s u b s e t
o(b)
A(P)b.
The e a r l i e r r e s u l t s o f t h i s s e c t i o n i d e n t i f y t h e symmetric t e n s o r a l g e b r a o f v a r i o u s power s e r i e s s p a c e s o f i n f i n i t e t y p e . In particular,
p r o p o s i t i o n 6.44
Proposition 6.56
S(S)
2 s
implies the following r e s u l t .
where
r a p i d l y d e c r e a s i n g s e q u e n c e s and
is t h e s p a c e of i s t h e symmetric t e n s o r
S(s)
s.
a l g e b r a of
56.5
EXERCISES
6.57* ___
Let
space.
s
E =
n
CXEA
where each
Ea
For each f i n i t e s u b s e t
J
of
i s a l o c a l l y convex
E A
'
let
E.
=
TI
CXEJ H(OE) is
If
H(OE ) i s regular f o r every J show t h a t J regular. Show t h a t H(Op,) is a regular inductive l i m i t .
6.58* ___
Let
b e a l o c a l l y convex s p a c e .
E
If
H(OE)
is
r e g u l a r and d o e s n o t c o n t a i n a n o n t r i v i a l v e r y s t r o n g l y conv e r g e n t s e q u e n c e , show t h a t metrizable subset
__ 6.59*
If
show t h a t ___ 6.60*
H(K) If
contains
K
of
i s r e g u l a r f o r e v e r y compact
E.
i s a compact s u b s e t o f a Fre'chet n u c l e a r s p a c e
K
convex space
K
H(K)
i s a ,333-2 s p a c e . i s a compact s u b s e t of a m e t r i z a b l e l o c a l l y
K E
and
show t h a t
U
i s an open s u b s e t of
E
which
351
Germs, surjective limits, E -products and power series spaces
is a locally
m
convex a l g e b r a .
is a locally
m
convex a l g e b r a f o r any open s u b s e t
6.61 __
( 8 (nE) ,.rW) n
/
E
If
is a distinguished Frechet space, E
E.
show t h a t
i s quasi-normable.
A l o c a l l y convex s p a c e
convergence c r i t e r i o n i f g i v e n
s a t i s f i e s t h e s t r i c t Mackey
E
B C E
c l o s e d convex b a l a n c e d bounded s u b s e t such t h a t
of
U
i s boundedly r e t r a c t i v e f o r every p o s i t i v e i n t e g e r
i f and o n l y i f
6.62 -
(H(U),.rw)
H e n c e show t h a t
E
and
bounded t h e r e e x i s t s a A
of
E
containing
i n d u c e t h e same t o p o l o g y on
EA
E
B.
Show t h a t a n i n j e c t i v e i n d u c t i v e l i m i t o f B a n a c h s p a c e s i s boundedly r e t r a c t i v e i f and o n l y i f it i s r e g u l a r and
s a t i s f i e s t h e s t r i c t Mackey c o n v e r g e n c e c r i t e r i o n . Show t h a t t h e f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t o n a
6.63* -
l o c a l l y convex space E . (a)
Given a z e r o neighbourhood
W
in
E
there exists
a c l o s e d convex balanced zero neighbourhood V C W,
and a bounded s e t
for all (b)
(c)
E
such t h a t
V C B + 6V
6>0.
c o n t a i n s a f u n d a m e n t a l s y s t e m of semi-norms
such t h a t
Ei
f o r every
a
E
B
V,
i n d u c e s t h e norm t o p o l o g y o n
in
r,
c o n t a i n s a fundamental
E$
s y s t e m o f semi-norms
s u c h t h a t t h e normed t o p o l o g y o n P(mE,) c o i n c i d e s w i t h t h e induced topology of ( Q ( m ~ ) , ~ f) o r e v e r y (d)
E
a
in
r, 6
in
r
.
i s a n o p e n s u r j e c t i v e l i m i t o f normed l i n e a r
spaces.
r
r
358
Chapter 6
6 . 6_ 4 _
Show t h a t a n y l o c a l l y c o n v e x s p a c e w h i c h s a t i s f i e s t h e
equivalent conditions of t h e preceding exercise i s quasi-
If
normable.
i s a F r g c h e t - S c h w a r t z s p a c e which s a t i s f i e s
E
show t h a t
the condition of exercise 6.63, 6.65* ___
N
.
i s a s u r j e c t i v e l i m i t of
E = lirn (Ea,na)
If
E ZC
c-
a EA
normed l i n e a r s p a c e s , show t h a t t h e r e i s a n a t u r a l o r d e r on (i.e.
if
al,a2
surjections such t h a t
II
then there exist
A
E
“l’a3
Ii
0
Ea
:
na
“lYa3
+
3
- na
3
Ea
1
and
a 3 e A
and continuous
n
: E a
“2’“3
a n d iIa
1
o
n
LY03
+ E
3 = 11
“2
a2
A
“ 2
Hence
).
d e d u c e t h a t a s u r j e c t i v e l i m i t o f Banach s p a c e s i s an o p e n surjective l i m i t .
6.66
If
E = l i m
i s a s u r j e c t i v e l i m i t and e a c h
(Ea,na)
t-
Ea
a has t h e approximation property,
show t h a t
has the
E
approximation property. __ 6.67
If
E = l i m
i s a s u r j e c t i v e l i m i t ofcomplete
(Ea,IIa)
t-
a
l o c a l l y convex s p a c e s ,
show t h a t
n
E = l i m
A
where
(Ea,Iia)
f-
A
f
i
na:E
+
and
Ea
Ii
for all a. __ 6.68
A surjective l i m i t
E = l i m
(Ea,Xa)
is directed i f
t-
aEr
it enjoys t h e p r o p e r t y d e s c r i b e d i n e x e r c i s e 6.65.
i s a n open s u r j e c t i v e l i m i t i f and o n l y i f i s a n o p e n s u r j e c t i o n f o r e a c h a,B E r , asB.
E
II
a,
Show t h a t
B:E
a
+
Eg
Show t h a t a
d i r e c t e d s u r j e c t i v e l i m i t of Frgchet spaces i s an open s u r j ective l i m i t .
6.69
If
uous b a s i s ,
E
i s a l o c a l l y convex space w i t h an e q u i c o n t i n -
show t h a t
E
i s a s u r j e c t i v e l i m i t o f normed
359
Germs, surjective limits, E -products and p o w e r series spaces l i n e a r s p a c e s , each o f which h a s an e q u i c o n t i n u o u s b a s i s .
Show
i s an open s u r j e c t i v e l i m i t o f l o c a l l y convex
E
also that
s p a c e s , e a c h o f which h a s an e q u i c o n t i n u o u s b a s i s and a d m i t s a c o n t i n u o u s norm. __ 6.70*
By c o n s i d e r i n g t h e s p a c e
show t h a t i n g e n e r a l
r
co(r),
uncountable,
bounded s u b s e t s o f
T~
H(E) = l i m E i , t 1
d o n o t u n i f o r m l y f a c t o r t h r o u g h some
e v e n when we a r e d e a l -
Ei
i n g w i t h an open and compact s u r j e c t i v e l i m i t .
6.71
-
Let
V
b e a R e i n h a r d t domain,
containing the origin,
i n a Banach s p a c e w i t h a n u n c o n d i t i o n a l b a s i s .
6.72* -
Let
E
/
be a Frechet-Schwartz
space.
Show t h a t
Let
K =
6
K. i s a compact s u b s e t of E. If T E H(K)' I that there exists T . E H(K.)' f o r each j such t h a t n J J T = T. where each
1.J = 1
_ 6 . 7_ 3*
K. J
j -1
show
J '
If
show t h a t
i s a compact s u b s e t o f a l o c a l l y convex s p a c e
K
H(K) = l i m
(H(V),T~).
&
VDK,V o p e n
___ 6.74
Show t h a t t h e t - p r o d u c t
a
space.
6.75"
If
E,
is a
d3F6
o f two
$38
spaces i s again
s p a c e a n d a n i n d u c t i v e l i m i t of
Banach s p a c e s w i t h t h e a p p r o x i m a t i o n p r o p e r t y v i a c o m p a c t mappings,
show t h a t
e v e r y compact s u b s e t 6.76*
Let
K
H(K) K
E.
b e a compact s u b s e t o f
i s p o l y n o m i a l l y convex i n
p o l y n o m i a l s on
has t h e approximation property f o r of
(CN
Cn
f o r each
and suppose
CN n
E
N.
a r e sequentially dense i n
6.77* Let A(P) b e a s t a b l e n u c l e a r Fre'chet a d m i t s a c o n t i n u o u s norm. Show t h a t
nn(K)
Show t h a t t h e H(K). s p a c e which
Chapter 6
3 60
_ 6 . 7_ 8*
q
m
an = n
a = ("n)n=l'
Let
are p o s i t i v e r e a l numbers.
Am(&),
where
6n
Let
A(P)
6.79* -
of weights P.
i f f o r each
P
(log(n+l))q
Show t h a t
where
(log(n+l))P+l(log log(n+l))q
=
03
(an)n,l
i s a Schwartz s p a c e i f and o n l y
A(P)
in
there exist
P
E
P
E C + such t h a t a (Un)n=l 0 n 6 u n a nt f o r a l l n . i s a Montel space i f and o n l y i f f o r each
and
6.80*
P
such t h a t
Let
HM(U) = H(U)
l i m inf = 0. a A j # O , j + m a' nj b e a F r e c h e t Montel space.
A(P)
f o r any open s u b s e t
U
of
Show t h a t a l o c a l l y c o n v e x s p a c e
6.82 -
topology
o(E,E')
Show t h a t m
( a n ) n = l ~A ( P )
(n.)? there exists J J=1 an.
and each subsequence of i n t e g e r s E
n.
be a sequence space with s a t u r a t e d system
Show t h a t
A(P)
m
and =
for all
m
(aA)n=1
p
(H(Am(a)b),~O)
Show t h a t
n(P)A.
E
w i t h t h e weak
i s an open s u r j e c t i v e l i m i t o f f i n i t e
dimensional spaces.
56.6
NOTES A N D REMARKS The c o m p l e t e n e s s o f
l o c a l l y convex space [503].
H(K),
H e showed t h a t
K
a compact s u b s e t o f a
was f i r s t i n v e s t i g a t e d b y J . M u j i c a
E,
H(K)
i s c o m p l e t e whenever
m e t r i z a b l e l o c a l l y convex s p a c e w i t h p r o p e r t y cises 6 . 6 2 and 6 . 6 3 ) .
K.D.
B i e r s t e d t and R.
(B), Meise
E
is a
(see exer[69]
proved
t h e same r e s u l t f o r c o m p a c t s u b s e t s o f a F r g c h e t S c h w a r t z s p a c e and s u b s e q u e n t l y P.
A v i l e s and J . Mujica
[41]
extended t h i s
r e s u l t t o quasi-normable m e t r i z a b l e l o c a l l y convex s p a c e s . general r e s u l t that
H(K)
i s complete f o r any compact
of a m e t r i z a b l e l o c a l l y convex space, theorem 6 . 1 , S . D i n e e n [ZOO].
The
subset
i s due t o
361
Germs, surjective limits, E -products and power series spaces
Proposition 6 . 2 is due to R. Soraggi [669] while corollFurther examples aries 6.3 and 6.4 are d u e to S. Dineen [200]. including proposition 6 . 2 9 , concerning t h e regularity of when
K
H(K)
is a compact subset o f certain non-metrizable
locally convex spaces are given in R . Soraggi [667,668,669].. From the viewpoint o f holomorphic germs and analytic functiona l s , the following result o f J . Mujica [ S o l ] is also o f interest:
if
K
is a compact locally connected subset o f the met-
rizable Schwartz space
E = lim En, fn
where each
En
is a
normed linear space and the linking maps are precompact, then for each continuous linear functional T o n H(K) there exists a sequence o f vector measures such that 00
(ii)
f (iii)
if
1 m!
-
=
in
1
Smf(x)pm(dx)
P(mE,,))' l/m
norm o f
satisfying (i) and (iii),
a s an element
pm
then for each
Conversely, given a sequence H(K)
for every
H(K);
1 1 ~ ~ 1 is1 the ~
of &(K;
K
m
(um)m=l
then (ii)
n,
o f vector measures defines a n element o f
'.
Proposition 6 . 7 is due to S . B . Chae [ 1 2 0 ] . Proposition Baernstein [42], in his work o n the representation o f holomorphic functions by boundary integrals. 6.8 was discovered by A .
Proposition 6 . 9 , theorem 6 . 1 0 and corollary 6.11 are due to. K-D. Bierstedt and R . Meise [ 7 0 ] . See also E. Nelimarkka [525] for a further proof o f proposition 6.9. R . Meise has recently shown that T~ = T on any open subset o f a Frgchet nuclear space and thus the basis assumption in corollary 6.11 is not necessary. Example 6.13 is d u e to M. Schottenloher [644] who used it to prove corollary 6.40. Corollary 6.40 is also d u e
Chapter 6
362 independently, L.
and by a d i f f e r e n t method,
[53].
Nachbin
to J.A.
B a r r o s o and
The p r o o f g i v e n h e r e i s s l i g h t l y d i f f e r e n t
from e i t h e r of t h e above. The r e g u l a r i t y a n d c o m p l e t e n e s s o f i n d u c t i v e l i m i t s i s
see f o r i n s t a n c e , t h e
extensively discussed i n the literature, recent survey of K.
of K-D.
Floret
B i e r s t e d t and R .
and t h e f i r s t f e w s e c t i o n s
[238],
Meise
[70],
and h a s l e d t o t h e d e f i n -
i t i o n o f many s p e c i a l k i n d s o f i n d u c t i v e l i m i t s . research
[SO31 h a s l e d h i m t o d e f i n e " C a u c h y r e g u l a r "
l i m i t s and t h i s c o n c e p t , R.
Meise
J. M u j i c a ' s
[70],
as p o i n t e d o u t by K - D .
inductive
B i e r s t e d t and
c o i n c i d e s with t h e concept o f boundedly r e t r a c t -
i v e i n d u c t i v e l i m i t s i n t h e case of an i n j e c t i v e i n d u c t i v e
l i m i t o f Banach s p a c e s .
H.
Neus
s h o w e d t h a t many o f
[527],
these concepts coincide for countable inductive limits of Banach s p a c e s , and p r o v e d p r o p o s i t i o n 6 . 1 6 ,
i s an a b s t r a c t v e r s i o n ,
-
due t o K-D.
[69], o f a r e s u l t of J . Mujica inductive l i m i t
B i e r s t e d t and R.
[503].
( z ( V ) n H ( U ) ,1 1
l i m
Proposition 6.15 Meise
The i d e a o f u s i n g t h e
11")
i s due t o J . Mujica
KCVC U
[SO31 who p r o v e d p r o p o s i t i o n 6 . 1 2
and used i t t o p r o v e propo-
s i t i o n 6 . 1 8 and c o r o l l a r y 6 . 1 9 . S u r j e c t i v e limits are due independently t o S. 1901 and E . L i g o c k a
examples and a p p l i c a t i o n s t o
i n f i n i t e d i m e n s i o n a l holomorphy a r e g i v e n in[190] Further references are P. [ 2 0 7 ] , Ph. [463,467]
and R.
Berner
S. Dineen,
Noverraz
[552], M.
Soraggi
due t o L.A.
d e Moraes
P r o p o s i t i o n 6.27 R.
[498],
i s due t o P .
and
[443].
S.
Dineen
Schottenloher
Schottenloher [640], M.C.
[669].
independently,
[58,59,60,61,62],
Ph. Noverraz and M.
lemma 6 . 2 4 a r e g i v e n i n S . D i n e e n r e s u l t i s due,
[189,
[ 4 4 3 ] , (who u s e s t h e t e r m i n o l o g y b a s i c
system). Their basic properties,
[186,189,191,193],
Dineen
Matos
Examples 6 . 2 1 , 6 . 2 2 , 6 . 2 3 [190].
and
Proposition 6.25 i s
while a p a r t i c u l a r case of t h i s to P.J.
Boland and S. Dineen[gi].
B e r n e r [ 6 1 ] a n d S . D i n e e n [194].
Soraggi proves proposition 6.29 i n
[669].
In studying vector valued distributions,
L.
Schwartz
[648]
363
Germs, surjective limits,E -products and power series spaces compensated f o r t h e absence of t h e approximation p r o p e r t y by defining M.
6 -
products (definition 6.30).
Schottenloher
[631] i n t r o d u c e d
as a tool
e-products
i n i n f i n i t e dimensional holomorphy. In
[639] h e
p r o v e d lemma 6 . 3 2 ,
propositions 6.33,6.34,
a r y 6.35 and gave example 6.31. 6.34 is due t o A .
Hirschowitz
coroll-
A weak f o r m o f p r o p o s i t i o n
p43
,
p r o p o s i t i o n 3.41
and
w e i g h t e d v e r s i o n s o f t h e same p r o p o s i t i o n a r e g i v e n i n K.
Bierstedt
[66,p.44
Theorem 6 . 3 6 i s new.
and 551.
The i d e a
o f u s i n g t e n s o r p r o d u c t s and t h e c o n n e c t i o n between t h i s
o f A . Grothendieck
theorem and p r o p o s i t i o n 1 5 , c h a p t e r 2 ,
was p o i n t e d o u t t o t h e a u t h o r b y D . counterexample,
[287]
Earlier a d i r e c t
Vogt.
which a p p l i e d t o t h e n u c l e a r power s e r i e s s p a c e
c a s e , was g i v e n b y S . D i n e e n
[202],
(see exercise 5.82).
It
would b e o f i n t e r e s t t o e x t e n d t h i s counterexample t o t h e t h a t t h i s i s p o s s i b l e ) and t h u s
g e n e r a l case ( i t i s our b e l i e f
give a completely self-contained
proof.
We d o n o t know i f t h e
b a s i s hypothesis i n theorem 6.31 i s necessary. 6 . 3 7 i s due t o S . Dineen 6.39 are due t o K-D. applications of
6
[202].
P r o p o s i t i o n 6.38 and c o r o l l a r y
B i e r s t e d t and R .
-products
Proposition
Meise
[69,70].
Further
i n i n f i n i t e d i m e n s i o n a l holomorphy
a n d k e r n e l t h e o r e m s f o r a n a l y t i c f u n c t i o n a l s may b e f o u n d i n K-D. B.
B i e r s t e d t and R. Perrot
Meise [69,70]
and i n J . F .
Colombeau and
[157,158,159,161,162].
A l l t h e r e s u l t s o f s e c t i o n 6.4 a r e due t o M.
Meise a n d D .
Bb'rgens,
Vogt and most o f them a r e c o n t a i n e d i n
comprehensive p a p e r , p a r t i a l l y summarised i n [95],
[96].
R.
This
contains
many f u r t h e r i n t e r e s t i n g e x a m p l e s o f s t r u c t u r e t h e o r e m s f o r H(Am(a)i).
T h e same a u t h o r s h a v e w r i t t e n a f u r t h e r a r t i c l e
[97] on t h e A - n u c l e a r i t y o f s p a c e s o f h o l o m o r p h i c f u n c t i o n s using refinements of t h e techniques developed i n The s y m m e t r i c t e n s o r a l g e b r a duced by A.
Colojoar?i
theorem 6.55 f o r
DF
[139].
[96].
( d e f i n i t i o n 6.54)
was i n t r o -
She proved an a b s t r a c t form of
nuclear spaces but did not establish a
c o n n e c t i o n between h e r r e s u l t s and holomorphic f u n c t i o n s .
was d o n e i n
[96] and d e t a i l e d i n [487].
This
3 64
Chapter 6 The r e s u l t s and methods o f s e c t i o n 6 . 4 a r e s t i l l i n t h e
p r o c e s s o f f i n d i n g t h e i r f i n a l form and v e r y r e c e n t d e v e l o p -
ments s u g g e s t t h a t t h e y w i l l p l a y a v e r y i m p o r t a n t r o l e i n t h e future of the subject. D.
Vogt
We s h a l l o n l y m e n t i o n t h a t R .
Meise and
[485,486] have r e c e n t l y obtained a holomorphic
c r i t e r i o n f o r d i s t i n g u i s h i n g open p o l y d i s c s i n c e r t a i n n u c l e a r power s e r i e s s p a c e s a n d h a v e shown i n [489] t h a t t h e t h r e e topologies
To
, ~ u and
spcace with a basis,
T &
on
H(A(P)),
A(P)
a fully nuclear
can a l l b e i n t e r p r e t e d as normal t o p o l o g -
ies i n t h e sense of G.
KGthe [ 3 9 7 ] .
Appendix I
FURTHER DEVELOPMENTS IN INFINITE DIMENSIONAL HOLOMORPHY
In t h i s appendix, we provide a b r i e f survey of some r e s e a r c h c u r r e n t l y being developed within i n f i n i t e dimensional holomorphy.
The t o p i c s we d i s -
cuss emphasise t h e a l g e b r a i c , geometric and d i f f e r e n t i a l , r a t h e r than t h e topological a s p e c t s o f t h e theory.
We hope t h i s i n t r o d u c t i o n w i l l i n s p i r e
t h e reader t o f u r t h e r readings and t o an o v e r a l l a p p r e c i a t i o n of t h e u n i t y of t h e s u b j e c t . THE LEI7 PROBLEM
We begin by looking a t a s e t of conditions on a domain convex space
U
i n a locally
E.
(a)
U
i s a pseudo-convex domain;
(b)
U
i s holomorphically convex.
(c)
U
i s a domain of holomorphy.
(d)
U
i s t h e domain of e x i s t e n c e of a holomorphic
function; (e)
The
(f)
If
a
3
problem i s solvable i n
U;
i s a coherent a n a l y t i c sheaf, then
H1(U;S) = 0. A l l these conditions a r e equivalent when
E
i s a f i n i t e dimensional
space (see L . Hormander [347] and R. Gunning and H. Rossi [294]) and t h i s equivalence may be regarded a s one of t h e h i g h l i g h t s o f s e v e r a l complex vari a b l e theory.
Note t h a t condition (a) i s m e t r i c , (b) geometric, (c) and (d)
a n a l y t i c , (e) d i f f e r e n t i a l and ( f ) a l g e b r a i c . c l a s s i c a l Cartan-Thullen theorem [118], and (c) a r e equivalent. equivalent f o r domains i n
In 1911, E . E . C2.
In t h e case of
E = Cn,
the
published i n 1932, a s s e r t s t h a t (b) Levi [441] asked i f (a) and (d) were
This became known a s t h e Levi problem and
was solved by K . Oka [558] i n 1942 and extended t o domains i n
3 65
Cn
by K. O k a
Appendix I
366 [559]
in 1953 and by F. Norguet [530] and H.Bremermann
The implication (f)
=>
(e)
[loll in 1954.
is due to P. Dolbeault [208], (b) => (f) is
due to H. Cartan [115] and (a)
=>
(e) is proved by L. Hormander [346].
Attempts to extend these conditions and to show their equivalence on arbitrary locally convex spaces have never been routine and have led to many interesting developments and results. We now describe the evolution of this line of research in infinite dimensions together with some related topics such as plurisubharmonic functions, envelopes of holomorphy, etc. H.J.Bremermann
[lo31 in 1957 was the first to consider pseudo-convex
domains, domains of holomorphy and plurisubharmonic functions (see proposition 4.12) in infinite dimensions. space to be pseudo-convex if the boundary o f
IJ)
He defined a domain U
-log dU
(dU(x)
in a Banach
is the distance from x
to
is plurisubharmonic and showed that this was equival-
ent to the finite dimensional sections of U
being pseudo-convex. In 1960
he showed that the envelope of holomorphy o f a tubular domain in a Banach space was equal to its convex hull [lo41 and afterwards [lo51 extended a number of his results to linear topological vector spaces. C.O. Kiselman [381] proved that the upper regularization of a locally bounded countable family of plurisubharmonic functions on a Frgchet space was plurisubharmonic (this is known as the convergence theorem) and this was extended to arbitrary families on complete topological vector spaces by P. Lelong [425], G. Coeurg [126,129] and Ph. Noverraz [536]. In [425] P. Lelong began a systematic study of the basic properties of plurisubharmonic functions and polar sets in topological vector spaces. This direction of research is developed in P. Lelong [426,428,429,430,434],
G. Coeur; [127,128,129], M. Herve/ [326] and Ph. Noverraz [536,537,538,545]. By using multiplicative linear functionals, H. Alexander [5] showed, in his thesis, that a domain U rJ
extension U,
spread over E,
in a Banach space E
erty that the canonical mapping of
(H(U),.ro)
ological isomorphism. He also noted that if and only if
fl
E
admits a holomorphic
which is maximal with respect to the propinto
(H(U),ro)
(H(??),T~)
is a top-
is a barrelled space
is finite dimensional and thus could not conclude that
was the natural envelope of holomorphy of
U.
J.M. Exbrayat [233] is
the only accessible reference for Alexander's unpublished thesis.
367
Further developments
The next contributions are due to G. Coeur6 [128,129].
He defined
pseudo-convex domains spread over Banach spaces by using the distance function and showed that a domain spread X is pseudo-convex if and only if the plurisubharmonic hull of each compact subset of X is also compact. Ihis result was extended to locally convex spaces by Ph. Noverraz [544]. To overcome the inadequacies of the compact open topology encountered by Alexander, Coeur; defined the T~ topology on domains spread over separable Banach spaces and showed that any holomorphic extension of a domain leads to a
T~
topological isomorphism of the corresponding space of holo-
morphic functions. This result was later extended to domains spread over arbitrary Banach spaces by A. Hirschowitz 1338,3431. G. Coeur; also proved in [129] that a suitable subset
5(X)
of the
could be endowed with
the structure of a holomorphic manifold spread over E a holomorphic extension of X holomorphy and
H(X)
spectrum of H(X),
T~
X a domain spread over a separable Banach space E,
and identified with
and that, furthermore, if X
separates the points of X
then X
is a domain of 5 (X).
In 1969, two important contributions were made by A. Hirschowitz [ 3 3 5 , 3361. In [335], he showed that the Levi problem had a positive solution and this result was subsequently extended to Riemann domains over CN by M.C. Matos [456] and to domains spread over A C , A arbitrary, by V. Aurich [33]. In his analysis, A. Hirschowitz for open subsets of CN
showed that any pseudo-convex open subset U
of CN had the form
-'(nn(U)) for some positive integer n where nn is the natural U = IIn projection from CN onto Cn. This result, together with factorization properties of holomorphic functions on CN given by A. Hirschowitz in [335] and also obtained independently by C.E. Rickart [605], led eventually to the concept of surjective limit (see 56.1) and to a technique for overcoming the lack of a continuous norm in certain delicate situations. V. Aurich used the bornological topology associated with the compact open topology in A his investigation of the spectrum of H(U), U a domain spread over C [331. In [336], see also [337] for details, A. Hirschowitz showed that the unit ball of &([O,R]),
fi
the first uncountable ordinal, is not the domain
of existance of a holomorphic function, i.e. (c) #> (d).
This counter-
example to the Levi problem and B. Josefson's [358] example of a domain in co(r),
r uncountable, which is holomorphically convex but not a domain
of
Appendix I
368
holomorphy, i.e. (b) #> (c), rely heavily on the non-separability of,&[O,n] and r respectively and, indeed, it appears that countability assumptions have always, and probably always will, enter into solutions of the Levi problem. We note in passing that A. Hirschowitz introduces bounding sets in [336] and that this concept had also arisen in H. Alexander’s work on normal extensions
,
in S. Dineen’s investigation of locally convex top-
ologies on spaces of holomorphic functions, [177,178], and in M. Schottenloher’s study [631] of holomorphic convexity. In three further papers 1338,340,3431, A . Hirschowitz looked at various other aspects of analytic continuation over Banach spaces. He showed, using germs of holomorphic functions, that every domain spread over a Banach space has an envelope of holomorphy. His investigation of vectorvalued holomorphic functions showed that whenever C
valued holomorphic
functions can be extended then so also can Banach valued holomorphic mappings and that conditions (b) and (c) , (resp (d)), remain unchanged when holomorphic functions are replaced by Banach (resp. separable Banach) space valued mappings. In 1970, S. Dineen 11761 replaced H(U)
by Hb(U),
the set of holo-
morphic functions on U which are bounded on the bounded open subsets of
E
contained in U and at a positive distance from the boundary of U.
Since Hb(U)
has a natural Frgchet space structure he was able, by suit-
ably modifying conditions (b) and (c), to obtain a Banach space version o f the Cartan-Thullen theorem. This approach was developed by M.C. Matos [457,459,460] who proved similar Cartan-Thullen theorems for various subalgebras of H(U). Independently of S . Dineen 11761 and A . Hirschowitz 13381, M. Schottenloher [631,632] was considering a much more general situation by defining regular classes (see also G. Coeur6 [129]) and admissible coverings for domains spread over a Banach space. F o r each regular class he proved a Cartan-Thullen theorem. By looking at all regular classes and by generalizing the classical intersection theorem for Riemann domains to infinite dimensions, he showed that the envelope of holomorphy could be identified with a connected component of the T~ spectrum. In [640], he extended this result to domains spread over a collection o f locally convex spaces which included all metrizable spaces and alld43w spaces, (see also K. Rusek and J. Siciak [618]). In later papers, [633,635,638,639] he
Further developments
3 69
considered various other topics on analytic continuation in infinite dimensions such as vector-valued extensions and the extension problem for Mackey holomorphic functions. We now digress a little to describe more recent results on the spectrum of H(U). In [352], M. Isidro showed that Spec(H(U),To) 2 U when U is a convex balanced open subset of a complete locally convex space with the approximation property and this result was extended to polynomially convex domains in quasi-complete spaces with the approximation property by J. Mujica, [502,505]. In [504], J. Mujica proved that Spec(H(U),T6) Q U when U is a polynomially convex domain in a Frgchet space with the bounded approximation property and the counterexample of B. Josefson [358] shows that this result is not true for every polynomially convex domain in Banach spaces with the approximation property. M. Schottenloher proved Spec(H(U),ro) = U for U pseudo-convex in a Frgchet space with basis. The study of the spectrum is the study of the closed maximal ideals and a few authors have also studied finitely generated ideals i n
H(U). J. Mujica shows in [505] that H(U), U a polynomially convex domain in a Frzchet space with the approximation property, is the T~ closure of the ideal generated by anyfinite family of functions in H(U) without common zero. In [277], B. Gramsch and W. Kaballo prove the following result: if
A is a Banach algebra with identity e, U is a polynomially convex domain in a a3-R space with Schauder basis and (fj)i=lCH(U) have the property that for every x 1;=1
(x) j aj,xf.
=
in
U
there exists (a,,X)j=ltA such that n e then there exists (aj) j=lC A such that
n
ajfj(x) = e for every x in U. In particular, this shows that the ideal generated by any finite family of holomorphic functions without common zero in U . is equal to H(U) (see also M. Schottenloher [646]). Further results and examples on analytic continuation, the spectrum of Cartan-Thullen theorems and the envelope of holomorphy are given in H(X), the book of G. Coeur6 [131]. We now return to our main theme. The following fundamental property of pseudo-convex domains in a locally convex space is due to S. Dineen, [183,186] and Ph. Noverraz [540, 5441; if U is a pseudo-convex (resp. finitely polynomially convex) open subset of a locally convex space E , p E cs(E), Il is the natural surjection from E onto E,ker(p), and n(U)
Appendix I
370 h a s non-empty i n t e r i o r t h e n sections of
R(U)
U = Il
-1
(n(U))
and t h e f i n i t e dimensional
are pseudo-convex ( r e s p . p o l y n o m i a l l y c o n v e x ) .
Various o t h e r forms and r e f i n e m e n t s o f t h e above a r e known and t h e y a l l o w one t o t r a n s f e r problems, such as t h e Levi problem, from
U
to
n(U)
and t o g e n e r a t e l o c a l l y convex s p a c e s w i t h p r e a s s i g n e d p r o p e r t i e s . I n [ 1 7 5 ] , S . Dineen r e p l a c e d t h e H(U)
and on showing
T~
=
T&
t o p o l o g y by t h e
T
t o p o l o g y on
T~
(theorem 4.38) o b t a i n e d a C a r t a n - T h u l l e n
theorem, i . e . ( b ) < = >( c ) , f o r b a l a n c e d open s u b s e t s o f a Banach s p a c e w i t h an uncoriditional b a s i s .
The f o l l o w i n g y e a r , S . Dineen and A . Hirschowitz
[203] improved t h i s r e s u l t by showing t h a t a domain
U
i n a Banach s p a c e
w i t h a Schauder b a s i s i s a domain of holomorphy i f i t s f i n i t e d i m e n s i o n a l s e c t i o n s are p o l y n o m i a l l y convex.
T h i s r e s u l t was extended t o s e p a r a b l e
Banach s p a c e s w i t h t h e p r o j e c t i v e approximation p r o p e r t y by Ph. Noverraz [540,543,546] t o m e t r i z a b l e and h e r e d i t a r y Lindel'df s p a c e s w i t h a n e q u i Schauder b a s i s by S . Dineen [186], t o
838
s p a c e s w i t h a b a s i s by N . Popa
[586] and t o v a r i o u s o t h e r s p a c e s by R. Pomes [583,584].
S. Dineen a l s o
showed i n [186] t h a t t h e c o l l e c t i o n o f s p a c e s f o r which t h i s r e s u l t was v a l i d was c l o s e d under t h e o p e r a t i o n o f open s u r j e c t i v e l i m i t . In [179], S . Dineen showed t h a t an open s u b s e t o f a Banach s p a c e w i t h
a Schauder b a s i s i s p o l y n o m i a l l y convex i f and o n l y i f i t s f i n i t e dimens i o n a l s e c t i o n s have t h e same p r o p e r t y .
T h i s r e s u l t was extended t o Banach
s p a c e s w i t h t h e s t r o n g a p p r o x i m a t i o n p r o p e r t y by Ph. Noverraz [540,544] and t o v a r i o u s o t h e r s p a c e s , i n c l u d i n g n u c l e a r s p a c e s , by u s i n g s u r j e c t i v e l i m i t s i n S . Dineen [183,186] and Ph. Noverraz [540,544].
A l l these
r e s u l t s a r e c o n t a i n e d i n t h e v e r y g e n e r a l r e s u l t o f M. S c h o t t e n l o h e r [643] who proved t h a t t h e same e q u i v a l e n c e was v a l i d i n any l o c a l l y convex s p a c e with t h e approximation p r o p er t y . We now look a t two c l o s e l y r e l a t e d q u e s t i o n s c o n c e r n i n g p o l y n o m i a l s , Runge's theorem and t h e Oka-Weil theorem. polynomials a r e dense i n s ubs et of
Cn,
(H(U),T~),
i f and o n l y i f
U
U
Runge's theorem s t a t e s t h a t t h e a h o l o m o r p h i c a l l y convex open
i s p o l y n o m i a l l y convex w h i l e t h e Oka-
Weil theorem s t a t e s t h a t a holomorphic germ on a p o l y n o m i a l l y convex
compact s u b s e t nomials.
K
of
Cn
can b e u n i f o r m l y approximated on
K
by poly-
37 1
Further developments I n [605], C . E . R i c k a r t proved an Oka-Weil theorem f o r
C*.
S. Dineen
[179] e x t e n d e d Runge's theorem t o Banach s p a c e s w i t h a Schauder b a s i s and i n c o l l a b o r a t i o n w i t h Ph. Noverraz [539,541] proved an Oka-Weil theorem f o r t h e same c l a s s o f s p a c e s .
C . Matyszczyk [469] showed t h a t t h e p o l y n o m i a l s
a r e s e q u e n t i a l l y dense i n
( H ( U ; F ) , T ~ ) when
open s u b s e t o f
E
and
E
approximation proper i y .
and
U
i s a p o l y n o m i a l l y convex
a r e Banach s p a c e s w i t h t h e bounded
F
The n e x t s e t o f c o n t r i b u t i o n s were made indepenS. Dineen [183,186],
d e n t l y by Ph. Noverraz [540,543,546],
S c h o t t e n l o h e r [31] and E . Ligocka [443].
R . Aron and M.
Noverraz proved Runge's theorem
and t h e Oka-Weil theorem f o r l o c a l l y convex s p a c e s w i t h t h e s t r o n g approxi m a t i o n p r o p e r t y , w h i l e R . Aron and M . S c h o t t e n l o h e r [31] proved a v e c t o r v a l u e d Runge theorem f o r domains i n Banach s p a c e s w i t h t h e a p p r o x i m a t i o n property.
Ligocka proved an Oka-Weil theorem f o r l o c a l l y convex s p a c e s
which c o u l d be r e p r e s e n t e d as a p r o j e c t i v e l i m i t o f normed l i n e a r s p a c e s w i t h a Schauder b a s i s and t h i s r e s u l t i n c l u d e d t h o s e o f Dineen.
E . Ligocka
a l s o showed t h a t any p o l y n o m i a l l y convex compact s u b s e t of a complete l o c a l l y convex s p a c e had a fundamental neighbourhood system o f p o l y n o m i a l l y convex open s e t s .
J . Mujica [502] p o i n t e d o u t t h a t L i g o c k a ' s p r o o f e x t e n d s
t o q u a s i c o m p l e t e s p a c e s and hence f o r t h i s c o l l e c t i o n o f s p a c e s t h e OkaWeil and Runge theorems a r e e q u i v a l e n t ( s e e a l s o Y . Fujimoto [ 2 4 9 ] ) .
In
/
[470], C . Matyszczyk proved a n Oka-Weil theorem f o r F r e c h e t s p a c e s w i t h t h e a p p r o x i m a t i o n p r o p e r t y and t h i s was extended t o h o l o m o r p h i c a l l y complete m e t r i z a b l e l o c a l l y convex s p a c e s by M . S c h o t t e n l o h e r [643].
In [502],
J . Mujica o b t a i n e d a v e r y g e n e r a l r e s u l t by p r o v i n g t h e Oka-Weil theorem
f o r q u a s i - c o m p l e t e l o c a l l y convex s p a c e s w i t h t h e approximation p r o p e r t y and a p p l i e d t h i s r e s u l t t o c h a r a c t e r i s e t h e p o l y n o m i a l l y convex.
E . Ligocka [443] i s s t i l l open;
H(U),
U
The f o l l o w i n g s u b t l e problem posed by if
s u b s e t o f t h e l o c a l l y convex s p a c e subset o f
spectrum o f
F u r t h e r a p p r o x i m a t i o n theorems a r e g i v e n i n C . Maty-
szczyk [470] and J . Mujica [504].
A
T~
E ( t h e completion o f
K
E
i s a p o l y n o m i a l l y convex compact
is
K
a p o l y n o m i a l l y convex compact
E)?
The s t u d y o f t h e Levi problem l e d d u r i n g t h i s p e r i o d t o t h e i n v e s t i g a t i o n o f c o n c e p t s such as holomorphic c o m p l e t i o n ( s e e s e c t i o n 2 . 4 ) , pseudo-completion,
spaces, e t c .
We refer t o Ph. Noverraz [540,543,544,
546,5471, M. S c h o t t e n l o h e r [633,637,645], [135] f o r d e t a i l s .
S . Dineen [184,186] and G . Coeurg
These topics and fundamental p r o p e r t i e s o f pseudo-
convex domains and p l u r i s u b h a r m o n i c f u n c t i o n s are s t u d i e d i n t h e t e x t o f
Appendix I
312 Ph. Noverraz [ 5 4 5 ] .
More r e c e n t a r t i c l e s on p l u r i s u b h a r m o n i c f u n c t i o n s
and p o l a r s e t s are S. Dineen [193,196], E . Ligocka [444], M . E s t g v e s and C . Her&
[231,232], S. Dineen and Ph. Noverraz [205,206], P . Lelong [438,
439,4401, B. A u p e t i t [32],Ph. Noverraz [554,557] and C . O .
Kiselman [388].
The n e x t r e s u l t on t h e e q u i v a l e n c e o f t h e v a r i o u s c o n d i t i o n s i s due t o Ph. Noverraz [543,546]. subsets of
&3/3
S. Dineen [190].
He proved t h e C a r t a n - T h u l l e n theorem f o r open
s p a c e s and t h i s was extended t o
$!y???-
s p a c e s by
L . Gruman [289,290] was t h e f i r s t t o g i v e a complete
H e used
s o l u t i o n t o t h e Levi problem i n an i n f i n i t e d i m e n s i o n a l s p a c e .
2
the solution t o the
problem i n f i n i t e dimensions and an i n d u c t i v e
c o n s t r u c t i o n t o show t h a t pseudo-convex domains i n s e p a r a b l e H i l b e r t s p a c e s a r e domains o f e x i s t e n c e o f holomorphic f u n c t i o n s .
The t e c h n i q u e and
r e s u l t o f L . Gruman have i n f l u e n c e d a l m o s t a l l l a t e r s o l u t i o n s t o t h e Levi problem.
He a l s o showed t h a t a f i n i t e l y open pseudo-convex s u b s e t o f a
vector space over
i s t h e domain o f e x i s t e n c e o f a G-holomorphic
C
f u n c t i o n ( s e e a l s o S. Dineen [186,187],
J . Kajiwara [365,366,367,368],
Y . Fujimoto [ 2 4 9 ] ) .
Kiselman [291] t h e n s o l v e d t h e
L . Gruman and C.O.
and
Levi problem on Banach s p a c e s w i t h a Schauder b a s i s and Y . H e r v i e r [329] e x t e n d e d t h i s r e s u l t t o domains s p r e a d .
I n [546] and [548] Ph. Noverraz
e x t e n d e d t h e s o l u t i o n o f t h e Levi problem t o Banach s p a c e s w i t h t h e bounded a p p r o x i m a t i o n p r o p e r t y and proved, f o r t h e s e s p a c e s , t h e f o l l o w i n g Oka-Weil theorems:
(i)
UCU'
is
then
H(U')
if T~
U
and
h u l l o f each compact s u b s e t of
U
a r e pseudo-convex domains w i t h
U'
dense i n
i f and o n l y i f t h e
H(U)
i s contained i n
pseudo-convex open s e t and t h e compact s u b s e t H(U)
h u l l t h e n e v e r y holomorphic germ on
by holomorphic f u n c t i o n s on
U.
K
K
(ii)
U;
of
U
H(U') if
U
is a
i s equal t o i t s
can be approximated on
K
Both ( i ) and ( i i ) were g e n e r a l i z e d t o
domains s p r e a d o v e r F r g c h e t s p a c e s and
33&'s p a c e s
w i t h f i n i t e dimension-
a l Schauder d e c o m p o s i t i o n s by M. S c h o t t e n l o h e r [ 6 4 0 ] .
Ph. Noverraz [548]
and R . Pomes [583,584] t h e n s o l v e d t h e Levi problem f o r 3 3 J s p a c e s w i t h a Schauder b a s i s .
The n e x t i m p o r t a n t development i s due t o M. S c h o t t e n l o h e r [ 6 3 6 , 6 4 0 ] . He combined r e g u l a r c l a s s e s , a d m i s s i b l e c o v e r i n g s , s u r j e c t i v e l i m i t s and
a s u b t l e b u t v e r y c r u c i a l m o d i f i c a t i o n o f L . Gruman's c o n s t r u c t i o n t o s o l v e t h e Levi problem f o r domains s p r e a d o v e r h e r e d i t a r y Lindelb'f l o c a l l y convex s p a c e s w i t h a f i n i t e d i m e n s i o n a l Schauder d e c o m p o s i t i o n .
This
Further developments
373
collection of spaces contains all Frgchet spaces and all & $ ? Y l a Schauder basis.
spaces with
Particular cases of Schottenloher's result are given in
S. Dineen, Ph. Noverraz and M. Schottenloher [ 2 0 7 ] .
M. Schottenloher [636,
6401 and P . Berner [59,60] obtained, independently, the following result:
is an open surjective limit and every pseudo-convex domain
if E = lim E - a
CXEA
spread over E , ~ E A , is a domain of holomorphy (resp. domain of existence) a
then every pseudo-convex domain spread over E
is a domain of holomorphy
(resp. domain of existence). In [ 3 6 ] , V. Aurich showed that domains of existence of meromorphic functions in Banach spaces with Schauder bases are domains of existence of holomorphic functions. In [154], J.F. Colombeau and J. Mujica solved the Levi problem for open subsets of
60 3"rz. spaces.
They reduced the Levi problem on 3 3 k
spaces to the Levi problem on a Hilbert space, where L. Gruman's result applied, by using surjective limits and by combining the fact, noticed by space is also open with previous authors, that any open subset of a
&3n
respect to a weaker semi-metrizable locally convex topology with Grothendieck's result [ 2 8 6 , 2 8 8 ] that for any sequence of neighbourhoods of O , ( U . ) . in a
0 X . IU 1. J
DF
space there exists a sequence of scalars
(Xj)j
1 1
such that
is also a neighbourhood of zero (see also corollary 2.30).
This
approach has been developed by J.F. Colombeau and J. Mujica 11561 in the r
study of Hahn-Banach extension theorems and convolution equations. In [ 5 0 6 ] , J. Mujica solves the Levi problem for domains in
(E',ro)
E a separable Frgchet space with the approximation property by using topological methods. Mujica also proves in [SO61 that a holomorphically convex domain in ( E ' , T ~ ) , E a separable Frgchet space, is the domain of existence of a holomorphic function and this result was extended, using
quite different methods, by M. Valdivia [691] to the case where E
is an
arbitrary Frgchet space. M. Valdivia obtains a number of interpolation theorems for vector valued holomorphic functions in [691]. See also M. Schottenloher [636] . This completes our survey of the Levi problem and the Cartan-Thullen theorem in infinite dimensional spaces. Our analysis has hopefully shown their central role in infinite dimensional holomorphy and their importance
Appendix I
374
in motivating new ideas and concepts. This direction of research still contains many open problems, e.g. the Levi problem has not been solved and no Cartan-Thullen theorem exists for arbitrary domains in separable Banach
spaces. Indeed the reader will no doubt have observed that all known positive results on the Levi problem involve an approximation property assumption and this excludes certain separable Banach spaces. Further references for the above topics are J. Horvath [349], W. Bogdanowicz [77] D. Burghela and A . Duma [110], E. Ligocka and J. Siciak [446], E. Ligocka [444,445], M. Her&
[325,326], J. Bochnak and J. Siciak [75], C.E. Rickart
[606], S. Baryton [54], I.G. Craw [170], S . Dineen [193], G. Coeur6 [132, 133,1341, G. Katz [372], J. Kajiwara [365], V. Aurich [34,37], Y . Hervier [330], L.A. de Moraes [495,496,497], A . Bayoumi [SS], M.G. Zaidenberg [719], S.J. Greenfield [279] and Y. Fujimoto [249].
a
operator can be In finite dimensions fundamental solutions of the obtained from the potential kernel, i.e. from a fundamental solution of the Laplacian. L. Gross [284] (see also P . Lgvy [442]) has studied infinite dimensional generalizations of the potential kernel and found, because of the absence of a translation invariant (i.e. Lebesgue) measure on an infinite dimensional locally convex space, that the natural setting for finding fundamental solutions of the Laplacian was an abstract Wiener space with its associated Gaussian measure. A triple (j,H,B) is called an abstract Wiener space if H
is a separable Hilbert space, B is a Banach space, j
is a continuous injection of H onto a dense subspace of B
and the norm
of B is, via j, a "measurable" norm on H (if for instance, H=B and j is a Hilbert-Schmidt operator with non-zero eigenvalues, then (j,H,H) is an abstract Wiener space). The canonical Gaussian "measure" on H leads to a true measure on C.J. Henrich
B
for any abstract Wiener space
[322] was the first to investigate the
5
(j,H,B). equation in
an infinite dimensional setting. His approach was influenced by the work of L. Gross [284] on the infinite dimensiondl Laplacian, by H. Skoda's research [662] on the finite dimensional
5 equation and by the work of
L. Hormander [346] on 'L estimates for partial differential operators. C.J. Henrich's work is very fundamental, quite delicate (even the statement of the equation and the interpretation of the solution necessitate a careful examination) and his ideas have influenced later developments. His main result is the following:
375
Further developments
if H
is a separable Hilbert space and
w
is an
(0,l)
form on H which factors through an abstract Wiener space as a closed form of polynomial growth, then there exists a
,Ad -
an
(*I
function of polynomial growth on H,a, such that
= w.
The condition on w abstract Wiener space ial growth on B
in
means the following: there exists an
(*)
(j,H,B),
a
3 closed
(0,l)
form
-
w
o f polynom-
such that the following diagram commutes j _ _ _ _ f
A0”(B)
B
Equivalently we may say that
(*)
is a solution to the
3
equation on a
dense subspace o f H. In [421], B. Lascar shows that Henrich’s solution can be extended to the whole space (i.e. to H) as a distributional solution to the
a equation.
A summary of the work of C . J . Henrich is given in [364] by J. Kajiwara. The formula for Henrich’s solution is very technical, mainly because Gaussian measures are not invariant under translation and this leads to complicated terms when differentiating under the integral s i p . In [187], S. Dineen used transfinite induction and sheaf cohomology to show that each infinitely dteaux differentiable closed (0,l) form on a finitely open pseudo-convex subset Q of a complex vector space is the image by 3 of an infinitely G&eaux differentiable function on 0. I n his study of the representation o f distributions by boundary values
of holomorphic functions, D. Vogt [701] encountered the vector valued 5 spaces (definition 5 . 3 8 ) . He proved the foll-
problem and discovered DN owing result [701]. are equivalent: 1)
If E is a 3 3 h s p a c e , then the following conditions
each E-valued distribution of compact support in R may be
Appendix I
376
r e p r e s e n t e d as t h e boundary v a l u e o f an element o f H(C\R;E),
a
(2) t h e mapping
is a
(3) E b
:.hm,(R2 ; E )
- - + g ( R 2 ; E ) is surjective,
space.
DN
$ a = w on a convex open s u b s e t
A.Rapp [601,602] s o l v e d t h e e q u a t i o n
o f a Banach s p a c e w i t h r e g u l a r boundary when t h e c l o s e d form
i s of
w
s u f f i c i e n t l y slow growth n e a r t h e boundary and E . Ligocka [444,445] o b t a i n -
&'
ed a solution f o r
f u n c t i o n s o f bounded s u p p o r t on .a Banach s p a c e .
Both used s t r a i g h t f o r w a r d g e n e r a l i z a t i o n s o f t h e f i n i t e dimensional method. Next, P . Raboin made a number o f i m p o r t a n t c o n t r i b u t i o n s by r e t u r n i n g t o t h e approach of C . J . he d e f i n e d t h e space
Henrich L'
of
and u s i n g Gaussian measures. (0,q)
9 i n t e g r a b l e w i t h r e s p e c t t o t h e Gaussian measure
H i l b e r t space
H
i~ on
t a t i o n f o r t h e adjoint of
T
9
the separable
5 t o L 2 was 9 9 After obtaining an i n t e g r a l represen-
and showed t h a t t h e r e s t r i c t i o n
a c l o s e d o p e r a t o r with dense range.
I n [587,589]
d i f f e r e n t i a l forms which a r e s q u a r e
of
T
and e s t a b l i s h i n g a p r i o r i e s t i m a t e s ( i n t h e
manner o f L . Hormander [346] f o r t h e f i n i t e dimensional c a s e ) t h a t each c l o s e d form i n
H e proved t h a t e a c h of
L2
3
was t h e form i n
a
L21
image o f a member o f
I n [589], Raboin showed t h e e x i s t e n c e o f a
problem f o r
&"
closed
(0,l)
L2q.
was t h e image o f an element
whose r e s t r i c t i o n t o a c e r t a i n d e n s e subspace o f
function. the
L2q+l
.g" c l o s e d
he proved
H
"&""
was a
l'al''
solution t o
forms, bounded on bounded s e t s , and
e x t e n d e d t h i s r e s u l t i n [593], ( s e e a l s o
[ 5 9 0 , 5 9 1 , 5 9 2 ] ) , t o pseudo-convex
domains i n a H i l b e r t s p a c e by u s i n g a g e n e r a l i z e d Cauchy i n t e g r a l formula for
&-
functions.
I n [137], G . Coeurg g i v e s a n example o f a the unit ball
B
5 1 f u n c t i o n on
R1 c l o s e d
(0,l)
o f a H i l b e r t s p a c e which i s n o t t h e image by
5
form on
o f any
B.
The n a t u r a l s t e p from H i l b e r t s p a c e s t o n u c l e a r s p a c e s , s u g g e s t e d by C . J . Henrich
[322], was t a k e n by P . Raboin i n [588,590,591,592,593].
[593], h e proved t h a t any
,&"
closed
(0,l)
form, s a t i s f y i n g a modest
t e c h n i c a l c o n d i t i o n on a pseudo-convex open s u b s e t
R o f a 83-Qs p a c e
In
377
Further developments
with a basis was the image by
5 of a
,(.' function on R .
In [164],
J.F. Colombeau and B. Perrot prove that every ,&"
closed ( 0 , l ) form on a E is the image by 5 of a 4" function on E (see also the remark by P. Kre'k in 56.0 of [418]) and in [166] they extend this result
83n space
to pseudo-convex domains in E by D. Nosske 15311).
(this result was also found, independently,
The initial version of J.F. Colombeau's and
B. Perrot's solution to the 2 problem [166] was considerably simplified by a result on hypoellipticity (due to P. Mazet [480]) which yields, as a particular case, the following: Any Ggteaux
A"
is a (Frgchet)
solution to the
a" solution.
2
problem which is locally bounded
Recently, R. Meise and D. Vogt [488], have shown that the solvability of the
a
problem on a nuclear Frgchet space E
property DN
implies that E has
(definition 5.38).
Applications of the infinite dimensional
a operator to natural
/
Frechet algebras are given in B. Kramm [398] and to convolution operators by J.F. Colombeau, R. Gay and B. Perrot in [148]. Application of the 5 operator to the Cousin I problem are discussed below. SHEAF THEORY Sheaf theory and sheaf cohomology play an important role in several complex variable theory and it is probable, see for instance B. Kramm [398], that the same remark will eventually apply to infinite dimensional holomorphy. Of key importance for finite dimensional holomorphy are theorems A and B of H. Cartan [115]. Theorem B states that HP ( X , 3 ) = 0 f o r any and any coherent analytic sheaf 3 on the Stein manifold X. Theorem B can be used to solve the a problem and to resolve the Cousin I p3 1
problem (also called the additive Cousin problem) on holomorphically convex domains in En.
Classically the Cousin I problem was to find a several
complex variable version of the Mittag-Lefflertheorem - which showed the existence o f a meromorphic function in any domain of Q: with preassigned poles. The several complex variables version sought to characterise within the collection of principal parts on a domain X in Cn those which gave rise to a meromorphic function on in Ells].
X. This problem was solved by H. Cartan
318
Appendix 1
In recent years, various authors (e.g. L. Hormander [ 3 4 7 ] , C.E. Rickart [605] and P. Raboin [588]) have assigned the terminology "Cousin I problem" to a more general collection of problems of which the following is typical : given a covering
(Ui)iEI
locally convex space E, such that
of a domain X, and hij
E
spread over a
H(Ui"U.)
for all
3
i,j € 1
for all i,j and k in 1,does there exist a family (hi)iEI, hi E H(Ui) such that hi-h. = h.. on U i n U . for all i and j in
1
I?
11
3
Using Cech cohomology we see that (**) has a solution for any set of data 1 { U . ,h. .I if H (X,@) = 0 where denotes the sheaf of holomorphic germs 1 1J on X. It is easy to show that a generalised Mittag-Leffler theorem is valid on X whenever H 1(X,t3) = 0 . Banach algebra considerations motivated the first examples of sheaf cohomology with values in a sheaf of holomorphic germs in infinitely many variables. In [12], R. Arens proved that H p ( g , @ I ) = 0 for any pzl A where A is a Banach algebra with continuous dual A' and spectrum ;6 and where 8,, is the sheaf of weak* holomorphic germs on A'. this result to show H1($,Z)
{xEA, x invertible} /;exp(x)
He applied
; XEA}
(see also R. Arens [13] and H. Royden [610]). In [605], C.E. Rickart proved that Hp(K,o) = 0 for any p>l and any polynomially convex compact subset K of CA, A arbitrary. C.E. Rickart [605] also states and solves a Cousin I problem on the set K and applies it to prove the Rossi local maximum modulus principle for Banach algebras.
P. Silici showsin
[659]
that theorems A
and
B
are valid f o r
379
Further developments
IC . H1(U,CG) = 0 A
compact p o l y d i s c s i n proved t h a t
By u s i n g t r a n s f i n i t e i n d u c t i o n , S . Dineen [187] f o r any f i n i t e l y open pseudo-convex domain
U
i n a complex v e c t o r s p a c e , where P G i s t h e s h e a f o f GGteaux holomorphic
8
germs, and used t h i s r e s u l t t o s o l v e t h e Levi problem and t h e f o r Gzteaux holomorphic and d t e a u x
8" f u n c t i o n s .
problem
J . Kajiwara [368]
extended t h i s r e s u l t t o the h i g h e r cohomology groups on f i n i t e l y open pseudo-convex domains i n p r o j e c t i v e s p a c e ( s e e a l s o Y . Fujimoto [ 2 4 9 ] ) .
In
[192], S. Dineen showed t h a t Cousin I i s n o t s o l v a b l e , and h e n c e H1(U,U) # 0
2
and t h e
problem i s n o t s o l v a b l e , f o r any domain
U
in a
l o c a l l y convex s p a c e which does n o t admit a c o n t i n u o u s norm and i n [35] V . Aurich proved t h a t a g i v e n f a m i l y o f p r i n c i p a l p a r t s on a S t e i n m a n i f o l d
spread over
CA,
a r b i t r a r y , g i v e s r i s e t o a meromorphic f u n c t i o n if and
A
o n l y i f t h e p r i n c i p a l p a r t s a l l f a c t o r t h r o u g h some
Cn.
The n e x t development i s due t o P . Raboin [ 5 8 8 ] who proved, u s i n g h i s s o l u t i o n t o the
2 problem, t h e f o l l o w i n g Cousin I r e s u l t ;
pseudo convex domain i n a F r g c h e t n u c l e a r s p a c e
E
n is a
if
w i t h a b a s i s and
t h e n f o r e a c h convex compact i s a s e t o f Cousin I d a t a on {$li,gij}i,j b a l a n c e d s u b s e t K o f E t h e r e e x i s t s a f a m i l y I f i E H ( Q . n EK)}i such that
g..
11
=
fi-fj
on
Q i o Q . n E Kf o r
space w i t h cl osed u n i t b a l l t o t h e topology o f
1
In
f o r any pseudo-convex domain i n v o l v e d a s o l u t i o n of t h e that
B 3-n
of unity.
and each
K
EK.)
i
fi
and
j.
1
(EK
in
a
&33n
i s t h e Banach
i s holomorphic with r e s p e c t
[593], P . Raboin proved t h a t
U
5
all
1
H ( Up) = 0
space with a b a s i s .
H i s proof
problem, t h e Oka-Weil theorem and t h e f a c t
A" p a r t i t i o n s a3-l s p a c e s by J . F .
s p a c e s a r e h e r e d i t a r y L i n d e l a f s p a c e s and admit T h i s r e s u l t was e x t e n d e d t o a r b i t r a r y
Colombeau and B. P e r r o t [164,166].
Theorems A and B o f H. C a r t a n have been extended t o v e c t o r v a l u e d holomorphic f u n c t i o n s on a f i n i t e d i m e n s i o n a l s p a c e by L . Bungart [ l o g ] . T h i s completes o u r d i s c u s s i o n o f c o n d i t i o n s ( a ) , ( b ) , ...,(f ) f o r i n f i n i t e dimensional spaces. DIFFERENTIAL EQUATIONS We now d i s c u s s c o n v o l u t i o n o p e r a t o r s and p a r t i a l d i f f e r e n t i a l o p e r a t o r s
on s p a c e s of holomorphic f u n c t i o n s o v e r l o c a l l y convex s p a c e s . A s t h i s s u b j e c t forms p a r t o f a book i n p r e p a r a t i o n by J . F . Colombeau, o u r
Appendix Z
380
presentation will be brief and concentrate mainly on the role played by this topic in the general development of infinite dimensional holomorphy. C.P. Gupta [295] was the first to consider convolution operators on spaces of holomorphic functions over locally convex spaces and his approach influenced many later workers in this area. The main finite dimensional considerations of C.P. Gupta were the results and techniques o f B. Malgrange [448] and A. Martineau [452 A simplified description of the basic approach used by C.P. Gupta goes A a locally convex translation invariant space of
as follows. Given
holomorphic functions on the locally convex space E , a convolution operator on A is defined as a continuous linear operator from A into itself which commutes with all translations. For operator has the form where
C"
an
&
=
H(C)
each convolution
dn
1 -
The Bore1 transform establishes a one-to-one correspondence between convolution operators on
&
, the elements of
and a space of holomorphic
functions of exponential type on E ' . The existence and approximation problem for convolution operators is then transposed and solved as a division problem for holomorphic functions on El. C.P. Gupta's [295] investigation of convolution operators on Banach spaces led him to HNb(E), the space of holomorphic functions of nuclear bounded type on E, and to the correspondence HNb(E);I = Exp(E;). He showed that every convolution operator on H (E) was surjective and that Nb solutions of the associated homogeneous equation could be approximated by exponential polynomial solutions. Extensions of this method to more general classes of locally convex spaces and to other collections of holomorphic functions are given in C.P. Gupta [296,297], L . Nachbin [511], P.J. Boland [79,80,81,82], M.C. Matos [458,463,464,467], S . Dineen [177], P.J. Boland and S. Dineen [ 8 8 ] , T.A.W. Dwyer [218,221,222,223,225], P . Berner [62], D. Pisanelli [580,58l], J.F. Colombeau and M.C. Matos [150,151], J.F. Colombeau and B. Perrot [163,167], F. Colombeau, T.A.W. Dwyer and B. Perrot [147], and J.F. Colombeau and J. Mujica [156].
38 1
Further developments A d i f f e r e n t approach i s t a k e n by T.A.
Dwyer [214,215,216,219]
(see
a l s o 0 . Bonnin [94]) i n s t u d y i n g p a r t i a l d i f f e r e n t i a l o p e r a t o r s on holomorHe d e f i n e s t h e Fock s p a c e s
p h i c Fock s p a c e s o f H i l b e r t - S c h m i d t t y p e . Yp(E)
on a H i l b e r t s p a c e
(and a f t e r w a r d s on c o u n t a b l y H i l b e r t s p a c e s
E
and o t h e r c l a s s e s o f l o c a l l y convex s p a c e s , see a l s o J . Rzewuski [621,622]) and shows t h a t
/I PflI P
2
11 Pm(I
partial differential operator
. /I f Ilp
P(D)
7
f o r any f i n (E) and any P m Pn(D). Using t h i s i n e q u a l i t y
=
Dwyer showed t h a t a l l such p a r t i a l d i f f e r e n t i a l o p e r a t o r s map
3P (E)
3 P (E)
onto
and g e n e r a l i s e d a number o f f i n i t e d i m e n s i o n a l r e s u l t s ( s e e F . N o t a b l e a s p e c t s of Dwyer's work, s e e t h e refer-
T r e v e s [686], c h a p t e r 9 ) .
are h i s c o n c r e t e r e p r e s e n t a t i o n o f
e n c e s c i t e d above and [224,226,227], c o n v o l u t i o n o p e r a t o r s by means o f
( V o l t e r r a ) k e r n e l s , e t c . and h i s
'L
r e c o g n i t i o n of a r e l a t i o n s h i p between c e r t a i n a b s t r a c t d i f f e r e n t i a l e q u a t i o n s i n l o c a l l y convex s p a c e s and problems i n c o n t r o l t h e o r y , a n a l y t i c b i l i n e a r r e a l i z a t i o n s , quantum f i e l d t h e o r y , e t c . ( s e e a l s o J . F . Colombeau and B . P e r r o t [158,162], J . F . Colombeau [145], P . Kr6e [401,410,417] and P . Kr6e and R. Raczka [ 4 1 9 ] ) . The long term r e l e v e n c e o f c o n v o l u t i o n
o p e r a t o r s i n i n f i n i t e l y many v a r i a b l e s may w e l l depend on t h i s k i n d o f r e c o g n i t i o n and i n s i g h t . The most r e c e n t developments i n t h i s g e n e r a l d i r e c t i o n a r e due t o J . F . Colombeau, R . Gay and B . P e r r o t [148].
They p r o v e , u s i n g a p r e p a r -
a t i o n theorem f o r holomorphic f u n c t i o n s on a Banach s p a c e due t o J . P . f&' ( Q ) =
holomorphic f u n c t i o n
on a c o n n e c t e d domain
nuc le a r space
E
s o l u t i o n s of t h e T
f
(Q)
Q
i n a quasi-complete dual
and a p p l y t h i s r e s u l t t o g e t h e r w i t h t h e e x i s t e n c e o f
5
problem on
i s a c o n v o l u t i o n o p e r a t o r on
p € H M ( E ) t h e n any s o l u t i o n t r a n s f o r m o f a n element
U
f
ajQ
A"
spaces t o prove t h e following:
Exp(E')
with c h a r a c t e r i s t i c function
of t h e equation
o f %'(E)
Ramis
f o r any non-zero Mackey (or S i l v a )
[598] ( s e e b e l o w ) , t h a t
f o r which
Tf = 0
pU = 0.
i s t h e Bore1 The f i n i t e
d i m e n s i o n a l a n a l o g u e s o f t h e s e r e s u l t s are due t o L . Schwartz [647] and R . Gay [254] r e s p e c t i v e l y .
The t h e o r y o f c o n v o l u t i o n o p e r a t o r s drew a t t e n t i o n t o t h e r o l e o f n u c l e a r p o l y n o m i a l s i n t h e g e n e r a l t h e o r y of holomorphic f u n c t i o n s i n i n f i n i t e l y many v a r i a b l e s and p r o v i d e d t h e f i r s t examples o f a f u n c t i o n space r e p r e s e n t a t i o n o f i n f i n i t e dimensional a n a l y t i c f u n c t i o n a l s .
The
if
382
Appendix 1
appearance of nuclearity motivated L. Nachbin [508,509] to introduce the concept of holomorphy type as a means of investigating holomorphic functions whose derivatives pertained to a certain class of polynomials (e.g. compact, Hilbert-Schmidt, nuclear, etc.) and whose Taylor series expansion satisfied growth conditions relative to the canonical semi-norms on the underlying spaces of homogeneous polynomials. The theory of holomorphy types has been developed in various directions by L. Nachbin [508, 509,511,512], S. Dineen [177], R.M. Aron [15,16,17,19], T.A.W. Dwyer L214, 216,221,222,2231, P.J. Boland [78,80,81], S.B. Chae [119,120] and L.A. de Moraes [495,496,497], and led, eventually, through the work of Boland, to the theory of holomorphic functions on nuclear spaces as outlined i n chapters 1 , 3 , 5 and 6. The Bore1 transform and the correspondence between analytic functionals on H ( E ) and holomorphic functions of exponential type on E' were almost totally developed within the framework o f convolution operators as outlined above (see the references listed previously and also T.A.W. Dwyer [217,220]), and although in this text we have more o r less exclusively preferred the representation of analytic functionals by holomorphic germs the motivations and guidelines arising from the exponential representation were always suggestive and significant. ANALYTIC GEOMETRY The first comprehensive treatments of analytic sets in infinite dimemsions are due to P.J. Ramis [594] and G . Ruget [613], both of whom worked only in Banach spaces but were aware that many of their results and techniques extended to locally convex spaces. Most of the other important developments in this area are due to P . Mazet [479]. As in the finite dimensional case (see for instance M. He&
[324]) the local theory is
first developed by studying the ideal structure of the commutative ring (the space of holomorphic germs at the origin in the locally convex space E), and then applied to obtain global results. The ring @(E) is
Q(E)
an integral domain and a local ring but is Noetherian if and only if E finite dimensional. Since the Noetherian property of W(cCn) plays a
is
crucial role in the finite dimensional theory, new methods in commutative algebra had to be developed and these appear to be o f independent interest.
383
Further developments
Two of the key results in both the finite and infinite dimensional theories are the Weierstrass Factorization and Preparation Theorems. Weierstrass Factorization Theorem If g # 0 E *(E)
then there exists a decomposition of E, El @Ce,
that the restriction of g
to Ce has order p # 0 and for any
there exists a unique polynomial U(E)
(J.P. Ramis [594] and P. Mazet [ 4 7 9 ] ) .
r of degree < p and a unique q
such f
E
@(E)
in
such that f = g.q+r.
A distinguished polynomial relative to the decomposition El @Ce a mapping of the form (zl,z)-zP
P-1
1
+
is
ai(zt)z i
i=O where ai(zf)
E
@(El)
and ai(0) = 0 for i
=
0,...,p-1.
The Weierstrass Preparation Theorem If 0 # g E O(E) has order pbl then there exists a decomposition of E, El @Ce, such that g can be written in a unique fashion, g = h.P, where h of @ ( E )
is an invertible element
and P is a distinguished polynomial of degree p
relative to
the decomposition El 8 Ce. Using the above theorems one shows [594,479], that U(E) is a unique factorization domain. In [ 2 3 6 ] , M.J. Field uses the Factorization Theorem to prove that a germ in @ ( E )
is irreducible if and only if its
restriction to some sufficiently large finite dimensional subspace is irreducible.
A subset X of a complex manifold modelled on a locally convex space is called analytic if for each x in X there exists a triple (Vx,fx,Fx) where Vx is an open neighbourhood of x, Fx is a locally convex space, fx E H(Vx;Fx) and X n V x = {YE Vx;fx(y) = 0 1 . If Fx can be chosen to be finite dimensional (respectively one dimensional) for each x in X then we say X is finitely defined (respectively a principal analytic set or a hypersurface). Thus, a finitely defined analytic subset is one which is locally defined by a finite number of scalar equations. An example of an analytic subset, not generally finitely defined, is the spectrum of a
384
Appendix I
commutative Banach algebra polynomial on If
A.
This s e t i s t h e zero s e t of a 2 homogeneous
(see J . P . Ramis [598,p.32] and B. Kramm [398]).
A'
i s an a n a l y t i c subset of a complex manifold
X
l o c a l l y convex space
E
e x i s t s a decomposition
then
of
in
U
and a biholomorphic mapping
in
E
such t h a t
p o i n t s of x.
each a n a l y t i c germ
The i d e a l
U
We l e t
of
a
onto a neighbourhood of
0
X*
denote t h e r e g u l a r
i s c a l l e d t h e (geometric) codimension of
a,
V(f)a
I(Xa)
X
at
i n Ua(E),
X
at
a.
To
t h e space
by l e t t i n g
i s an i r r e d u c i b l e germ.
Xa
V
I(Xa) = { f ~ @ ~ ( E ) ; =f \0~) . a i s equal t o i t s r a d i c a l and i s a prime i d e a l i f and only
I(Xa)
associate the
where
of
$
we a s s o c i a t e an i d e a l
Xa
i f there
X
w i l l denote t h e a n a l y t i c germ of t h e a n a l y t i c subset
Xa
of holomorphic germs a t
if
modelled on a
an open neighbourhood
E,
$(XnV) = $(V)ftE1.
and dim(E2)
X
i s a regular point of
a E X
El 8 E 2
U
To each i d e a l
9
i n Ua(E)
we may
"object"
i s t h e a n a l y t i c germ a t
r e p r e s e n t a t i v e of t h e germ ( f o r example, i f
dim(E) <
m)
a
9
If
f.
then
defined by t h e zero s e t of some
&
i s generated by
V(3) =
ifl,.
V(fi)a
. .,fn}
i s an a n a l y t i c
germ of a f i n i t e l y defined a n a l y t i c subset b u t , unfortunately, f o r a r b i t -
9
trary
the object
V(3)
i s n o t i n general an a n a l y t i c germ.
The main problem f o r t h e l o c a l theory i s t h e following: ideals
9
i n W(E),
given t h a t
V(3)
i s an a n a l y t i c germ a t
f o r what 0,
do we
have Rad(3)
=
I(V(g))?
The f i r s t r e s u l t on t h i s problem, a N u l l s t e l l e n s a t z f o r p r i n c i p a l i d e a l s ,
i s due t o J . P . Ramis [598,p.29] (see a l s o M . J . F i e l d "2361 and t h e d i v i s i o n theorem of C . P . Gupta [295,proposition 131) and t h i s may be s t a t e d as follows : if
E
i s a Banach space,
glv(f) = 0 such t h a t
f
and
g
E
W(E)
and
then t h e r e e x i s t s a p o s i t i v e i n t e g e r gm
generated by
m
belongs t o t h e p r i n c i p a l i d e a l i n @(E) f.
385
Further developments In o r d e r t o i d e n t i f y c e r t a i n i d e a l s with germs o f a n a l y t i c s u b s e t s ,
J . P . Ramis introduced t h e concept of geometric i d e a l and proved t h e followIf
ing result:
i s a Banach space and
E
then t h e following a r e e q u i v a l e n t :
9
i s a prime i d e a l i n B(E)
i s a geometric i d e a l ( i . e . t h e r e e x i s t s a f i n i t e l y
(a)
1C I ( V ( 9
generated i d e a l
9
in &(E)
where
X
i s an i r r e d u c i b l e germ of a f i n i t e l y
(b)
= I(X)
such t h a t %
))
defined a n a l y t i c s u b s e t . (c)
ht(
is f i n i t e (the height of
)
, ht(]
is the
)
length of a maximal proper chain of prime i d e a l s j o i n i n g and
(0)
(d)
8
)
1.
has a normal decomposition ( i . e . t h e r e e x i s t s a
decomposition o f such t h a t @ ( E l ) n
1
with dimension
E1&E2,
E,
and
= (0)
V() )
n E2
=
(E2)<"
(0)).
The equivalence of (b) and (c) i s a N u l l s t e l l e n s a t z f o r prime i d e a l s of f i n i t e height.
9
at
The dimension o f
( i n f a c t it i s equal t o 0).
i n condition (d) i s an i n v a r i a n t of
E2
ht(Q ) )
and i s ca(1ed t h e codimension of
X
I t i s n o t d i f f i c u l t t o s e e t h a t t h i s concept of codimension
coincides with t h a t given e a r l i e r f o r r e g u l a r p o i n t s of an a n a l y t i c s u b s e t . I t i s n o t known i f every prime i d e a l of f i n i t e h e i g h t i n f i n i t e l y generated.
is
@(E)
An e s s e n t i a l p a r t o f t h e proof of t h e above r e s u l t
uses ramified coverings and t h i s concept i s a l s o important i n t h e l a t e r works o f P . Mazet [475,477,479]
and V. Aurich [ 3 8 ] .
The next developments a r e due t o P . Mazet [471,472,473] J . P . R a r n i s [598] and G . Ruget [613,614]).
(see a l s o
Mazet proved t h e following
result: if
3
i s an i d e a l i n O(E),
by
n
elements
have height
E
a Banach space,which i s generated
then t h e minimal prime i d e a l s containing
3
all
,
I n t h i s way, s i n c e
Rad(3)
i s then t h e i n t e r s e c t i o n o f a f i n i t e
number o f prime i d e a l s o f f i n i t e type, Mazet obtained a N u l l s t e l l e n s a t z f o r f i n i t e l y generated i d e a l s .
Afterwards, he introduced [474,476,479]
the
Appendix I
386
concept of a C.M,
ring which generalized the notion of a Cohen-Macauley
ring (see 0 . Zariski and P. Samuel [ 7 2 2 , Appendix 61).
He proved, using
ramified coverings and techniques from homological algebra, that @ ( E ) a
C.M,
ring for any locally convex space E
is
and subsequently obtained
the Nullstellensatz for finitely generated ideals in @ ( E ) .
8
If 9
9c 1
i s an ideal in LJ (E)
finitely generated ideal E l @ E2
we let ht(
9)
=
inflht(3 ) ;
and for any analytic subset X we let codim Xa [473] and
9
in 13 (E)
Codim V ( 9 )
=
3 prime and
ht(I(Xa)).
Any
admits a normal decomposition
i s , in this case, also equal to the dimen-
sion of E 2 . Finally we list some properties and examples of analytic subsets which follow (some not exactly immediately) from the above results. Many of the
results we quote are to be found in J . P . Ramis [598] and apply to analytic subsets of a Banach space. For results on locally convex spaces we refer to P . Mazet [479] where the reader will find applications of infinite dimendimensional analytic geometry to the theory of finite dimensional analytic
If X
k
is a finitely defined analytic germ then
X = U X . where each Xi is a finitely defined 1=1 1 irreducible analytic germ. This decomposition of X is unique modulo a permutation of the X.'s given that no proper inclusions are allowed (see P. Mazet [ 4 7 3 ] ) . The set of regular points X* of a finitely defined analytic subset X is dense in X. If X
is an analytic subset of an infinite dimensional
Banach manifold U
and Xx
does not contain a principal
analytic germ at any point x in X (or equivalently if inf Codim Xx 3 2 ) then every C-valued holomorphic XE
x
function on (see also M . J .
U\X
extends to a holomorphic function on U
Field [ 2 3 6 ] ) .
A locally compact analytic subset of an infinite dimensional
locally convex space has infinite codimension at each point. If m
is a meromorphic function on a complex Banach
387
Further developments manifold
m
then t h e s e t o f p o l e s of
U
a n a l y t i c subset o f
m
minancy of
U
is a principal
and t h e s e t o f p o i n t s o f i n d e t e r -
i s an a n a l y t i c s u b s e t of codimension 2 a t
each p o i n t . (vi)
i s an a n a l y t i c s u b s e t of a Banach manifold and codim X, < then x E X * . I f X E X* then If
X
codim Xx = l i m i n f (Codim X ) . YEX* Y
(vii)
A f i n i t e l y defined a n a l y t i c s u b s e t o f a Banach manifold i s
i r r e d u c i b l e i f and o n l y i f (viii)
i s a f i n i t e l y defined a n a l y t i c s u b s e t o f a complex
X
If
Banach manifold of
i s connected.
X*
U
X\ X* = S(X) , t h e s i n g u l a r s e t
then
i s an a n a l y t i c s u b s e t of
X,
and
U
1 + Codim Xx ,< Codim S(X)x
x
f o r every (ix)
If
in
S(X).
pX
Codim XX = p } .
{xEX;
=
a l l points.
-X =
Let
Then
U
Each p o i n t o f
p=l i n f i n i t e codimension ( a s a p o i n t o f i s an a n a l y t i c s u b s e t .
-X
X)
U
is a finitely
p,
X\upX '.
defined a n a l y t i c subset o f
if
upX
i s an a n a l y t i c s u b s e t o f a complex Banach manifold
X
we l e t
k
at has
"X
but it i s n o t known
The decomposition
X = opXymX p=l
i s known a s t h e canonical decomposition o f t h e a n a l y t i c s u b s e t X .
(x)
Let on
fEH(U;V), and
E
F
U
and then If
into
F
U.
If
f o r every
dim Coker ( d f ( x ) ) <-)
(i.e.
(xi)
E
V
f-'(K) f(X)
X
complex Banach manifolds modelled
X
r e s p e c t i v e l y , and l e t
analytic subset of from
and
df(x) x and
be a f i n i t e l y defined
i s a Fredholm o p e r a t o r
in
flX
U
(i.e.
dim k e r ( d f ( x ) ) <
i s a proper mapping
i s compact f o r every compact s u b s e t
K
i s a f i n i t e l y defined a n a l y t i c s u b s e t o f
i s a f i n i t e dimensional a n a l y t i c space,
E
of
X)
V.
is a locally
Appendix I
388 convex s p a c e ,
i s an open s u b s e t o f
Q
p r o p e r holomorphic mapping o f
X
into
If
and
U
and
R
then
are complex Banach manifolds and
V
is a
$I
@(X)
is a
a.
f i n i t e dimensional a n a l y t i c s u b s e t o f (xii)
E
H(U,V)
f E
i s a p r o p e r Fredholm mapping, t h e n t h e s e t o f c r i t i c a l v a l u e s of
f
is a f i n i t e l y defined a n a l y t i c subset o f
V.
The r e s u l t s o f (x) and ( x i ) are d i r e c t image theorems and g e n e r a l i s e a well known f i n i t e dimensional r e s u l t o f R . Remmert [ 6 0 3 ] .
(x)
i s due
t o J . P . R a m i s and G . Ruget [600] ( s e e a l s o J . P . R a m i s [ 5 9 8 ] and G . Ruget [613]).
( x i ) i s due t o D . Barlet and P. Mazet [475] ( s e e a l s o P . Mazet
[477,479])
and g e n e r a l i s e s a H i l b e r t space result of G . Ruget [613] and
a Frgchet s p a c e r e s u l t o f B. Saint-Loup [623].
( x i i ) i s due t o V. Aurich
[38] and u s e s a d i r e c t image theorem i n i t s p r o o f . Aurich [38] i s v e r y r e c e n t and v e r y i n t e r e s t i n g .
The a r t i c l e o f V . I t contains an i n f i n i t e
dimensional v e r s i o n o f t h e Remmert Graph Theorem, a l o c a l d e s c r i p t i o n o f holomorphic Fredholm mappings and shows how i n f i n i t e dimensional holomorphy may b e used t o u n i f y and extend r e s u l t s a r i s i n g i n o t h e r c o n t e x t s ( s e e a l s o D . Abts [ l ] , H . Arker [ 1 4 ] ) .
F u r t h e r r e f e r e n c e s t o i n f i n i t e dimensional a n a l y t i c geometry and a n a l y t i c sets a r e H. C a r t a n [116,117],
I . F . Donin
[209], A. Douady [210],
Zaidenberg [719] and J . P . R a m i s [594,595,596,597,599].
M.G.
J.P.
Ramis
[599] i s a s u r v e y a r t i c l e and c o n t a i n s a l i s t of open problems and conjectures. HOMOGENEOUS SPACES Domains
and
U
V
i n l o c a l l y convex s p a c e s
p h i c a l l y e q u i v a l e n t , we w ri t e mapping f-'
E
f
from
H(V;U))
.
U
onto
V
U
E
and
F
are holomor-
V , i f t h e r e e x i s t s a biholomorphic
(i.e.
f EH(U;V),
f
i s b i j e c t i v e and
The problem o f c l a s s i f y i n g h o l o m o r p h i c a l l y e q u i v a l e n t
domains i n l o c a l l y convex s p a c e s h a s l e d , mainly due t o t h e e f f o r t s o f L . A . Harris, W .
elegant r e s u l t s .
Kaup and J . P . Vigue', i n r e c e n t y e a r s t o many deep and T h i s branch o f i n f i n i t e dimensional holomorphy u s e s t h e
methods o f d i f f e r e n t i a l geometry,
C*
a l g e b r a s and Lie groups.
Before
d e s c r i b i n g b r i e f l y t h e i n f i n i t e dimensional t h e o r y , w e s k e t c h t h e r e l e v a n t p o r t i o n s o f t h e f i n i t e dimensional t h e o r y and refer t o S. VLgi [689],
389
Further developments
M. Koecher [392], B.A. Fuks [250,chapter 51 and S. Kobayashi [389] for further details. The Riemann mapping theorem states that any two proper simply connected domains in C
are equivalent. By proving that the domains
are not equivalent, H. Poincar6 [585] showed that the Riemann mapping theorem does not extend to En
and opened the way for a classification
theory in several variables. A short elementary proof of Poincare's result is given in H. Alexander [6]. The problem of holomorphically classifying all simply connected domains in Cn proved unwieldy and so attention focused on more "manageable'? classes of domains which still contained the more interesting examples - the homogeneous, symmetric and Siegel\domains. Let
A (U)
denote the set of all biholomorphic mappings from U
itself. The elements of
4 (U)
into
are called (holomorphic) automorphisms of
U. A domain U in a locally convex space E is called homogeneous if for each a,bE U there exists a $ in A (U) such that $(a) = b, i.e. 4 (U) acts transitively on U. A domain U is said to be symmetric if for each a in U there exists a $ in 4 (U) such that $ 2 = @ . $ = Id and a is the unique fixed point o f @. (U) In 1935, H. Cartan [114] prove that is a Lie group when U is a bounded domain in an. Using this result and structure theorems for Lie groups, E. Cartan [112] classified all irreducible bounded symmetric domains in Cn (a domain is irreducible if it is not equivalent to a non-trivial product of domains, any symmetric domain is equivalent to a finite product of irreducible symmetric domains). He showed that there are four classical (i.e. corresponding to "classical" Lie groups of matrices)classes of domains, now called Cartan domains of type I,II,III and IV, each of which can be represented by matrix inequalities (see below) and also exceptional domains in C16 and C2'. An
Appendix I
390
important r o l e , i n E . Cartan's and o t h e r r e s e a r c h e r s ' work i n t h i s a r e a , i s played by pseudo-metrics on
4 (U)
which a r e i n v a r i a n t under
U
d(U)
e q u i v a l e n t l y , f o r which each element of
or,
i s an isometry (see below
for details). A completely new approach was i n i t i a t e d by M. Koecher [390,391,392],
who discovered t h e r e l a t i o n s h i p between Cartan domains and Jordan algebras. This approach was adopted by L.A. H a r r i s [309] who introduced t h e concept of
J*
Let
algebra.
H
and
K
Banach space of a l l bounded l i n e a r operators from a l g e b r a i s a closed subspace &L A
E
&(A*
of
A(H;K)
i s t h e a d j o i n t of
d(K;H)
E
L(H;K)
be complex H i l b e r t spaces and l e t
A).
H
to
such t h a t By t a k i n g
K.
A
AA*A
E
H
and
denote t h e J*
& whenever K
finite
dimensional, one o b t a i n s t h e c l a s s i c a l Cartan domains a s t h e open u n i t b a l l b a l l s of t h e following
a l g e b r a s ( t h e Cartan F a c t o r s ) ,
J*
Type 1
CL =
d
Type I1
=
{A
€ X ( H ; H ) ,A*
Type 1 1 1
GL &
=
{A
E
Type I V
& i s a s e l f - a d j o i n t subalgebra of J ( H ; H ) AL
in Every
,&*
(H;K)
&H;H),
A* = -A}
.&*
such t h a t
i s a s c a l a r m u l t i p l e of t h e i d e n t i t y f o r each
a.
algebra i s a
J*
A
algebra and r e c e n t l y , L . A . H a r r i s [315]
has shown t h a t an a l g e b r a i c theory f o r for
= A}
J*
algebras, s i m i l a r t o t h a t known
a l g e b r a s , e x i s t s and includes a s p e c t r a l decomposition theorem f o r
s e l f - a d j o i n t o p e r a t o r s and a f u n c t i o n a l c a l c u l u s . We now t u r n t o t h e i n f i n i t e dimensional theory.
The f i r s t r e s u l t
obtained i n t h i s a r e a ( S . J . Greenfield and N.Wallach [ 2 8 1 ] ) i s s i m i l a r i n s p i r i t t o t h e f i r s t r e s u l t o f H. Poincare/ i n t h e f i n i t e dimensional theory. I t s t a t e s t h a t t h e open u n i t b a l l
B
of an i n f i n i t e dimensional H i l b e r t
space i s n o t holomorphically equivalent t o
BxB.
I n a f u r t h e r paper [282]
Dn i s t h e open u n i t b a l l of t h e i n f i n i t e dimensional Cartan I Factor $(Cn,a2) ( t h e case n = l was t h e same authors c h a r a c t e r i z e A ( D n )
where
a l s o found, independently, by A. Renaud [604]) and they show t h a t
,D
a r e holomorphically equivalent i f and only i f
n=m.
Dn
and
391
Further developments Harris showed t h a t t h e open u n i t b a l l o f a
I n [309], L . A .
J*
algebra
i s a homogeneous symmetric domain and any biholomorphic mapping between t h e
CL,
open u n i t b a l l s LoTg
where
TB
and
and
&ao,
B
L
a0
J* a l g e b r a s
of the
i s a s u r j e c t i v e l i n e a r isometry o f Go i n t o i t s e l f ,
i s a Mobius t r a n s f o r m a t i o n o f T* (A)
=
& and $ h a s t h e form
CL o n t o 63
i.e.
1 1 2 2 (I-BB*) (A+B)( I + B * A ) -(I-B*B) ~
.
I n t h e same p a p e r , h e showed t h a t no i n f i n i t e dimensional C a r t a n domain o f type I - I V i s holomorphically equivalent t o a n o n - t r i v i a l product o f b a l l s . I n a more r e c e n t p a p e r [314], L . A . Harris shows t h a t t h e i n f i n i t e dimens i o n a l analogues o f t h e classical C a r t a n domains of d i f f e r e n t t y p e s a r e n o t holomorphically e q uivalent. Schwarz's lemma.
A l l o f t h e above a u t h o r s make e x t e n s i v e u s e o f
I n [377], W . Kaup and H . Upmeier show t h a t Banach s p a c e s
w i t h b i h o l o m o r p h i c a l l y e q u i v a l e n t u n i t b a l l s a r e i s o m e t r i c a l l y isomorphic ( s e e a l s o L . A . Harris [ 3 1 3 , p . 3 8 8 ] ) . C . E a r l e and R. Hamilton [228]
i n t r o d u c e d t h e concept o f i n v a r i a n t
m e t r i c f o r holomorphic f u n c t i o n s i n i n f i n i t e dimensions i n o r d e r t o prove a f i x e d p o i n t theorem ( s e e e x e r c i s e 4 . 5 4 ) and i n r e c e n t y e a r s , i n v a r i a n t
metrics have been s t u d i e d and a p p l i e d i n i n f i n i t e dimensional holomorphy by L . A . Harris [313,314], J . P . Viguk [696,697,698,700], W. Kaup [375], M.
H e r d [327,328], S . J . G r e e n f i e l d and N.Wallach [281] and T . Hayden and T. S u f f r i d g e [319].
A d e t a i l e d and v e r y r e a d a b l e account o f Schwarz-Pick
systems o f pseudometrics can b e found i n L . A . Harris [313] ( t h i s p a p e r a l s o c o n t a i n s a set o f e x e r c i s e s and a l i s t o f open problems) and fundamental p r o p e r t i e s o f t h e Caratheodory metric are g i v e n i n J . P . Vigug [696, Appendix].
The c o r r e s p o n d i n g t h e o r y f o r
a l g e b r a s i s developed by L.A.
J*
Harris i n [314].
The f u n c t i o n PD(Z1,z2)
where
z1,z2
=
tanh-'
z1- 2 2
lie in the unit disc /
metric, t h e P o i n c a r e metric, on
D
=
D
log
z
1z1-z21+ 11-z 1 2 I
o f t h e complex p l a n e d e f i n e s a
with t h e u s e fu l p ro p e rt y t h a t
Appendix 1
392
Consequently, any holomorphic automorphism of D is a p D isometry. p D is not a Euclidean metric but is equivalent to the Euclidean metric on D inherited from C.
A system ( L . A . Harris [313,p.356]), which assigns a pseudo-metric t o each domain in every normed linear space is called a Schwarz-Pick system if the following conditions hold: (i) (ii)
the pseudo-metric assigned to D if
and
p1
and V
p2
are the pseudo-metrics assigned resp. to U
then p2(f(z),f(y))
f e H(U;V)
is the Poincare metric,
and x,y
E
< pl(x,y) for all
U.
The Caratheodory pseudo-metrics, Pc,
defined on U
by
form a Schwarz-Pick system and are the smallest of all pseudo-metrics
p
which satisfy (the Schwarz-Pick inequality) for all
fEH(U;D)
and all x,y
in U
On the other hand, the Kobayashi pseudometric PK, domain U by
pK(x,y)
where p;(x,y)
=
=
n inf{Ci=l~i( X ~ - ~ , X ~x)i;e u
inf {PD(z,w);z fEH(D;U)
= f(x),
all
defined on a
i, xo=x, xn=y}
w=f(y)},
also form a Schwarz-Pick system and are the largest pseudometrics which satisfy
393
Further developments p ( f ( z ) , f ( w ) ) 8 pD(z,w) If
Z,WE
D
and
i s a pseudo-metric on a bounded domain
p
f E H(D;U) U
i n t h e normed l i n e a r
a r i s i n g from a Schwarz-Pick system, then t h e r e e x i s t p o s i t i v e
E
space
for a l l
numbers
A
and
tanh-'
A
and consequently norm of
such t h a t
B
p
i s t o p o l o g i c a l l y equivalent t o t h e metric given by t h e
E.
The above pseudo-metrics, a s well a s t h e i n f i n i t e s i m a l F i n s l e r pseudometrics and t h e i r i n t e g r a t e d forms s t u d i e d i n [228] and [313], a r e an Their
important t o o l i n t h e works of L . A . Harris W . Kaup and J . P . Vigu6.
r o l e , however, i s not apparent from our b r i e f o u t l i n e h e r e s i n c e i n v a r i a n t m e t r i c s u s u a l l y appear i n t h e proof r a t h e r than i n t h e statement of r e s u l t s e . g . J . P . Vigu6 [696] shows t h a t U
i s a complete metric space when
(U,pc)
i s a bounded homogeneous domain i n a Banach space and then uses t h i s
r e s u l t t o show t h a t bounded homogeneous domains a r e domains o f holomorphy f o r bounded holomorphic functions. In [317] L . A . H a r r i s and W . Kaup prove t h a t t h e group of a l l l i n e a r i s o m e t r i e s of a homogeneous u n i t b a l l i n a Banach space i s a (Banach) Lie group (see P . de l a Harpe [303]), and using t h i s r e s u l t , J . P . Vigu'e [696] showed t h a t A(U) Banach space.
i s a Lie group f o r
with open u n i t b a l l J . P . Vigu;
U
a bounded symmetric domain i n a
J . P . Viguk [696,698] gives an example of a Banach space
Eo
such t h a t
A (Eo)
i s n o t a Lie group.
E
In [696],
proved t h a t bounded symmetric domains i n Banach spaces a r e
homogeneous - t h e converse i s n o t t r u e even i n f i n i t e dimensions - and holomorphically equivalent t o balanced domains. [696] include t h e endowing of
A
(U),
The methods of J . P
.
Vigu;
bounded, with t h e s t r u c t u r e of a
U
uniform and topological space ( c a l l e d t h e l o c a l uniform topology, a f i l t e r
3
-+
Vx
of
f E ~ ( U ) i f and only i f f o r each x
such t h a t
31vcLlvx
x
in
U
uniformly on
3
a neighbourhood Vx)
and showing t h a t
X
A(U)
i s a complete topological group (see a l s o W . Kau? [375] and J . M .
Isidro [554]).
He a l s o s t u d i e s t h e s e t
a t i o n s of a bounded domain
U
g(U)
o f i n f i n i t e s i m a l transform-
( i . e . t h e s e t of holomorphic v e c t o r f i e l d s
Appendix I
394
a r i s i n g f r o m the group homomorphisms o f +$(t)x
E.
a n a l y t i c ) and shows t h a t
U
a Lie a l g e b r a and a Banach s p a c e .
R
into
g(U)
fc(U)
with
(t,x)
E
RxU
has the natural structure of
The Lie a l g e b r a r e s u l t i s a l s o due
i n d e p e n d e n t l y t o H . Upmeier [688].
E
t r i p l e or a h e r m i t i a n J o r d a n t r i p l e ,
J*
A
and a mapping (a)
(5 , x , y ) + Z( ~ , x , y ) i s symmetric and CC
t h e mapping
for all
5, n, 5,
for all
5
x
x
and
-t
A morphism o f
and
antilinear
C
E
E
XEE
and a l l
e x p ( i t Z ( 5, 5,x))
t E
t h e mapping
R
i s an isometry o f
t r i p l e s i s a mapping
J*
f:(E,Z)
for a l l
f(Z(x,y,z)) = Zl(f(x),f(y),f(z)) J*
y
5,
in
[c)
i s a Banach s p a c e
such t h a t
Z;E3 + E
linear i n the variables
(b)
(E,Z),
E.
(E1,Z1)
+
x,y,z
in
such t h a t E.
t r i p l e s were i n t r o d u c e d by W . Kaup [375] who proved t h e deep r e s u l t
t h a t t h e c a t e g o r y o f s i m p l y c o n n e c t e d , symmetric, complex Banach m a n i f o l d s with base point i s equivalent t o t h e category of a l g e b r a may be endowed w i t h t h e s t r u c t u r e o f a of a
J* If
(E,Z)
is a E
I n [697], J . P . Vigu; J*
Every
J*
The d e s c r i p t i o n
triple
J*
t r i p l e and
5
E
E
let
denote t h e closed
E5
g e n e r a t e d by
shows t h a t t h e s i m p l y connected domain a s s o c i a t e d w i t h
(E,Z)
i s h o l o m o r p h i c a l l y e q u i v a l e n t t o a bounded domain
i f and o n l y i f t h e r e e x i s t s a p o s i t i v e r e a l number
5EE
triples.
triple.
t r i p l e g i v e n h e r e i s t a k e n from J . P . Vigu/e [679].
r e a l s u b s p a c e of
the
J*
J*
the restriction t o
eigenvalue i n t h e i n t e r v a l
E5
o f t h e mapping
-k
(-a,
11 5 11
)
.
x
+
k
such t h a t f o r a l l
Z(5,S,x) h a s a r e a l
395
Further developments R e c e n t l y , J . P . Vigug [700] h a s o b t a i n e d a s u f f i c i e n t c o n d i t i o n f o r t h e c o n v e x i t y o f c e r t a i n domains. homogeneous domain positive integer each of d e g r e e
U
No
and a f a m i l y
E
i s convex i f t h e r e e x i s t s a
(Pi)iEI
o f homogeneous polynomials,
for all
i EI}.
such t h a t
No,
( X E E ; Ipi(x)I
=
H e h a s shown t h a t a b a l a n c e d bounded
i n a Banach s p a c e
U
I 1
F u r t h e r r e s u l t s on homogeneous and symmetric domains i n normed l i n e a r s p a c e s may b e found i n W. Kaup [373,374 3761, L . L .
Stacho [670], A. Douady
[210], R . Braun, W . Kaup and H . Upmeier [ 9 9 , 1 0 0 ] , W. Kaup and H. Upmeier [378], S.B. Chae [121], J . P . Vigug [699
and L . A . Harris
3121.
A p p l i c a t i o n s o f i n f i n i t e dimensional bounded symmetric domains t o t h e o r e t i c a l p h y s i c s are g i v e n i n I.A. S e r e s e v s k i i [655]. The o n l y r e s u l t s w e know on t h e holomorphic c l a s s i f i c a t i o n o f domains i n non-normed l o c a l l y convex s p a c e s a r e v e r y r e c e n t and due t o R. Meise and D . Vogt [485,486].
These r e s u l t s a r o s e from t h e i r i n v e s t i g a t i o n o f holo-
morphic f u n c t i o n s on p o l y d i s c s i n n u c l e a r power s e r i e s s p a c e s ( s e e 5 6 . 4 ) . I n [ 4 8 5 ] , t h e y prove t h e f o l l o w i n g r e s u l t .
series s p a c e .
a = (an)n
If
and
be a n u c l e a r power Let A,(a) b = ( b n ) n are sequences o f p o s i t i v e ~
r e a l numbers such t h a t
open p o l y d i s c s i n Al(a)b t h e n Da and D,, 1 1 E A1 ( a ) ;i and b A1 ( a ) &. I n a l l y equivalent i f
are
show t h a t t h e p o l y d i s c s
and
in
b
a r e n o t holomorphic[486], t h e a u t h o r s
( r e s p . Am(a)I ) B m o r p h i c a l l y e q u i v a l e n t i f and o n l y i f t h e r e e x i s t s a b i j e c t i o n Da
Db
Al ( a )
n a t u r a l numbers such t h a t (a) and
1
fi
aj
6
I
Ma.
1
f o r some
M 21
are h o l o of t h e
Appendix I
396
and j=l
belong t o
Al(cx)i
j =1
A s an immediate consequence, one sees t h a t t h e r e e x i s t uncountably many open p o l y d i s c s i n
R1(ct)i
which a r e p a i r w i s e h o l o m o r p h i c a l l y i n e q u i v a l e n t .
This concludes our survey o f f u r t h e r t o p i c s i n i n f i n i t e dimensional holomorphy.
A number o f o t h e r t o p i c s such as C . E .
Rickart's algebraic
approach t o i n f i n i t e d i m e n s i o n a l holomorphy [ 6 0 7 ] , t h e Lorch t h e o r y o f a n a l y t i c f u n c t i o n s i n i n f i n i t e l y many v a r i a b l e s (B. G l i c k f i e l d [255,256], S. Baryton [ 5 4 ] ) , s p e c t r a l t h e o r y i n an i n f i n i t e number o f v a r i a b l e s (G. E g u e t h e r and J . P . F e r r i e r [230] and J . P . F e r r i e r [ 2 3 5 ] ) , a p p l i c a t i o n s of p l u r i s u b h a r m o n i c i t y t o t h e s t u d y o f Banach a l g e b r a s (B. A u p e t i t [ 3 2 ] ) , J . L . T a y l o r ' s [683] a p p l i c a t i o n o f holomorphic mappings between Banach s p a c e s t o t w i s t e d p r o d u c t s o f Banach s p a c e s , o r even some o f t h e a p p l i c a t i o n s o f i n f i n i t e d i m e n s i o n a l holomorphy t o t h e o r e t i c a l p h y s i c s c o u l d a l s o have been i n c l u d e d h e r e and would p e r h a p s have g i v e n a more b a l a n c e d s u r v e y .
A t any
r a t e , we hope t h e r e a d e r enjoyed o u r sample o f r e s u l t s and t h a t t h e i n a c c u r a c i e s ( t h e r e a r e always some) are minor.
ADDED-IN PROOF A f t e r t h e f i n a l d r a f t o f t h i s t e x t was completed, t h e a u t h o r became aware o f t h e book "Holomorphic maps and i n v a r i a n t d i s t a n c e s " by T . Franzoni and E . V e s e n t i n i , North Holland Mathematical S t u d i e s , 40, 1980, pp.226. T h i s s e l f - c o n t a i n e d t e x t i s a good i n t r o d u c t i o n t o holomorphy on Banach s p a c e s and h a s some o v e r l a p w i t h 1 1 . 1 , 5 1 . 2 ,
S2.1
and
14.1.
The main
t o p i c i n t h i s book i s i n v a r i a n t m e t r i c s on domains i n Banach s p a c e s and H i l b e r t spaces.
By u s i n g S a t z 3 . 2 and p r o p o s i t i o n 7 . 3 o f a recent p r e p r i n t of D. Vogt "Frechetraume, zwischen denen j e d e s t e t i g e l i n e a r e Abbildung
b e s c h r g n k t i s t " one can remove t h e b a s i s h y p o t h e s i s i n theorem 6.36.
Appendix I1
DEFINITIONS AND RESULTS FROM FUNCTIONAL ANALYSIS, SEVERAL COMPLEX VARIABLES AND TOPOLOGY
We p r o v i d e h e r e ,
f o r the benefit of the non-specialist,
a s h o r t l i s t o f d e f i n i t i o n s and r e s u l t s which are e i t h e r frequently used o r quoted without proof i n the t e x t . 1.
Let X be a Hausdorff t o p o l o g i c a l space;
X is
completely regular i f and o n l y i f f o r each c l o s e d s u b s e t A o f X and each compact s u b s e t K o f X w i t h AnK =
there exists a
r e a l v a l u e d c o n t i n u o u s f u n c t i o n f on X'such t h a t f(A) = o and f ( K ) = 1.
L o c a l l y convex s p a c e s are completely r e g u l a r .
X i s c a l l e d a k-space
i f e v e r y f u n c t i o n on X w h i c h
c o n t i n u o u s on compact s e t s ( i . e . A subset A of
a
k-space
is
hypocontinuous) i s continuous.
i s open ( r e s p . c l o s e d ) i f and o n l y
i f i t s i n t e r s e c t i o n w i t h e a c h compact s e t i s open ( r e s p . closed) i n t h e induced topology.
A c l o s e d s u b s e t (and a n
open s u b s e t ) o f a k-space i s a k-space.
X i s a LindeZ6f
s p a c e i f and o n l y i f e v e r y open c o v e r o f X a d m i t s a c o u n t a b l e subcover. Kn
X i s s a i d t o b e u-compact i f X =
i s compact.
I f , moreover,
el
Kn
where each
t h e s e q u e n c e (Kn)n c a n b e c h o s e n
s o t h a t e a c h c o m p a c t s u b s e t o f X i s c o n t a i n e d i n som e K n t h e n we s a y X i s hemicornpact.
a-compact--j
We a l w a y s h a v e h e m i c o m p a c t
+
Lindelb'f and t h e r e v e r s e i m p l i c a t i o n s are n o t
i n general true.
The f o l l o w i n g d i a g r a m i l l u s t r a t e s a
number o f examples and c o u n t e r e x a m p l e s i n l o c a l l y convex s p a c e t h e o r y ( a l l t h e l o c a l l y convex s p a c e s a r e assumed t o b e i n f i n i t e dimensional). 397
39 8
Appendix II
+ rechet o n t e l space
4 k - s p a c e Lindelb':
1-compacl
hemi c o m p a c t
ual of r g c h e t Monte1 (DFF Space
b .
+
+
+
FM x D F M
Von s e p a r a b 1e R e f l e x i v e Banach Space
R e f l e x i v e Banach s p a c e w i t h weak toPologY
t+
Lemma
If X and
Y a r e topoZogicaZ
s p a c e s , X compact and
Y Hausdorff, t h e n a c o n t i n u o u s b i j e c t i v e mapping of X o n t o Y is a t o p o Z o g i c a Z homeomorphism.
A s c o l i ' s Theorem and Z e t F C & ( X ) .
L e t X be a H a u s d o r f f t o p o Z o g i c a Z s p a c e Then F is a compact s u b s e t o f & ( x ) ,
399
Definitions and results
endowed w i t h t h e compact o p e n t o p o t o g y , if and onZy if {f(x), f
E
each x in
F} is a bounded s e t of c o m p l e x numbers for
x.
f ( x ) f o r every If ( f a ) a E r E F a n d f a ( x ) t h e n f E F f i . e . F is p o i n t w i s e c l o s e d ) .
in X
LI:
F i s e q u i c o n t i n u o u s o n t h e compact s u b s e t s of X f i . e . if K i s a compact s u b s e t of X, x E X and 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. <
For each
-.
n
choose a
The s e t
m
n=1
i s t h e r e q u i r e d neighbourhood o f z e r o .
T h i s r e s u l t may be compared w i t h
c o r o l l a r y 2.30 and i s due t o J . F . Colombeau and J . Mujica [ 1 5 6 ] . t h a t any e n t i r e f u n c t i o n on a
d? z q
I t says
s p a c e may b e f a c t o r e d t h r o u g h a normed
l i n e a r s p a c e as a n e n t i r e f u n c t i o n o f bounded t y p e .
T h i s r e s u l t may b e
combined w i t h e x t e n s i o n theorems o f R . Aron and P . Berner [26] c o n c e r n i n g holomorphic f u n c t i o n s o f n u c l e a r bounded t y p e on Banach s p a c e s t o g i v e a f u r t h e r p r o o f o f P . B o l a n d ' s [83] Hahn-Banach theorem f o r holomorphic f u n c t i o n s on
~ 8 3 Qs p a c e s
(corollary 5.50).
The r e s u l t s i n [156] are a l s o
u s e f u l i n s t u d y i n g c o n v o l u t i o n o p e r a t o r s i n i n f i n i t e l y many v a r i a b l e s , u n i f y i n g r e s u l t s o f C . P . Gupta [295], P . Boland 1791, J . F . M . C . Matos [150], and M . C . Matos [ 4 6 7 ] .
Colombeau and
I t i s n o t known i f t h e r e s u l t o f
t h i s e x e r c i s e e x t e n d s t o a r b i t r a r y open s u b s e t s o f b j m s p a c e s .
CHAPTER THREE 3.66 -
The g e n e r a l r e s u l t i n t h i s exercise i s due t o K . Noureddine and
J . Schmets [535], w h i l e t h e a p p l i c a t i o n t o holomorphic f u n c t i o n s on l o c a l l y
convex s p a c e s i s due t o Ph. Noverraz [553].
S e e a l s o J . Schmets [627,629].
3.69 __
See Ph. Noverraz [553].
3.70 -
T h i s e x e r c i s e as w e l l as e x e r c i s e s 3 . 7 1 and 3.72 a r e due t o Ph
Notes on some exercises
Noverraz [553].
419
In o u r applications we have an absolute basis and so we
do not need these more general results which apply to locally convex spaces with an equischauder basis. 3.73 __
See N. Kalton [371].
3.75
First show that
and
See J . M . Ansemil and S. Ponte [lo] for further details.
F.
3.76
(ExF)~= EtxFt
for any locally convex spaces E
This result, proved by J.M. Ansemil and S. Ponte in [lo], follows
from exercise 3.75 and from results of A. Grothendieck [288, chapter 4, part 21 on the equicontinuity of bilinear mappings on
DF
spaces.
3.77 The space F of this exercise is known as a DFC space. Fundamental properties of DFC spaces are given in A. Grothendieck [288, p.158-1641 and they arise in infinite dimensional holomorphy in the papers of M. Schottenloher [639], J. Mujica [SO61 and M. Valdivia [691]. DF
spaces and
DFC
exercise 3.76 cannot be applied here. that
DFC
Although
spaces have much in common, the method used to solve For this exercise one uses the fact
spaces are hemicompact k-spaces. See J . M . Ansemil and S. Ponte
[lo]. 3.78
The space E
of this exercise is hemicompact but not a k-space.
Indeed, the compact open topology on
E
is the k-topology associated with
the weak topology (see M. Schottenloher [639]) and thus this example shows that the k-space property of
F,
in exercise 3.77, is necessary. To prove
this exercise, begin by examining carefully the inductive limit definition of the
T~
topology. F o r further details, see J.M. Ansemil and S. Ponte
[lo].
3.80 -
See S. Dineen [185].
__ 3.83
See N. Kalton [371].
3.84 -
Use the estimate given in proposition 1.41.
3.85
A more general result is proved in chapter
4.
Appendix I l l
420
3.86
A c a r e f u l r e a d i n g o f example 3.47 s h o u l d h e l p i n s o l v i n g t h i s
exercise.
See a l s o e x e r c i s e 3.87. The method o u t l i n e d i n t h i s e x e r c i s e , t o g e t h e r w i t h t r a n s f i n i t e
3.88 __
i n d u c t i o n , i s used i n S. Dineen [179] and Ph. Noverraz [545] t o c o n s t r u c t t h e envelope o f holomorphy o f 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 space.
3.89
T h i s r e s u l t i s g i v e n w i t h o u t proof i n T . Abuabara [ 3 ] .
See a l s o
S. Dineen [177]. 3.90 __
This r e s u l t appears i n J . M .
__ 3.91
If
2 1 pa( C )
P
Ansemil and S . Ponte [ l o ]
t h e n t h e r e e x i s t s a s e t o f scalars
(aij)ijeI
such t h a t
The f i n i t e sums i n t h i s expansion can be i d e n t i f i e d with c o n t i n u o u s p o l y -
t1 and t h e s e converge u n i f o r m l y on t h e compact s u b s e t s o f
nomials on to
P.
See M.C. Matos [462] and J . A .
C1
Barroso and L . Nachbin [53] f o r
further details.
3.92
The s p a c e
func ti on space
CN
h a s many d i f f e r e n t s t r u c t u r e s and c o n s e q u e n t l y t.he
H((CN) may b e s t u d i e d from a number o f d i f f e r e n t a n g l e s .
One approach i s t o n o t e t h a t each holomorphic f u n c t i o n on
factors
EN
t h r o u g h a f i n i t e dimensional s u b s p a c e (example 2 . 2 5 ) and t h i s h a s been used by J . M .
Ansemil [ 8 ] , V . Aurich [ 3 3 , 3 5 ] , C . E .
R i c k a r t [605], J . A .
[ 4 7 ] , P . Berner [59,61] and A . H i r s c h o w i t z [ 3 3 5 ] . which we d i s c u s s i n c h a p t e r 5 , i s t o view w i t h an a b s o l u t e b a s i s .
CN
Barroso
An a l t e r n a t i v e approach,
a s a Frgchet n u c l e a r space
E i t h e r approach g i v e s a s o l u t i o n t o t h i s exercise.
A f u l l d i s c u s s i o n o f t h i s problem, i n c l u d i n g a r e p r e s e n t a t i o n o f a n a l y t i c f u n c t i o n a l s on Ansemi1 I S ] .
CN
by f u n c t i o n s o f e x p o n e n t i a l t y p e , i s g i v e n i n J.M.
P . Berner [61] g i v e s a g e n e r a l r e s u l t which i n c l u d e s a
solution t o t h i s exercise. 3.95
I f each
isomorphic t o
C")
( c o r o l l a r y 3.65).
En
i s a f i n i t e dimensional s p a c e , t h e n
and hence
(H(C(N)),~o)
C:=lEn
is
i s a Frkchet n u c l e a r space
I n c h a p t e r 4 ( p r o p o s i t i o n 4 . 4 1 ) w e show t h a t
42 I
Notes on some exercises is complete (and in fact quasi-complete) if and only if (H(Cn,lEn),~o) each En is a finite dimensional space.
3.96
An alternative definition of holomorphic functions of nuclear
type is proposed by L . Nachbin [508,509,511]. This exercise, taken from
S. Dineen [177], shows that the two definitions do not coincide. One can take E
to be a separable Hilbert space in constructing a counterexample.
3.97 This exercise, as well as exercises 3.98,38913.100 and 3.101 all concern holomorphic functions of exponential type. These functions first arose in infinite dimensions in solving convolution equations on locally convex spaces and provide an alternative description of the space of analytic
functionals. Papers which discuss functions of exponential type in
infinite dimensions are J.M. Ansemil [a], P. Boland [79,80,81], C.P. Gupta [295,296], L. Nachbin [511,514], P. Lelong [431,437], M.C. Matos [458], T.A.W. Dwyer [216] and Y. Fujimoto [249]. can be found in A. Martineau [450,452]. found in P.J. Boland [80,81].
The finite dimensional theory Exercises 3.97 and 3.101 may be
Exercise 3.98 is the crucial part of the
division theorem used to prove existence theorems for convolution operators (see Appendix I). 3.102
Exercises 3.98 and 3.100 are due to C.P. Gupta [295].
This result is due to K-D. Bierstedt and R. Meise [69,70].
See
also proposition 6.9. 3.103
This is a special case of a result proved, using Cauchy
estimates and Ascoli's theorem, by J.F. Colombeau and D. Lazet in [149]. 3.104
The Schwartz property f o r
(H(U),.ro)
and
(H(U),.rw)
has been
investigated by.various authors, e.g. R. Pomes [584], K - D . Bierstedt and
R. Meise [69,70], P. Aviles and J. Mujica [41], J.F. Colombeau [146], J.F. Colombeau, R. Meise and B. Perrot [153], J.F. Colombeau and B. Perrot [159, 1611, J.F. Colombeau and R. Meise [152], Y. Fujimoto [249], A. Benndorf [56] and E . Nelimarkka [526].
Using operator ideals, E . Nelimarkka proved
a general result from which one may deduce theorem 3.64 for entire functions and also the present exercise.
422
Appendix III
CHAPTER FOUR 4.46
The result of this exercise may be interpreted as a form of
Schwarz's
lemma and it is due, as are most generalizations of Schwarz's
lemma, to L. Harris [304]. 4.47
The proof of this result (see L. Harris [308]) involves the sub-
harmonicity of the spectral radius, a result proved by E . Vesentini [692]. Further maximum theorems involving the spectral radius are given in E . Vesentini [693,694] and B. Aupetit [32]. This result is due to J. Globevnik [258].
4.48 ___
Globevnik also shows
that the condition f(0) = 0 may not be replaced by the weaker condition
1 f(O)Ja
=
0. The proof uses subharmonicity of the spectral radius.
4.50 See L. Harris [306]. This result shows that the numerical range can be used as an alternative to the sup norm, to obtain Cauchy estimates for the terms in the Taylor series expansion of a holomorphic function. __ 4.51
This result is due to I. Shimoda [656, theorem 41, (see also L.
Harris [305,313 corollary 321, S.J. Greenfield and N.R. Wallach [282, theorem 2.1, corollary 2.11, A. Xenaud [604] and W. Kaup and H. Upmeier, [3771). 4.52
This result is due to L. Harris [307].
of the iteration method of H. Cartan [113].
The proof is an elaboration
An application of this result
gives a further proof of theorem 4.3.
~
4.53
This result is due to T.L. Hayden and T.J. Suffridge [319].
The
mapping considered is a Mb'bius transformation, and this, together with Schwarz's
lemma, plays an important role in L. Harris' work on
B*
and J*
algebras. See also S . J . Greenfield and N.R. Wallach [282, theorem 4.11,
A. Renaud [604] and Appendix I. 4.54
This is known as the Earle-Hamilton fixed point theorem. The
original proof, which involves the construction of a Finsler metric on U satisfying the Schwarz-Pick condition, is quite difficult and can be found in [ 2 2 8 ] .
A deeper analysis of the constructions used in [228] and a
further proof are given i n L. Harris [313].
A more elementary proof,
Notes on some exercises
423
t o g e t h e r w i t h a r e s u l t c o n c e r n i n g t h e dependence o f t h e f i x e d p o i n t on a parameter can be found i n M. Hervk [328].
Various o t h e r f i x e d p o i n t
theorems f o r holomorphic mappings on H i l b e r t s p a c e s a r e proved i n T.L. Hayden and T . J . S u f f r i d g e [318,319] and i n S . J . G r e e n f i e l d and N . R .
Wallach
[282].
4.57
T h i s r e s u l t i s due t o A . Hirschowitz [ 3 4 0 j .
4.59
The r e q u i r e m e n t o f uniform c o n v e x i t y i n p r o p o s i t i o n 4 . 1 6 can b e
r e p l a c e d by t h e weaker h y p o t h e s i s t h e r e i s a compact s e t that
l\yl\ = 1,
s a t i s f i e d by
K
" f o r every
such t h a t f o r e v e r y
(Ix+y 113 2- 6 i m p l i e s co(r)
x
Y E
K+EB".
in
with
E
/IxI(= 1
there is a
E > O
6>0
such
This c o n d i t i o n i s n o t
and so t h i s e x e r c i s e shows t h a t we do n o t have n e c e s s -
a r y and s u f f i c i e n t c o n d i t i o n s f o r s t r i c t i n e q u a l i t y i n p r o p o s i t i o n 4 . 1 6 . See C . O .
Kiselman [387] f o r f u r t h e r d e t a i l s .
4.63
Use t h e f a c t t h a t t h e c l o s e d bounding s u b s e t s of
4.64
The n e c e s s i t y f o l l o w s from theorem 4 . 2 8 ( c ) .
F
are compact.
The s u f f i c i e n c y
f o l l o w s from a theorem o f R. Aron and P . Berner [26].
A survey o f extension
r e s u l t s f o r holomorphic mappings on Banach s p a c e s i s g i v e n i n R . Aron [ 2 5 ] .
4.68
Use t h e f a c t t h a t
& ,
[0,1]
h a s t h e polynomial D u n f o r d - P e t t i s
p r o p e r t y , R . Ryan [619], and t h a t t h e u n i t v e c t o r b a s i s of weakly t o z e r o .
4.69
t2
tends
See R . Aron [25] f o r f u r t h e r d e t a i l s .
Any c o n t i n u o u s l i n e a r mapping from
co
to
L1
i s compact.
See
R . Aron [25]. 4.70 -
See S . Dineen [177].
4.71
T h i s r e s u l t , due t o L . Nachbin [509],
shows t h e inadequacy o f t h e
compact open t o p o l o g y f o r holomorphic f u n c t i o n s on Banach s p a c e s .
See a l s o
H . Alexander [5] and e x e r c i s e 2.103. 4.72 T h i s r e s u l t i s due t o P . Lelong [431]. R e l a t e d r e s u l t s o f i n t e r e s t are t o b e found i n J . S i c i a k [657] and P . Lelong [433]. See a l s o lemma 1.19 e x e r c i s e 1 . 8 1 , and t h e n o t e s and remarks o f c h a p t e r 1.
424
Appendix III
4.73 __
This result is due to R. Aron and M. Schottenloher [31]
4.75 __
This exercise is not difficult. However, to show that
is complete is much more difficult (see J. Mujica [499,503]) and this problem in chapter 6. 4.76 __
See S.B. Chae [121].
4.77
See Ph. Noverraz [552] and exercise 2.83.
__ 4.78
This is known as Patil's problem.
~
It was first posed by D. Patil
at the conference on Infinite Dimensional Holomorphy in Kentucky during June of 1973. Extensive work has been done on this problem by a number of different authors notably J. Globevnik. With this problem as motivation,
R. Aron, J. Globevnik and M. Schottenloher [29] studied interpolation sequences and found new proofs of some classical theorems. The problem for separable Banach range spaces was solved independently by R. Aron [223 (who reduced the problem to the case where the range space was ,co and then used cluster sets and Blaschke products) by J. Globevnik [261] (whose approach involved a generalization of the Rudin-Carleson interpolation theorem to vector valued functions) and by W. Rudin [612]. A counterexample showing that the result does not extend to arbitrary Banach spaces is given by B. Josefson in [360].
Bo
let
Josefson proved the following result:
be the open unit ball of co(r),
r uncountable,
and suppose f E H(Bo;co(r)) connected bounded subset U
-
f(Bo) 3 U (open)
then
then there exists an open of co(r) such that if 1 + - Bo. f(Bo)+U 10
Further extensions of this counterexample are due to J. Globevnik [274] who showed that if a Banach space contains a non-separable analytic image of the unit ball of co(r), r uncountable, then it contains an isomorphic copy o f co(r') where r ' is uncountable. Further readings on the above problem are to be found in J. Globevnik [262,266,267,268,270,271,275].
4.80 ___
See J. Globevnik [269]
425
Notes on some exercises
4.81
For further examples of (holomorphic) determining sets see J.
Chmielowski [123,124], J. Chmielowski and G. Lubczonak [125], P. Boland and
S. Dineen [91], S. Dineen [200], L . A . de Moraes [498], L. Waelbroeck [712] and proposition 6.25. This result is due to L. Waelbroeck [712] and involves an
4.82 -
application of the closed graph theorem. CHAPTER FIVE mm First show that for any sequence (u )m=l, there exists a sequence in Q;, (u~);=~, such that
5.56
=
m m (Un)n= 1 ’
+
in Q~
For further details, see P.J. Boland and S. Dineen [91]. See P.J. Boland and S. Dineen [91].
5.57 -
Related to this exercise is
the open problem of whether or not A-nuclear spaces are Mackey spaces.
5.59
A similar argument to that used in lemma 5 . 4 3 suffices
5.65 -
This result is due to J.F. Colombeau and R . Meise [152].
5.66
This is a generalization of theorem 5.21 and is proved in a similar
fashion. A full proof is given in P.J. Boland and S. Dineen [go]. 5.68 -
See S. Dineen [ 2 0 2 ] .
u
m
{ ( z ~ ) ~ =E ~E; sup lznanl n Consider the function 5.69
Let
f(z)
If K m
=
1
=
m d
(azIm
=
is a compact subset o f
sn
<-,
n n=l m
(N)
and
V
U,
1 1-z.a.
___
<
1)
and let T
E
Spec(H(U),T6).
on U.
1 1
there exist
6 = (6 )
m
n n=1’ a neighbourhood of zero such that
6 n > 1 and
426
Appendix III
6 (K+V) C U .
Hence f
E
Hence
H(U)
where b = T(z ) n n
c
Since
1
and
and b
(N)lambml
Ianbnl < 1.
If T
5.70 T
E
E
1
( ~ ~ ) iE=' ~then
(lim
H(V),T~)',
and
so
T
E
H(U).
M Since H(U ) '
5.71 __
See K-D.
Bierstedt and R . Meise [ 6 9 , 7 0 ] .
5.73
If f(z)
=
1
(Nl
m
amz
E
HHU(U)
show that
mEN
-
1-zw
-
<
B'
M
all V 3 U M
Z
lnnbnl
U an open polydisc in E '
apply corollary 5.35.
-
a,
1
all n
E
V X I ,V open
M
this shows
l-\ctnbn(
If
--f
H(V )
<
(N)/N%ml
In particular, we have
and ln=l(Nnbnl < m .
n
mE N
m
n=l
m
1
=
(bn)n=l.
=
fi
=
mcN
SUP
(,)lT((nz)m)l
mE N
Z
m
(2 .) 1-znwn
Show also that the mapping
n=l
?'
HHy(U)
then
we may
427
Notes on some exercises
z
+
1-zw
z
H(U).
E
For f u r t h e r d e t a i l s c o n s u l t P . Boland and S. Dineen 1911. 5.74
S e e R . S o r a g g i [669].
A q u o t i e n t mapping i s a n open mapping.
Show t h a t t h e c a n o n i c a l mapping from
H(OE)
using t h e d e f i n i t i o n o f inductive l i m i t . that
i s r e g u l a r if H(OE)
H(OF)
onto
If
i s r e g u l a r and t r a n s f e r r i n g t h i s t o t h e
d u a l s p a c e we o b t a i n t h e f o l l o w i n g :
i s a c l o s e d subspace o f a
M1
if
on M2 and T~ = T o,b In c e r t a i n c a s e s , f o r i n s t a n c e when
H(M2)
f u l l y nuclear space H(M1).
can r e p l a c e
T
by ,
~
~T
~
t h e n T~ = T on 0 ,b i s a F r g c h e t s p a c e , one
M2
.
5.75 __
S e e J . F . Colombeau and R . Meise [152].
5.76 -
See P . J .
Boland and S. Dineen [ 9 1 ] .
(p(nE),~o) and
and hence
i s an
E
((@(nE),~o)A)A
(@ ("E;)
, T ~ )
with 5.77 -
T
0
(6("E)
w e have
and
T~
#
T~
'i' (($(nEA)
,T:*)
By c o r o l l a r y 5 . 2 6 ,
A - n u c l e a r space a r e both
(@("E),T;*)
A-nuclear s p a c e s , a l l w i t h t h e same a b s o l u t e b a s i s . lemma 5 . 1 .
i s c o n t i n u o u s by
H(OF)
i s f u l l y n u c l e a r t h i s shows
E
( 8("E)
Since
,T,);)
.g @("Ei;).
, T ~ ) ;
& ''
Now u s e
i s the b a r r e i l e d t o p o l o g y a s s o c i a t e d
-cw
by example 5 . 3 6 ( b ) .
See a l s o P. Berner [ 6 1 ] . We see by u s i n g example 5.36
See P . J . Boland and S . Dineen [ 9 1 ] .
t h a t an u n n e c e s s a r y e x t r a h y p o t h e s i s was i n c l u d e d i n t h e o r i g i n a l p r o o f
given i n [91].
5.78
See K-D.
5.79 -
Since
E
B i e r s t e d t and R . Meise [ 7 0 ] . i s f u l l y n u c l e a r , it i s isomorphic t o
a f u l l y n u c l e a r spcace. same a b s o l u t e b a s i s as
(F',T~)
F;
where
is
F
i s a n u l t r a b o r n o l o g i c a l s p a c e and h a s t h e
Since
( F ' , T ~ ) = F;.
E = Fb
is a b a r r e l l e d space
one can complete t h e p r o o f i n a number o f d i f f e r e n t ways. 5.81
See S . Dineen [202].
t h a t f o r any i n f i n i t e s u b s e t t h e r e e x i s t s a n open p o l y d i s c
T
If J V
of
in
and
(H(E),T~)I
E
N"),
E;
with
T(zm) = b
bm # 0
such t h a t
each
m
m
show in
J,
Appendix 111
428
1mEJ
By comparing exercise.
1 CmeJ ____ m
and
6m
one p r o v e s one h a l f of t h e
Ib," (Iv
The o t h e r h a l f i s proved u s i n g A - n u c l e a r i t y , a f u r t h e r compar-
i s o n o f a c o n v e r g e n t and d i v e r g e n t s e r i e s and e x e r c i s e 5 . 8 0 . If
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 , one may a l s o
U
prove t h e f o l l o w i n g : i n f i n i t e subset
lmEJbmwm I
to
of
J
T
N")
E
( H ( U ) , T ~ , ~ and )~
then every
c o n t a i n s an i n f i n i t e s u b s e t . J '
Wn = t ( m , n ) 1 , m = l , 2 , . . . l .
f o r any complex number
hl(n)
= bm
T(zm)
such t h a t
H(U").
E
Let
__ 5.82
if
C
Since
(C,l,l,
...,l , l , ...)
belongs
one c a n show t h a t
1
I i s f i n i t e f o r each a . z J i n H ( A ~ ( ~ ) ) .If la(m,n) 'b(m,n) 1 jEK(N) I i s any s t r i c t l y i n c r e a s i n g sequence of p o s i t i v e i n t e g e r s and (ni)Yz1 i s a sequence o f i n t e g e r s , t h e n t h e sequence
belongs t o 'i
la(mi,ni)
H(A1(a)). y6
6>0
1
b
m
one t h e n shows t h a t
1
i s f i n i t e f o r each
(mi,ni)
j EN
ajzJ
in
An a p p l i c a t i o n of e x e r c i s e s 5.80 and 5 . 8 1 now shows t h a t
continuous. and
By u s i n g t h e sequence
Al(cx).
m
(ni)i=l
n1 = 1 such t h a t
If
was
T
T~
c o n t i n u o u s t h e n t h e r e would e x i s t
T
is
c>O,
a s t r i c t l y i n c r e a s i n g sequence of p o s i t i v e i n t e g e r s w i t h ] b m /6 c / j ~ ~ l / ~ ~ ( ~ f. o) r q aol l
each p o s i t i v e i n t e g e r
n
1
1=1
m
in
N").
t h e r e would e x i s t a p o s i t i v e i n t e g e r
Hence f o r
mn
such
429
Notes on some exercises
for all mzmn. To
An analysis of this inequality will show that T is not
continuous. A full proof of this exercise is given in S. Dineen [202].
It is
based on an examination of the proof of theorem 5.42 which showed that any counterexample of this type must violate condition (**) in that proof.
It
would be interesting to extend this exercise to non-DN Frgchet nuclear spaces with a basis.
A s it stands, theorem 5 . 4 2 and exercise 5 . 8 2 together
state the following: if E
is a nuclear power series space then
and only if E
To
=
is of infinite type.
6
on H ( E )
if
The full converse to theorem 5 . 4 2 is proved, by.other methods, in chapter 6 . 5.83
Holomorphic functions on (the space of rapidly decreasing functions) arose in the work of P. Krge [402,410] concerning boson fields
in the mathematical founaations of quantum field theory. It is possible that the subject of this exercise, anticommutative forms, will play an analogous role in the theory of fermion fields. This exercise shows that topological vector space properties of H ( E )
are inherited by
HA(E).
For
further details see S. Dineen [195,201]. 5.84 -
See R. Ryan [620]
CHAPTER 6 __ 6.57
This result is due to R. Soraggi [669], and can either be proved
directly o r obtained as a corollary of proposition 6 . 2 9 . __ 6.58
This result is also due t o R. Soraggi [668] and is a natural con-
sequence of the approach adopted in proposition 6.2. By generalising the construction in example 2.54 and using proposition 3.50, R. Soraggi [667] has recently been able to drop the very strongly convergent sequence hypothesis in this exercise for certain fully nuclear spaces. 6.59
The proof of this exercise depends on the following non-trivial
430
Appendix IIl B i e r s t e d t and R. Meise [ 6 9 , 7 0 ] :
r e s u l t due t o K-D. E,F
and
G be two H i l b e r t s p a c e s and a normed l i n e a r s p a c e
respectively.
Let
Ql
Let
open s u b s e t o f
E
Suppose t h e l i n e a r mapping
G.
t h e l i n e a r mapping suppose
b e a n open s u b s e t o f
P : F-tE
II(R2)CR1.
is
and l e t
Then t h e mapping
R : H"(Q1)
Let +
b e a bounded
i s compact and
G+F
0:
summable.
R1'7
R2
Il = p o u
Hm(Q2),
and
R(f) =
fan/
i s an a b s o l u t e l y summing mapping.
*L
6.60
See J . Mujica [499,503].
__ 6.63
A l o c a l l y convex s p a c e which s a t i s f i e s any o f t h e e q u i v a l e n t con-
d i t i o n s o f e x e r c i s e 6 . 6 3 i s s a i d t o have p r o p e r t y
(B)
[47], J . Mujica [ 4 9 9 ] , P . A v i l e s and J . Mujica [41], R . Meise [ 6 9 , 7 0 ] ) .
(see J . A .
Barroso
B i e r s t e d t and
K-D.
I t was used by v a r i o u s a u t h o r s t o e x t e n d r e s u l t s from
Banach s p a c e s t o a more g e n e r a l class o f s p a c e s b u t h a s become o b s o l e t e w i t h t h e development o f f i n e r t e c h n i q u e s .
This r e s u l t i s given i n P .
A v i l e s and J. Mujica [41]. We refer t o S . Dineen [190] f o r e x e r c i s e s 6 . 6 5 t o 6 . 6 9 .
6.65
~
6.64
f o l l o w s from t h e f a c t t h a t t h e i n v e r s e system o f neighbourhoods forms a b a s i s and n o t j u s t a s u b b a s i s and from an a p p l i c a t i o n o f t h e open mapping theorem between Banach s p a c e s .
Exercise 6.69 i s u s e f u l i n extending the
c l a s s of l o c a l l y convex s p a c e s f o r which a s o l u t i o n t o t h e Levi problem holds. 6.70 __
See Ph. Noverraz [552].
___ 6.72
T h i s r e s u l t i s due t o K-D.
B i e r s t e d t and R . Meise [ 7 0 ] .
I t s proof
r e l i e s on an open mapping theorem ( p r o p o s i t i o n 6 . 8 ) , t h e Banach-Dieudonne theorem ( s e e J . Horvath [348]) and t h e f a c t t h a t when
K
H(K)
i s a S i l v a space
i s a compact s u b s e t o f a Frgchet-Schwartz s p a c e .
6.73 -
See K-D.
B i e r s t e d t and R . Meise [ 7 0 ] .
6.75 -
See K-D.
B i e r s t e d t and R . Meise [ 7 0 ] .
6.76
Apply t h e f i n i t e d i m e n s i o n a l Oka-Weil approximation theorem.
See
Notes on some exercises
43 1
K-D. Bierstedt and R. Meise [70] and M. Schottenloher [643] f o r details __ 6.77
This result is due to M. Borgens, R. Meise and D. Vogt [96]. See
also P. Berner [61], J.M. Ansemil [8] and proposition 6.28. 6.78
~
This result follows from theorem 6.51 and is due to M. Borgens, R.
Meise and D. Vogt [96]. 6.79
These are classical results (see G . K6the [397]) and are analogous
to the Grothendieck-Pietsch criterion for the nuclearity of a sequence space (proposition 5.4). 6.80 __
This result is new. A particular case is given in S. Dineen [194]..
Since A(P) is a Fr6chet Monte1 space A(P)E, has a Schauder basis and m m A(P)E, = lim Em where each Em = { ( ~ ~ ) ~ = ~ ; s u pBI x1 < -1 for some sequence n n -+ n m
o f scalars
(B"'):=~.
Moreover, the unit ball of Em,B is a relatively m' compact subset of A(P);( and Bm n {(x~):=~; lim x Bm = 01 is dense n n f o r each m. If (with respect to the topology o f A ( P ) ; ( ) in n-B m f E HM[U) we may now apply theorem 4.28 to show that f E H(U). This is an interesting example of an application of a counterexample to prove a positive result.
6.81
Use example 5.45 and the method of proof given in proposition 6.37.
This Page Intentionally Left Blank
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INDEX
................................. A-nuclear space. reflexive ...................... Abel's theorem .................................. Absolute basis .................................. Absolute decomposition .......................... Abstract Wiener space .......................... Admissable coverings .......................... Algebraic dual ................................. Algebraic hyperplane ............................ Analytic bilinear realization ................... Analytic functional ................ 156. 2 3 6 . A-nuclear space
226. 425 229 102
228. 229 114 374 368. 372 1
215 381 292. 296. 334. 361 363. 381. 382. 420
Analytic set .................................... Analytic set. codimension of .....................
383 386
Analytic set. finitely defined ................... Analytic set. germ o f ............................ Analytic set. irreducible ........................ Analytic set. principal Anticommutative forms Approximation property
383 384 384. 386
........................ ........................
.........
40.
383 428 4 6 . 139. 2 0 9 . 289. 328
333. 3 5 9 . 369. 370. 371. 404
Ascoli's theorem ........................ Associated barrelled topology................ Associated sequence ......................... Associated topology ......................... 48 1
131. 155. 3 9 8 . 435. 421 112. 300
337 7 4 . 110.
146. 153
482
Index
................................... .........................................
B-continuous function
411
B-nuclear space
267
B, property ............................................. Baire space ............................................ Baire theorem .......................................... Banach-Dieudonng theorem .............................. Banach-Lie group ......................................
..................................
Banach-Stone theorem Barrelled space ....................................... Basis ...............................................
..............................
Boundary values (of holomorphic function) ............ Boundedly retractive inductive limit . . . . . . . . . . . . . . . . Bounding set ........................................
..............................
C* algebra .......................................... Calculus of variations .............................. Caratheodory metric ................................. Cartan domain ....................................... Cartan factor ....................................... Cartan-Thullen theorem ............................ Category .......................................... C a t e g o r y , first-
subset
........................
163
2 1 8 , 229 2 1 9 , 404
2 2 9 , 2 7 8 , 2 9 3 , 3 6 9 , 404
................................
Boundedness, radius of
39 3
209
Borel measurable function ....................... Borel transform ................................. Bornological space ..............................
Bornological space, DF
6 8 , 399 412
9 4 , 183, 404
Basis, monotone
Bilinear mapping
411
2 4 , 112, 400
Basis, absolute ...................................... Basis, equicontinuous (equi-Schauder) ................
.................................... Basis, Schauder ....................... 218, Basis, shrinking ................................ Basis, unconditional ............................ Biholomorphic mapping ...........................
4 30
Category of locally convex spaces ................. Category of topological spaces ....................
293 183, 2 8 9 , 404 2 0 5 , 2 0 6 , 3 8 4 , 388 2 , 190, 406 3 8 0 , 3 8 1 , 382 3 1 , 137, 249 16, 1 1 1 , 400 131 375 3 1 2 , 357 173, 2 0 2 , 2 0 3 , 368 166, 206 390 101
39 2 389 390 3 6 5 , 3 6 8 , 3 7 0 , 372 415 207 1 6 , 400 1 6 , 5 4 , 399
48 3
Index
......................... ................................... Cauchy-Hadamard formula ............................ Cauchy inequalities ................................ Cauchy integral formula ............................ Cauchy . Riemann equations ......................... Cesaro sums ........................................ Closed forms ....................................... Codimension (of analytic set) ...................... Coherence (of Taylor series expansions) ............. Coherent Sheaves ................................... Compact mapping .................................... Category, second-subset Cauchy estimates
Compact-open topology .............................. Compact operator ...................................
................................... Control theory ................................... Convergence, Mackeydriterion .................... Convergence, pointwise ............................. Convergence, strict Mackey-criterion .............. Convergent, very strongly ............ 81, Convergent, very weakly.................... Convolution operators...................... Curve of quickest descent .................. a problem ............................... DF ........................................ DFC ,...................................... DFM ....................................... DFN ....................................... DFS ....................................... decomposition, absolute ................... decomposition, equi-Schauder .............. Cousin I problem
decomposition, ./:absolute
..................
decomposition, RSchauder .................. decomposition, Schauder................... decomposition, shrinking................... determining manifold ....................... determining set .......................... direct image theorem .....................
43 9 0 , 3 0 1 , 422 165, 338 5 7 , 408 2 3 7 , 3 7 6 , 408 5 4 , 103 196 375 386 9 0 , 301 377 9 3 , 152 2 3 , 7 1 , 399 4 30 104, 378 38 1 62 9 6 , 148, 399
357 9 7 , 149, 2 8 1 , 3 2 1 , 325 82,
97
104, 3 8 0 , 4 1 8 , 421 101
3 6 5 , 3 7 1 , 3 7 4 , 379 18, 131, 147, 3 0 7 , 403 419 14 17 15
1 I4 1 I4 1 I4
114 114, 294 114, 147 414 211, 319, 425 388
48 4
Index
distinguished Frschet space
............... a .............
25,
-
distributional solution of division theorem ........................... Dixmier-Ng theorem
3 4 , 357 375
3 8 0 , 3 8 4 , 421
.........................
417
domain of existence......................... domain of holomorphy ...................... domain, polynomially convex ................ domain, pseudo-convex
3 6 5 , 3 6 7 , 372 365 2 1 3 , 3 5 9 , 369
......................
domain spread .............................. dominated norm (DN) space .......
6 4 , 3 3 5 , 365 367 262, 288, 334, 375, 377, 429
Dunford-Pettis property ............... eigenvalues ...................................
.45, 293, 4 1 3 , 423 374, 394, 413
........................ ............................
envelope of holomorphy
3 6 6 , 368 329
evaluation mapping exponent sequence..............................
336
exponent sequence, nuclear.....................
336
exponent sequence, stable...................... exponential polynomial solutions.. .............
336 380
exponential type, functions of .................
.151, 1 5 6 , 4 2 0 , 421
exponential type, functions of nuclear ......... extreme points
................
......................... factorization lemma .......................... factorization properties ...................... factorization theorem ......................... finitely open topology ........................
152 161, 204, 205, 211, 405 105
factorization, global
finitely polynomially convex domain............
................................ theorem ...........................
Finsler metric fixed point
Fock space .................................... Frgchet space ................................. Fredholm operator ............................. fully nuclear space............................ fully nuclear space, with basis ............... functional calculus ..........................
11,
63,
98 367 296
16,
53,
9 2 , 3 7 9 , 411 369 393, 422 2 0 6 , 3 6 9 , 422 38 1 1 2 , 400 387 3 3 , 1 3 9 , 229 229 35 6
485
Index
G holomorphic function
.............................................
GPteaux holomorphic function
.......................................
Gaussian measures .....................................
159 385 8 4 , 250 9 1 , 255
germ. of analytic set ................................ germ, nuclear holomorphic ............................
384 138
Grothendieck-Pietsch criterion ........................
Hartogs' theorem
2 2 2 , 431
..................................
.
........ .........................
holomorphic function
401 202. 215. 296. 418
holomorphic
hemicompact space ........................ holomorphic completion ....................
54 9
5 9 . 103. 409. 415 397. 419 203. 215. 371
............................
57
holomorphic function of nuclear bounded type .....
386
............ -function .................. germ ............................... nuclear -germs ..............
156. 421
holomorphic function of nuclear type
54
holomorphic. G
holomorphic holomorphic. holomorphic vector field
54 215. 374
geometry of Banach spaces ............................. geometric ideal ...................................... germ, holomorphic .................................... germ, hypoanalytic ...................................
Hahn-Banach theorem Hahn-Banach theorem
54
8 4 . 250 136
........................
393
holomorphically convex domain .................... holomorphy type .................................
311. 365. 409 5 1 . 382
..............................
389
homogenous polynomial homogeneous subspace ............................ hypoanalytic function ...........................
196
homogeneous domain
...........................
3 6 0 . 319
hypoanalytic germ .............................. hypocontinuous function ......................... hypocontinuous ideal. maximal
homogeneous polynomial
91. 255 397
............
13
.....................................
.................................... 2. 13. induction ................... induction. transfinite .................
369
ideal. geometric
385 2 8 . 39.
194. 266. 343 112. 379. 420
48 6
Index
inductive limit, boundedly retractive
.....................
inductive limit, locally convex-topology inductive limit, regular ................... inductive limit, strict
..............
inductive tensor product
17,
...................
........................................... infrabarrelled space................... ..................... intersection theorem ....................................... invariant metric .......................................... iteration method ........................................... irreducible domain ......................................... irreducible analytic set ................................... *
J triple
.......................... ..........................
Jordan algebra
k
-
................................ Krein-Milman theorem ..........................
kernel theorem
Liouville's theorem
81,
104; 3 3 2 , 3 9 7 , 4 1 5 , 4 1 9 Ill
334, 363 211, 4 0 6
157
.....................
local (sheaf) topology ................... local uniform topology ..................
.............
389 3 8 4 , 386
212 4 5 , 6 8 , 6 9 , 9 5 , 2 9 0 , 3 7 2 , 3 7 9 , 397
local maximum modulus principle . . . . . . . . . .
locally m convex algebra
39 3 422
39 3
.............
locally bounded function..............
368
39 4
local boundedness........................ local connectedness
40 1
68, 1 0 4 , 2 0 4 , 2 1 4 , 3 6 5 , 372
Levi problem
..................
263
39 0
....................... 14, space .................................
LindelGf space
406
394
.......................
........................... Lie algebra ............................ Lie group ............................. Lifting theorems ....................... limited set ............................
9 8 , 2 7 8 , 405
1 6 3 , 2 1 1 , 3 9 0 , 422
space
Kelley
34,
16, 399
...................
infinite matrix
J* algebra
16, 400
86, 9 7 , 1 4 1 , 2 5 3 , 304
....................
inductive limit topology
3 1 2 , 357
1 1 0 , 1 1 1 , 408
413 30 1 378 7 5 , 308 393 1 0 , 5 8 , 7 7 , 1 0 4 , 1 9 9 , 2 5 8 , 290
9 8 , 9 9 , 3 5 5 , 417
M closure topology Mackey
. Arens
Index
487
........................................ ....................................
402
15
theorem
Mackey continuous
.........................................
14
Mackey convergence criterion...............................
Mackey convergent sequence ................................ Mackey holomorphic function (Silva) ....................... Mackey space .......................... Mackey
.
strict-convergence
criterion
.................................
mapping, bilinear mapping, compact ..................................
................................. holomorphic ..............................
mapping, diagonal
mapping, nuclear mapping. symmetric
...............................
............................... ...............................
maximum modulus theorem
...........................
meromorphic function............................... Mittag-Leffler theorem ............................ Mobius transformation..............................
Modular hull ....................................... Modularly decreasing set .......................... Montel space ...................................... Montel theorem ....................................
.................................... multiplicative linear functional .................. multiplicative polar ............................. Morera theorem
Noetherian ring .................................. Normal decomposition ............................. normal mapping
14
61 35. 291. 402. 425
...........
topology......................... mapping, biholomorphic ............................
mapping. mapping, n-linear
62
..................................
normal topology .................................. nowhere dense set ................................ nuclear. dual-space ........................ nuclear exponent sequence .......................
nuclear. fully - space ....................... nuclear function of . exponential type ..........
357 35 384. 388 2 , 184. 406 93.
152 3
57 1 2 1 . 402 n
1
408 104. 373. 377. 386 103. 377 211. 391. 422 223. 230 2 3 0 ; 288 14. 402 155. 408. 416 102 106. 290. 366 290 382 385 94 364 177 21 336 33. 229 152
488
Index
nuc 1ear A nuclear mapping
space
.....................................
nuclear polynomial nuclear.
s
........................
....................................
157. 363 21. 402 21
................... ............................... space
nuclear sequence space nuclear space ........................................ nuclearly entire functions ............................
Nullstellensatz ...................................... numbering function ....................................
....................................... Oka-Weil theorem ...................................... Open mapping theorem ................................ Orlicz spaces ....................................... numerical range
56 222 21. 403 136 3 8 4 . 385 340 205. 211. 422 3 1 1 . 3 7 0 . 379 278. 307. 405. 430 214
paracompact spaces.................................. partial differential operator ...................... Patil problem ......................................
3 7 4 . 379
plurisubharmonic functions
170. 366
.........................
PoincarG metric .................................... polar set .......................................... polar. multiplicative .............................. polarization formula ...............................
polydisc ........................................... polynomial ......................................... polynomial. bounded on equicontinuous sets
45.
424 391 2 1 5 . 366 290 4 . 320 230 3
.........
32
polynomial. continuous ............................. polynomial growth .................................. polynomial. hypocontinuous .........................
polynomial. Mackey continuous ..................... polynomial. homogeneous .......................... polynomial. nuclear .............................. polynomial. weakly compact ....................... polynomially convex domain ......................... pointwise convergence...............................
....................................
ported topology ported semi-norm .................................. power series space .................................
95
10
375 13 14 3 21 45 2 1 3 . 3 5 9 . 369 1 4 8 . 399 24.
72 72
2 6 8 . 2 8 9 . 336
489
Index
........................... principal parts ............................... product. E ............................... product. tensor ............................... projective tensor product ...................... proper mapping ................................ property (B) ................................ property ( S ) ................................ pseudo-convex topologies .......................
383
Preparation theorem
Q family
377 328. 407 I . 4 9 . 328. 334. 406
406 387 430 34.
.....................................
........................... quasi-normable space ........................... radius of boundedness .......................... radius of pointwise convergence ................
quantum field theory
.................. Radon- Nikodp Property ........................ rapidly decreasing sequence.................... ramified coverings ............................ reflexive space .............................. regular classes .............................. regular inductive limit ................ regular point of analytic set .......... Reinhardt set ..........................
110
157. 295. 381 133. 3 1 3 . 358 166. 206 166 103. 166
radius of uniform convergence
Remert graph theorem
..................
................ residue theorem ........................ resolvent functidn ..................... removable singularities
................
Riemann mapping theorem Rotund Banach space .................... Rudin-Carleson interpolation theorem Runge's theorem ................................ Russo-Dye theorem ..............................
............
/-absolute
decomposition
.....................
/-Schauder decomposition .....................
62 411
178 262. 291. 356. 429 385 40 1 368. 372 86. 9 7 . 141. 2 5 3 , . 3 0 4
230
.
384 240. 359 388 103 102 211 389 161 424 104. 370 163 114 114
490
Index
Schauder basis ........................
218, 229, 278, 293, 404
............... Schwarz lemma ........................ Schwarz-Pick system ......................
114, 294
Schauder decomposition
161, 2 1 1 , 3 9 1 , 422 39 2
Schwarz-Pick inequality (condition) .......
3 9 2 , 422
.......................... semi-Monte1 space ........................
3 4 , 1 5 2 , 4 0 2 , 421
Schwartz space
14, 402
semi-Reflexive space ..................... separately holomorphic ...................
1 4 2 , 2 5 9 , 401 54,
5 9 , 148, 409, 415
....................... sequence space, nuclear .................. sequential compactness ...................
22 1
sequence space,
222 177 178, 2 0 7 , 212
sequential convergence sheaf cohomology ........................ sheaf (local) topology ................... shrinking decomposition .................. Silva holomorphic function ............... Solid set ................................ space, barrelled
377 7 5 , 308 1 1 4 , 147 61 2 30
........................
2 4 , 112, 400
space, bornological ...................... space, dispersed .........................
1 6 , 1 1 1 , 400 46
space, distinguished Frechet .............. I
............... ....................... ...................... 14,
25,
space, dominated norm (DN)
2 6 2 , 2 8 8 , 3 3 4 , 3 7 5 , 376
space, hemicompact space, k
397, 419 8 1 , 8 1 , 104, 3 3 2 , 3 9 7 , 4 1 5 , 419
space, Kelley ............................. space, Lindel'df ................. 45, 68, 69, space, Mackey
...................
space, Monte1 ................... space, nuclear ................... space, paracompact .............. space, quasinormable ............ space, Schwartz .................
.............. space, semi-reflexive ........... space, superinductive ........... space, semi-Monte1
space, ultra bornological
.......
3 4 , 357
111
9 5 , 2 9 0 , 3 6 5 , 3 7 2 , 397 35, 291, 4 0 2 , 425 14, 402 21, 403 45,
95
1 3 3 , 3 1 3 , 358 3 4 , 152, 402 14, 402 142, 2 5 9 , 401 15,
68
2 4 , 111, 4 0 0
49 1
Index
space
.
w
spectral radius
105
..........................
2 0 4 . 2 1 1 . 422
spectral decomposition theorem............ strict Mackey convergence criterion
390
...........
strict inductive limit .........................
.......................... ................ strictly convex Banach space .................. strong topology ............................... subharmonic function .......................... superinductive space .......................... surjective limit ..........................
357 17.
34.
9 8 . 278. 405 9 9 . 418
strictly compact set
161
strictly c convex Banach space
surjective limit. open
161 2 2 . 401 422 15.
6 8 . 415
316. 362. 367. 373
........................
316
surjective limit, compact ..................... surjective limit. directed .................... symmetric domain
316 358
.............................
symmetric tensor algebra symmetrization opesator
T.S. completeness
389
......................
2 9 6 . 354. 363
......................
2
..........................
...................... tensor products ....................... topology.associated ................... topology. compact open ................ topology. finitely open ............... topology. Kelley ..................... topology. local (sheaf) ................
128. 148 5 4 . 120
Taylor series expansion
topology. local uniform ............... topology. Mackey ...................... topology of pointwise convergence............ topology of the M closure..................... topology. ported
..............................
topology. strong topology. ‘6 topology. T n topology. Tw
.............................
............................. ............................. ..............................
1. 4 9 . 328. 334. 4 0 6 . 4 1 3
16.
53.
110.
146. 153
23.
7 1 . 399
9 2 . 3 7 9 . 411 I l l 7 5 . 308 393 35 96.
148. 399 15 24.
72
2 2 . 401 73 92 24.
72
Index
492
.............. unconditional basis ..................... uniform boundedness principle .......... unzform convexity ..................... uniform factoring ..................... unique factorization domain ............ unit vector basis ...................... universally measurable ................
24.
ultra hornological spaces
very strongly convergent sequence
.......
........ .......................
very weakly convergent sequence Vitali's theorem
................... weak holomorphy ........................ weak* sequentially compact ............. weakly compactly generated Banach space ...... weak conditionally compact ................... Weierstrass Factorization theorem ............ Weierstrass Pxeparation theorem ............... weights ..................................... weak Asplund
1 1 1 . 400
183. 289. 404 13.
50
171. 423 319 383 179. 221 414 81.
97.
149. 281.
321. 325 82.
97
155. 416 212
space
414 178. 207.
212
178. 207.
212 179 383 383
221.
263
