Differential Calculus and Holomorphy: Real and Complex Analysis in Locally Convex Spaces

DIFFERENTIAL CALCULUS AND HOLOMORPHY

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NORTH-HOLLAND MATHEMATICS STUDIES

64

Notas de Matematica (84) Editor: Leopoldo Nachbin Universidade Federal do Rio de Janeiro and University of Rochester

Differential Calculus and Holomorphy Real and complex analysis in locally convex spaces JEAN FRANGOIS COLOMBEAU University of Bordeaux Talence, France

1982 NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM

NEW YORK

OXFORD

0

North-Holland Puhlishing Company, I982

All rights reserved. N o part of thispuhlication may he reproduced, stored in a retrievulsystem, or transmitted, in any f o r m or by any means, electronic, mechanical, photocopying, recording or otherwise. without the prior permission of lhe copyright owner.

ISBN: 0 4 4 4 8 6 3 9 7 4

1’ublbher.s: NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM NEW YORK OXFORD Sole disrribirtor~for the U.S.A.and C’aizrtrf[i ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 52 VANDERBILT AVENUE NEW YORK, N.Y. 10017

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Colombeau, J e a n F r a n $ o i s , 1947D i f f e r e n t i a l c a l c u l u s and holomorphy. (Nortp-Holland mathematics s t u d i e s ; 64) ( N o t a s d e matematica ; 84) B i b l i o g r a p h y : p. Includes index. 1. L o c a l l y convex s p a c e s . 2 . C a l c u l u s , D i f f e r e n t i a l . 3 . Holomorphic f u n c t i o n s . I T i t l e . 11, S e r i e s . 111. S e r i e s : Notas d e m a t e d t i c a (North Holland Publishring Company) ; 84. QA322. c63 515.7’3 82-3523 ISBN 0-444-86597-4 ( E l s e v i e r ) AACR2

PRINTED IN THE NETHERLANDS

Dedicated to LEOPOLDO NACHBIN on the occasion of his sixtieth birthday ( 7 January 1982)

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FOREWORD FIRST P A R T T h i s book p r e s e n t s t h e T h e o r y of D i f f e r e n t i a l C a l c u l u s a n d H o l o m o r p h y in i t s m o d e r n s e t t i n g of infinite d i m e n s i o n a l (locally c o n v e x ) s p a c e s . Although t h i s s u b j e c t h a s v e r y o l d r o o t s s t e m m i n g back to t h e l a s t c e n t u r y , it h a s grown v e r y r a p i d l y in t h e r e c e n t y e a r s , and m o s t of the r e s u l t s included in t h i s book w e r e a v a i l a b l e u p to now only in a r t i c l e f o r m a n d , t h e r e f o r e , not r e a l l y a c c e s s i b l e to the non s p e c i a l i s t . F u r t h e r m o r e one m a y r e a s o n n a b l y believe that t h e g e n e r a l a s p e c t of the T h e o r y will c e r t a i n l y not change fundamentally in t h e f u t u r e . S e v e r a l books on D i f f e r e n t i a l Calculus o r on Holomorphy in l o c a l l y convex s p a c e s a l r e a d y e x i s t , but they do not contain m o s t of t h e r e s u l t s p r e s e n t e d h e r e , which n e v e r t h e l e s s have

a fundamental c h a r a c t e r in the T h e o r y . T h e r e a d e r i s only r e q u i r e d to have a n e l e m e n t a r y knowledge of finite d i m e n s i o n a l Differential Calculus a n d H o l o m o r p h y , of G e n e r a l Topology, a n d s o m e f a m i l i a r i t y with H i l b e r t s p a c e s , n o r m e d s p a c e s a n d l o c a l l y convex s p a c e s a t t h e u s u a l l e v e l of a g r a d u a t e student in M a t h e m a t i c s . T h i s w o r k i s m a i n l y a n e l a b o r a t i o n of m y l e c t u r e s given a t t h e S t a t e U n i v e r s i t y of C a m p i n a s d u r i n g the l o c a l w i n t e r t e r m s 1978 and 1980, which w e r e m a d e p o s s i b l e by financial s u p p o r t from F . A . P. E. S. P. a n d F.I. N. E . P.. On t h e p e r s o n a l s i d e it is a g r e a t p l e a s u r e to e x p r e s s m y g r a t i t u d e to t h e m a n y f r i e n d s who a i d e d m e in p r e p a r i n g t h i s book. I a m p a r t i c u l a r l y indebted to P r o f e s s o r Leopoldo Nachbin f o r h i s k i n d e n c o u r a g e m e n t s a n d f o r a c cepting m y text a s p a r t of h i s s e r i e s Notas d e M a t e m a t i c a . V a r i o u s p o r t i o n s of t h e m a n u s c r i p t have been r e a d a n d a m e n d e d by J. A r a g o n a , H . A . Biagioni,

J . E . Gal4, C . O . K i s e l m a n , P . Lelong, M . C . M a t o s , R . M e i s e , A. M e r i l , V . B. M o s c a t e l l i , J . Mujica, 0. W . P a q u e s , B. P e r r o t . To a l l h e l p e r s , n a m e d a n d u n a m e d , I extend m y w a r m e s t t h a n k s .

J.-F.COLOMBEAU vii

viii

Foreword

SECOND P A R T The a u t h o r e x p e c t s t h i s book to be u s e f u l f o r t h e following r e a s o n s . P r e s e n t l y , it i s g e n e r a l l y c o n s i d e r e d t h a t D i f f e r e n t i a l C a l c u l u s in n o r m e d s p a c e s , i n t e r m s of i t s b a s i c definitions and p r o p e r t i e s , i s a 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 of t h e finite d i m e n s i o n a l c a s e . H o w e v e r m o s t s p e c i a l i s t s , c o n s i d e r that i t s extension to locally convex s p a c e s cannot be a t t a i n e d by a t r i v i a l t r a n s f e r of a r g u m e n t s ( s e e f o r i n s t a n c e Av e r b u c k - S m o l y a n o v [ I ] p . 2 0 2 ) . The a u t h o r ' s p e r s o n a l opinion i s that t h i s situation s t e m s f r o m t h e fact that l o c a l l y convex s p a c e s w e r e not well u n d e r s t o o d f r o m t h e viewpoint of t h e i r bounded s e t s in connection with Differential C a l c u l u s : indeed, if we u s e S i l v a ' s definition, a s done in t h i s book, w e a p o s t e r i o r i a s c e r t a i n that Differential Calculus in locally convex s p a c e s i s , in s o far a s we c o n s i d e r definitions a n d i m m e d i a t e basic p r o p e r t i e s , a 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 of the n o r m e d s p a c e s c a s e . T h e r e s u l t i n g t h e o r y h a s v e r y s i m p l e a n d c l e a r foundations and t h e r e f o r e one m a y c o n c e n t r a t e o n e ' s e f f o r t s on d e e p e r r e s u l t s s u c h a s t h o s e p r e s e n t e d in t h i s book. In t u r n , the t h e o r y i s v e r y r i c h and a d e e p u n d e r s t a n d i n g of it opens the d o o r s f o r new applications to o t h e r b r a n c h e s of M a t h e m a t i c s a n d to T h e o r e t i c a l a n d Mathematical Physics, and, a s a consequence, such applications a r e p r e s e n t l y e x p e r i e n c i n g a r a p i d growth. At f i r s t t h e a u t h o r w a s m a i n l y i n t e r e s t e d in Differential C a l c u l u s , but b e c a m e convinced t h a t , l i k e in t h e finite d i m e n s i o n a l c a s e , Differential C a l c u l u s a n d Holomorphy a r e i n d i s s o c i a b l e . Indeed, like in the finite d i m e n s i o n a l c a s e , H o l o m o r p h y i s a t o o l f o r Diff e r e n t i a l Calculus

(for i n s t a n c e a P a l e y - W i e n e r - S c h w a r t z t h e o r e m a n d

division of i m a g i n a r y - e x p o n e n t i a l - p o l y n o m i a l s l e a d to e x i s t e n c e of

C

a.

solutions f o r finite d i f f e r e n c e p a r t i a l d i f f e r e n t i a l e q u a t i o n s , s e e c h a p t e r 1 3 ) a n d , a t t h e s a m e t i m e , Differential C a l c u l u s is a tool for Holornorphy ( c o n s i d e r f o r i n s t a n c e the powerful m e t h o d s coming f r o m t h e r e s o l u t i o n of the

2

equation, s e e c h a p t e r 1 5 , 1 6 ) . T h e r e f o r e both t h e o r i e s a r e jointly

studied i n t h i s book, in the g e n e r a l s e t t i n g of l o c a l l y convex s p a c e s , a n d then t h e g e n e r a l p i c t u r e b e c o m e s r a t h e r s i m i l a r to the f a m i l i a r one of t h e finite d i m e n s i o n a l c a s e , while Differential C a l c u l u s a n d Holomorphy in t h e Banach s p a c e s c a s e give p a r a d o x i c a l l y t h e i m p r e s s i o n to be p o o r e r t h e o r i e s .

CONTENTS

1

GENERAL INTRODUCTION PART 0 : A E V I E W OF-THE LINEAR BACKGROUND

3

Introduction

9

0.1 0.2 0.3 5 0.4 0 0.5 4 0.6 9 0.7 § 0.8

0 6

Locally convex s p a c e s Bornological v e c t o r s p a c e s E l e m e n t s of duality C o m p a c t a n d n u c l e a r mappings i n n o r m e d s p a c e s Schwartz and n u c l e a r s p a c e s A f e w c l a s s e s of infinite d i m e n s i o n a l s p a c e s Compact a n d n u c l e a r s u b s e t s of F r g c h e t s p a c e s Multilinear mappings a n d p o l y n o m i a l s ,

4

10 15 19 22 25 29 35

PART I : BASIC D I F F E R E N T I A L CALCULUS AND H O L O M O R E Y 41

Introduction C h a p-ter 1 : 1.0 1.1 9 1.2 0 1.3 4 1.4 $ 1.5

5

$ 1.6

_Differentiable m a p p i x s , b a s i c p r o p e r t i e s

45

Definition of differentiability i n n o r m e d s p a c e s Definition of (Silva) differentiable mappings Definitions of Cn a n d Co3 mappings Mean value t h e o r e m a n d T a y l o r ' s f o r m u l a s Cco mappings i n the e n l a r g e d s e n s e Mappings which a r e "locally differentiable between n o r m e d s p a c e s " Cco mappings of u n i f o r m bounded type

47 48 52 55 61

69 72

C h a p t e r 2 : Holomorphic m a p p i n g s , b a s i c p r o p e r t i e s

77

Q 2.1

79 83

5

2.2

4 2.3 2.4 5 2.5 2.6 $ 2.7 $ 2.8

Gateaux a n a l y t i c mappings Silva holomorphic mappings Holomorphic mappings a n d Silva h o l o m o r p h i c mappings i n t h e e n l a r g e d s e n s e Cco differentiability of h o l o m o r p h i c m a p p i n g s An example S e r i e s of homogeneous polynomials Holomorphic mappings of u n i f o r m bounded type Holomorphic r e p r e s e n t a t i o n of F o c k s p a c e s of Boson F i e l d s ix

88 92 98 101 105 109

Contents

X

C h a p t e r 3 : C l a s s i c a l p r o p e r t i e s of holomorphic mappings

$ 3.1 $ 3.2 3.3

$ 3.4

V e c t o r valued holomorphy v e r s u s scalar v a l u e d holomor p hy Zorn's t h e o r e m Hartogs' theorem Montel's t h e o r e m

C h a p t e r 4 : Lopolopies on -

$ 4.1 $ 4.2

4

4.3 4.4

4

a@, F ) and 3Cs(n,F )

N a t u r a l topologies on c$(n,F) a n d K ( 0 , F ) C o m p l e t e n e s s of X S ( n , F ) a n d d(D,l?) Schwartz p r o p e r t y of 3C (fi,F) a n d d ( Q , F ) Reflexivity of K,(n,F) Sand r$(Q,F)

113

115 118 123 126 129 131 134 138 142

C h a p t e r 5 : Approximation a n d density r e s u l t s

143

4

A density r e s u l t in 3-C (0,F ) A density r e s u l t i n STQ,F) c03 p a r t i t i o n s of unity

145 149 154

C h a p t e r 6 : € - p r o d u c t and k e r n e l t h e o r e m s

157

5.1 $ 5.2 $ 5.3

$ 6.1

0 6.2

$ 6.3

Schwartz e - p r o d u c t in s p a c e s of holomorphic functions Schwartz € - p r o d u c t in s p a c e s of CO3 functions Approximation p r o p e r t y i n Ks(Q) a n d a(n)

C h a p t e r 7 : T h e F o u r i e r - B o r e 1 a n d F o u r i e r transfor-m2

$ 7.1

0

7.2 $ 7.3 0 7.4

P r e l i m i n a r y r e s u l t s on the F o u r i e r - B o r e 1 transform The F o u r i e r - B o r e 1 i s o m o r p h i s m Holomorphic g e r m s T h e F o u r i e r t r a n s f o r m a n d the P a l e y - W i e n e r Schwartz i s o m o r p h i s m s

C h a p t e r 8 : N u c l e a r i t y of s p a c e s of h o l o m o r p h i E C m mappings

0

8.1 8.2 $ 8.3

4

N u c l e a r i t y of ws(Q) Strong n u c l e a r i t y of Ks(Q) Non n u c l e a r i t y of d(D)

PART I1 : CONVOLUTION AND

159 164

16 6

-167 169 176 182 18 6 192

193

2 01 205

EQUATIONS

Introduction

206

C h a p t e r 9 : Convolution equations i n p(E)

208

$ 9.1

9

9.2 $ 9.3 $ 9.4

F o r m a l power s e r i e s a n d duality A division r e s u l t T h e convolution o p e r a t o r s on P ( E ) E x i s t e n c e of solutions

209 213 217 222

Contents

xi

C h a p t e r 0 : Convolution e q u a t i o n s i_n s p a c e s of e n t i r e functions of exponential type

$ 10.1 $ 10.2

8

10.3

The convolution o p e r a t o r s on 3 lck ( 0 ) Approximation of the solutions E x i s t e n c e of solutions

C h a p t e r 11 : Division of d i s t r i b u t i o n s

$ $ $ $ $ $

11.1 11.2 11.3 11.4 11.5 11.6

The Wei'erstrass preparation t h e o r e m Division by a complex polynomial Division of d i s t r i b u t i o n s by holomorphic f u n c t i o n s Application t o e x i s t e n c e of solutions Division by r e a l a n a l y t i c functions of finite type I m p o s s i b i l i t y of the division by r e a l polynomials

223 224 229 235 245 246 2 52 262 266 2 68 272

C h a p t e r 12 : Convolution equations in s p a c e s of h o l o m o r p h i c functions 2 77 -

$ 12.1 $ 12.2 $ 12.3 $ 12.4 $ 12.5

8 8

12.6 12.7

$ 12.8

T h e convolution o p e r a t o r s on K s ( E ) E n t i r e functions of n u c l e a r bounded type o n a Banach s p a c e T h e convolution o p e r a t o r s on Xu, b(E) T h e convolution o p e r a t o r s on XN, (E) A division r e s u l t Existence a n d approximation r e s u l t s in KN, b(E) E x i s t e n c e a n d a p p r o x i m a t i o n r e s u l t s in Xu, b ( E ) a n d X (E) Convo?ution o p e r a t o r s of finite type

C h a p t e r 13 : L i n e a r finite d i f f e r e n c e p a r t i a l d i f f e r e n t i a l equations in 8 ( E )

8

13.1

$ 13.2

8

13.3

8

13.4

A division r e s u l t by i m a g i n a r y exponential polynomials A Paley-Wiener-Schwartz t h e o r e m and a division r e s u l t E x i s t e n c e a n d a p p r o x i m a t i o n of solutions i n normed spaces E x i s t e n c e of solutions i n locally convex s p a c e s

279 283 293 3 01 3 08 317 321 324 326 327 337

3 42 3 46

C h a p t e r 14 : Pseudo-conve? d o m a i n s a n d a p p r o x i m a t i o n results

$ 14.1 $ 14.2 $ 14.3 $ 14.4

G l i m p s e at pseudo-convexity a n d d o m a i n s of h o l o m o r phy T h e LCvi p r o b l e m T h e Runge a p p r o x i m a t i o n t h e o r e m An a p p r o x i m a t i o n t h e o r e m

C h a p t e r 15 : T h e

6

15.1

$ 15.2

347

5

equation

-

3 48 3 54 3 62 372 3 76

Differential f o r m s a n d b op r a t o r A r e v i e w of H o r m a n d e r ' s L e s t i m a t e s and existence t h e o r e m s

P

377 3 79

xii

Contents

$ $ $ $ $

15.3 15.4 15.5 15.6 15.7

6

15.8

A review on i n t e g r a t i o n in H i l b e r t s p a c e s A b a s i c existence r e s u l t An hypoellipticity r e s u l t The 5 equation in a s c a l e of H i l b e r t s p a c e s E x i s t e n c e of Cm solutions in pseudo-convex open s u b s e t s of DFN s p a c e s E x i s t e n c e of C m solutions in n u c l e a r A . c . s .

Chapter>& :

$ 16.1 $ 16.2

6

16.3

Some applications

-

of the b

equation

Solution of the f i r s t Cousin p r o b l e m A counterexample On solutions of s o m e homogeneous convolution equations

383 388 3 98 4 01 404 407 409 410 412 416

Bibliographic Notes

423

B i bli og r aphy

431

Index

453

GENERAL INTRODUCTION

H i s t o r i c a l l y t h e s t u d y of Infinite 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 n d Holomorphy g o e s b a c k t o t h e e n d of t h e last c e n t u r y : V o l t e r r a 1887

.. .

[ 11

( s e e the h i s t o r i c a l s u r v e y of A v e r b u c k a n d Smolyanov [ 2 ] ) .

Among t h e p i o n e e r s i n t h i s f i e l d let u s also quote H i l b e r t [l], H a d a m a r d

[l

1, Gateaux [ l , 21

a n d F r C c h e t [1, 2, 3 1 . During t h e f i r s t d e c a d e s of t h e

20th c e n t u r y t h e t h e o r y in the s e t t i n g of B a n a c h s p a c e s w a s s t u d i e d i n d e t a i l a n d now its b a s i c f a c t s a r e quite c l e a r . T h e c a s e of g e n e r a l l o c a l l y convex topological v e c t o r s p a c e s h a s b e e n s t u d i e d s i n c e t h e m i d d l e t h i r t i e s a n d h a s given r i s e t o a c o n s i d e r a b l e a m o u n t of l i t e r a t u r e ( s e e the s u r v e y p a p e r s by A v e r b u c k a n d Smolyanov

[ 21,

Nashed [l] and their bibliographies)

At t h e s a m e t i m e , v a r i o u s T h e o r e t i c a l a n d M a t h e m a t i c a l P h y s i c i s t s i n t r o d u c e d a n d u s e d , m o r e o r l e s s r i g o r o u s l y , Infinite 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 n d Holomorphy a s a tool in Q u a n t u m F i e l d T h e o r y (Bogoliubov a n d Shirkov

[ 11,

Berezin

[ 11, . . . ) .

At p r e s e n t , both t h e o r i e s of D i f f e r e n t i a l C a l c u l u s a n d of Holom o r p h y i n l o c a l l y convex s p a c e s a r e i n t e n s i v e l y developed by M a t h e m a t i c i a n s i n m a n y c o u n t r i e s a s b r a n c h e s of P u r e M a t h e m a t i c s , while a p p l i c a tions to o t h e r f i e l d s a p p e a r f r o m t i m e t o t i m e .

A s i d e f r o m Banach s p a c e s , t h e m o s t " i n t e r e s t i n g " a n d "usual" s p a c e s c o n s i d e r e d nowadays a r e l o c a l l y convex s p a c e s with s o m e c o m p a c t n e s s or n u c l e a r i t y a s s u m p t i o n . In t h e s e s p a c e s , i t h a s b e e n n o t i c e d r e c e n tly, t h e T h e o r y of Differential C a l c u l u s a n d H o l o m o r p h y b e c o m e s c o n s i d e r a b l y c l e a r e r a n d d e e p e r . Many "deep" r e s u l t s f r o m t h e f i n i t e d i m e n s i o n a l c a s e r e m a i n v a l i d , in g e n e r a l with non t r i v i a l new p r o o f s , i n v a r i o u s s u b c l a s s e s of t h e s e s p a c e s . 1

General Introduction

2

T h i s book g i v e s a n e l e m e n t a r y a c c o u n t of t h e s e r e c e n t r e s u l t s , s o f o r a v a i l a b l e i n a r t i c l e f o r m a n d only a c c e s s i b l e t o s p e c i a l i s t s . I t s

aims a r e a l s o t o c l a r i f y t h e s i t u a t i o n of D i f f e r e n t i a l C a l c u l u s i n l o c a l l y convex s p a c e s a n d t o be a convenient r e f e r e n c e book f o r s p e c i a l i s t s a s w e l l a s a convenient tool f o r M a t h e m a t i c i a n s working i n o t h e r f i e l d s a n d f o r Mathematical P h y s i c i s t s . T h e c o n t e n t s of t h i s book s h o u l d be e a s i l y a c c e s s i b l e t o s t u d e n t s in M a t h e m a t i c s a n d t o T h e o r e t i c a l P h y s i c i s t s , s i n c e the r e q u i r e d m a t h e m a t i c a l knowledge i s , on t h e one hand,the c l a s s i c a l f i n i t e d i m e n s i o n a l Differential Calculus and Holomorphy and, on the o t h e r , s o m e f a m i l i a r i t y with the r u d i m e n t s of G e n e r a l Topology, H i l b e r t s p a c e s , n o r m e d s p a c e s a n d l o c a l l y convex s p a c e s . T h e t h e o r e m s a n d p r o o f s w e r e c h o s e n a s being both t h e s i m p l e s t and the m o s t r e p r e s e n t a t i v e o n e s . Short Bibliographic Notes on each c h a p t e r a r e g i v e n a t t h e e n d of t h e b o o k : t h e y c o n t a i n t h e s o u r c e of t h e r e s u l t s e x p o s e d in the book a n d r e f e r t o o t h e r r e s u l t s about w h i c h p r e c i s e r e f e r e n c e s a r e g i v e n . In t h i s way we hope t o have r e c o n c i l e d s i m p l i c i t y a n d c l e a r n e s s with s o m e c o m p l e t e n e s s . T h e b a s i c knowledge o n infinite d i m e n s i o n a l s p a c e s i s r e v i e w e d i n P a r t 0 , t h e n the book is m a d e of P a r t I on t h e g e n e r a l t h e o r y of differ e n t i a b l e a n d h o l o m o r p h i c m a p p i n g s a n d P a r t I1 o n s o m e p a r t i a l d i f f e r e n t i a l a n d convolution e q u a t i o n s , T h e r e a d e r is r e f e r e d t o t h e i r r e s p e c t i v e i n t r o d u c t i o n s . In p a r t i c u l a r t h e b a s i c p r o b l e m of t h e c h o i c e of the d e f i n i t i o n s of d i f f e r e n t i a b l e a n d h o l o m o r p h i c m a p p i n g s i n l o c a l l y convex s p a c e s i s d i s c u s s e d i n t h e I n t r o d u c t i o n of P a r t I .

PART 0 A REVIEW OF THE LINEAR BACKGROUND

INTRODUCTION

T h i s p a r t 0 is devoted t o c l a s s i c a l d e f i n i t i o n s a n d p r o p e r t i e s of l o c a l l y convex s p a c e s

( a .c . s.

f o r s h o r t ) a n d convex b o r n o l o g i c a l v e c t o r

s p a c e s ( b . v . s . f o r s h o r t ) , of l i n e a r , m u l t i l i n e a r m a p p i n g s a n d polynom i a l s . It is not a s k e t c h of t h e T h e o r y of Infinite D i m e n s i o n a l S p a c e s , but j u s t a s u m up of m a i n d e f i n i t i o n s a n d r e s u l t s which a r e of a c o n s t a n t u s e i n t h e book. M o s t of r e s u l t s a r e g i v e n without proof b u t with c l a s s i c a l r e ferences. We do not r e c a l l b a s i c definitions a n d e l e m e n t a r y p r o p e r t i e s of B a n a c h a n d H i l b e r t s p a c e s , which a r e a s s u m e d t o be known. I n $ 0 . 1 a n d 0 . 2 we define

L . c . s . , b . v . s . , t o p o l o g i e s a n d b o r n o l o g i e s . I n 6 0 . 3 we

p r e s e n t e l e m e n t s of dua1ity;some of t h e s e r e s u l t s a r e not c l a s s i c a l . In

0 . 4 we e x p o s e definitions a n d p r o p e r t i e s of c o m p a c t a n d n u c l e a r m a p pings between n o r m e d s p a c e s , a n d i n $ 0 . 5 we i n t r o d u c e S c h w a r t z a n d nuclear R . c . s .

and b.v.s.

0 . 6 i s c o n c e r n e d with a d e s c r i p t i o n of v a -

r i o u s c l a s s e s of s p a c e s we n e e d ( F r k c h e t , S i l v a , b o r n o l o g i c a l , E 3 s p a c e s

...).

$ 0 . 7 d e a l s with u s e f u l p r o p e r t i e s of t h e " c o m p a c t bornology" a n d

of t h e " r a p i d l y d e c r e a s i n g bornology" of a n y F r k c h e t s p a c e , t h a t w i l l be u s e d thoughout all t h i s book. Then in $ 0 . 8 we c o n s i d e r continuous m u l t i l i n e a r m a p p i n g s a n d continuous p o l y n o m i a l s on A . c . s . , bounded m u l t i l i n e a r m a p p i n g s a n d bounded p o l y n o m i a l s o n b . v . s . , All t h e s e c o n c e p t s a r e t r e a t e d in d e t a i l s i n c l a s s i c a l books t o which we r e f e r : B o u r b a k i €13, G r d t h e n d i e c k [3], a n d M o s c a t e l l i [l]

, Horwath[l],

Nachbin [Z, 43, P i e t s c h [l], T r e v e s [l]

. .,

Hogbk-Nlend [l), HogbB-Nlend

Kelley a n d N a m i o k a

ill,

Kdthe [I],

R o b e r t s o n a n d R o b e r t s o n [l], S c h a e f e r [l],

3

4

A review of the linear baekgroutd

8

0 . 1 , 1 Definition

-

K ( K = R or

.A

CC

0 . 1 LOCALLY CONVEX SPACES

L e t E be a l i n e a r s p a c e o v e r t h e field topology on E is s a i d t o be a l i n e a r topology ( o r a

v e c t o r topology) f a d d i t i o n a n d s c a l a r m u l t i p l i c a t i o n a r e continuous m a p p i n g s ( f r o m E x E t o E and IK x E t o E

respectively).

W e c a l l a topological l i n e a r s p a c e a n y l i n e a r s p a c e E equipped with a l i n e a r topology. T h e n the s y s t e m of n e i g h b o r h o o d s of a point x is obtained by t r a n s l a t i o n f r o m the s y s t e m of o - n e i g h b o r h o o d s : if

i s a f u n d a m e n t a l s y s t e m of o - n e i g h b o r h o o d s , t h e n f o r e v e r y point x

EE

the f a m i l y

? ( x ) = { x - tV

is a

where

I

V

fundamental s y s t e m of n e i g h b o r h o o d s of x ( w e a l s o s a y : a b a s e

of n e i g h b o r h o o d s of x ) .

0. 1 . 2 Definitions.if

A s u b s e t A of a l i n e a r s p a c e is said to b e convdx

e x t By E A w h e n e v e r x , y E A and

with

~ r t .

B , fl

a r e positive r e a l n u m b e r s

p = 1.

A subset A

of a l i n e a r s p a c e i s s a i d t o be b a l a n c e d if

w h e n e v e r x E A and

Ih

I

E A

5 1 (X E IK).

A s u b s e t A of a l i n e a r s p a c e is said t o be a b s o r b i n g if f o r e v e r y x E A

there exists some

0. 1. 3 Definition. -

E

0 with

6x

E A.

A topological l i n e a r s p a c e E i s c a l l e d a l o c a l l y

c o n v e x s p a c e ( 1 . c . s . f o r s h o r t ) if i t i s Hausdorff and i f a n y o - neighborhood c o n t a i n s a c o n v e x o -ne ighborhood

.

Locally convex spaces 0.1.4

PROPOSITION.

-

(a)

Let

5

E be a 1 . c . s .

a homotethy i n v a r i a n t f u n d a m e n t a l s y s t e m

Then there exists

V of o - n e i g h b o r h o o d s

which a r e c o n v e x , b a l a n c e d , a b s o r b i n g and s u c h t h a t : (i)

forevery V

(ii)

for e v e r y x

(b)

Conversely let E

,V 6

?J

E E, x

#

1

2'

there i s V

3

E ? J w&V

t h e r e is V

0,

E

3

CV p V 1 2

with x

?J

be a l i n e a r s p a c e o v e r JR 01

6V

.

C and l e t

v

be a homotethy i n v a r i a n t f a m i l y of c o n v e x , balanced, a b s o r b i n g s u b s e t s

of E

f o r which

(i) and

(ii) huld.

T&

i s f u n d a m e n t a l system

y

of o-neighborhoods f o r a (Hausdorff ) l o c a l l y convex topology on E.

F c r a proof s e e Ktithe

0. 1 . 5 PROPOSITION. -

[ 11

5

15 and

E v e r y 1. c . s.

5

18 o r S c h a e f e r

has a

[ 11 8

1

.

f u n d a m e n t a l s y s t e m of

o-neighborhoods c o n s i s t i n g of c l o s e d , c o n v e x , balanced s e t s ,

F o r a proof s e e S c h a e f e r

[ 11 5

1.

0. 1. 6 S e m i - n o r m s defining t h e topology. on a 1. c. s .

E

{ p > o such that x C P U

is a convex, balanced

n o r m a s s o c i a t e d to U" "gauge of U".

A s e m i w o r m p defined

is continuous if and only if it i s of t h e f o r m

p(x) = inf

where U

-

p i s called the " s e m i -

o-neighborhood.

o r t h e "Minkowski functional of U" o r t h e

T h e topology of E is thus d e t e r m i n e d by t h e c o l l e c t i o n of

all continuous s e m i - n o r m s . A locally convex topology on a l i n e a r s p a c e E c a n a l s o be c o n s t r u c t e d by m e a n s of a s y s t e m P with the following p r o p e r t i e s if

(ii)

if x EE, x

:

there is p

pl, p 2 c p

(i)

f

0,

of s e m i - n o r m s

E P with p.(x) 5 p(x) i f

t h e r e i s p E b with

p(x)

>

0.

i = 1,2.

6

A review of the linear background

T h i s is j u s t a r e f o r m u l a t i o n of (0. 1.4). T h e sets

u k P for p

E

'5

= Ix

1

E E s u c h that p ( x ) < h

X > 0 f o r m a b a s e of convex balanced closed

and

o-neighborhoods.

0. 1.7

L i n e a r s u b s p a c e s a n d finite d i m e n s i o n a l 1.c. s . -

s u b s p a c e of a 1. C .

E is a l s o a 1.c. s .

S.

Each linear

with the induced topology.

E v e r y n-di:nensional 1. c. s . o v e r M is topologically i s o m o r p h i c to Mn with i t s n a t u r a l topology ( s e e K6the [ l

] f3 15. 5 ) .

So a n y finite n d i m e n s i o n a l s u b s p a c e of a 1. c . s . is topologically i s o m o r p h i c to M f o r s o m e n EN.

-

0. 1.8 Quotient s p a c e s .

If F is a c l o s e d l i n e a r s u b s p a c e of a (not

n e c e s s a r i l y Hausdorff) 1. c. s. quotient s p a c e Q = E / F

E, a l o c a l l y convex topology on the

is obtained f r o m the s e m i - n o r m s

* .

p ( x ) = inf I p ( x f y ) when y

E F

1

i f x i s the i m a g e of the e l e m e n t x E E u n d e r the c a n o n i c a l quotient mapping E

-

.

E/=

S e e KBthe

[ I]

0. 1. 9 Notation.-

5

Let E

o-neighborhood.

Let p

pv

i.e.

is a norm,

14 o r S c h a e f e r

V

be a 1 . c . s .

[ 13

1.2, for instance.

and V a convex balanced

denote t h e s e m i - n o r m a s s o c i a t e d t o

V. If

if V d o e s not c o n t a i n a n y s t r a i g h t l i n e , w e

the l i n e a r s p a c e E n o r m e d by p If V c o n t a i n s a V V' s t r a i g h t l i n e , i.e if p is n o t a n o r m b u t only a s e m i - n o r m , w e denote

denote by E

V

Locally coilvex spaces

V

by E V the q u o t i e n t s p a c e

-1

P,

, which is , of c o u r s e , a n o r m e d

(0)

space.

0. 1.10 E m b e d d i n g of a 1 . c . s .

i n a p r o d u c t of n o r m e d s p a c e s .

~

defining the topology of E (0. 1.6). quotient s p a c e ( E , p,)/p:

L e t u s denote by E

the n o r m e d

a *

If x E E l e t u s denote by x

(0).

under the quotient mapping E

- (Xs)*c* a

Let E

a n d l e t ( p ) nEA be a f a m i l y of continuous s e m i - n o r m s

be a 1.c.s.

in E

-

-

E

@

. Let

f r o m E to the p r o d u c t s p a c e

0. A

Q

its image

I denote the mapping

E a . Since E i s

Hausdorff, I i s i n j e c t i v e and t h u s E m a y be i d e n t k e d with a topological s u b s p a c e of the p r o d u c t

p.

E

a'

We s a y that E i s t h e p r o j e c t i v e limit

and we w r i t e E = lim E U ClcEA tions (tnd r e s u l t s , s e e Kathe [l] Q 18 and rj 19. of t h e n o r m e d s p a c e s (E

u aEA

. F o r defini-

A s u b s e t B of a 1. c . s o E is called bounded

0. 1. 1 1 Bounded s e t s . if f o r e a c h o-neighborhood

U

there is

E 3

o such that

6B e U

.

It

follows f r o m (0. 1.5) that the c l o s u r e of a bounded set is a g a i n bounded. The collection @ bornology" of E.

of a l l bounded s u b s e t s of E is c a l l e d the "Von Neumanr, It i s i m m e d i a t e to c h e c k the following p r o p e r t i e s

covers E, i.e.

3EER

VxEE

s u c h t h a t x E B.

1)

i3

2)

if B

3)

if B

4)

if B

5)

if

6)

i f B € 0 t h e n the convex, balanced h u l l of B ,

E

A

and B ' c B

then B '

1'

B E B then B y B e B 2 1 2

1'

B E l3 then B fB E B 2 1 2

B E 0 and

s t i l l a n e l e m e n t of 8

€a

EX t h e n A B E B denoted by

TB,

is

(we have

r 5 =I

s u c h t h a t x . E B and finite

7)

:

n o s t r a i g h t l i n e IfSx (x

f

0)

is bounded.

2 / A . 11.

1,.

8

A review of the linear background

0 . 1.12 D u a l s p a c e .

-

T h e topological d u a l ( r e f e r e d to a s t h e d u a l

E is the l i n e a r s p a c e E' which c o n s i s t s

throughout the book) of a 1 . c . s.

of allcontinuous l i n e a r f o r m s on E ,

i.e.

continuous l i n e a r mappings f r o m E

to

E' i s t h e l i n e a r s p a c e of a l l

lK.

It is a c o n s e q u e n c e of the (fundamentally i m p o r t a n t ) H a h n - B a n a c h t h e o r e m that if

E F',

and if u

F i s a n y topological l i n e a r s u b s p a c e of the 1. c . s . E t h e n u m a y be extended a s a n e l e m e n t of

E is s e p a r a t e d by i t s d u a l E' ( i . e .

c o n s e q u e n c e , a n y 1 . c . s. with x

#

0

3 u E E ' s u c h that

0 . 1. 13 P o l a r i t y .

-

El. A s a

U(X)

#

VxE E

0).

i s a s u b s e t of E we define i t s p o l a r ACE' a s

If A

the s e t

= IuEE'

0 . I . 1 4 Strong d u a l topology. of a 1.c. s .

-

s u c h that

1.

l u ( x ) IS1 f o r e v e r y x 6 A

T h e s t r o n g tcpology on Lhe duai E '

E i s defined a s the topology of u n i f o r m c o n v e r g e n c e on the

bounded s u b s e t s of E .

When topologized in t h i s w a y , the d u a l E ' is

denoted by EtB.The 1. c. s.

E'

B

h a s a s a b a s e of o - n e i g h b o r h o o d s the

s e t of p o l a r s of bounded s u b s e t s of E.

0. 1.15 D u a l w e a k topolopy. 1. c . s .

E

The w e a k topology on the d u a l E '

of a

is defined i n t h e s a m e way a s the s t r o n g d u a l topology but

h e r e the finite s u b s e t s of E

play the r o l e

w e a k topology is denoted by

u ( E ' , E ) a n d we w r i t e E'

Gf

the bounded s u b s e t s . T h e (5

f o r the d u a l E '

u n d e r i t s w e a k topology.

0. 1 . 1 6

Cauchy s y s t e m s .

i s a s e t of e l e m e n t s x

a

-

B

in A , a r e l a t i o n

{x }

a

in a 1.c.s.

E

E E which a r e uniquely a s s o c i a t e d with the

e l e m e n t s of a n index s e t A and

A directed system

s u c h t h a t for c e r t a i n p a i r s of

u 26

indices a

i s defined with t h e following p r o p e r t i e s :

9

LocaIZy convex spaces

f o r finitely m a n y i n d i c e s

(2)

with

a 2 - Ql , . . . , a > a n

.

8

l,. . .,a n 6 A

t h e r e is a n index

S e q u e n c e s a r e the d i r e c t e d s y s t e m s obtained

when A

a€ A

is t h e s e t

of n a t u r a l n u m b e r s ,

A d i r e c t e d s y s t e m (x ) ct

f o r e a c h neighborhood x

P

E U

if

U

a€ .4

of x ,

c o n v e r g e s to a n e l e m e n t x € E if

t h e r e i s a n index 6 €A 0

such that

a>ao.

We c a l l a d i r e c t e d s y s t e m (x ) e a c h o-neighborhood

a aEA

a Cduchy s y s t e m if f o r

U , t h e r e is a n index a E A 0

with x -x E U

a p

a , @2 a 0 .

if

E a c h c o n v e r g e n t d i r e c t e d s y s t e m is a Cauchy s y s t e m . If the c o n v e r s e i s a l s o t r u e , t h e n the 1.c. s . is c a l l e d c o m p l e t e . F o r e a c h 1. c . s .

E t h e r e is a c o m p l e t e 1. c. s .

-

E,

uniquely

d e t e r m i n e d up t o a topological i s o m o r p h i s m , w h i c h c o n t a i n s E a s a dense linear space.

E is called the c o m p l e t i o n of E ; s e e Ktjthe [ 1 ]

A d i r e c t e d Cauchy s y s t e m (x ) of e l e m e n t s x

0’

for

@€A,

a aEA

i s bounded

6 18.

is c a l l e d bounded if t h e s e t

. A 1. c . s.

is c a l l e d q u a s i -

c o m p l e t e if e a c h bounded Cauchy s y s t e m in i t i s c o n v e r g e n t .

10

A review of the linear background

5

0.2

0.2. 1 Definitions.

A family W

BORNOLOGICAL VECTOR S P A C E S

-

L e t E be a l i n e a r s p a c e ( o v e r K = R o r C),

of s u b s e t s of E satisfying the p r o p e r t i e s

1) to 7 ) l i s t e d

in (0. 1. 11) i s s a i d to define a convex l i n e a r bornology on E.

The pair

( E , @ ) is called a convex bornological v e c t o r s p a c e ( b . v . s . f o r s h o r t ) . T h e e l e m e n t s of

b

a r e c a l l e d bounded s e t s

. A family

8 ' of bounded

s e t s i s called a b a s e of bounded s e t s ( o r a f u n d a m e n t a l s y s t e m of bounded s e t s ) if f o r e a c h B

0.2. 2 Examples,

-

(1)

E 63,

t h e r e is B ' E B ' with B e B'.

L e t E be a l i n e a r s p a c e . The f a m i l y of s u b s e t s

of E which a r e cnntained i n a finite d i m e n s i o n a l l i n e a r s u b s p a c e of E and bounded t h e r e f o r m a bornology, chlled the finite d i m e n s i o n a l bornology of E.

( 2 ) Let E be a 1.

C.

s.

T h e f a m i l y of its bounded

s e t s defined in ( 0 . 1. 11) is called the Von Neumann this b.v.s.

bornology of E and

s t r u c t u r e is denoted by BE. (3)

L e t E be a q u a s i - c o m p l e t e 1. c . s . .The

f a m i l y of r e l a t i v e l y c o m p a c t s u b s e t s of E f o r m a bornology the c o m p a c t bornology of E

, called

( i n a q u a s i - c o m p l e t e 1.c. s . the c l o s e d ,

convex, balanced hull of a c o m p a c t s e t i s still c o m p a c t , s e e Kothe [ l ]

4 20.6). (4)

dual

El

L e t E be a 1. c . s.

and l e t u s c o n s i d e r in its

the s e t s which a r e contained in the polar of a ( v a r i a b l e )

o-neighborhood of E

( i . e. the equicontinuous s e t s ) . T h e y f o r m a b o r n o -

l o g y , called the equicontinuous b,ornology of

El.

11

Bomvlvgical vector spaces

0 . 2 . 3 Bornological dual.

-

L e t E be a b . v. s. .We denote by E "

the l i n e a r s p a c e of a l l l i n e a r f o r m s on E which a r e bounded on e v e r y bounded s u b s e t of E .

T h e s e f o r m s a r e c a l l e d the bounded l i n e a r f o r m s

on E and E X is called,.the b o r n o l o g i c a l d u a l of E .

W e s a y t h a t E Y s e p a r a t e s E ( o r that the duality between E and

Ex is s e p a r a t i n g ) if f o r e v e r y x E E , x form u on E

s u c h t h a t u(x)

#

0.

#

0 ,

t h e r e i s a bounded l i n e a r

In Rornology

,

a n a n a l o g u e of t h e

Hahn-Bandch t h e o r e m i s k n o w n t o b e f a l s e i n g e n e r a l and t h e r e a r e s e v e r a l e x a m p l e s of b.v. s. (even of n u c l e a r b.v. s . , defined below) which a r e not s e p a r a t e d by t h e i r d u a l ( s e e Hogbk-Nlend [ 2 b. v. s .

3 Chap.

XI). Nevertheless all

c o n s i d e r e d in t h i s book a r e a s s u m e d t o be s e p a r a t e d by t h e i r

d u a l s . T h i s a s s u m p t i o n is e s s e n t i a l f o r m a n y r e s u l t s a n d we s h a l l not r e p e a t i t . A l l "usual" b.v. s. a s w e l l a s all"usua1" s p a c e s c o n s t r u c t e d f r o m t h e m hdve this p r o p e r t y .

0.2.4

Notations and bornological inductive l i m i t s . -

convex, b a l a n c e d , bounded s u b s e t of a b.v.s. v e c t o r s p a n of B ( i . e .

EB

u

E.

L e t B be a

W e denote by E

B

the

n B) n o r m e d by t h e gauge p of B n E IN 0 such t h a t x E AB , which is a n o r m s i n c e B

B ( p (x) = inf i k B c o n t a i n s no s t r a i g h t line.

Let b.v. s E

(Bi)i.I

I)

be a f u n d a m e n t a l s y s t e m of bounded s e t s in t h e

which a r e convex and balanced. Denote by E

.

the normed i T h e n o r m e d s p a c e s Ei a r e d i r e c t e d u n d e r i n c l u s i o n in the

space E i' s e n s e that f o r e v e r y p a i r of i n d i c e s i ,

j E I,

t h e r e e x i s t s a n index kEI

with Ei c E k and E . c Ek. By definition ( 0 . 2. 1) a s u b s e t of E is J hounded if and only i f it is contained in s o m e E. a n d bounded t h e r e . W e e x p r e s s t h i s f a c t by s a y i n g that E is the b o r n o l o g i c a l ( i n j e c t i v e ) inductive l i m i t of the n o r m e d s p a c e s E . , i E I ,

and we w r i t e E = lim 4

Ei.

For

i €1 the g e n e r a l definition of b o r n o l o g i c a l inductive l i m i t s s e e Hogbe-Nlend c h a p . 11.

[ 1]

12

A review of the linear background

0.2.5

The M a c k e y - c l o s u r e topology on E ,

define a topology T E

7E.

-

If E is a b.v. s.,

we

called t h e M a c k e y - c l o s u r e topology , a s

follows : a subset

n

c E i s open f o r

T E

if

n

o r if f o r e v e r y x E R

=

and e v e r y bounded s u b s e t B of E t h e r e is a n

E>

0 such that

x t G B c Q .

E i i n the s e n s e of ( 0 . 2 . 4 ) , t h i s definition a m o u n t s

If E = l i m i21 to : 0

n

E i i s open in the n o r m e d s p a c e E . f o r e v e r y i

I

.

T h i s topology i s not a l i n e a r topology in g e n e r a l ( f o r d e t a i l s s e e Hogbe-Nlend [ 2

0.2.6

]

c h a p . 11).

Bornivorous sets. -

A s u b s e t P of E is s a i d t o be

b o r n i v o r o u s if f o r e v e r y bounded s u b s e t with

E

B of E t h e r e is a n

B c P. If E = lim E i a s in (0.2.4), iri I

this amounts t o : PnEi

is a o-neighborhood i n t h e n o r m e d s p a c e E . for e v e r y index i

Then

ncE, 0

#$

> CI.

F:

E I

, is open f o r the topology 7 E ( 0 . 2 . 5 )

.

i f and

only i f f o r e v e r y x E R t h e r e is a b o r n i v o r o u s s e t P s u c h t h a t

xtPcR

.

0 . 2 . 7 The bornological topology T E . a b.v. s.

E,

-

We define a n o t h e r topology on

called the bornological topology a n d denoted by T E ,

as

follows :

Q c E is open f o r T E if 0 = $ o r if f o r e v e r y x E R t h e r e is a convex, balanced, b o r n i v o r o u s s e t Q with x t Q c Q.

It is i m m e d i a t e to c h e c k t h a t T E is a Hausdorff 1. c . s. ( r e c a l l that E is a s s u m e d to be s e p a r a t e d by i t s d u a l ) , t h a t T E is f i n e r than

13

Bontological vector spaces

T E and that E X= ( T E ) ' a l g e b r a i c a l l y . A b a s e of 0-neighborhoods f o r

T E c o n s i s t s of the f a m i l y of a l l convex balanced b o r n i v o r o u s s u b s e t s of E. 0.2.8

Remark. -

F o r r e a d e r s who a r e acquainted with topological

inductive limits l e t u s m e n t i o n that T E is the topological inductive l i m i t of the topologies of the n o r m e d s p a c e s

E . ( i n the c a t e g o r y of topological

s p a c e s , not v e c t o r s p a c e s ) a n d t h a t T E i s the l o c a l l y convex topological inductive l i m i t of the n o r m e d s p a c e s KBthe

0.2.9

[ 1 ] 9 19

( i n the c a t e g o r y of 1 . c . s ) . See

f o r inductive l i m i t s of 1.c. s.

P o l a r b . v . s.

-

A b.v.s.

of a n y bounded s u b s e t of

a polar b.v.s.

E.

.

E is s a i d to be p o l a r i f the c l o s u r e

E f o r the topology T E is bounded. T h e r e f o r e

a d m i t s a b a s e of convex, b a l a n c e d , bounded s e t s which

a r e closed for T E.

F o r u s t h e i m p o r t a n c e of t h i s c o n c e p t c o m e s f r o m

the f a c t that i t i s a good s e t t i n g f o r a p p l i c a t i o n s of t h e m e a n s value t h e o r e m (chap. I ) .

0.2.10

Complete b. v . s o -

A b.v. s . E i s s a i d t o be c o m p l e t e i f t h e r e

e x i s t s a b a s e of convex, b a l a n c e d , bounded s e t s ( B . ) . 1

normed space E a "Banach disc".

Bi

la

s u c h that e a c h

is a Banach s p a c e . Such a bounded s e t B .

is c a l l e d

E a d m i t s a b a s e of convex,

C l e a r l y a polar and c o m p l e t e b. v. s .

b a l a n c e d , Banach d i s c s that a r e c l o s e d for T E .

0.2.1 I

Bornological subspaces.

F be a l i n e a r s u b s p a c e of E.

-

L e t E be a b . v . s . a n d l e t

It is i m m e d i a t e l y s e e n that the i n t e r

s e c t i o n s with F of the bounded s u b s e t s of E struGture.

-

e q u i p F with a b . v . s .

F equipped with t h i s s t r u c t u r e is c a l l e d a bornological

s u b s p a c e of E . One a l s o p r o v e s e a s i l y that on a f i n i t e d i m e n s i o n a l l i n e a r s p a c e

A review of tlic linear background

14

t h e r e is a unique b . v . s .

0.2.12

s t r u c t u r e (which is s e p a r a t e d by its dual).

Mackey-convergent sequences.

-

L e t E be a b . v . s . . A

s e q u e n c e ( x ) c E is said to be Mackey c o n v e r g e n t o r bornologically n c o n v e r g e n t ( M - c o n v e r g e n t f o r s h o r t ) t o somlj e l e m e n t x of E (and

M

___> x ) i f t h e r e e x i s t s o m e bounded s u b s e t B of E n and s o m e null sequence ( E > 0) s u c h that xn - x E c n B . n

w e write x

This m e a n s that if E =

lim

E . i n the s e n s e of (0.2.4),

the

+

set

x

n

lxn -+

lnEW u

x in E .

1

lx

1

i E I is contained in s o m e n o r m e d s p a c e E . and

.

0

0

If S i s a s u b s e t of E topology 7 E if and only if

one c h e c k s e a s i l y t h a t S is c l o s e d f o r the :

x E S n =,

x n-

M

X E S .

j X

In this c a s e we s a y that S i s Mackey c l o s e d . For d e t a i l s on Mackey c o n v e r g e n c e s e e Hogbe-Nlend [ Z j c h a p I1

0.2.13

Remark.

-

When dealing with b . v . s. one h a s to be

v e r y carefii

b e c a u s e t h e r e a r e s e v e r a l t r a p s : not only t h e Hahn-Banach t h e o r e m i s not t r u e in g e n e r a l in the c a s e of b . v . s . (0.2.3),

b u t a l s o t h e (bornologi

-

g i c a l ) completion of a p o l a r b . v . s. is not p o l a r i n g e n e r a l ( C o l o m b e a u , Grange', P e r r o t

c I),

Moscatelli

[ I])

and the M a c k e y - c l o s u r e o f a

l i n e a r s u b s p a c e is not obtained via M - c o n v e r g e n t s e q u e n c e s ( C o l o m b e a u , Lazet, Perrot [I

1, HogbC-Nlend

[ 2 1,

Perrot

[ 1 , 2 1.. . .

.

15

Elements of duality

5

E L E M E N T S O F DUALITY

0.3

0.3. 1 N a t u r a l l y r e f l e x i v e 1. c . s .

-

L e t E be a 1. c. s . and l e t

E ' be its d u a l , We e q u i p E ' with i t s equicontinuous bornology, i . e . the a r e a b a s e of bounded s e t s i n E '

set

o-neighborhoods in E.

E ' X

IV

1

is a b a s e of

E ' i s now a b . v . s. and we m a y c o n s i d e r i t s Clearly E c E ' x

bornological d u a l E l x . If E =

if

algebraically.

a l g e b r a f c a l l y we s a y that E is a n a t u r a l l y r e € l e x i v e

1.C.S.

L e t u s e q u i p ElX with the tOpOlOgy of u n i f o r m c o n v e r g e n c e on the bounded s e t s of

El,

i . e . the p o l a r s i n E ' X

a b a s e of o-neighborhoods i n E ' Y

.

If E i s a n a t u r a l l y r e f l e x i v e 1. c . s.,

0

of the s e t s V e

then t h i s topology on

c o f n c i d e s exactly with t h e topology of E , h e n c e t h e equality E = holds not only algebrai'cally

form

El

Elx El*

but a l s o topologically (.this follows f r o m 0

t h e bipolar t h e o r e m (0.3.6) : V = balanced )

.

?

if V

is closed convex and

R e m a r k on v a r i o u s notions of reflexivity.

0.3.2

-

T h e notion

of

n a t u r a l r e f l e x i v i t y i s not quite c l a s s i c a l , but i t i s the r i g h t one f o r u s . C l a s s i c a l l y one d e f i n e s s e m i - r e f l e x i v e and r e f l e x i v e s p a c e s ( s e e Kdthe [l]

5

2.3) , One m a y e a s i l y p r o v e that a n a t u r a l l y r e f l e x i v e 1. C . S .

is

a l w a y s s e m i - r e f l e x i v e , but not a l y a y s r e f l e x i v e . So we p r e f e r to s p e a k of " n a t u r a l l y reflexive" s p a c e s r a t h e r that of " c o m p l e t e l y r e f l e x i v e " s p a c e s a s i n Hogbg-Nlend

[ 1]

chap. 6.4.

A " n a t u r a l l y reflexive" s p a c e

w a s called "completely s e m i - r e f l e x i v e " i n K r 6 e [ 2 ]

. Naturally reflexive

1. C . s . a r e not in g e n e r a l " q u a s i - b a r r e l e d " h e n c e not r e f l e x i v e in t h e c l a s s i c a l s e n s e ( s e e Kdthe

[ 1] 3 23 for instance).

16

A review oftlie linear background

0 . 3 . 3 Reflexive b . v . s .

-

S t a r t i n g w i t h a b.v. s .

E one e q u i p s i t s

d u a l EX with the topology of 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 i . e . the p o l a r s of t h e bounded s u b s e t s of E

of E ,

o-neighborhoods in E x . Now

f o r m a b a s e of

E x i s a 1.c. s . and w e m a y e q u i p i t s

topological dual Ex' with the equicontinuous bornology.

.

and E c E"

E i s said to be a r e f l e x i v e b . v . s .

Ex " is a b. V . s .

if E = Ex' a l g e b r a y c a l l y and

bornologically. F o r t h i s kind of r e f l e x i v i t y s e e Hogb6-Nlend

[ 11 c h a p , 6 .

L e t u s j u s t r e m a r k t h a t it follows from the b i p o l a r t h e o r e m ( 0 . 3 . 6 ) t h a t if E

( 0 . 2 . 9 ) and if E = E x '

is a p o l a r b . v . s.

algebralcally, then this

identity is a l s o b o r n o l o g i c a l .

0. 3 . 4 D u a l p a i r s .

-

T w o l i n e a r s p a c e s L 1 and

L

2

over K

f o r m a d u a l p a i r ( o r a r e in a s e p a r a t i n g d u a l i t y ) w h e n a n e l e m e i i t of denoted by < u , x > ,

i s a s s o c i a t e d with e v e r y p a i r

if

2

Then L of

L

2

L1 in

is b i l i n e a r f r o m L

i s fixed a n d x # 1 i s fixed and u # 0 , t h e r e i s x E L

if x E L

u EL

L2

Y L to K, 2 1 0 , t h e r e is u E L2 with<'u,x>{ 0 and

s u c h a way t h a t : ( i ) the f o r m c ,> (ii)

(u,x)

K,

1

withCu,x,#

0

.

m a y be identified with a s u b s p a c e of the a l g e b r a i c d u a l

1

and c o n v e r s e l y .

Examples.

-

If E i s a Hausdorff 1. c .

S .

with d u a l E ' ,

<, 2 i s the d u a l i t y b r a c k e t < u , x > = u(x) , u E E ' , x E E ,

and if t h e n E and

E ' f o r m a d u a l p a i r (by t h e H a h n - B a n a c h , t h e o r e m s e e (0. 1 . 1 2 ) ) . If E i s a b.v.s.

E'

( s e p a r a t e d by its d u a l , as we a l w a y s a s s u m e ) t h e n E and

a l s o f o r m a dual pair. If L1 and

L

2 define i t s p o l a r S c L

f o r m a d u a l p a i r and if

0

every x

E S\

.

2

as the s e t

S is a s u b s e t of

] u C L 2 such that

L1' w e

lcu,x>lCl for

Elements of duality

0.3.5

Let L

Weak toDoJoevL-

and

17

L

be a d u a l p a i r . We define 2 a s the topology of u n i f o r m c o n v e r -

1

L ) on L 1' 2 1 gence on the finite s u b s e t s of L i . e . the p o l a r s of t h e finite s u b s e t s of 2' L a r e a b a s e of o-neighborhoods f o r o ( L 1 , L ). We define s i m i l a r l y 2 2 the w e a k topology CT ( L L ) on L2. U ( L 1 , L 2 ) and U ( L 2 , L 1 ) a r e 2' 1 c l e a r l y H a u s d o r f f 1 . c . s.. A p a r t i c u l a r c a s e w a s a l r e a d y c o n s i d e r e d in

the w e a k topology

cr(L

(0.1.15).

0 . 3 . 6.

-

and L be a d u a l p a i r and l e t 1 2 If M c L2 d e n o t e s the p o l a r of M, then t h e

Bipolar theorem.

M be a s u b s e t of 0

M,

p o l a r of

L

1'

00

denoted by M ,

Clearly M c M T h e bipolar

00

is a s u b s e t of

of a s u b s e t M of L

M

convex, balanced h u l l of M.

F o r a p r o o f , s e e KSthe

[ 11 4

1

i s the weakly c l o s e d ,

20

Topologies c o m p a t i b l e with a d u a l p a i r .

0.3.7.

L

called the b i p o l a r 1 The bipolar t h e o r e m a s s e r t s :

.

00

of M.

Let L

0

-

be a d u a l p a i r . T h e n a ( H a u s d o r f f ) l o c a l l y convex topology said to be c o m p a t i b l e with the d u a l i t y b e t w e e n L dual pair

0.3.8.

( L 1 , L 2 ) ) if

Mackey's t h e o r e m .

, L 2) 1

-

'k:

Let L1 and L

%

on L

1

have the s a m e bounded s e t s

Neumann bornology on L1.

0.3.9

and L 1 2 on L is 1 ( o r with the

and L 1 2 L2 m a y be identified with the topological d u a l

l o c a l l y convex topologies

(L

Let L

-

be a d u a l p a i r . T h e n a l l 2 c o m p a t i b l e w i t h t h e duality

,

S e e Ktithe

[ I]

5 20.

and L be a d u a l p a i r . T h e n w e 1 2 a s the topology of unifarm define t h e Mackey topology 'r ( L L 2 ) on L 1 c o n v e r g e n c e on t h e weakly c o m p a c t , convex, balanced s u b s e t s of L 2' Mackey topology.

Let L

i. e . they define the s a m e V o n

A review of the linear backgrvund

18

7 ( L 1 , L ) i s c l e a r l y a Hausdorff 1. c . s. topology on L 2 1' than the weak topology U ( L 1 , L2).

L e t L4 and

Mackey-Arens theorem. -

0.3.10

which is f i n e r

p a i r . Then a l o c a l l y convex topology

on L

1

L

be a d u a l 2 is compatible with

the d u a l pair ( L1, Lz) if and only i f it is f i n e r than ( o r equal to) the weak topology

T(L 1 ' L2)

topology

0. 3. 1 I

o ( L L ) and c o a r s e r than ( o r equal to) the Mackey 1' 2 See Kdthe

.

PROPOSITION.

-

[ 11 §

2 1.

L e t E be a n a t u r a l l y r e f l e x i v e 1. c . s. and

l e t E ' be i t s d u a l equipped with the equicontinuous bornology. T h e n the topologies

Proof.

-

-

T E ' (0.2.7) and E '

Algebravcally

m e a n s that T E ' and E.

( 0 . 3.8),

B

P

( 0 . 3 . lo), TE'

( 0 . 1.14) a r e i d e n t i c a l .

( T E ' ) ' = E''

( 0 . 2.7),

h e n c e ( T E ' ) ' = E,

which

is a topology compatible with the duality between E!

C l e a r l y (El )' El

B

c (TE')'

= E.

T h e r e f o r e , by ( 0 . 3 . l o ) , ( 0 . 3 . 9 )

coyncides with the Mackey topology

?(E',E).

i s c o a r s e r than T ( E ' , E ) . T h u s T E ' = El

B

By

.I

19

Compact and nuclear mappings

d

0.4 COMPACT AND NUCLEAR MAPPINGS IN NORMED SPACES

0.4.1

-

Compact mappings.

A l i n e a r mapping f r o m a n o r m e d s p a c e E

t o a n o r m e d s p a c e F i s c a l l e d p r e c o m p a c t if the i m a g e of t h e c l o s e d unit b a l l of E is a p r e c o m p a c t s u b s e t of F (a s u b s e t S c F is c a l l e d precompact if, for every

E

Z 0,

t h e r e e x i s t s a finite n u m b e r x

of e l e m e n t s of S s u c h t h a t , if B(xi, S

then 5 e

i = l

B(xi, F),

where

11 ]IF

E ) =

/x E F s u c h t h a t

IIx-x.

1

1'

.., x

1''F< E 1,

d e n o t e s the n o r m i n F).

A l i n e a r mapping f r o m a n o r m e d s p a c e E to a n o r m e d s p a c e F is called c o m p a c t i f the i m a g e of t h e c l o s e d unit b a l l of E is contained in a c o m p a c t s u b s e t of F ( i . e . is r e l a t i v e l y compact).

F is a Banach s p a c e t h e n e a c h p r e c o m p a c t l i n e a r m a p p i n g

If

f r o m E into F is c o m p a c t ( s i n c e c o m p l e t e and p r e c o m p a c t imply compact)

.

-

L e t E a n d F be n o r m e d s p a c e s , l e t T : E

-

F be a l i n e a r

mapping and l e t t T : F' E' be its t r a n s p o s e d ( i . e . t T ( y ' ) (x) = y' (Tx) if y ' E F' and x E E ) . T h e n , if T is p r e c o m p a c t , tT is c o m p a c t (a proof is in HogbC-Nlend S c h a e f e r [ I ] chap. 111

0.4.2

0

Nuclear mappings.

[ 11 chap.

7 lemma 2 o r

9.4).

-

A linear mapping T f r o m a normed space

E into a n o r m e d s p a c e F i s c a l l e d n u c l e a r if it a d m i t s the r e p r e sentation

1 n x' n (x) y n

Tx = n

20

A review of the linear background

IXnI < t

for every x E E , where

a3

(i.e.

( 1n ) E R

1

1,

E'

n

E a c h n u c l e a r mapping is p r e c o m p a c t ( s e e P i e t s c h F o r e a c h n u c l e a r mapping T , nuclear (see Pietsch

0.4.3

c 1 1 p r o p 3.1.8).

5).

t h e t r a n s p o s e d mapping t T is

Q u a s i - n u c l e a r mappings. -

T f r o m a normed space E

[ 1Jprop 3. I.

A continuous l i n e a r mapping

into a n o r m e d s p a c e F is called q u a s i -

n u c l e a r i f and only if t h e r e is a n o r m e d s p a c e

G containing F a s a

topological s u b s p a c e s u c h t h a t T i s n u c l e a r a s a m a p p i n g f r o m E into

G.

T h u s we have the d i a g r a m :

fC

nuclear

,/

U

i n c l u s i o n with induced topology

q u a s i -nuc l e a r

The i m p o r t a n c e of t h i s concept c o m e s f r o m t h e following : THEOREM.

-

T h e c o m p o s i t i o n p r o d u c t of two q u a s i - n u c l e a r m a p p i n g s

is n u c l e a r ( s e e P i e t s c h [ I ] th. 3 . 3 . 2 ) .

0.4.4

F a c t o r i z a t i o n of a n u c l e a r mapping. -

normed space

, l e t F be a B a n a c h s p a c e and l e t T be

mapping f r o m E into F.

L e t E be a a nuclear

Then there exists a separable Hilbert space

H such that T m a y be f a c t o r i z e d through H a s the p r o d u c t T

2

o T

1

21

Compact and nuclear mappings

of t w o c o n t i n u o u s l i n e a r m a p p i n g s , w i t h T

2

i n j e c t i v e , a c c o r d i n g to t h e

d i a g r a m below :

E

T

H T h e p r o o f is e a s y by d e f ( 0 . 4 . 2 ) ; s e e a l s o S c h a e f e r

[ 11 chap.

I11

5 7.3.

22

A review of the linear background

§ 0.5

0.5.1

S C H W A R T Z AND NUCLEAR S P A C E S

S c h w a r t z and n u c l e a r 1. c . s .

-

A 1. C . S .

E is c a l l e d a

S c h w a r t z 1. c , s , ( r e s p e c t i v l e y a n u c l e a r 1. c . s . ) if f o r e v e r y c o n v e x , balanced o-neighborhood balanced

V in E t h e r e e x i s t s a smaller convex,

o-neighborhood

U i n E s u c h t h a t the c a n o n i c a l mapping

( s e e ( 0 . 1. 9)) i s p r e c o m p a c t ( r e s p . n u c l e a r ) between the n o r m e d s p a c e s

EU and E V . A n u c l e a r 1. c. s . i s a f o r t i o r i a S c h w a r t z 1. c . s..

0.5.2

S c h w a r t z and n u c l e a r b . v . s .

-

L e t E be a comp1eteb.v. s .

W e s a y that E is a S c h w a r t z b.v. s . ( r e s p e c t i v e l y , a n u c l e a r b . v . s ) i f f o r e v e r y convex, b a l a n c e d , bounded s u b s e t

B of E t h e r e e x i s t s a

l a r g e r convex, balanced, bounded s u b s e t B ' of E

such that the

inclusion mapping : i

EB

BB' BEBI

is c o m p a c t ( r e s p . n u c l e a r ) between the n o r m e d s p a c e s E B and E

(note that E

B

and E B , m a y be c h o s e n t o be B a n a c h s p a c e s ) .

B'

A n u c l e a r b . v . s . is a f o r t i o r i a S c h w a r t z b . v . s .

0.5.3

Co-Schwartz and c o - n u c l e a r 1. c . s .

-

A 1.c.s.

is

called a c o - S c h w a r t z 1. c. s. ( r e s p e c t i v e l y a c o - n u c l e a r 1. c . s . ) if its Von Neumann bornology is a S c h w a r t z ( r e s p . , a n u c l e a r ) bornology.

23

Schwartz and nuclear spaces

0.5.4

Subspaces.

-

E v e r y s u b s p a c e of a S c h w a r t z 1. c . s .

( r e s p e c t i v e l y a n u c l e a r 1. c . s . ) is a S c h w a r t z 1. c. 1.c.

S.

(resp., a nuclear

S.).

E v e r y Mackey c l o s e d s u b s p a c e of a S c h w a r t z b.v. s . ( r e s p e c t i v e l y a n u c l e a r b . v . s . ) i s a S c h w a r t z b.v. s . ( r e s p . , a n u c l e a r b . v . s . ) .

T h e p r o o f s a r e e a s y - u s e th. ( 0 . 4 . 3 ) in the n u c l e a r c a s e .

0.5.5.

THEOREM.

-

Let E -

be a b . v . s . a n d l e t E x

d u a l equipped with i t s n a t u r a l topology defined in (0.3.3).

denote its

If E

S c h w a r t z b.v. s . ( r e s p e c t i v e l y , a n u c l e a r b . v . s . ) then E X is a S c h w a r t z 1. c . s. ( r e s p . a n u c l e a r 1.c. s . ) . T h e proof follows f r o m p r o p e r t i e s ( 0 . 4 . 1) and ( 0 . 4 . 2 ) of t r a n s posed m a p p i n g s

L e t E be a 1 . c . s . T h e n E -

0.5.6.

THEOREM. -

1.c.s.

( r e s p e c t i v e l y a n u c l e a r 1 . c . s . ) if and only i f i t s d u a l E '

is a S c h w a r t z

equipped

with t h e equicontinuous bornology is a S c h w a r t z b.v. s. ( r e s p . a n u c l e a r b . v . s.). T h e proof t h a t i f E is a S c h w a r t z 1. c. s, t h e n E' i s a S c h w a r t z b. v. s. and the a n a l o g o u s proof i n t h e n u c l e a r c a s e a r e similar to t h o s e i n (0. 5. 5)

.

F o r t h e c o n v e r s e u s e that E i s a topological s u b s p a c e of

EIx ( f r o m the b i p o l a r th. (0. 3 . 6 ) )

. For

a d e t a i l e d proof i n t h e S c h w a r t z

c a s e , s e e HogbC-Nlend [ 13 chap. VII. 0.5.7.

THEOREM.

strong dual E '

0

t h i s last c a s e E

-

L et E -

be a q u a s i - c o m p l e t e 1 . c . s . I f s

is a S c h w a r t z 1.c. s . ( r e s p e c t i v e l y a n u c l e a r 1. C . s . - in is c l a s s i c a l l y called a d u a l n u c l e a r 1 . c . s . ) t h e n E

a c o - S c h w a r t z ( r e s p . , a c o - n u c l e a r ) 1. c. s..

2

24

A review of the linear background

F o r a proof,identify E equipped with its Von N e u m a n n bornology with a Mackey c l o s e d b o r n o l o g i c a l s u b s p a c e of (El ) I equipped with the

P

e q uic ontinuous bornology.

0.5.8.

H i l b e r t i a n s t r u c t u r e of n u c l e a r b . v . s . and 1. c .

n u c l e a r b.v.s.

S.

-

If E is a

it follows f r o m (0.4.4)t h a t t h e r e i s a b o r n o l o g i c a l

r e p r e s e n t a t i o n E = l i m E . ( i n the s e n s e of ( 0 . 2 . 4 ) ) w h e r e t h e s p a c e s -

i t I

1

E . a r e separable Hilbert spaces. F o r t h e same r e a s o n , if E is a n u c l e a r 1. c . s . t h e r e is a b a s e of

o-neighborhoods

E

,

'i

0.5.9

( V .).

1 1

t I

s u c h t h a t the a s s o c i a t e d n o r m e d s p a c e s

( 0 . 1.9), a r e s e p a r a b l e p r e - H i l b e r t s p a c e s .

THEOREM.

reflexive 1.c.

S.

-

A c o m p l e t e S c h w a r t z 1. c . s. i s a n a t u r a l l y

(definition ( 0 . 3 . 1)).

A proof i s in Hogbk-Nlend € 1 ] ( 7 . 2 . 4 ) . 0.5.10.

THEOREM.

-

A Schwartz b.v.s.

(definition ( 0 . 3. 31.

A proof i s in Hogbk-Nlend

[

11 ( 7 . 2 . 4 ) .

i s a reflexive b . v . s .

Infinite dimensional spaces

5

0.6.1

0.6

25

A FEW CLASSES O F INFINITE DIMENSIONAL S P A C E S

-

B o r n o l o g i c a l 1.c.s.

If E

is a 1. c . s . ,

we denote by BE

Von N e u m a n n bornology (defined in ( 0 . 1. 1 1 ) ( 0 . 2 . 2 ) ) .

its

We s a y that E is

a bornological 1. c . s . if t h e topological e q u a l i t y T B E = E

( s e e dkf. ( 0 . 2 . 7 ) ) holds. T h i s a m o u n t s to : e v e r y bounded l i n e a r mapping f r o m E

into

a n a r b i t r a r y 1. c . s . is continuous. M o s t of the "usual" s p a c e s e n c o u n t e r e d in d i s t r i b u t i o n t h e o r y ( S c h w a r t z

0.6.2.

[ 23)

M e t r i z a b l e and F r k a h e t 1. c . s.

a d m i t s a countable b a s e of

have t h i s p r o p e r t y .

-

A 1. c . s. is m e t r i z a b l e if it

o-neighborhoods. A F r C c h e t s p a c e is a

c o m p l e t e m e t r i z a b l e 1. c. s . E v e r y b o r n i v o r o u s s e t in a m e t r i z a b l e 1 . c . s . i s a

PROPOSITION 1.-

o-neighborhood ( h e n c e it c o n t a i n s a convex balanced b o r n i v o r o u s set].

A s a c o n s e q u e n c e TBE = T B E = E. The proof is e a s y .

PROPOSITION 2 . -

If_ (Bn)n

is a s e q u e n c e of bounded s e t s ( r e s p e c -

t i v e l y of c o m p a c t s e t s ) in a F r k c h e t 1.

c >O n

such t k t t

s u b s e t of E .

u

C.

s.

then t h e r e is a s e q u e n c e

E ~ B ,is a bounded ( r e s p . a r e l a t i v e l y c o m p a c t )

n EN

The proofs a r e easy.

A review of the linear background

26

PROPOSITION 3 . -

Any null s e q u e n c e in a m e t r i z a b l e 1 . c . s .

E

is

Mackey-convergent to 0 for the Von Neumann bornology of E. The proof is e a s y .

-

THEOREM.

0.6.3.

A Frgchet 1.c.s.

E is a S c h w a r t z 1 . c . s .

( r e s p e c t i v e l y , a n u c l e a r 1. c . s . ) if and only i f i t is a c o - S c h w a r t z 1. c. s . ( r e s p . , a c o - n u c l e a r 1 . c . s ) . In o t h e r w o r d s a F r k c h e t s p a c e E is a S c h w a r t z ( r e s p e c t i v e l y , a n u c l e a r ) 1. c . s. i f and only if s o i s i t s s t r o n g d u a l E'

__

i?'

[ 11

In the S c h w a r t z c a s e a proof i s in Hogb6-Nlend nuclear case, see Pietsch

0.6.4

c3 spaces.-

e l] 4.3.3

7 . 3 . In the

and 4.4. 14.

25 s p a c e , o r " s t r i c t inductive limit of

A

F r g c h e t s p a c e s ' ' i s m a d e u p in the following way : l e t (E ) be n n €IN ) of F r g c h e t s p a c e s s u c h t h a t , f o r a n i n c r e a s i n g sequence (E c E n nt1 (therefore, E is induces on E the topology of E e v e r y n, n n n En+ I a closed s u b s p a c e of E n t l ) . T h e ( l o c a l l y convex) inductive limit of t h e s p a c e s E n m a y be d e s c r i b e d a s the "union" of t h e s p a c e s E n ,

bearing

i n mind the i n c l u s i o n s E

cE a n d it is a 1.c. s . with the following n nt1' topology : a b a s e of o-neighborhoods is m a d e up of t h e convex, balanced subsets

R of E s u c h t h a t , f o r e v e r y n 6 IN,

fl

n

En is a

o-neighborhood in E n . A c l a s s i c a l e x a m p l e is the ( S c h w a r t z ) s p a c e functions with c o m p a c t s u p p o r t in a n open s e t PROPOSITION.

-

E v e r y bounded s u b s e t of a

u)

a(W)

c IR

n

.

of

CcD

S 5 s p a c e is contained and

bounded in s o m e E n ( s e e S c h a e f e r ( 1 1 c h a p I1 $ 6 . 5 and Kdthe €13 PROPOSITION. -

s p a c e is c o m p l e t e ( s e e S r h a e f e r

Every 25

c h a p . I1 5 6 . 6 and Kbthe [ I 1

$'

19).

5

[ 11

19).

27

Infinite dimensional spaces

0.6.5

Topological b . v . s.

b.v.

if

S.

-

We s a y t h a t a b.v. s.

E is a topological

BT E = E bornologically ( s e e def.(0.6. 1)).

Since, if F i s a n y 1. c . we obtain

S .

,then B T B F = B F ,(immdiate proof),

:

A b.v. s . E is a topological b . v . s . i f and only if its bornology is the Von Neumann bornology of s o m e locally convex s p a c e topology o n the underlying l i n e a r s p a c e E . CLassical e x a m p l e s of non-topological b.v. s. a r e t h e s p a c e s c ( E ) and

s ( E ) defined below in ( 0 . 7 . 6 ) and (0. 7. 7) if E is a n y infinite

d i m e n s i o n a l Banach s p a c e ( s i n c e it is i m m e d i a t e t o p r o v e t h a t T c ( E ) and T s(E) coyncide with the given B a n a c h s p a c e topology of E ) .

0.6.6.

THEOREM.

-

A p o l a r b . v . s . with a countable b a s e of

bounded s e t s i s a topolugical b.v. s.. F o r a proof s e e K6the [ I ]

0.6.7

Silva s p a c e s .

-

9

2 9 . 5 (4)

A Silva s p a c e is a S c h w a r t z b . v . s .

with a

countable b a s e of bounded s e t s .

T h e r e f o r e , a Silva s p a c e E m a y be r e p r e s e n t e d b o r n o l o g i c a l l y

-

as E = lim

En,

where the spaces E

n

a r e B a n a c h s p a c e s s u c h that,

n EN for every n

E N , E n is contained i n E

nt 1

with a c o m p a c t i n c l u s i o n .

By ( 0 . 6 . 6 ) , a Silva s p a c e i s a topological b . v . s . 0.6.8.

THEOREM.

-

(a)

.

E is a Silva s p a c e then E x , equipped

with its n a t u r a l topolovy of 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

of E

, i s a Frkchet-Schwartz space

.

28

A review of the linear background

(b)

If E -

is a F r C c h e t - S c h w a r t z s p a c e t h e n E l ,

equipped with

i t s equicontinuous bornology, is a Silva s p a c e .

F o r a p r o o f , s e e Hogbe-Nlend

[ 1]

c h a p VII. S i n c e they a r e

topological b. v . s . ,Silva s p a c e s a r e a l s o called "Strong d u a l s of F r C c h e t S c h w a r t z s p a c e s " ( D F S s p a c e s f o r s h o r t ) , (E = E

X I

i f E i s a Silva

s p a c e , from (0. 5. l o ) ) . 0.6.9

THEOREM.

-

If E

is a Silva s p a c e , t h e Mackey c l o s u r e

topology T E coi'ncides with the bornolovical topology T E . F o r a proof s e e Hogbe-Nlend € 1

0.6. 1 0 N u c l e a r Silva s p a c e s .

1 7.3.1

-

t h . 1.

A n u c l e a r Silva s p a c e is a

Silva s p a c e which is a n u c l e a r b . v . s . , i. e . bornologically r e p r e s e n t e d a s E = lim 4

a b.v. s.

E that m a y be

En, where the spaces E

nE N a r e Banach s p a c e s such t h a t , f o r e v e r y n ,

En

En+l

n

with a n u c l e a r

inclusion.

0.6.11

THEOREM,

-

(a)

I f _ E i s a n u c l e a r Silva s p a c e t h e n ??,

equipped with its n a t u r a l topolovy, is a n u c l e a r F r e c h e t s p a c e . (b)

Jf-E

is a n u c l e a r F r C c h e t s p a c e then El,

equipped with

i t s equicontinuous bornology, is a n u c l e a r Silva s p a c e .

A proof follows f r o m ( 0 . 6 . 8 ) a n d ( 0 . 4 . 2 ) ; s e e a l s o P i e t s c h

[ 1)

4. 4. 1 3 . N u c l e a r Silva s p a c e s a r e a l s o c a l l e d " s t r o n g d u a l s of n u c l e a r F r k c h e t s p a c e s " (D. F . N .

s p a c e s f o r s h o r t ) , s e e 0. 6. 9.

.

29

Subsets of Frichet spaces

9

0.7

COMPACT AND NUCLEAR SUBSETS O F A F R E C H E T S P A C E

T h e following t h e o r e m i s a t the b a s i s of m a n y r e s u l t s p r o v e d i n t h i s book.

0.7.1

THEOREM.

-

Let

be a F r e ' c h e t s p a c e , l e t R

E

b

s

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 and l e t K be a c o m p a c t s u b s e t of

E c o n t a i n e d in 0. T h e n t h e r e e x i s t s a n u l l s e q u e n c e (x ) of n n G N points in R s u c h that

Befm-e giving t h e p r o o f , we r e c a l l t h a t

f-

denotes the \ n n @N c E , i . e . the

/x

c l o s e d , convex, b a l a n c e d h u l l of t h e s e t

fxnJnEN i n t e r s e c t i o n of all c l o s e d c o n v e x b a l a n c e d s u b s e t s of E t h a t c o n t a i n

r 1p = i ( n=1 Proof.-

y )EE where n n

S i n c e K is c o m p a c t i n R

K c ( I - E ) 0. L e t ( U n I n E N o-neighborhoods i n E

n

, t h e r e is a n

c > 0 such that

be a b a s e of c l o s e d , c o n v e x , b a l a n c e d

such that U n + I C U n

f o r e v e r y n and

U c E R . K is c o m p a c t , h e n c e p r e c o m p a c t a n d , t h e r e f o r e , t h e r e i s a 1 finite s e t B of points in K s u c h t h a t 1 K c B

1

+ -21

Ye

30

A revim ofthe linear background

We s e t :

A

1

= (K-B ) 1

1 n u1 2

E v e r y point i n K may be w r i t t e n a s

1

'

y1 t z1

where

1

zyl E B

1

and

€ A 1 . A 1 is c o m p a c t , h e n c e p r e c o m p a c t , a n d t h e r e f o r e , t h e r e is a 1 finite s e t B c A such that : z

2

1

A

1 c B 2 t - U

1

2

22

'

We s e t

A

If x E K ,

-2- y 2

E B

2

x =

2

1 y 2

and

c

where

z1

y1

E B

1

= ( A - B ) n 71 l 2 2

2

t z

1

z2E A2.

.

U2

1 y 2 t z2, 2

a s before and now z 1 = 2Therefore

where

1 1 x = - y t - y t z 1 2 2 2 2 2 1 1 y 2 E B 2 and z

2

A straightforward

2

A2

induction g i v e s a f i n i t e s u b s e t B

s u c h that

A n - l c Bn t 2 - n U

A

n

= (A - B ~ n ) n-1

n

,

'n

n

of A n - 1

31

Subsets of Frichet spaces

and s u c h t h a t e v e r y point x € K m a y be w r i t t e n a s

1

\

x =

where 2-r

Yr

therefore z

if 1 5 r

Br

-

-. 0 if n

n

2 - r yr t z n ,

1SrSn

=n

and w h e r e

z

n

E An.

An c 2-n

‘n’

and

m

r = l

Since t h e s e t s Bn a r e f i n i t e , w e m a y define a s e q u e n c e

tco

u

{x 1 P PEN

If x E K , the s e t of points that every y

( I ) becomes

1

/x by : P PEN

n=l

{yr

1

fx \ p pel" s o

i n (1) is contained i n

m a y be w r i t t e n a s x :

2n B n .

n

r

f o r s o m e index n

E IN.

Therefore

tm

(2)

Since B hence x

n

P

cA +

x = > r = l

n-1

0 if p

H o w e v e r , the s e t w r i t e (2) a s

c 2

- ( n - 1) 4

t m

IXP ‘,,IN

2 - r xn

1 ,

r

w e h a v e t h a t ZnB

n

e 2 Un-l

F r o m ( 2 ) i t follows t h a t K c

r,

1

and ixp I p E N

is not n e c e s s a r i l y contained i n h .

We now

.

32

A review of the linear background X

x =

(3)

X .~

-rt1

( I - E )

n

r

€2c

r r 2 X

x

x

n n

E 2B 1

C

1

ZK,

hence

E 2 r Br c 2 U r - l

2

nl W

C

e

E

If r > 1 we have

xn

c 2 ~ 0 ,h e n c e E

n.

c Q

2E

.

Since

i f we s e t

= 1,

r r2

I

Ixf p PEN

ZnB

2B 1

=2(1-E)U

rA l

w e have f r o m ( 3 ) t h a t K c

2

ix'

u ..*' -2e

E

1

and

P FEN

txl

1

. . . a

P PEN

c Q

.

L e m m a ( 0 . 7 . 2 ) below c o m p l e t e s the proof.

0.7.2

Then

Proof.

LEMMA.

r fxn InEN -

- && =

rl

(xnIn ixn

IN be a null s e q u e n c e i n a F r e ' c h e t s p a c e .

lnEIN .

Clearly f

F o r the converse

l1 it s u f f i c e s to p r o v e that

i s c o m p a c t . F o r t h i s we c o n s i d e r

ta3 a sequence y

k

= n = 1

n,k

xn with

5 1.

For e v e r y

n = 1

l i e s in the c l o s e d unit ball of K = R o r "n, k kEN So we m a y e x t r a c t f r o m it a c o n v e r g e n t s u b s e q u e n c e . A c l a s s i c a l and n the s e q u e n c e

e a s y c o n s t r u c t i o n g i v e s a point y~

r, f x n I,,

IN

and a n infinite

C.

Subsets of Frechet spaces

s u b s e q u e n c e of ( y )

33

which c o n v e r g e s t o Y. The end of t h e p r o o f i s

left to the r e a d e r .

0.7.3

-

COROLLARY of th. ( 0 . 7 . 1 ) .

F r C c h e t s p a c e i s a S c h w a r t z bornology

_P- r o o f . -

T h e c o m p a c t bornology of a

.

F o r a n y given c o m p a c t s u b s e t K of a F r C c h e t s p a c e E w e

have f r o m ( 0 . 7 . 1 ) t h a t K c in E .

(Xn)nEN

s u c h that p

n

-

Let t cu i f n

r.t

'xn ' n c m

Em

be a s e q u e n c e of positive r e a l n u m b e r s

t

- -1

--L

03

f o r s o m e null s e q u e n c e

and p x n n

-

-

0 i n E if n

t cu

# 0 where d denotes a distance n defining the topology of E). T h e n , by (0. 7 . 2 ) , K' = / p nxn nE is 1 (for i n s t a n c e ,

In

= (d(xn,

if x

0))

I

ra

a c o m p a c t s u b s e t of E and K is c o m p a c t i n the B a n a c h s p a c e EKI'

0.7.4

T h e s t r i c t l y c o m p a c t bornology of a c o m p l e t e b. v.

be a c o m p l e t e b.v. s. and l e t E = l i m 4

i € I

-

Let E

E , bornologically, w h e r e t h e 1

s p a c e s E . a r e Bdnach s p a c e s . A s u b s e t K of E 1

S.

is s a i d t o be s t r i c t l y

c o m p a c t if K i s contained i n s o m e Ei and c o m p a c t t h e r e . t h e v e c t o r s p a c e E equipped with t h e sc E ; this means that " s t r i c t l y c o m p a c t bornology" of the c o m p l e t e b.v.s. We denote by E

a s u b s e t of E

is bounded in E

s t r i c t l y c o m p a c t s u b s e t of E.

SC

if and only if i t is contained i n s o m e

34

A review o f t h e linear background

Clearly E

SC

is a c o m p l e t e b . v . s . and from c o r o l l a r y ( 0 . 7. 3) i t

follows i m m e d i a t e l y t h a t E

sc

i s a S c h w a r t z b.v. s.

.

One m a y r e m a r k t h a t E x = E X a l g e b r a i c a l l y and t h a t the sc topologies 7E and 7 E a r e the s a m e . sc 0 . 7 . 5 Rapidly d e c r e a s i n g s e q u e n c e s .

-

of n nEN e l e m e n t s of a F r k c h e t s p a c e E i s said to be r a p i d l y d e c r e a s i n g i f , f o r k is a n u l l e v e r y n a t u r a l n u m b e r k E iN, the s e q u e n c e (n x ) n nE N s e q u e n c e in E . By ( 0 . 6 . 2 ) P r o p . 3 , f o r e v e r y

A s e q u e n c e (x )

k

E IN, the

sequence (n x ) n n€N is a null s e q u e n c e i n s o m e B a n a c h s p a c e E where B is a s u i t a b l e k 'k Now, by ( 0 . 6 . 2 ) P r o p . 2 , convex, balanced, bounded s u b s e t of E k

.

t h e r e i s s o m e convex, b a l a n c e d , bounded s u b s e t B of E

s u c h that

(x ) i s a rapidly d e c r e a s i n g sequence i n the Banach s p a c e EB. n

0.7.6

-

The n u c l e a r bornology of a Ere'chet s p a c e .

We define on a

F r k c h e t s p a c e E a new bornology, denoted by s ( E ) , in t h e following w a y : a s u b s e t K of E is bounded for the bornology s ( E ) if

K c

r,t

f o r s o m e 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 (x ) in E. n nEN

'xn ' n c N

THEOREM.-

If E

is a F r C c h e t s p a c e , then

T h e proof follows f r o m Hogbe-Nlend

0.7.7.

Exercise.

-

s(E) i s a nuclear b.v.s..

1 2 3 VIII 7 and (0.7.5).

L e t E be a F r k c h e t s p a c e and l e t c ( E )

denote i t s c o m p a c t bornology. Then t h e topologies

T E , Tc(E), T s ( E ) ,

T E , T s ( E ) a n d T c ( E ) coihcide with the given topology of E. Furthermore,

(c(E))'

= s(E))'

= E'

algebral'cally.

35

Multilinear mappings and polynomials

5

MULTILINEAR MAPPINGS AND POLYNOMIALS

0.8

-

0.8. 1 M u l t i l i n e a r mappings.

c

IK = IR or

-

~

f I, .

l X

n

o ,

1,

E and F b e l i n e a r s p a c e s o v e r

A n - l i n e a r mapping f r o m E"

tten

, O1

.. . , xU

) f o r e v e r y p e r m u t a t i o n u of the s e t n

A i s s a i d t o be s y m m e t r i c . F o r a n y n - l i n e a r

mapping A we define i t s s y m m e t r i z a t i o n A

where

sn

to F is a mapping

A ( x l . . .xn) E F which i s l i n e a r i n e a c h v a r i a b l e x . . If

n A ( x l , . ,. , x ) = A(x n *

Let

is the s e t of

by :

! p e r m u t a t i o n s of the s e t f 1,. .

n

n

1.

Continuous m u l t i l i n e a r m a p p i n g s . If E a n d F a r e 1 . c . s . n we denote by 2 ( E , F) the l i n e a r s p a c e of continuous n - l i n e a r n m a p p i n g s f o r m En to F. We denote by 2 ( E , F ) the l i n e a r s u b s p a c e 0.8.2.

of

L ( ~ E , F )of a14 s y m m e t r i c n - l i n e a r m a p p i n g s . If n =

o

we s e t

c(OE,F) = C. (OE,F) = F. A n - l i n e a r mapping is continuous iff i t is continuous at" t h e o r i g i n ; s e e f o r i n s t a n c e D i n e e n [ 1 ,

1

0.8. 3. A

Continuous polynomials.

-

E L ("E,F) a n d x E E we w r i t e Ax

is r e p e a t e d n n-homogeneous

t i m e s and A x polynomial

0

n

L e t E and F be 1 . c . s . . If

to denote A ( x , .

= A E F i n c a s e n = 0.

..

,x) w h e r e x

A

continuous

P f r o m E to F is a mapping

P:E

4

F

f o r which t h e r e i s s o m e A E L(nE, F) s u c h t h a t Px = Axn f o r e v e r y

xEE.

In o r d e r t o denote t h a t P c o r r e s p o n d s to A in t h i s way w e s h a l l

write P

=A. We s h a l l denote by

P("E,F)

the v e c t o r s p a c e of a l l

continuous. n - h o m o g e n e o u s polynomials f r o m E to F. A 6'(OE,F) = F and A = (As)

.

Obviously

A review of the littear background

36

If E and F a r e 1 . c . s . , a continuaus polynomial

P from E

to F i s a finite s u m of continuous n - h o m o g e n e o u s polynomials f o r n = 0, 1, 2 , . .

.

We s h a l l denote by P ( E , F ) the v e c t o r s p a c e of a l l

continuous polynomials f r o m E to F

0.8.4

PROPOSITION.

-

.

T h e mapping A E d i s ( n E , F )

-, A E f J ( n E , F )

i s a v e c t o r s p a c e i s o m o r p h i s m and we have the " p o l a r i z a t i o n f o r m u l a " :

E

For d e t a i l s s e e Nachbin [2]

0.8.5

PROPOSITION. -

respective norms

11 I I E

P E @ ( ~ E , F )w, e s e t

a nd

T h e n we have

:

6

3.

If

E

and

11 [IF)

n

= & I

F a r e n o r m e d s P a c e s (with n and i f AE e( E , F ) and

Multilinear mappings and polynomials F o r a proof s e e Nachbin 1 2 3

0.8.6

PROPOSITION.-

3.

IfA

following f o r m u l a h o l d s

A(x1,

5

37

E a. ( k E,F)*xoEE,

t h e n the

:

7k-(

...,xk )=i ),, k!

(-1)

t . . . f Ek) 1 A(x

0

t . . . t c x). k k 1 1

€1 = O o r I

k

Proof.

g(xl, .

-

A O or 1

L e t us denote the r i g h t - h a n d s i d e of the a b o v e f o r m u l a by

. . ,xk). We h a v e ,

by definition ,

1

A(x t 0

where (ao+

E

x t . . . t C

1 1

0

x ) =A(x t k k 0

= I and (

... ta,)

E

j ) '=

1 if j

"1 S

=

joS

... t j k

E

x t.. t 1 1

EkXk'.

.. , x o s

E

x t.

1 1

..t e kxk ) =

= 0 ( t h i s comes f r o m the f o r m u l a n!

.

j, I . . . j k !

(a,)

j0

...(a,)

jk

,

where

n

the s y m b o l s a . a r e e l e m e n t s of s o m e c o m m u t a t i v e a l g e b r a ; e a s y proof by induction on n).

Therefore

38

A review of the linear background

A. (1)

g(xl

.*o,X

Jo'

k1 =

j,!

*

a I

jk

o..jk!

j o t . . ,t j = k k

A(xo,.

,x o

...,

. .. , Xk)

Xk'

--vj terms 0

j terms

k

where

A s s u m e j = O f o r s o m e s 2 1. S

Without l o s s of g e n e r a l i t y we m a y

a s s u m e s = 1 , Then :

E2,

Therefore A .

Jo,. . But then j =.. . = j

1

by (1) and (2).

0 . 8.7

k

9

...,Ek=

jk

0

m a y be d i f f e r e n t f r o m z e r o only if j , 1

= 1 , j,

= 0 because j t. . . t j k = k 0

.

. . . , jk#

Thus g

0.

=A

I

Bounded m u l t i l i n e a r mappings and polynomials. -

If E and F

a r e b . v . s., w e define s i m i l a r l y the bounded n - l i n e a r m a p p i n g s f r o m E t o F, the bounded n-homogeneous p o l y n o m i a l s and t h e bounded polyn o m i a l s f r o m E t o F ( n o t i c e t h a t , a s i n the l i n e a r c a s e , "bounded" m e a n s , in f a c t , "bounded on t h e bounded s e t s " ) . T h e i r r e s p e c t i v e s p a c e s

39

Multilinear mappings and polynomials

n a r e denoted by L ( n E , F ) ( L ( E , F ) f o r s y m m e t r i c n - l i n e a r m a p p i n g s ) ,

P ( n E ; F ) and P ( E , F ) .

0.8. 8 The n a t u r a l bornology of L ( n E , F ) . -

If E a n d F a r e b.v. s.

t h e s p a c e L ( n E , F) i s equipped with the following " n a t u r a l " bornology : a subset M

c L ( n E , F ) is s a i d t o be bounded

i f f o r e v e r y bounded s e t B

in E the s e t

is a bounded s u b s e t of F.

T h e s e bounded s e t s a r e s o m e t i m e s called

t h e "equi-bounded" s u b s e t s of L ( n E , F ) . T h u s L ( n E , F) is a b . v . s . which i s s e p a r a t e d by its d u a l ( i f F is ).

PROPOSITION.

-

F o r e v e r y p a i r of b.v. s.

n a t u r a l n u m b e r s n, p E IN,

E , F and f o r e v e r y p a i r of

the a l g e b r a i c a l and b o r n o l o g i c a l e q u a l i t y

holds , ( T h e proof i s i m m e d i a t e ) ,

0 . 8 . 9 PROPOSITION.

then

-

Let E

d

F be b . v . s . I f F

is c o m p l e t e ,

L(nE, F) i s a c o m p l e t e b.v. s . f o r e v e r y n E N .

Proof.-

L e t ( B i ) i c I be a b a s e of bounded s e t s in E and l e t

(c.) J j EJ

is convex, j is a B a n a c h s p a c e . L e t M be a bounded

be a b a s e of bounded s e t s in F.

We m a y a s s u m e t h a t e a c h C

balanced and s u c h that EC n j s u b s e t of L( E , F). F o r e v e r y index i E I t h e r e e x i s t s a n index j ( i ) E J s u c h that M((Bi)n) e C . .

J(1).

We s e t

A review of the linear background

40

-

n n M = { u < L ( E , F ) s u c h t h a t , f o r e v e r y i € 1 , u((Bi) ) c C , ) 'j(1) N

.

N

T h e n M c M and i t is e a s y to p r o v e t h a t M i s a convex balanced n boundedsubset of L ( E , F ) f o r which t h e a s s o c i a t e d n o r m e d s p a c e ( L ( n E , F ) ) N is a Banach s p a c e . M

0 . 8 . 1 0 PROPOSITION

.-

L ( n E , F ) is a p o l a r b.v.5. Proof.-

L and F be b . v . s . -e t_E _ for every n E

IN

.

If

F i s p o l a r then

-

L e t M be a bounded s u b s e t of L ( n E , F ) and l e t M be i t s

c l o s u r e f o r t h e bornological topology T ( L ( n E , F)). We w a n t t o p r o v e that

-

(Bi)i.I

be b a s e s of bounded and ( C . ) . J JEJ s e t s in E and F r e s p e c t i v e l y . We c h o o s e C convex balanced and j c l o s e d in T F ( s i n c e F is p o l a r ) . F o r e v e r y index i E I t h e r e is

M i s bounded in L ( n E , F ) .

Let

is T o p r o v e that n bounded i n L ( n E , F ) i t s u f f i c e s to show that S ( ( B i ) )c C . f o r e v e r y j(l) index i E I. T h i s is done by c o n t r a d i c t i o n : i f not , t h e n t h e r e a r e a n n index i e I, a n x E (Bi) a n d a u ( with U ( X ) $ C . , Since C . . J(1)' J(') is c l o s e d in T F , t h e r e is a convex, balanced, b o r n i v o r o u s s u b s e t D of

a n index j ( i )

E J s u c h that M((Bi)n) e C .

j(1)'

F with

We s e t

V = { v E L ( n E , F ) such that F o r e v e r y v E V , u(x) t v ( x ) (u t V ) r'l M =

of u

L(nE, F ) ,

E

P.

Since V

f!

C.

,

J(1)

V(X)

hence u t v

ED

#M

1

. and, therefore

is a convex, balanced b o r n i v o r o u s s u b s e t n h e n c e a o-neighborhood of T ( L ( E , F ) ) , t h i s p r o v e s t h a t

and thus w e get a c o n t r a d i c t i o n .

PART I BASIC DIFFERENTIABLE CALCULUS AND HOLOMORPHY INTRODUCTION

T h e p u r p o s e of P a r t I i s t o i n t r o d u c e the definitions a n d b a s i c p r o p e r t i e s of d i f f e r e n t i a b l e and h o l o m o r p h i c m a p p i n g s i n infinite d i m e n s i o n a l s p a c e s . C o n c e r n i n g t h e b a s i c p r o b l e m of t h e c h o i c e of d e f i n i t i o n s , t h e p r e e m i n e n c e of t h e s o - c a l l e d S i l v a ' s definitions of

Co3

and holomorphic

m a p s will follow f r o m s e v e r a l r e s u l t s a n d r e m a r k s o b t a i n e d t h r o u g h o u t t h e book, a n d is e x p l a i n e d i n t h e s e c o n d p a r t of t h i s i n t r o d u c t i o n . But i t i s i m p o r t a n t t o n o t i c e t h a t in f a c t v a r i o u s d e f i n i t i o n s g e n e r a l l y c o i h c i d e i n "usual"

spaces. Main b a s i c p r o p e r t i e s of

Co3

and holomorphic m a p s a r e proved

in c h a p t e r s 1, 2 a n d 3. I n c h a p t e r s 1 and 2 s e v e r a l u s e f u l a n d n a t u r a l c o n c e p t s of

Co3

and holomorphic m a p s a r e introduced a n d t h e i r r e l a t i v e

connections a r e d i s c u s s e d . In c h a p t e r 4 we e q u i p t h e s p a c e s p e c t i v e l y t h e Silva

&(n,F )

and

xS (n, F)

(of r e s -

a n d t h e Silva h o l o m o r p h i c maps) with t h e i r

Cm

" n a t u r a l " t o p o l o g i e s , a n d we p r o v e t h a t , u n d e r g e n e r a l a s s u m p t i o n s o n t h e

E

spaces

and

F , these s p a c e s a r e complete Schwartz J . c . s . ,

f o r e "naturally reflexive

a(n,F )

and

sCs(Sa, F )

A.c . s.

(in t h e s e n s e 0 . 3 . 1 ) .

there-

Topologies on

a r e c o n s i d e r e d a s a tool f o r t h e s e q u e l of t h e book

s o we do not d i s c u s s v a r i o u s t o p o l o g i e s o n the s p a c e

x(hl,F) of h o l o m o r -

phic ( = G a t e a u x - a n a l y t i c a n d continuous) m a p s ; t h i s t o p i c is e x p o s e d i n t h e r e c e n t book by Dineen

[1

3,

s e e a l s o the Bibliographic N o t e s .

C h a p t e r s 5 a n d 6 d e a l with v a r i o u s p r o p e r t i e s of t h e s p a c e s S

(0, F )

and

s(n,F )

which a r e u s e d in t h e s e q u e l , m a i n l y a p p r o x i m a t i o n 41

42

Basic differential calculus and holomorphy

B(n)

a n d d e n s i t y r e s u l t s , the S c h w a r t z ' s € - p r o d u c t s

cW

E

F

and

sc,(n)

E

F

I

p a r t i t i o n s of unity. T h e F o u r i e r Bore1 i s o m o r p h i s m i s obtained i n c h a p t e r 7 , a n d is

a p p l i e d in c h a p t e r 8 f o r a proof of n u c l e a r i t y of t h e s p a c e s

X(0,F)

. In the

and

K,(n,F)

r e a l c a s e , which i s m o r e c o m p l i c a t e d , we p r e s e n t in

c h a p t e r 7 two P a l e y - W i e n e r - S c h w a r t z t h e o r e m s ( s e e a l s o c h a p t e r 1 3 ) . T h e concept of " b o r n o l o g i c a l v e c t o r s p a c e s " i s i n d i s p e n s i b l e t o obtain m a n y r e s u l t s i n t h e i r n a t u r a l s e t t i n g , a n d s o a i d s v e r y m u c h a t c l e a r n e s s a n d s i m p l i f i c a t i o n . Though not quite c l a s s i c a l , t h i s s t r u c t u r e i s e x t r e m e l y s i m p l e a n d e l e m e n t a r y (and is r e v i e w e d i n $ 0 . 2 ) . In o r d e r not t o d i s t u r b t h e h u r r i e d r e a d e r , who p o s s i b l y d o e s not know t h i s s t r u c t u r e , we a l w a y s avoid it in t h e A b s t r a c t of the c h a p t e r s . Now we explain o u r c h o i c e s of definitions of

and holomorphic

Co3

maps. L e t u s c o n s i d e r the h o l o m o r p h i c c a s e c o m p l e x , Hausdorff J . c . s .

n

a n d if

f i r s t . If

is a n o p e n s e t i n

f i n e n a t u r a l l y a h o l o m o r p h i c mapping

n* F

f :

E

and

F

are

E , we m a y d e -

a s being a mapping

which is both G a t e a u x - a n a l y t i c a n d c o n t i n u o u s , G a t e a u x a n a l y t i c ( G - a n a l y t i c f o r s h o r t ) m e a n s t h a t t h e r e s t r i c t i o n of s u b s p a c e of

E

f

t o any finite dimensional

i s h o l o m o r p h i c with v a l u e s i n t h e .l .c.s. F

. There

are

s e v e r a l e q u i v a l e n t definitions ( s e e Nachbin [3, 51 ) a n d t h i s g i v e s r i s e t o a good m a t h e m a t i c a l t h e o r y which n e v e r t h e l e s s s u f f e r s f r o m s o m e s e r i o u s defects. The space

K(Q, F )

of t h e s e h o l o m o r p h i c m a p p i n g s , e q u i p p e d

with n a t u r a l t o p o l o g i e s , is not c o m p l e t e i n g e n e r a l e v e n in t h e u s u a l c a s e s a n d many v e r y s i m p l e b i l i n e a r m a p p i n g s a r e not c o n t i n u o u s (for i n s t a n c e , let

E

denote a n o n - n o r m a b l e

a . c . s . and let

E'

B

be i t s dual e q u i p p e d

with the s t r o n g dual topology, t h e n i t i s i m m e d i a t e t o p r o v e t h a t t h e d u a l i t y b i l i n e a r function E x

x

,

E

P

x'

'

u

C X ' k )

43

Introduction

i s n e v e r c o n t i n u o u s ) . A v e r y n a t u r a l definition of h o l o m o r p h i c m a p p i n g s which d o e s not s u f f e r f r o m the above d e f e c t is t h a t of a Silva h o l o m o r p h i c m a p p i n g . A mapping

f : 0

3

F

is "Silva h o l o m o r p h i c in t h e e n l a r g e d

-

s e n s e " if, f o r e v e r y convex, b a l a n c e d , bounded s u b s e t of

f ,nnEB

restriction

f

to

Q

n

E

B

R n EB

:

( i . e . G - a n a l y t i c a n d c o n t i n u o u s ) . H e r e we r e c a l l t h a t

B , E

B

F

EB

E

of

the

is holomorphic

d e n o t e s the

n B , n o r m e d by the Minkowski f u n c t i o n a l n m IlxllB= inf { h r 0 s u c h t h a t x E h B ] a n d O n E B i s r e g a r d e d

l i n e a r s p a n of of

u

B

,

=

a s a n open s e t i n t h e n o r m e d s p a c e

EB

. We

s h a l l define "Silva h o l o m o r -

phic m a p p i n g s " andl'silva h o l o m o r p h i c m a p p i n g s i n t h e e n l a r g e d s e n s e " . T h e s e two notions coincide i n all "usual" c a s e s . We denote by

X

S

(n,F )

the s p a c e of all Silva h o l o m o r p h i c m a p p i n g s e q u i p p e d w i t h i t s n a t u r a l t o pology (which coi'ncides with t h e c o m p a c t - o p e n topology i n t h e " u s u a l " c a s e s ) . If

E

dual n u c l e a r (if

n

is a c o - n u c l e a r A . c . s .

0.5 : a quasi-complete

(defined i n

A. c . s . is c o - n u c l e a r ) a n d if

n

h a s s o m e Runge p r o p e r t y

is b a l a n c e d , f o r i n s t a n c e ) t h e n we p r o v e ( i n $ 5.1) t h a t

a d e n s e s u b s p a c e of

KS ( n , F ) . If,

one p r o v e s e a s i l y t h a t

x(0,F )

c o m p l e t i o n of

furthermore,

F

Ks(Q, F) is c o m p l e t e . T h u s

K ( Q , F ) is

is a complete A . c . s .

K,(n, F )

i s the

f o r the c o m p a c t - o p e n topology. T h e r e f o r e t h e

notion of a Silva h o l o m o r p h i c mapping i s d e e p l y r e l a t e d t o t h a t of a ( c o n t i nuous) h o l o m o r p h i c m a p p i n g . Many r e s u l t s i n t h i s book a r e b a s e d upon the c o m p l e t e n e s s a n d the " n a t u r a l r e f l e x i v i t y " ( i n the s e n s e defined i n

$ 0 . 3 ) of t h e s p a c e

KskJ, F )

. Furthermore

m a n y r e s u l t s i n the t h e o r y of

( c o n t i n u o u s ) h o l o m o r p h i c m a p p i n g s a r e c o r o l l a r i e s of m o r e g e n e r a l r e s u l t s c o n c e r n i n g Silva h o l o m o r p h i c m a p p i n g s . F o r all t h e s e r e a s o n s we u s e s y s t e m a t i c a l l y the concept of Silva h o l o m o r p h y w h e n e v e r i t i s u s e f u l , but " h o l o m o r p h i c " will a l w a y s m e a n " G - a n a l y t i c a n d continuous",

since

t h i s is the t e r m i n o l o g y i n u s e . T h e real c a s e is m o r e c o m p l i c a t e d a n d f o r s i m p l i c i t y we only consider

C

00

-

m a p p i n g s . T h e r e a r e v a r i o u s d e f i n i t i o n s of d i f f e r e n t i a b l e

m a p p i n g s ( s e e A v e r b u c k - S m o l y a n o v [ Z ] ), but m a n y of t h e m s u f f e r f r o m

44

Basic dijytrential calculus and holomotphy

t h e d e f e c t s a l r e a d y e n c o u n t e r e d i n the h o l o m o r p h i c c a s e ; t h o s e which do not a r e v e r y c l o s e to the definition of S e b a s t i z o e Silva [l, 2 , 31 , L e t u s j u s t give the definition of a C

W

- m a p p i n g i n S i l v a ' s e n l a r g e d s e n s e , which

E

h a s a particularly simple a s p e c t . Let

E

open s e t i n

. Let

F

of

/

(F, pv)

V , We denote by

s

p(:O) V

,

F , we denote by

F

Cw

is

V

V

n

an

denotes a

FV t h e n o r m e d

where

f :Q

F

i,

V

in

.

V

is

i n Silva's enlarged s e n s e

Co3

if : f o r e v e r y c o n v e x , b a l a n c e d , bounded s e t

of! E B -+

and

i s t h e Minkowski f u n c t i o n a l pv the c a n o n i c a l s u r j e c t i v e mapping F 4 F

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

b a l a n c e d 0-neighbourhood

a .c. s .

. c . s . If

be a r e a l o r c o m p l e x

convex, b a l a n c e d 0-neighbourhood i n quotient s p a c e

be a r e a l

B

E

in

F , t h e mapping

and e v e r y convex,

svo f

/nn EB:

( b e t w e e n t h e s e n o r m e d s p a c e s a n d in t h e u s u a l

( F r 6 c h e t ) s e n s e ) . If

E

is a c o m p l e x A . c . s .

t h e above definition c o i n c i d e s

w i t h t h a t of a Silva h o l o m o r p h i c mapping i n t h e e n l a r g e d s e n s e . A s i n t h e c o m p l e x c a s e , we define a l s o the notion of a Silva C

W

- m a p p i n g , which in

t h e "usual" c a s e s c o i n c i d e s with the above notion. We denote by

C

t h e s p a c e of these

03

-mappings f r o m

t u r a l topology (defined i n $ 4 . 1 ) . f l e x i v e if

E

and

F

to

Q

&(n,F )

s(n,F)

F , equipped with i t s n a -

is c o m p l e t e a n d " n a t u r a l l y " r e -

have s o m e s t a n d a r d p r o p e r t i e s and t h i s is b a s i c

f o r m a n y of o u r r e s u l t s . A f i n i t e - t y p e continuous polynomial on a 1 . c . s . , v a l u e d in a 1 .c. s . F , i s a f i n i t e l i n e a r c o m b i n a t i o n of m a p p i n g s

... x',(x)

y , where

xE E , x i E E '

and

yE F

. Now if

E

x

A .

XI(.)

is a co-

1

n u c l e a r & , c . s . , t h e s p a c e of all f i n i t e - t y p e continuous p o l y n o m i a l s is d e n s e in

d(0,F)

. Since a

f i n i t e - t y p e continuous p o l y n o m i a l is

Cw ,

s(n, F )

a c c o r d i n g t o a n y " r e a s o n a b l e " definition of d i f f e r e n t i a b i l i t y ,

i s the c o m p l e t i o n of t h e s p a c e s o b t a i n e d with o t h e r definitions of

C""-

m a p p i n g s , In s h o r t , t h e s i t u a t i o n is s i m i l a r t o t h a t encountered i n t h e c o m W

plex c a s e . W e s h a l l s y s t e m a t i c a l l y u s e t h e notion of S i l v a C - m a p p i n g s , which will s i m p l y be c a l l e d " C

W

-mappings".

CHAPTER 1 DIFFERENTIAL MAPPINGS, BASIC PROPERTIES

ABSTRACT. - T h e a i m of t h i s c h a p t e r i s t o s t a t e t h e d e f i n i t i o n of d i f f e r e n t i a b i l i t y u s e d in t h e book -i. e . S i l v a ' s d e f i n i t i o n - and two c l o s e l y r e l a t e d c o n c e p t s which h a v e t h e i r own i m p o r t a n c e : "Silva d i f f e r e n t i a b l e m a p p i n g s i n t h e e n l a r g e d s e n s e " and " l o c a l l y d i f f e r e n t i a b l e b e t w e e n n o r m e d s p a c e s " m a p p i n g s . Let u s j u s t s t a t e i n t h i s A b s t r a c t t h e following d e f i n i t i o n : L e t E and

and l e t U be a n o p e n s u b s e t of E .

F be 1 . c . s .

Then a mapping f f r o m U

t o F is

C

W

in Silva's enlarged sense if

f o r e v e r y convex balanced bounded s u b s e t B of E and e v e r y c o n v e x balanced

o-neighborhood

V i n F,

t h e r e s t r i c t i o n of f t o U n E

v a l u e d i n t h e n o r m e d s p a c e F V ,is Cm (here U p E and F = V

B

B' i n the u s u a l ( F r C c h e t ) s e n s e

i s c o n s i d e r e d a s a n o p e n s e t i n the n o r m e d s p a c e

(F, Pv)

where p

P i 1 ( 0)

d e f i n i t i o n of Silva

C

CD

is t h e gauge of V - s e e ( 0 . 1 . 9 ) ) .

V

EB The

m a p p i n g s is a t r i v i a l g e n e r a l i z a t i o n of t h e u s u a l

( F r k c h e t ) d e f i n i t i o n of t h e n o r m e d s p a c e s s e t t i n g , i f o n e knows t h e c o n c e p t of " b o r n o l o g i c a l v e c t o r s p a c e s " . So we d o not sbate i t i n t h i s A b s t r a c t . L e t us j u s t m e n t i o n ( t h ( 1 . 4 . 7 ) ) t h a t i f E 1 ; c . s . ( s e e (0.5. 3 ) , in p a r t i c u l a r S c h w a r t z 1. c. s ) , t h e

C

W

is a c o - S c h w a r t z

E m a y be a n y q u a s i c o m p l e t e d u a l

m a p p i n g s i n S i l v a ' s e n l a r g e d s e n s e f r o m fi

t o F c o i h c i d e w i t h t h e Silva

C

co

mappings from

two d e f i n i t i o n s coi'ncide i n m o s t u s u a l 1.c.

S.

0 to F.

So t h e s e

. T h e s u p e r i o r i t y of t h e

d e f i n i t i o n i n Silva ( u s u a l ) s e n s e c o m e s from th. ( 1 . 4 . 8 ) which p r o v e s (in the

Coo c a s e ) t h a t t h e e n l a r g e d d e f i n i t i o n is j u s t a p a r t i c u l a r a s p e c t

of t h e u s u a l d e f i n i t i o n ( s e e ( 1 . 4 . 8 ) f o r m o r e c l e a r n e s s ) , a s w e l l a s f r o m o t h e r r e s u l t s obtained l a t e r o n .

45

Difyerentiable mappings,basic properties

46

We p r o v e m a i n b a s i c p r o p e r t i e s of t h e s e d i f f e r e n t i a b l e mappings, i n p a r t i c u l a r t h e Mean Value T h e o r e m a n d T a y l o r ' s f o r m u l a s . In define t h e

C

a3

F

into

1. 6 we

m a p p i n g s "of u n i f o r m bounded type" a n d p r o v e t h a t if

is a Silva s p a c e a n d

E

4

F

a FrCchet s p a c e , then any

Co3

E

mapping f r o m

i s of u n i f o r m bounded t y p e , which h a s i m p o r t a n t a p p l i c a t i o n s

in the s e c o n d p a r t of t h e book. When one u s e s both bornology a n d topology a t t h e i r r i g h t p l a c e , m a n y b a s i c p r o p e r t i e s of t h e s e d i f f e r e n t i a b l e m a p s a r e t r i v i a l g e n e r a l i z a t i o n s of the c o r r e s p o n d i n g p r o p e r t i e s of t h e n o r m e d s p a c e c a s e . But we s h a l l a s c e r t a i n i n t h e s e q u e l of the book t h a t , a t t h e s a m e t i m e a s a c o n s i d e r a l l y g r e a t e r g e n e r a l i t y , i n d i s p e n s i b l e f o r m a n y a p p l i c a t i o n s of t h e Theory

-

which a r e a l r e a d y o b t a i n e d o r p r e s e n t l y i n m e a t i o n

-

the locally

convex s p a c e c a s e b r i n g s a lot of new p e c u l i a r i t i e s a n d d i f f i c u l t i e s . T h e n o r m e d s p a c e case is now q u i t e c l a s s i c a l a n d m a y be found f o r instance in C a r t a n

[ 13,

DieudonnC [l] , Lang

[ 1, 21,

H i l l e - P h i l i p s [ l ] , Nachbin [4], C o r w i n - S z c z a r b a

H i l l e [l] ,

[ l l , . ..

. The reader

is r e f e r e d t o t h e B i b l i o g r a p h i c Notes f o r r e f e r e n c e s a n d a s k e t c h of

t h e o r i e s of i m p l i c i t f u n c t i o n s a n d C a u c h y p r o b l e m s f o r o r d i n a r y d i f f e r e n t i a l e q u a t i o n s , which a r e not t r e a t e d i n t h i s book.

41

Differentiability in nomied spaces

5

R e c a l l on the definition of d i f f e r e n t i a b i l i t y i n n o r m e d s p a c e s

1. 0

L e t E be a n o r m e d s p a c e (of n o r m

E and F a n o r m e d s p a c e (of n o r m

1 . 0 . 1 DEFINITION. -

i n E to

1.0.2

I! I/F) .

11 11 E ),

U a n open s u b s e t of

A mapping r f r o m a o-neighborhood V

F i s s a i d to be a " r e m a i n 2 . e r " i f r ( o ) = o and if

A mapping f f r o m U

DEFINITION. -

d i f f e r e n t i a b l e a t a point a 6 U

to F

i s s a i d to be

if:

f(ath) = f(a) t f'(a). h t r (h) w h e r e f l ( a ) i s a l i n e a r continuous m a p p i n g f r o m E X i s a r e m a i n d e r in t h e s e n s e of

f point of

a

(1. 0. 1).

i s said t o be d i f f e r e n t i a b l e i n U

if i t is d i f f e r e n t i a b l e a t a n y

U.

1.0.3. and

F and w h e r e r

When E and

F a r e normed s p a c e s , the spaces

L(nE,F) ( s e e (0.8.2)(0.8.

g("E,F)

7) and ( 0 . 8 . 8 ) ) ) c o i h c i d e and t h e i r

topology ( o r equivalently t h e i r bornology) is given by t h e n o r m :

1.0.4.

Since f '

maps

U into L ( E , F ) , o n e m a y c o n s i d e r the

d i f f e r e n t i a b i l i t y of f ' . F r o m ( 0 . 8 . 8 ) one d e f i x e s by induction h i g h e r o r d e r

d e r iva tive s

.

DiffereritiabIe mappings, basic properties

48

1. 1

Let E of E

1

D e f i n i t i o n of (Si1va)differentiable m a p p i n g s

and E 2 b e two b . v . s .

P

and

a bornivorous subset

1

1'

A mapping r f r o m P

1. 1 1 DEFINITION. -

to E i s s a i d to 12 be a r e m a i n d e r if f o r e v e r y convex b a l a n c e d bounded s u b s e t B of E 11 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 B of E such that 22 a)

r(EB1) c B 2 f o r s o m e

c)

r(o) =

E

>o

s m a l l enough

0.

It i s c l e a r t h a t i f f u r t h e r m o r e r is l i n e a r t h e n r =

-

0.

a mapping f f r o m a t P to 11 if t h e r e e x i s t s a E 2 i s said to be d i f f e r e n t i a b l e a t the point a EE 1 l i n e a r bounded m a p p i n g f l ( a ) f r o m E to E (i.e. fl(a) E L(E1,E2)) 12 such that the application r f r o m P to E defined by : a12

1. 1 . 2 DEFINITION.

If a ___

EE

r ( h ) = f(a t h )

-

f(a)

-

f'(a) , h

is a r e m a i n d e r .

It i s c l e a r t h a t f ' ( a ) is unique i s a r e m a i n d e r , h e n c e is null.

I

-r if not f l l ( a ) - f ' (a) = r 2 2,a l,a

T h e two above definitions a r e i m m e d i a t e extensi0r.s t o t h e b o r n o l o g i c a l c a s e of t h e c l a s s i c a l definitions 1.0.1

and 1 . 0 . 2 of t h e n o r m e d s p a c e

c a s e . T h e c o n c e p t of cor.tir.uity of differer.tiable m a p p i n g s , developped below, is slightly m o r e s u b t l e .

49

(Silva) differentiable mappings

1. 1 . 3 Mackey-continuity. -

-

M

If f

is d i f f e r e n t i a b l e a t the point a and

a , s e e ( 0 . 2 . 12), t h e n f(xn) is defined f o r n l a r g e enough n M and f(xn) f(a).

if x

-

1. 1 . 4 Composition of d i f f e r e n t i a b l e m a p p i n g s . -

be t h r e e b.v. s . , l e t P

Let El, E

and E 3

P

be b o r n i v o r o u s s u b s e t s of E and E 2 1 2 r e s p e c t i v e l y and l e t a be a n e l e m e n t of E Let f f r o m a t P to E 1' 1 2 and g f r o m f ( a ) t P2 to E be d i f f e r e n t i a b l e m a p p i n g s a t t h e points 3 a and f ( a ) r e s p e c t i v e l y . T h e n

1

and

t h e r e e x i s t s a b o r n i v o r o u s s u b s e t P' C P ( i n E ) s u c h 1 1 1 t h a t the c o m p o s e d mapping g o f i s defined on a P' valued i n E 1' 3 and a)

b)

+

g o f

i s d i f f e r e n t i a b l e a t the point a and

(g o f ) ' ( a ) = g ' ( f ( a ) ) o

Proof.

-

:

fl(a).

F r o m def. ( 1 . 1 . 2 ) t h e r e is a b o r n i v o r o u s s e t PI1 c P 1

such t h a t f(a t P ' ) c f ( a ) t P 1 2' It s u f f i c e s now t o w r i t e

T h e r e f o r e g o f is defined on a t PI

1

.

to obtain the r e s u l t .

1. 1 a 5 DEFINITION.

-

L e t E l -E

b e a n open s e t f o r t h e topology T h e n a mapping f f r o m

T E

to E

5-

it is d i f f e r e n t i a b l e a t a n y point of

1

2

U

andlet U cE 1 1 of t h e M a c k e y - c l o s u r e , s e e ( 0 . 2 . 5 ) .

2

be b . v . s .

is s a i d t o be d i f f e r e n t i a b l e in U

1-

1'

Now w e i n v e s t i g a t e t h e continuity p r o p e r t i e s of d i f f e r e n t i a b l e mappings.

if

50

Differentiable mappings, basic properties

I , 1.6 Continuity of d i f f e r e n t i a b l e mappings f o r t h e M a c k e y - c l o s u r e topologies.

Let

PROPOSITION. -

El

and E 2

be b. v. s. and U

1

a n open s e t f o r t h e

If f i s a d i f f e r e n t i a b l e mapping f r o m U to E then f 121' equipped with t h e topology induced bv TE 1 , is continuous f r o m U 1 ' to 7 E 2 . topology

TE

-

is a b o r n i v o r o u s s u b s e t of 2 t h e r e is a b o r n i v o r o u s s u b s e t P of E such that E 2 and if a E U 1 1 1 f ( a t P 1 ) c f ( a ) t P which is i m m e d i a t e f r o m def. ( 1 . 1 . 2 ) . I 2

Proof.

1. 1 . 7

It s u f f i c e s t o c h e c k that i f

P

Continuity of d i f f e r e n t i a b l e m a p p i n g s defined on 1. c . s . L e t E and F be two 1.c. s.

. Let

B E and

B F denote t h e

Von Neumann bornologies of E and F r e s p e c t i v e l y , ( 0 . 2 . 2 ) , and l e t U be a n open s e t in E . When w e s a y that a mapping f f r o m U to F is d i f f e r e n t i a b l e w e c o n s i d e r i m p l i c i t e l y t h a t t h e 1.c. s .

E and F a r e

BE and B F r e s p e c t i v e l y . T h e n i t follows

r e p l a c e d by t h e b. v. s .

immediately f r o m (1.1.6) (since TBF is clearly finer than the given topology of F) t h a t any d i f f e r e n t i a b l e mapping f f r o m U to F is continuous on U ,

equipped with the topology induced by

TBE,

with

F.

v a l u e s in t h e 1 . c . s .

In a few u s u a l c a s e s the topology T B E c o f n c i d e s with t h e topology of the 1. c . s . m e t r i o a b l e 1.c. s

-

E.

T h i s i s in p a r t i c u l a r t r u e if E

is a

s e e ( 0 . 6 . 2 ) - or a Silva s p a c e - s e e (0.6.7)

and ( 0 . 6 . 9 ) -

So in t h e s e c a s e s any d i f f e r e n t i a b l e mapping i s continuous. However t h i s is f a r f r o m being t h e c d s e in g e n e r a l : s e e f o r i n s t a n c e the b i l i n e a r duality E

x

Elp

-I

IK ( c o n s i d e r e d in Introduction of P a r t I).

(Silva) differentiable mappings

1.1.8

L e t E be a 1 . c . s. with a s u b s e t A e E

Counterexample. -

and a point a

51

(the c l o s u r e of A i n E ) which is not the topological

l i m i t of any s e q u e n c e of points of A ( t h i s s i t u a t i o n is v e r y f r e q u e n t in non m e t r i z a b l e s p a c e s ) . L e t f (f(x) = 1 i f x

be t h e c h a r a c t e r i s t i c function of t h e s e t A

A and o o t h e r w i s e ) .

f is null on a s e t a t P w h e r e

P is s o m e b o r n i v o r o u s s u b s e t of E . T h e r e f o r e f is d i f f e r e n t i a b l e at t h e point a but not continuous a t t h i s point.

1. 1 . 9

Counterexample. -

A n e x a m p l e of a

Co3 function defined on

the S c h w a r t z ' s s p a c e J(IRn) and which is not continuous i s given ir, Averbuck-Smolyanov

E 21.

T h i s p r o v e s in p a r t i c u l a r t h a t the M a c k e y

c l o s u r e topology 7aRn)of

E(IRn) is s t r i c t l y f i n e r t h a n its u s u a l

l o c a l l y convex topology (which coi'ncides with the b o r n o l o g i c a l topology TD(IRn),

s i n c e L(lRn) is a bornological 1. c. s.).

52

Differentiable mappings,basic properties

$

1. 2

Definitions of

C

n

a n d Cco m a p p i n g s

Let E

1. 2 . 1 Mackey continuous m a p p i n g s . -

1

and E

2

be two

a n open s e t f o r T E A mapping f f r o m U to E 1 1' 1 2 is s a i d t o be Mackey-continuous (M-continuous f o r s h o r t ) a t a point .

b . v . s . and a 6 U1

U

i f f o r e v e r y convex balanced bounded s u b s e t B

e x i s t s a convex balanced bounded s u b s e t B

f(a t €B ) c B for some & > o 1 2

a)

f

of E

2

M-continuous in U

is s a i d t o be

2

1

of E

such that

there

1 :

small enough

if i t is a t e v e r y point of U

1

Any M-continuous mapping in U 1 is c l e a r l y continuous f r o m equipped with t h e topology

7E

1'

to

TE

t w o b . v . s . and

and E be 21A differentiable mapping f

U

Let E

a n open s e t f o r TE 1 1 ' is s a i d to be continuously d i f f e r e n t i a b l e in U

from U

to E 1 - 2 mapping f ' f r o m U

to L ( E 1 , E ) is M-continuous in u1 12 i s equipped with i t s n a t u r a l bornology defined in ( 0 . 8 . 8 ) ) .

1.2.3

R e c u r r e n t definitions of Let E

topology TE

1

1'

and E n

ul'

2'

1. 2 . 2 Continuously d i f f e r e n t i a b l e m a p p i n g s . -

E IN

2

.

Cn and C

be two b.v. s . ,

00

mappings.

1-

if t h e

-

U 1 be a n open s e t f o r the

to E i s s a i d to be ( n t 1 ) - t i m e s 1 2 ( r e s p e c t i v e l y i n U ) i f i t is n - t i m e s d i f f e r e n t i a b l e a t a point a E U A mapping f f r o m U

1 and if t h e mapping

differentiable in U from

u 1 - to

1 f(n) l t h e nth d e r i v a t i v e of f)

1 L ( ~ 1E, E 2 ) i s d i f f e r e n t i a b l e a t t h e point a

( r e s p e c t i v e l y i n U ).

1

EU

1

1'

53

C arid C a, mappings

from

A mapping f differentiable in U mapping

U 1 -t o2 E

i s s a i d to be

n - t i m e s c o n t i n u o u s k-

if i t i s n - t i m e s d i f f e r e n t i a b l e i n U

and if t h e 1 t o L ( n E 1 , E 2 ) is M-continuous in U from U 11

kn)

1

.

A mapping f f r o m U1 f o E tiable in U

1 (C" f o r e v e r y n EN

U 1 f o r s h o r t ) i f i t is n - t i m e s d i f f e r e n t i a b l e i n U 1

L

.

i s said to be infinitely d i f f e r e n -

2

Any d i f f e r e n t i a b l e m a p p i n g i n U

1

i s M-continuous in U

so 1' is n - t i m e s c o n t i n u o u s l y

any ( n t 1 ) - t i m e s differentiable mapping i n U 1 d i f f e r e n t i a b l e in U

1' C

1 . 2 . 4 C o m p o s i t i o n of

and U

U1

three b.v. s.,

7E2 respectively,

g :U

n

-

be o p e n s e t s f o r t h e t o p o l o g i e s

2

.

IN u

n

L e t E 1, E 2 and E

mappings. -

Let f : U

TE

1

be

3

and

E

and 2 n-times differentiable (respectively n-times

1

-, E be 2 3 c o n t i n u o u s l y clifferentiable) m a p p i n g s . L e t a E U

be such that f(a)E U2. 1 T h e n t h e r e e x i s t s a n open neighborhood U ' of a f o r t h e topology T E 1' 1 w i t h U ' c U1 , s u c h t h a t t h e c o m p o s e d m a p p i n g g o f i s defined on U i l . Furthermore g o f

is n-times differentiable (resp.

n - t i m e s continu

-

-0usly d i f f e r e n t i a b l e ) i n U' 1' T h e proof is i m m e d i a t e .

1 . 2 . 5 PROPOSITION.

Let E and 1-

-

open s e t f o r t h e topology

TE1.

Let f : U

E

-

b e b . v . s . and

U

an 1E 2 be a n - t i m e s d i f f e r e n -

2

1 T h e n for e v e r y a E U 1 t h e d e r i v a t i v e tiable mappinp in U 1' f (n)(a) E L(nE1, E z ) is s y m m e t r i c (i. e . i(")(a)E L ~ ( ~E E ~ )~) ,

.

Proof. the s e t

-

If h l , /hl,.

. . . , hn

EE

. . , hn 1 c E 1 ,

and

if

d d e n o t e s t h e l i n e a r s p a n of

we consider the restriction f

/uln

(atdl

54

Differentiable mappings, basic properties

If y '

EZ

t h e n u m e r i c a l function y '

d i f f e r e n t i a b l e in U c a s e , y ' [ f ( n)(a)hl

1

n (atd).

.. . h n ]

f/uln ( a t d l

O

is n - t i m e s

F r o m t h e r e s u l t i n t h e finite d i m e n s i o n a l

is s y m m e t r i c i n h l , .

. . , hn . Since

E

is 2 a s s u m e d ( a s a l w a y s in the book) to be s e p a r a t e d by its d u a l w e h a v e the

I

result.

1 . 2 . 6 Gateaux-differentiable m a p p i n g s .

IR

or

C) and

-

fi a s u b s e t of E s u c h that

finite d i m e n s i o n a l l i n e a r s u b s p a c e 6 is finitely open). L e t F be a b.

V.

L e t E be a l i n e a r s p a c e

nns

(over

is open i n 8 f o r any

of E (one s a y s c l a s s i c a l l y t h a t R

s. or 1. c . s. and

i n t o F. M'e s a y t h a t f is G a t e a u x - d i f f e r e n t i a b l e in

f a mapping f r o m 0

n

i f the r e s t r i c t i o n

of f to the i n t e r s e c t i o n of 62 and a n y one d i m e n s i o n a l s u b s p a c e of E

is

differentiable (in s o m e s e n s e , f o r i n s t a n c e in S i l v a ' s s e n s e ) . We s h a l l not u s e this concept of d i r e c t i o n a l d e r i v a t i v e , s i n c e t h e following s t r o n g e r concept i s s i m p l e r and m o r e adequate f o r o u r p u r p o s e . Note t h a t u n d e r n a t u r a l a s s u m p t i o n s it m a y be proved f r o m t h e m e a n value t h e o r e m t h a t Gateaux-differentiable m a p p i n g s a r e d i f f e r e n t i a b l e i n m u c h s t r o n g e r s e n s e s ( s e e Colombeau

[ 11 p 37 for instance).

1.2. 7 Finitely differentiable mappings.

-

If E , hl

and

F a r e a s before

w e s a y that f is a finitely d i f f e r e n t i a b l e mapping f r o m 61 i n t o F if f o r e v e r y finite d i m e n s i o n a l l i n e a r s u b s p a c e 6 of E , to R

n

8 , with v a l u e s i n F,

the r e s t r i c t i o n of f

is Silva d i f f e r e n t i a b l e . T h i s is e x a c t l y

t h e concept of Silva d i f f e r e n t i a b i l i t y if we e q u i p E with its f i n i t e d i m e n s i o n a l bornology 0 . 2 . 2 .

.

We s a y that f is finitely n - t i m e s d i f f e r e n t i a b l e (n 6 IN '2 if f is n - t i m e s Silva d i f f e r e n t i a b l e in

n,

u

00)

in

w h e n we equip E with

i t s finite d i m e n s i o n a l bornology.

1.2.8

Remark.

-

In the c o m p l e x c a s e it follows f r o m H a r t o g s ' t h e o r e m

and r e s u l t s i n c h a p 2 that the G a t e a u x - d i f f e r e n t i a b l e m a p p i n g s and the finitely d i f f e r e n t i a b l e mappings a r e equivalent c o n c e p t s .

5:

Mean vnlue theorem and Taylor's fomzu1ns

$

1.3

Mean value t h e o r e m and T a y l o r ' s f o r m u l a s

1.3.1 DEFINITION.

-

Let E

k

a 1.c.s.

& I

a n open i n t e r v a l

of the r e a l line lR. A mapping f f r o m I into E is s a i d t o be d i f f e r e n t i a b l e in t h e e n l a r g e d s e n s e in I if f o r e v e r y a E I t h e quotient f(a t h ) h

-

f(a)

t e n d s to a limit f l ( a ) 6 E when h

-

- IR (h f

o in

0).

C l e a r l y if BE d e n o t e s t h e Von N e u m a n n bornology of E ( 0 . 2 . 2 ) and if f i s Silva d i f f e r e n t i a b l e in I w i t h v a l u e i n B E

(1. 1.2),

f

is

d i f f e r e n t i a b l e in t h e e n l a r g e d s e n s e i n I w i t h value in E . We identify E and L(IR,E) by t h e m a p p i n g u

u E E and t E

1.3.2

R,

SO

- (t

t u ) if

f ' ( a ) is c o n s i d e r e d a s a n e l e m e n t of E .

Mean v a l u e t h e o r e m .

Let E

-

open i n t e r v a l of t h e r e a l line

IR. k t

t h e e n l a r g e d s e n s e from ] a , b[ i n t e r v a l contained in ] a , b[

.

H e r e w e r e c a l l that if A

&a

1. c . s . a n d l a , b [

an

f be a d i f f e r e n t i a b l e mappinq in

into E .

k t [ a ,B

]

be a c l o s e d

Then

is a s u b s e t of E w e denote by FA

the

c l o s e d convex balanced h u l l of A .

Proof.

-

Let us a s s u m e that, for every t E [ a , B ] , fl(t) E B

w h e r e B is a c l o s e d convex s u b s e t of E. balanced

0-neighborhood in E .

Let V b e a given c o n v e x

We s e t :

3 = it E [ a , p ] s u c h t h a t

agct

=$

f(g)-f(g)c(c -a)T(BtV)

1

.

56

Differentiable mappings, basic properties

Y

sup

to p r o v e

3.

.

3 t h e n t € 3 W e set f ( a ) E ( y - a ) ( B S V ) . We

If a h t b t ' a n d t ' e

ClearIy B €3.

By continuity f ( Y ) -

Y = ,B

differentiable a t

.

Y <

and f o r t h i s l e t us a s s u m e

a r e going

Since f i s

Y,

T h e r e f o r e if E P o is s m a l l enough,

f(Yt&)-f(a)E

f

( B t V ) t (Y-a) F ( B t V ) c ( Y - a t & ) f ( B t V ) .

T h i s c o n t r a d i c t s the a s s u m p t i o n t h a t

B u t if V

FB

= fl

V

'f = s u p

r a n g e s o v e r the convex balanced

5

.

Therefore

Y = @ and

o-neighborhoods in E ,

F(BSV) and t h e r e f o r e

f(F)

1 . 3 . 3 Remark.

-

If A

-

f(a) E ( P - b ) F B

i s a s u b s e t of a 1 . c . s .

E w e denote by ;(A)

t h e closed convex hull of A , i. e . the i n t e r s e c t i o n of a l l c l o s e d convex s u b s e t s of E

containing A . T h u s < ( A ) c f ( A )

and i s s m a l l e r in g e n e r a l .

T h e n it follows f r o m t h e proof of ( I . 3 . 2 ) t h a t in f a c t we have t h e b e t t e r result

However we s h a l l not u s e it in the s e q u e l . F o r a still m o r e g e n e r a l r e s u l t s e e Bucher-Frolicher [ l ]

.

Mean value theorem and Taylor's formulas

In c a s e E

57

i s a b . v . s . and u s i n g ( 1 . 3 . 2 ) w i t h t h e b o r n o i o g i c a l

topology T E ( 0 . 2 . 7) one o b t a i n s i m m e d i a t e l y :

1.3.4

COROLLA R Y

[Q,$1 c ] a ,

b[

. Then

into E

where, if A

.-

&f

-E

be a b . v . s . ,

] a , b [ cIR,

be a ( S i l v a ) d i f f e r e n t i a b l e m a p p i n p f r o m

i s a s u b s e t of E ,

3 a , b[

FA d e n o t e s t h e T E - c l o s e d c o n v e x

b a l a n c e d h u l l of A .

Remark.

-

If E

is a p o l a r b . v . s . ( 0 . 2 . 9 ) and A

a bounded s u b s e t of

E t h e n FA is s t i l l bounded i n E , which s h o w s t h a t t h e s e t t i n g of p o l a r b. v .

i s w e l l s u i t e d to a p p l y t h e m e a n v a l u e t h e o r e m .

S .

The m e a n value t h e o r e m i s m a i n l y applied i n the following

1.3.5

COROLLA,RY, -

be a n o p e n s e t f o r t h e topology

.

Let TE1

E

1

. Let

be two b . v . s . and l e t U 1 2 f be a d i f f e r e n t i a b l e m a p p i n g E

into E , L e t a and h be two p o i n t s i n E 1- LA 1 i s c o n t a i n e d in U Then segment ( a t t h S t S 1 1' -

from U

form:

such that the

,1

f ( a t h ) - f ( a ) - f ' ( a ) . h e y / ( f ' ( a t t h ) - f l ( a ) ) . h 105tS 1

F

(where

Proof.

-

d e n o t e s t h e c l o s e d c o n v e x b a l a n c e d h u l l f o r t h e topology TE ). 2

F o r t h e f i r s t f o r m u l a a p p l y (1. 3. 4) with

g(t) = f ( a t t h ) . F o r

t h e s e c o n d one a p p l y t h e first f o r m u l a with g b ) = f ( x ) - f ' ( a ) . x .

1

58

1.3.6

Differentiable mappings, basic properties

PROPOSITION.

-

Let E

-

1We a s s u m e E

and E

2

be two b . v . s . and U

T E -open s u b s e t of E is a p o l a r b . v . s . 1 1' 2 n - t i m e s d i f f e r e n t i a b l e mapping f r o m U to E If a 6 U 1 1 2.we s e t :

r(n)(h) = f(ath)

-

f(a)

-

f'(a).h

a 1 L e t f be a

and nE RT

-. . . - -1 f (4(a) "!

h

.

of E t h e r e is a 11 such that :

T h e n f o r e v e r y convex balanced bounded s u b s e t B convex balanced bounded s u b s e t B

r(") (EB1) c B

Proof. tion o n n

of E

2-

for

2-

2 &

>o

s m a l l enough

T h e c a s e n = 1 is definition ( 1 . 1.1). T h e proof is by induc-

. W e set

:

q(h) = f(ath) if a t h E U 1 .

-

f(a) - f'(a).h

-. .. -

--n1l f (n)( a ) h n

cp i s d i f f e r e n t i a b l e in U -a and 1 ci.'(h) = f ' ( a t h )

-

fl(a)

- ,.,

( r e c a l l f (PI(a)is symmetric from ( I . 2.5)). Since L ( E 1 , E z ) is a p o l a r b . v . s . (0.8.10) u e m a y apply t h e induction a s s u m p t i o n of o r d e r n - 1 to t h e mapping f ' . s u b s e t M of

F o r given B1 t h e r e is a convex balanced bounded

L(E

1'

E ) such that : 2

q ' ( r B ) c M f o r € > o small enough 1

59

Mean value theorem and Taylor's formulas

I t s u f f i c e s now t o apply ( 1 . 3. 5) t o the m a p p i n g rp :

It s u f f i c e s to take

1. 3. 7.

B

-

r

= 2

M (B1)

-

M(B ) = { u(x)JU, M . 1 x E B1

where

LAE

and E be two b. v. s . a n d 2 11 s U be a n open set f o r t h e topology T E Let f be a n-times 1 1' d i f f e r e n t i a b l e m a p p i n g f r o m U to E L e t a a n d h b e two p o i n t s i n 2' 1 E s u c h t h a t the s e g m e n t [ a th ) i s contained i n U Then 1 OLtSl 1' Taylor's formulas.

-

+

f(a t h )

- f(a)- f'(a). h-. . . .

f(a t h)-f(a) where

proof.

-

r

- f'(a)h

-

f("-l(a) h n - ' C ~ F {f ( n ) ( a t t h ) h n ) o4tSl

- (q!

-. . . - !-f ( n ) ( a )hng &

[[f(")(a t t h )

n!

d e n o t e s t h e convex b a l a n c e d T E

2

- f(")(a)] hn) 0 ht

- closed

hull

sl

.

We set R ( h ) = f(a +h) - f(a) - f l ( a ) . h-.

Setting a t h = y we define a m a p p i n g

If z = a +th, with o i t r l b a l a n c e d s u b s e t of E

2

,

. . . . --1

f(n-l(a) hn-l

by :

b e a T E - c l o s e d convex 2 2 f ( n ) (a t t h ) h n € B 2 if t E [ O , l ] . T h e n

y - z = ( l - t ) h. L e t B

such that

(n-I)!

60

Differen tiab le mappings, basic properties

F r o m (1. 3 . 4 ) , i f 0 s t 4 t 1 2

1

,

Dividing the r e a l s e g m e n t L O , 11 i n i n t e r v a l s the d i a m e t e r s of which t e n d to o we obtain :

Since R ( h ) = ep (a)- cp ( a t h ) we obtain t h e f i r s t f o r m u l a . T h e s e c o n d one is obtained f r o m t h e first one by a c l a s s i c a l change of f u n c t i o n , a s i n t h e p r o o f of (1. 3 . 5). See 1. 3. 8

-

Remark.

C a r t a n [ l ] for d e t a i l s .

Since t h e s e r e s u l t s a r e b a s i c we give f u l l p r o o f s ,

but in f a c t they m i g h t be deduced f r o m c l a s s i c a l r e s u l t s of t h e one d i m e n s i o n a l c a s e . L e t E be a 1. c. s . , A a s u b s e t of E a n d x E E , L e t u s

x Er A that i f

I

I

.

1

I

Is

s u p y ' ( c ) f o r e v e r y y ' t E' T h e n w e c l a i m SEA : it f o l l o w s f r o m t h e g e o m e t r i c form of t h e H a h n - B a n a c h t h e o r e m

a s s u m e that

y'(x)

x$!F A t h e r e i s y'E E ' with y ' ( x ) > 1 a n d

)I

I y'(FA)I 5

1 hence

sup y'(c 5 1 (the g e o m e t r i c f o r m of t h e Hahn-Banach t h e o r e m m a y €A be found i n Kothe (13 $ 17 ) F r o m t h e one d i m e n s i o n a l m e a n value

5

.

t h e o r e m applied to y'o f one h a s :

T h u s one o b t a i n s ( 1 . 3. 2)

1.3.9 Remark.

-

.

V a r i o u s c l a s s i c a l a p p l i c a t i o n s of t h e m e a n value t h e o r e m ,

c o n c e r n i n g p a r t i a l l y d i f f e r e n t i a b l e mappings defined on a p r o d u c t , o r Gateaux d i f f e r e n t i a b l e m a p p i n g s , .

. . a r e not d i s c u s s e d

h e r e ; t h e y m a y be

found in Colombeau [ I ] i n the g e n e r a l s e t t i n g of b . v . s .

.

61

C (u mappings in the enlarged sense

5

1. 4

Coo m a p p i n g s i n t h e e n l a r g e d s e n s e

n

In t h i s s e c t i o n we define t h e c o n c e p t s of C

and C

co

m a p p i n g s in

S i l v a ' s e n l a r g e d s e n s e , which i n the f r a m e w o r k s of 1. c. s. look n a t u r a l t h a n t h e c o n c e p t s i n S i l v a ' s s e n s e defined i n But th. "usual"

1. 4. 8. below s h w s t h a t t h e s e two c o n c e p t s

cases for C

og

m a p p i n g s and th. 1. 4. 9

of S i l v a ' s definition (in c a s e of C

mappings)

A mapping f f

E U

to F

1. 2

1. 1. a n d

coihcide

.

in

shows t h e p r e e m i n e n c e

.

L e t E be b. v. s . , U a n o p e n s e t f o r

1. 4. 1 DEFINITION. -

F a 1. c. s .

03

4

more

7

E

U

is said to be differentiable in

U in the enlarged sense if for every a € U, h c E with a t h E U one m a v w r i t e :

f ( a t h ) = f ( a ) t f ' ( a ) . h t r (h) a w h e r e f ' ( a ) is l i n e a r bounded f r o m E into F a n d w h e r e der

"

i n the following s e n s e :

f o r e v e r y convex b a l a n c e d bounded s u b s e t B

of

r

a

i s a " r e m a i n --

E a n d e v e r y continuous

s e m i - n o r m p CF

Remark.

-

T h i s c o n c e p t of a r e m a i n d e r i s w e a k e r t h a n i n (1.1.1) w h e r e

the e x p r e s s i o n

\I

c o n v e r g e s t o o b o r n o l o g i c a l l y i n F while B

i n t h e definition j u s t above it c o n v e r g e s only topologically , Ir, f a c t t h e s e r-d (\h'l coihcide in m a n y two c o n c e p t s of c o n v e r g e n c e f o r t h e e x p r e s s i o n l ' h / IR u s u a l c a s e s f o r t h e p a i r of s p a c e s ( E , F ) ; s e e C o l o m b e a u [ I 7 1 $ 2 . Since m we s h a l l uniquely be c o n c e r n e d with C m a p p i n g s , t h i s d i f f e r e n c e w i l l in

-

f a c t be m e a n i n g l e s s f r o m th. 1 . 4 . 7 ,

1.4.8 below.

62

1. 4. 2

Differentiable mappings, basic properties

It is i m m e d i a t e t o check t h a t

Equivalent definition.-

def (1. 4. 1) i s equivalent t o : If E , U and F a r e a s i n (1. 4. 1) with F c o m p l e t e ( i n o r d e r to define

f’(a).h a s l i m i t of a Cauchy s e q u e n c e ) or F a 1. c. s with a continuous algebraically) , a V f r o m U into F i s d i f f e r e n t i a b l e i n t h e e n l a r g e d s e n s e

n o r m ( i n t h i s l a s t c a s e one m a y c h o o s e F = F mapping f

in U i f for e v e r y convex b a l a n c e d bounded s u b s e t B i n E a n d e v e r y convex balanced o-neighborhood V i n F the m a p p i n g with domain U n E

considered in the normed space E

B

r a n g e the n o r m e d s p a c e F V = (F’ ’V) -1 surjection mapping F

-. FV)

(sv

(0)

svo f

T o p o l o g y of

d e n o t e s the c a n o n i c a l

is “ d i f f e r e n t i a b l e

L(nE,F).-

nEB

and

B

in U

to the u s u a l definition (1. 0. 2 ) i n n o r m e d s p a c e s . 1. 4. 3

/u

L e t E be a b. v. s.

n

E

B

according

a n d F a 1.c. s.

If n €IN we denote by L ( n E , F) the l i n e a r s p a c e of n - l i n e a r bounded n into F ( a c c o r d i n g t o notation (0. 2. 2) a n d (0. 8. 7) m a p p i n g s f r o m I? n we should denote t h i s s p a c e L ( E, B F ) but the above notation L ( nE, F) i s shorter)

.

n T h i s s p a c e L ( E, F) i s n a t u r a l l y equipped with the topology of n uniform c o n v e r g e n c e on t h e bounded s u b s e t s of E , i. e. t h e s e t s

?J

(B, V) = [ u E L ( n E , F) s u c h t h a t u (Bn) C V

a r e a b a s e of o-neighborhoods

when B r a n g e s o v e r a

b a s e of

bounded s e t s i n E a n d V r a n g e s o v e r a b a s e of o - n e i g h b o r h o o d s i n F

.

It i s i m m e d i a t e t o c h e c k the a l g e b r a i c and topological equality:

L(~EL , ( P ~ , ~ = ) L) ( (n i f n, p

E IN

F)

63

C m mappings in the enlarged sense

1. 4. 4

Higher o r d e r d e r i v a t i v e s in the e n l a r g e d s e n s e .

-

It is now obvious f r o m (1. 4. 1) a n d (1. 4. 3 ) t o define t h e n t i m e s d i f f e r e n t i a b l e and n t i m e s continuously d i f f e r e n t i a b l e mappings i n the e n l a r g e d s e n s e . C l e a r l y t h i s a m o u n t s to : L e t E be a b. v. s.

1. C - S .

.

, U a n open set for

7 E and F a complete

A m a p p i n g f f r o m U to F i s n t i m e s

(n

d i f f e r e n t i a b l e i n the e n l a r g e d s e n s e in U i f f o r e v e r y boundet s u b s e t B i n E a n d e v e r y convex b a l a n c e d i n F the m a p p i n g s V the n o r m e d s p a c e FV

€IN U [a)) convex b a l a n c e d

0- neighborhood

V

with d o m a i n U n E C E and range f/UnE B B i s n B t i m e s d i f f e r e n t i a b l e i n the s e n s e of n o r -

m e d spaces. T h e definition of n t i m e s continously d i f f e r e n t i a b l e m a p p i n g s

is exactky similar. 1. 4. 5

Remark.

i s a b. v. s . ,

-

It follows i m m e d i a t e l y f r o m t h e definitions t h a t i f E

U a n open set f o r T E , F a 1. c. s.

any mapping f : U

a n d nEIN U {a], then

F n t i m e s d i f f e r e n t i a b l e i n U i n the s e n s e of

-t

$ 1 a n d $ 2 is n times d i f f e r e n t i a b l e i n U i n t h e e n l a r g e d s e n s e . Same result for 1. 4. 6

'I

continuously d i f f e r e n t i a b l e

s c a l a r valued C

i n p l a c e of "differentiable".

In (2. 5. 2 ) below we s h a l l give a n e x a m p l e of a

Example. a3

"

m a p p i n g i n t h e e n l a r g e d s e n s e which i s not C

a3

in

t h e s e n s e of fj 2 (however see ( 1 . 4 , s ) b e l o w ) . S e v e r a l r e s u l t s c o n c e r n i n g connections between n t i m e s d i f f e r e n t i a b l e m a p p i n g s in the e n l a r g e d s e n s e and i n S i l v a ' s u s u a l s e n s e

-

thdt c o h c i d e i n m a n y u s u a l c a s e s f o r the p a i r ( E , F ) - a r e given i n C o l o m beau

[ 1 7 1 $2.

S i n c e t h e s e r e s u l t s a r e m a r g i n a l f o r t h e p u r p o s e of t h i s

book, w h e r e only C

a3

m a p p i n g s a r e u s e d , w e d o not e x p l a i n t h e m . In th.

1 . 4 . 7 and 1.4.8 below we only c o n s i d e r t h e c a s e of is p a r t i c u l a r l y n e a t .

C

00

m a p p i n g s , which

Differentiable mappings, basic properties

64

- Let

E

Eand F a 1. c. s.

.

1. 4. 7

THEOREM.

s e t for

T

be a

Schwartz

;i.v.

If f is a m a p p i n g f r o m U into F , p r o -

- --

perties (i) and (ii) below a r e equivalent : ( i ) f i s infinitely d i f f e r e n t i a b l e in U i n t h e

s e n s e of

6

2

differentiable.in U in the enlarged sense.

(ii) f i s infinitely

i s a p a r t i c u l a r c a s e of th.

Th. 1. 4. 7

U an open set

s.,

E s c = E i f E is a S c h w a r t z b. v. s.

1. 4. 8

below,

since

,

L e t u s r e c a l l t h a t i f E i s a c o m p l e t e b. v. s .

we denote by

E

the l i n e a r s p a c e E equipped with t h e bornology of t h e s t r i c t l y s . c. c o m p a c t s u b s e t s of E , s e e (0. 7 . 4 ) .

1. 4.8 T

E * F

THEOREM.

-

_a 1. c. s.

and ( i i ) below

Let E -

. If

be a c o m p l e t e b. v. s . , U a n open s e t f o r

f i s a mapping from

U i n t o F , p r o p e r t i e s (i)

a r e equivalent :

( i ) f i s infinitely differentiable

(considering U

in U in the enlarged sense

as contained i n the b. v. s. E ) .

(ii) f i s infinitely d i f f e r e n t i a b l e i n U (in t h e s e n s e of $ 1. 2 ) i f U

i s c o n s i d e r e d i n t h e b. -

I. 4.9

V.S.

E

s. c.

F u n d a m e n t a l CommentslLL

,

When E i s a c o m p l e t e b. v. s . ,

w h i c h i s the c a s e i n the a p p l i c a t i o n s , th.

1. 4. 8

s h o w s that t h e c o n c e p t

of Cm m a p p i n g s i n the e n l a r g e d s e n s e i s e x a c t l y a p a r t i c u l a r a s p e c t of the concept of C

00

m a p p i n g s (in S i l v a ' s s e n s e adopted i n t h i s book ).This

r e s u l t is one of o u r basic t h e o r e t i c r e a s o n s to c o n s i d e r t h a t S i l v a ' s d e f i n i tlon a s given ir, Y 1. 2 i s the m o r e b a s i c concept. It w i l l r e m a i n t r u e , with a s i m p l e r p r o o f , in the h o l o m o r p h i c c a s e (th. 2 . 3 . 4 below).

1.4.10 cow.

The proof of th. 1.4.8 u s e s two l e m m a s t h a t w e s t a t e and p r o v e

C

LEMMA 1.

-

L e t E b e a c o m p l e t e b. v. s. a n d F a 1. c. s . . T h e n t h e

two s p a c e s L ( n E , F ) a n d L ( n E nologically. proof.

65

mappings in the enlarged sense

, F) a r e e q u a l a l g e b r a f c a l l y a n d b o r -

s C.

T h e i n c l u s i o n L ( n E , F)c L( nE

.

, F)

is trivial F o r the c o n v e r s e n L( E F) and u $ L( nE, F) .

S.C.

l e t u s a s s u m e by a b s u r d t h a t t h e r e is a u

E

SC'.

T h e r e f o r e t h e r e i s a c o n v e x b a l a n c e d bounded s u b s e t B of E a n d a c o n v e x n i s not balanced o-neighborhood V i n F s u c h that u ( B ) = [u(X)]

.

bounded i n t h e s e m i - n o r m e d s p a c e ( F , p ) T h u s t h e r e i s Xac B e que nc e V n X € B , s u c h t h a t p (u(X )) 0 3 . We m a y a s s u m e (xp)p€" p V P - -1 2 p v ( u ( X ) ) # 0 f o r e v e r y pElN a n d w e s e t Y = [ p (u(X ) ) ] X ;fY ) P P V P P PPEN n n a n d p v ( u ( Y p ) ) tm. i s a n u l l s e q u e n c e i n t h e n o r m e d s p a c e (E ) = (EB)

-+

+

(yp)i, G I N

(E

i s a bounded s u b s e t of

a s s u m p t i o n that

n u E L( E

S.C.

)"

Bn

and t h i s c o n t r a d i c t s the

, F).

sc.

T h e p r o o f of t h e b o r n o l o g i c a l e q u a l i t y i s q u i t e s i m i l a r : we n a s s u m e by a b s u r d t h a t t h e r e i s a bounded s u b s e t M of L ( E , F) t h a t i s SC n not bounded i n L ( E , F). T h u s t h e r e a r e s e q u e n c e s ( u ) u EM, and P PEm' P X EBn s u c h t h a t p u ( X ) ) + t m We g e t a c o n t r a d i c t i o n a s (xp)pcN' p V P P

.

.

before.

LEMMA 2.

-

L e t E be a B a n a c h s p a c e (with n o r m d e n o t e d by

11 11 )

and l e t (h ) be a null sequence i n E . T h e n t h e r e e x i s t s a convex n nEN b a l a n c e d c o m p a c t s u b s e t K of E s u c h t h a t hn(\ L hnll ) 2 / 3 f 0 r a l l n. K

/I

Proot.

We set k

n

=

(11 h n \ \ ) - 2 / 3 hn

i s a null sequence in E hn = ( (Ihnll )2'3k n h e n c e

.

K =

-

if h

I? { k n ]

n

II KS:( I/h n I I )

J'hn

n

#

0

and

(11

k =O n

i f h = O;(k ) n n nEIN

E i s~ a c o m p a c t s u b s e t of E

.

2/3

.

66

Differentiable mappings, basic properties

1. 4. 1 I (i)

=)

P r o o f of t h (1. 4. 8).

-

(ii). L e t f be GO3 i n U

CI

L e t u s now p r o v e t h e i m p l i c a t i o n

E i n the e n l a r g e d s e n s e . F i r s t let u s

p r o v e f is one time d i f f e r e n t i a b l e i n U c E

BF.

, valued i n t h e b. v.

s. c.

S.

L e t u s notice that t h e m e a n value t h e o r e m a n d T a y l o r ' s f o r m u l a s

of $ 3 a r e still valid f o r d i f f e r e n t i a b l e m a p p i n g s i n t h e e n l a r g e d s e n s e s i n c e they follow f r o m (1. 3. 2). If { a t t h )

ostsl

c U we have :

L e t K be a convex b a l a n c e d s t r i c t l y c o m p a c t s u b s e t of a n d let

E

> 0

be s u c h t h a t a t

c o m p a c t in t h e 1. c. s.

F.

KCU

.

E

T h e s e t [ f"(a th)'jhE E K i s

2

L( E, F) h e n c e bounded i n t h i s 1. c. s. a n d t h u s

bounded f o r t h e n a t u r a l bornology of L ('E, F), (0. 8. 8 ) , 2 2 a n d bounded a l s o i n L ( E , F) . L e t M be a c l o s e d (for T L ( E ,F)

this

s e t is

S.C.

S.C.

which i s a p o l a r b. v. &. f r o m (0. 8. 10)) convex b a l a n c e d bounded s u b s e t ' of L ( 2 Es. , F) s u c h t h a t t h e s e t f f"(a th)) hEE K i s c o n t a i n e d i n M

.

Let B be a c l o s e d convex b a l a n c e d bounded s u b s e t of F such' that

M(EK

E K )C B . F r o m (1)

F r o m def (1. 1. 1 ) a n d (1. 1. 2 ) t h i s p r o v e s t h a t f i s d i f f e r e n t i a b l e i n

U cE

s. c.

. let

. Now let u s p r o v e h i g h e r o r d e r d i f f e r e n t i a b i l i t y

f(n) : U

L(nE, F) b e t h e nth d e r i v a t i v e of f

.

W e set :

A s before (3)

Ra,,(h)

E

2

[ f(nt2)(a+th) h ) 0

g t rl

of f

. If nElK

mappings in the enlaced sense

C

in the 1. c. s.

L ("F, F)

.

The set [ f(nf2)(a t h )

61

lhE

is compact in

n the 1. c. s. L( E, F ) hence, a s before, bounded in the b. v. s. 2 L (nf2)E L (nE , F ) = L( E F)) s. c. s. c. , s. c . ,

.

that f ( n )

, F) and the proof s. c. i s differentiable (in Silva's usual sense (1. 1. 2 ) from

U c E

to L ( nE

Now B becomes

s. c.

a bounded

s. c.

, F)

subset of L ( n E

i s similar to the above c a s n = 0

.

Now l e t us prove the implication (ii) 3 (i) .Let f be Coo (in the sense of $ 2 ) in U c E

s. c.

.

Let B be a closed convex balanced

bounded Banach disc in E and let V be a convex balanced o-iieighborhood in F

. It

Cco from

U nEB c E

suffices to prove that the mapping sv

first prove that

B

to the normed space

FV'

/unEB

If a E U let

us

is

By absurd let us a s s u m e the existence of a null sequence (h ) in n nElN the Banach space E B such that

Let K be obtained from E

(1.4. 10).

B

and (hn)nEIN according to lemma 2 in

F r o m formula (1) above there is some C

Pv(ra(hn)) for every nElN

.

5

0 such that

c (I1 hnll

Therefore from lemma 2 :

)

K

2

Differentiable mappings, basic properties

68

which c o n t r a d i c t s ( 4 ) above. T h e r e f o r e f i s one t i m e d i f f e r e n t i a b l e

n

(h) be defined by ( 2 ) . ( 3 ) holds EB to FV. If nEIN l e t R from U a, n , F) = L ( nE, F) i f I' i s c o n s i d e r e d in t h e p o l a r b. v. s. L ( n E s .c . (nt-2) bornologically from l e m m a 1 i n (1. 4. 10) T h e set { f (a t h ) I h E

.

i s bounded i n L ( ( n t 2 ) E , F) i f a t n d e n o t e s the n o r m of L ( E gives that

l , 4 . 12 C

1

Further results,

E

K i s compact in U

n EB. If

q

F ) the s a m e proof a s above i n c a s e n = 0

B' V

-

S e v e r a l o t h e r r e s u l t s on c o i h c i d e n c e of

o r C n m a p p i n g s i n t h e e n l a r g e d s e n s e and i n S i l v a ' s ( o r d i n a r y )

s e n s e a r e given i n Colombeau

[ 17]prop

1, 2 and 3

.

69

"Locally difyerentiable between nomed spaces"

4

1. 5 Mappings u h i c h a r e " l o c a l l y d i f f e r e n t i a b l e b e t w e e n n o r m e d s p a c e s " A n o t h e r c o n c e p t of

C

n

mappings

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

"in a l o c a l s e n s e

s t r u c t u r e s of t h e s p a c e s

I'

a n d which i s a n a t u r a l e x -

t e n s i o n of t h e d e f i n i t i o n i n c a s e of n o r m e d s p a c e s i s t h e following o n e , o b t a i n e d by "patching

1. 5. 1.

together" differentiable mappings in normed spaces.

-

DEFINITIONS.

Let

E &F

U C E be a n o p e n set f o r t h e topology from

U

2

m e d spaces" s e t B of -

B' o f F

be two b. v. s. a n d l e t 7

E. I f p c l N

u

{a], a mapping f

F i s s a i d t o be ' ' l o c a l l y p t i m e s d i f f e r e n t i a b l e b e t w e e n n o r a t a point a € U

E there exist

E

i f f o r e v e r y c o n v e x b a l a n c e d bounded s u b -

>O

a n d a c o n v e x b a l a n c e d bounded s u b s e t

such that

2 ) t h e r e s t r i c t i o n of f k a t

E

( U h. B ) i s p t i m e s O
differentiable from a t

E

(U 1 B)

( e q u i p p e d with t h e n o r m e d s t r u c t u r e

0<1<1

of t h e a f f i n e s p a c e a t E

)

B

to the n o r m e d s p a c e F B I .

f i s s a i d t o be locally p - t i m e s differentiable between n o r m e d

s p a c e s i n U i f it i s a t e v e r y point of U

.

We d e f i n e a n a l o g o u s l y t h e

l o c a l l y p - t i m e s c o n t i n u o u s l y d i f f e r e n t i a b l e m a p p i n g s . T h i s c o n c e p t is v e r y R a t u r a l i n the f r a m e w o r k of b o r n o l o g i c a l v e c t o r s p a c e s , and i s v e r y m u c h e m p h a s i z e d by th. 2 . 2 . 3 of c h a p . 2 t h a t s t a t e s t h a t a n y Silva h o l o m o r p h i c m a p p i n g is l o c a l l y

C

00

b e t w e e n n o r m e d s p a c e s . Ir? the s e q u e l of t h i s

9

we c l a r i f y t h e c o n n e c t i o n s b e t w e e n t h i s c o n c e p t and t h e p r e v i o u s l y defined c o n c e p t s of Silva d i f f e r e n t i a b l e m a p p i E g s .

70

Differentiable mappings, busie properties

C l e a r l y it follows f r o m ( 1 . 1. 1) a n d (1. 1. 2 ) t h a t i f f i s locally one t i m e d i f f e r e n t i a b l e between n o r m e d s p a c e s at a point a t h e n f i s one t i m e d i f f e r e n t i a b l e a t a 1. 5 . 2

PROPOSITION.

-

L e t U be a TE-open f : U

of

-t

F d

9 2),then

E &F

set.

rf pEIN,

,

. b e two b. v. s .

with F polar.

p r l , and if the mapping

p t1) -times continuously d i f f e r e n t i a b l e i n U (in t h e s e n s e f i s l o c a l l y p - t i m e s continuously d i f f e r e n t i a b l e b e t w e e n

normed spaces in U . proof.

Let f : U

and let a E U

-t

F be (p t1) t i m e s continuously d i f f e r e n t i a b l e i n U

.

L e t B be a convex b a l a n c e d boundet Subset of E. i If 2 S i 5 p t1, f ( i ) i s M a c k e y - continuous (1. 1. 3 ) from U to L ( E , F) i so t h e r e a r e a bounded s u b s e t M. of L ( E, F) a n d q > 0 w i t h (i)(a t q B ) c M i L e t B’b e a T F - c l o s e d convex b a l a n c e d bounded f

.

s u b s e t of F which c o n t a i n s

lJ f 2 < i4 p t l

(i) i ( a t qB).B

.

Then, f r o m the m e a n

value t h e o r e m (1. 3. 5) , the r e s t r i c t i o n of f t o a t q (u 1 B) c a t E B O<X
1. 5. 3

B’

Example.

is p - t i m e s continuously d i f f e r e n t i a b l e .

-

In

4

,

‘

1

2. 5 above w e s h a l l give a s c a l a r valued

m a p p i n g which i s locally Ca3 between n o r m e d s p a c e s a n d w h i c h i s cn not c (in Silva’s s e n s e of 4 2 )

.

Remark. -

L e t E be a c o m p l e t e b. v. s. and F a 1. c.

U c E be a .rE-open s e t a n d f : U s p a c e s . T h e n c l e a r l y f i s Cm h e n c e f r o m th. (1. 4. 8) , f i s C

4

F b e locally C

a l

8..

Let

between n o r m e d

from U to F in the enlarged sense, a3

(in S i l v a ’ s s e n s e ) f r o m U c o n s i d e r e d

71

“Locallydifferentiable between normed spaces”

1. 5 . 4

Example.

-

to F which i s C

03

T h e r e is a 1. c. s. F a n d a m a p p i n g $ i n R a n d not l o c a l l y C

00

from R

between n o r m e d s p a c e s

i n EX : a n e x a m p l e i s given i n Colombeau [4], 9 6 , e x a m p l e 5 . 1. 5. 5

Further resulte.

-

T h e i n t e r e s t e d r e a d e r i s r e f e r r e d to

Colombeau [43 a n d [ 173 f o r v a r i o u s r e s u l t s on c o i h c i d e n c e o r i m p l i c a tions between l o c a l l y p - t i m e s (continuously o r not) d i f f e r e n t i a b l e m a p pings between n o r m e d s p a c e s a n d q t i m e s (continuously o r not) differentiable mappings.

72

Differentiable mappings, basic properties

0 1. 6 . 1

Cm m a p p i n g s of u n i f o r m bounded type

1. 6

If E a n d F a r e 1. c . s.

m a p p i n g s of u n i f o r m type.-

Cm

f : E

COD m a p p i n g

convex balanced

4

a

F i s s a i d t o be of u n i f o r m type i f t h e r e i s a

o-neighborhood

V in E s u c h that f m a y be facto-

r i z e d i n t h e following way :

E

v

N

with f

a C

Remark. -

03

m a p p i n g f r o m t h e n o r m e d s p a c e E V i n t o t h e 1. c. s.

in F = 'Jc

type

.

(a)

a n d we p r o v e i n (2. 7. 2 ) t h a t i t i s not C a, of u n i f o r m

T h e r e f o r e the C

( e x c e p t obviously when

1.

6. 2

c o n s i d e r e d i n ( 2 . 7. 2 ) i s

F r o m ( 2 . 4. 3) t h e m a p p i n g

Cm

F.

m

m a p p i n g s a r e n o t , i n g e n e r a l , of u n i f o r m type i s a n o r m e d s p a c e , see a l s o (1. 6. 3) below )

Cm m a p p i n g s of u n i f o r m bounded type. -

If

F is a n o r m e d space

N

and i n t h e SLtme conditions a s (1.6. l ) , i f f and a l l i t s s u c c e s s i v e d e r i v a t i v e s a r e bounded on a n y s e t nV,

f o r n EN,

(the spaces L ( E V , F )

being equipped with t h e i r u s u a l n o r m ) w e s a y t h a t f u n i f o r m bounded type. If E Cm

-

F is a

Cm

d

C

03

m a p p i n g of

is a n o r m e d s p a c e we only s a y t h a t f

m a p p i n g of bounded t y p e . If

f :E

is

F is a b . v . s .

o r a 1.c.s.

is a

we s a y that

m a p p i n g of u n i f o r m bounded t y p e i f t h e r e i s a N

bour.ded d i s c B ir. F s u c h t h a t f m a y be f a c t o r i z e d a s f = i o f w h e r e B a3 FB is C ig:FB F i s the c a n o n i c a l i n j e c t i o n m a p and w h e r e f : E N

-

of u n i f o r m bounded type a s defined a b o v e ; i . e . we h a v e t h e d i a g r a m i n th. 1.6.3

below.

A r e s u l t which s h o w s t h e r e l e v a n c e of t h i s c n n c e p t is

:

1. 6. 3

THEOREM. -

Let E

balanced o-neighborhood

B

in

be a S i l v a s p a c e , F a m e t r i z a b l e

(Silva) Coo m a p p i n g from E

and f a

73

mappings of uniform bounded type

C

V

-&

E

a F

.

1. c . s .

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

and a c o n v e x

b a l a n c e d bounded s e t

F s u c h t h a t f m a y b e f a c t o r i z e d a c c o r d i n g t o t h e following

diagram :

E

with

N

)F

2 Cm m a p p i n g of bounded type on E

f

V’

the n a t u r a l inclusion m a p p i n g .

and where i

B-

denotes

B e f o r e t h e proof we n e e d :

I. 6. 4

LEMMA. -

and f : E

+

Let E b e a S i l v a s p a c e ,

1. c. s .

F a l o c a l l y bounded m a p p i n g on E (i. e. bounded on a

neighbourhood of e a c h point of E ) o-neighborhood

.

Then t h e r e e x i s t a convex balanced

V i n E a n d a c o n v e x b a l a n c e d bounded s u b s e t B of

F s u c h t h a t f o r e v e r y n €IN t h e r e i s a pEIN

Proof.-

F a metrizable

Let

(K,)

Ew

with

be a n e x h a u s t i v e s e q u e n c e of c o n v e x b a l a n c e d

c o m p a c t s u b s e t s of E , ( 0 . 6 . 7 )

. For e a c h n

that : n K

n

c K

m(n)

*

n c l N , let m(n)ElN be s u c h

74

Differentiable mappings, basic properties

F r o m a s s u m p t i o n on f t h e r e i s a convex, b a l a n c e d o-neighborhood

V

n

i n E a n d a convex b a l a n c e d bounded s u b s e t B

f

We

n

in F s u c h t h a t

(Kn t Vn ) c Bn.

set : V = n (Knt n

1 ; V ). m(n)

V is convex, b a l a n c e d a n d i s a b o r n i v o r o u s s u b s e t of E, so V i s a o-neighborhood f o r t h e topology of E (= T E )

.

Furthermore :

and therefore

F r o m prop.2

of (0.6. 2) t h e r e i s a s e q u e n c e

E

n

> 0 such that

i s again a bounded s u b s e t of F. T h e n

1. 6. 5

P r o o f of th

1. 6. 3

.-

F o r e v e r y nElN

L ( nE, F) i s a m e t r i e a b l e s p a c e B a bounded set i n F , w e 1~ (V", B) =

.

the

space

If V is a o-neighborhood

i n E and

set u EL

(%,

n F) s u c h t h a t u (V ) c B

1,

C a, mappings of uniform bounded type If u E L ( n E , F ) , from ( 0 . 6 . 9 ) be a b a s e of convex balanced balanced

o-neighborhood V

r €IN with

P

75

u i s continuous on E

n

.

Let (W ) P PEN o-neighborhoods in F. T h e r e is a convex in E

such that for every q

E IN t h e r e

is

S i n c e E is a Silva s p a c e it is t h e s t r o n g d u a l of a F r e c h e t s p a c e (0.6.8) and it follows by p o l a r i t y f r o m p r o p . 2 of ( 0 . 6 . 2 ) t h a t , f o r e v e r y s e q u e n c e

( V p ) of o-neighborhoods in E , t h e r e is a s e q u e n c e

v=np v P

p

/A

P

>o

such that

p

is a g a i n a o-neighborhood i n E. T h e r e f o r e f o r e v e r y p , q E N t h e r e is r EN with

u((q v)”) c r

wP

.

Since (W ) is a b a s e of o-neighborhoods in F t h i s p r o v e s t h a t f o r e v e r y P q E N , u((q V)”) is a bounded s u b s e t of F. We j u s t p r o v e d t h a t a n y n u E L ( n E , F ) is i n s o m e s e t v ( V , B). T h e s a m e proof applied t o a n C L ( n E , F) g i v e s t h a t t h i s f a m i l y i s equicontinuous f a m i l y ( u . ) . 1 1EI n contained in s o m e set %(V , 13) c L(nE, F). T h u s a n y bounded s u b s e t of

L ( n E , F) i s contained in s o m e s e t v ( V n , B ) . Now l e t u s c o n s i d e r t h e mapping f( n) : E

-

L(nE,F).

Let us

denote by (w ) a b a s e of convex balanced o-neighborhoods i n P P N L ( n E , F ) and by s : L ( n E , F ) L(nE,F)w the associated canonical P P

-

s u r j e c t i o n m a p p i n g s . L e m m a ( 1 . 6 . 4 ) applied t o s o f ( n ) g i v e s t h a t P in E s u c h t h a t for e v e r y q EN t h e r e is t h e r e is a o-neighborhood v P a r c l N with

76

Differentiable mappings. basic properties

Since E is a Silva s p a c e w e obtain a s a b o v e t h a t independent of p EN

. So t h e r e

V

P

is a convex balanced

m a y be c h o s e n o-neighborhood

V

in E and a convex, balanced bounded s u b s e t bin in L ( n E , F ) s u c h t h a t ,

n

f o r e v e r y q EN , t h e r e is r EN with

8 of t h e f o r m ?(Vn,B), f o r s o m e n and s o m e bounded s u b s e t B in F. T h u s w e

F r o m above we m a y a s s u m e o-neighborhood V in E

in E and n in F s u c h t h a t , f o r e v e r y q E N

obtain that t h e r e e x i s t a convex balanced o-neighborhood a convex balanced bounded s e t B there is r

E IN with

n

T h i s inclusion holds f o r e v e r y n EN

>o

o such that V =

. As

n pnVn

V

before there a r e sequences

is a g a i n a o-neighborhood in n E and t h a t B = Ll EnBn is a g a i n a bounded s u b s e t i n F. A s a c o n s e n quence, for every (n,q)E N N t h e r e is a r f IN with

bn

and

E

n

We m a y a s s u m e B i s c l o s e d in F ,

hence in T B F = F (0.6.2). Then

t h e m e a n value t h e o r e m ( 1 . 3.5) p r o v e s the d e s i r e d f a c t o r i z a t i o n of f .

B

CHAPTER 2 HOLOMORPHIC MAPPINGS, BASIC PROPERTIES

ABSTRACT. In t h i s c h a p t e r we study d i f f e r e n t i a b i l i t y i n t h e p a r t i c u l a r c a s e of c o m p l e x s p a c e s , i . e . Holomorphy. E x c e p t in

$’ 2 . 6

w h e r e s e r i e s of

homogeneous polynomials and r e a l a n a l y t i c m a p p i n g s a r e s t u d i e d , a l l s p a c e s considered h e r e a r e complex. L e t E be a l . c . s . ,

R to F

mapping f f r o m

Sl

a n open s e t i n E and F a 1 . c . s .

i s s a i d to be h o l o m o r p h i c i n 0 if i t is

Gateaux-analytic and continuous in R.

A rndpping f f r o m R

s a i d to be Silva holornorphic if i t is Gdteaux-analytic i n following p r o p e r t y : f o r e v e r y x t h e r e is E Y O

with x t e B

A

cn

E n

to

F is

and h a s the

and e v e r y bounded s u b s e t B i n E

and f bounded on x t EB.

We obtain : l e t E be a c o - S c h w a r t z 1 . c . s . ( s e e ( 0 . 5 . 3 ) , f o r i n s t a n c e E rndy be 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 1 . c . s . ) , s e t in E and F a q u a s i - c o m p l e t e 1 . c . s .

.

n

a n open

T h e n i f f i s a mapping f r o m

to F the following p r o p e r t i e s a r e e q u i v a l e n t : (i)

f i s Silva h o l o m o r p h i c in 62

(ii)

f o r e v e r y convex balanced bounded s u b s e t B of E

r e s t r i c t i o n of f

to

R nE

B

the

( c o n s i d e r e d a s a n open s u b s e t of the norrned

s p a c e E B ) valued in t h e 1. c . s .

F is h o l o m o r p h i c ; in t h i s c a s e we s a y

t h a t f is h o l o m o r p h i c in t h e e n l a r g e d s e n s e i n

R

.

is d i f f e r e n t i a b l e in C l

(iii)

f

(iv)

f is infinitely d i f f e r e n t i a b l e i n

R

Holomotphic mappings, basic properties

78

(v)

f is Cm i n t h e e n l a r g e d s e n s e i n il'

(vi)

f is locally Coo between n o r m e d s p a c e s in h2

(vii)

f

is "locally analytic between n o r m e d s p a c e s " in F1

( t h i s l a s t concept is defined i n $ 2). We a l s o develop m a i n b a s i c p r o p e r t i e s of holomorphic m a p p i n g s , s u c h a s Cauchy's i n t e g r a l f o r m u l a , T a y l o r s e r i e s e x p a n s i o n s ,

.. . .

W e a l s o study s e r i e s of homogeneous polynomials (in both t h e r e a l a n d

the complex c a s e ) and p r o v e i n $ 2 . 6 :

THEOREM. - Let nClN

E

of

E

.

be a Banach s p a c e a n d

F

a 4, c. s.

-

. For

every

be a given continuous homogeneous F valued polynomial tco If t h e s e r i e s f c o n v e r g e s pointwise i n a n a b s o r b i n g s u b s e t n=O then i t c o n v e r g e s n o r m a l l y (definition 2 . 6 . 2 ) a t the o r i g i n of E

let

on

E

fn

c

.

If t h e series convergesin a 0-neighbourhood of

E , t h e n i t s s u m is conti-

nuous a t the o r i g i n . In $ 2 . 7 we prove t h e i m p o r t a n t t h e o r e m 2 . 7 . 4 t h a t any h o l o m o r phic mapping o n a Silva s p a c e , valued i n a F r 6 c h e t s p a c e is of " u n i f o r m bounded type". Finally i n

8

2 . 8 we d e s c r i b e b r i e f l y how m a t h e m a t i c a l P h y s i c i s t s

u s e infinite dimensional holomorphic functions t o r e p r e s e n t F o c k s p a c e s of Boson fields a n d o p e r a t o r s t h e r e i n .

79

Gateaux analytic mappings

9 2.1 2 . 1. 1 F i n i t e l y open s e t s . of E.

W e say that

linear subspace 6

Gateaux analytic mdppings.

-

L e t E be a l i n e a r s p d c e and

$1 a s u b s e t

is f i n i t e l y open i f for e v e r y f i n i t e d i m e n s i o n a l

I)

of E ,

R

d

fi

is open in 6 ( 6 is equipped with i t s

unique topology of finite d i m e n s i o n a l 1.c. s . (0. 1 . 7 ) ) . T h i s definition a m o u n t s to : if we e q u i p E with the bornology of the finite d i m e n s i o n a l bounded s e t s (0.2.2) topology

TE

then

is open f o r t h e

(0.2.5).

2 . 1 . 2 Gateaux a n a l y t i c m a p p i n g s ( f i r s t definition).

c o m p l e x l i n e a r s p a c e and complex 1.c.s.

-

fl a finitely o p e n s u b s e t of

A mapping f from

I)

g

L e t E be a E. Let F b e

a

F is s a i d t o be Gdteaux

analytic (G-analytic f o r s h o r t ) if f o r e v e r y x E

n

and h

E E the

quotient

F when t h e c o m p l e x n u m b e r

h a s a l i m i t i n the 1 . c . s .

2.1.3

PROPOSITION.

-

z

tends to

0.

L e t E be a c o m p l e x l i n e a r s p a c e and

a finitely open s u b s e t of E . a mapping f r o m fl L F .

Let F

be a c o m p l e x 1.c. s.

n

dnd l e t f

T h e n the two following p r o p e r t i e s a r e

equivalent : (i)

-

s u c h t h a t , when z

o (z

Y' (ii)

and

f o r e v e r y x €61

[

t?

C,

e

f(x t zh)

for every x 6

Z

# -

h E E t h e r e i s s o m e A = a ( x , h ) EF

o), f(x)

for every y'

- 4 J -

F'

o

fl and h EE t h e r e is s o m e R E F s u c h

80

Holomorphic mappings, basic properties

that w h e n z

4

.______--._I

o the quotient f(x t zh) - f ( x ) Z

converges bornologically to b o r n o l o g y of

F (0.2.2)).

Consequence.

-

.t

( 0 . 2 . 12) ( f o r t h e Von N e u m a n n

%F

( 2 . 1 . 3 ) p r o v e s t h a t i n def ( 2 . 1 . 2 ) one m a y w e a k e n o r

s t r e n g t h e n t h e c o n c e p t of c o n v e r g e n c e without c h a n g i n g t h e c o n c e p t of G-analytic mappings.

2. 1 . 4 P r o o f of ( 2 . 1 . 3 ) .

-

The implication (ii)

s u f f i c e s to p r o v e (i) 3 (ii). If x E s u c h t h a t x t zh 6

R if

y ' E F' the f u n c t i o n z

-

Iz

I
1

( i ) i s t r i v i a l s o it

E t h e r e is s o m e a

>o

s i n c e 0 is f i n i t e l y open. F o r e v e r y

y' [f(xtzh)J f r o m

Iz I <

h e n c e it m a y b e developped f o r

a

R and h

2

a

Iz ]
in c o n v e r g e n t p o w e r s e r i e s :

= ~ ' ( 1h)e n c e

y'[f(xtzh)

-

f(x) -

2.t

J

n-2

= z

n2 2

If o < a ' < a w e s e t k ( y ' , a

I)

=

1 n 2 2

Ian/ ( € I ' ) ~ < - ~t m

. T h e n if Izl
81

Gateaux analytic mappings

We s e t :

I

A = /y 6 F such t h a t , f o r e v e r y y ' E F', y ' ( y )

A is bounded in O ( F , F ' )

( s e e 0.1.15),

I

S k(y',

1

a')

.

therefore f r o m Mackeyls

t h e o r e m (0.3.8) and Mackey A r e n s t h e o r e m ( 0 . 3 . l o ) ,

A is bounded i n F.

Since

f ( x t z h ) - f(x)

if

1 z1 S

a

w e obtain (ii)

I,

-

zi?

E ( Iz

1)"

A

.m

2 . 1.5 G a t e a u x - a n a l y t i c m a p p i n g s ( g e n e r a l definition). -

F r o m ( 2 . 1.3)

the following definition extends ( 2 . 1 . 2 ) :

Let E

Let

of E .

F.

F

be a c o m p l e x l i n e a r s p a c e and a complex b.v.s.

Then we say that f

R a finitely open s u b s e t

and l e t f be a mapping f r o m 0 to

is G-analytic in

if f o r e v e r y x E

nand

-

o in

h E E ,the quotient

f(xtzh) - f(x) Z

c o n v e r g e s bornologically ( 0 . 2 . 1 2 ) in the b . v . s . with

z

f

o

.

F when z

C

2. 1 . 6 Uniqueness of analytic continuation. A s in t h e f i n i t e d i m e n s i o n a l o r i n the n o r m e d s p a c e c a s e t h e u n i q u e n e s s of a n a l y t i c continuation is a b a s i c f e a t u r e of ar.alytic m a p p i c g s

:

a2

Holomorphic mappings, basic properties

PROPOSITION.

-

Let E -

be a c o m p l e x l i n e a r s p a c e t h a t we equip with

t h e finite d i m e n s i o n a l bornology ( s e e 0 . 2 . 2 ) and l e t

n

be a finitely open

s u b s e t of E ( s e e 2. 1. I ) , t h a t we a s s u m e to be connected f o r t h e topology TE.

Let F

be a c o m p l e x b . v . s .

G-analytic mappings f r o m with

a E

n

L e t f and g be two

F which coi'ncide on a s u b s e t a t P,

R and P s o m e b o r n i v o r o u s s u b s e t of E. T h e n

Proof.-

f = g

& I fl

.

If x E 0 we s h a l l p r o v e f(x) = g(x) and it s u f f i c e s t o p r o v e

t h a t y'(f(x)) t h a t the b.v.s.

y'(g(x)) f o r e v e r y y ' EFX ( s i n c e w e a s s u m e a s a l w a y s F is s e p a r a t e d by its dual). Since h2 is connected

t h e r e is a continuous c u r v e L of e x t r e m i t i e s x and a ,

which is

contained in 0 and in s o m e finite d i m e n s i o n a l s u b s p a c e of E . Considering t h e r e s t r i c t i o n s of y ' o f and y ' o g t o a neighborhood of L i n this finite d i m e n s i o n a l s u b s p a c e the d e s i r e d r e s u l t follows i m m e d i a t e l y f r o m the c l a s s i c a l u n i q u e n e s s of a n a l y t i c continuation in the f i n i t e d i m e n sional case.

Silva holomophic mappings

83

$ 2 . 2 Silva h o l o m o r p h i c m a p p i n g s

2.2.1 i-2

a

-

h - l o c a l l y bounded m a p p i n g s . 7

L e t E and F be b . v . s .

and

E - o p e n set. A mapping f f r o m '2 t o F is s a i d t o be b o r n o l o g i -

c a l l y l o c a l l y bounded ( b - l o c a l l y bounded f o r s h o r t ) a t a point a E 0 if f o r e v e r y bounded s u b s e t B of E t h e r e is f ( a t & B ) is bounded i n F. is a t e v e r y point of

Example.

-

E

>o

s u c h t h a t the s e t

f is said to be b - l o c a l l y bounded i n 62 if it

0.

It follows i m m e d i a t e l y f r o m def (1. 1. 2) t h a t a n y d i f f e -

r e n t i a b l e mapping i n '2 is b - l o c a l l y bounded i n 0 .

2 . 2 . 2 Silva h o l o m o r p h i c m a p p i n g s . b . v . s . and

-

Let -

Eand F be c o m p l e x

R a'rE - o p e n s e t . A mapping f f r o m R t o F is s a i d t o

be Silva h o l o m o r p h i c in R if it is C - a n a l y t i c (2. 1 . 5 ) and b - l o c a l l y bounded ( 2 . 2 . 1 ) i n 62

.

If E and F a r e n o r m e d s p a c e s t h i s definition is

Remark. -

e x a c t l y a c l a s s i c a l o n e , s e e Hille

[ 1] for

i n s t a n c e , It follows immediately

f r o m C a u c h y ' s f o r m u l a t h a t in c a s e of n o r m e d s p a c e s i t is equivalent to C a n a l y t i c and continuous.

2 . 2 . 3 THEOREM. with F R to F

-

Let E and -

F be two c o m p l e x b. v. s .

- 7 E-open

p o l a r and l e t R c E be a

s e t . If f i s a mapping f r o m

then the following conditions a r e e q u i v a l e n t :

(i)

f is Silva h o l o m o r p h i c in

(ii)

f is d i f f e r e n t i a b l e i n

n

( i i i ) f is l o c a l l y h o l o m o r p h i c between n o r m e d s p a c e s i n 0

.

84

Holomorphic mappings, basic properties

C l e a r l y ( i i i ) m e a n s thdt : f o r e v e r y a f balanced bounded s u b s e t B of E t h e r e is

&

> o and a c o n v e x balanced

bounded s u b s e t B ' of F s u c h t h a t f(a t & B ) c F tion of f to a t

E

(

u

AB) valued in

o
and e v e r y c o n v e x

B'

and t h a t the r e s t r i c -

F i s holomorphic in the B'

s e n s e of t h e s e n o r m e d s t r u c t u r e s .

2 . 2 . 4 P r o o f of t h ( 2 . 2. 3 ) . -

t o prove (i)

* (iii) .

If x

E 0 and if B is a c o n v e x b a l a n c e d bounded

t h e r e is a c o n v e x b a l a n c e d TF-closed bounded s u b s e t B '

s u b s e t of E

F and t h e r e i s

a 9 - 0 s u c h t h a t f(x t U B ) C B ' .

Ilh/lB < a , if h '

E EB

function z - + y '[f(x t h Therefore

C l e a r l y (iii) 3 (ii) 3 ( i ) , s o i t s u f f i c e s

with f

l\h' [ / B


If h E E B w i t h

IIhIl' B and i f y ' f F W , the

z h ' ) ] E d; is h o l o m o r p h i c in

{ Iz

I< 1 1

:

Since f is G - a n a l y t i c we s e t : .t = l i m

f(xthtzh')

- f(xth)

Z

z - + o Therefore

yi[f(xthtzhf) - f(xth)

-

zA]=

n-2 an(4

(z) n 2 2

if

IzI<

1. F r o m C a u c h y ' s f o r m u l a :

c C.

of

85

Silva holornovhic mappings

Since

I/ht Sh'1lB

y'(B')

'

$a,

f(xtht6h')

E

B ' and

a E-

1

(.In

m)'w h e r e

d e n o t e s t h e c l o s u r e i n CC of t h e s e t y ' ( B ' ) . T h e r e f o r e

y'[f(xthtzh') -f(xth)

1z I
Choosing

closed for

z.E]E( [ z

(

Iz l)n-2(r)-n)

a:

y")

we have

y'[f(xthtzh')

for some m >o

-

-

f(xth)-zk]

E m( 1 z

i n d e p e n d e n t on y ' .

T F ( h e n c e u ( F ,F

y " ) a:

S i n c e F X = ( T F ) ' and s i n c e B ' i s

) closed f r o m Mackey and M a c k e y - A r e n s '

t h e o r e m s - s e e ( 0 . 3 . 8 ) , (0. 3. l o ) ) , i t follows f r o m t h e b i p o l a r t h e o r e m

(0.3.6) that B' =

(B')O0.

Since the above f o r m u l a holds f o r e v e r y

y ' E F Y = (TF)' ,

f(zthtzh')

-

f ( x t h ) - z.4 E (

This proves that f is

1z

m B'

G-analytic f r o m x t o (

.

u

o< A i l

XB) t o FBI.

f i s holomorphic f r o m xta(UXB) to o< A< 1 i n t h e s e n s e of t h e s e n o r m e d s t r u c t u r e s .

S i n c e f u r t h e r m o r e f(xtCLB) C B ' ,

FB,

2 . 2 . 5 C o m p o s i t i o n of h o l o m o r p h i c m a p p i n g s . d i f f e r e n t i a b l e m a p p i n g s i s d i f f e r e n t i a b l e (1. 1,4),

S i n c e t h e c o m p o s i t i o n of i t follows f r o m

t h . (2.2 , 3 ) t h a t t h e c o m p o s i t i o n of h o l o m o r p h i c m a p p i n g s is holomorphic. T h i s r e s u l t m a y a l s o be p r o v e d d i r e c t l y i n a q u i t e i m m e d i a t e w a y .

Hotomorphic mappings, basic properties

86

2.2.6 and

Cauchv's i n t e g r a l f o r m u l a .

0 a 7E-open set.

-

L e t E be a compbex b. v . s .

L e t F be a p o l a r b . v . s .

and l e t f : 0 - I F be

a Silva holomorphic mapping. T h e n f r o m (2.2. 3) and the theor; of h o l o m o r p h i c functions of one v a r i a b l e , f o r e v e r y x balanced bounded s u b s e t B c E , bounded s u b s e t B '

i f h E B and

contained in

t h e r e is

0>o

E0

and e v e r y convex

and a convex balanced

F such t h a t

of

I < & , w h e r e T' is a " s i m p l e " c l o s e d r e c t i f i a b l e c u r v e , { 16 I< a 1 c C around z E CC and o r i e n t e d i n t h e positive 12

A

s e n s e . The i n t e g r a l is defined in t h e B a n a c h s p a c e F B I , of the n o r m e d s p a c e F B I . In p r a c t i c e

r

In c a s e F i s a 1 . c . s . dnd f a to F,

1 6 -z 1

completion

w i l l be a c i r c l e .

G-analytic mapping f r o m

the i n t e g r a l is t a k e n in the c o m p l e t i o n F of the 1. c . s .

F

,

n

for

s m a l l enough, and Cauchyls i n t e g r a l f o r m u l a s t i l l h o l d s .

P r o o f of Cauchyls i n t e g r a l f o r m u l a . i s a b . v . s . and f

-

First w e consider the c a s e F

is Silva h o l o m o r p h i c . The i n t e g r a l

A m a k e s s e n s e in t h e B a n a c h s p a c e F B I s i n c e w e m a y i n t e g r a t e , like in

t h e c a s e of s c a l a r valued f u n c t i o n s , a continuous function defined on a f i n i t e r e a l i n t e r v a l and valued in a B a n a c h s p a c e , Application of t h e classical Cauchy's integral formula to y' o f ,

for e v e r y y 1 E ( F B , )

since f(xtzh) E F

B"

I

.

I

w i t h y' E ( F B , ) , g i v e s :

A

T h u s f o r m u l a (1) holds in F

h e n c e in F B' ' B' T h e c a s e F is a 1. c . s . is q u i t e s i m i l a r .

87

Silva holomophic mappings

2 . 2 . 7 P o w e r s e r i e s expansion of a Silva h o l o m o r p h i c m a p p i n g .

Now u n d e r t h e a s s u m p t i o n s i n ( 2 . 2 . 6 ) and if F s p a c e , we have i n t h e Banach s p a c e

FB

f

IT s u c h

that

16

I =r < a

is a Banach

'

( t h e proof i s s i m i l a r t o t h a t of ( 2 . 2 . 6 ) ) .

r = 15

B'

-

If we c h o o s e

and if

Iz l a - ,

we o b t a i n (with p o s i -

t i v e o r i e n t a t i o n of t h e path of integration)

Since f ( x t BB) c B' t h i s s e r i e s i s u n i f o r m l y c o n v e r g e n t i n for every h E B if

Therefore

I z 1.-

r'
.

If n EN we s e t u s u a l l y :

b of ( x , h ) = f ( x ) and

n 20 F r o m the c l a s s i c a l f i n i t e d i m e n s i o n a l c a s e r e s u l t s i t follows t h a t n the mapping h -+ 6 f(x, h) i s polynomial homogeneous of d e g r e e n in h ( t h e proof i s s i m i l a r t o t h a t of (1.2.5)).

F r o m formula (3) this l a s t

mapping is a l s o bounded on e v e r y bounded s u b s e t of E , t h e r e f o r e i t is i n P("E, F).

Holomorphic mappings, basic properties

88

2 . 3 H o l o m o r p h i c m a p p i n g s and Silva h o l o m o r p h i c m a p p i n g s in t h e e n l a r g e d s e n s e

2. 3. 1 H o l o m o r p h i c m a m i n e s . -

R

1. c . s . and

E and F be c o m p l e x

Let

to F

a n open s u b s e t of E . A m a p p i n g f f r o m 0

to be holomorphic in 0 i f it i s G - a n a l y t i c and continuous in

R e m a r k 1.

-

If E

is a I . c . s . dnd

E'

the bilinear mapping

i s c l e a r l y Silva h o l o m o r p h i c ( i f we e q u i p

B

fi

.

i s said

i t s s t r o n g d u a l (0. 1 . 1 4 ) then

E and E '

N e u m a n n b o r n o l o g i e s ) , It is e a s y t o p r o v e t h a t if E

P

with t h e i r Von i s not a n o r m e d space

t h i s b i l i n e a r m a p p i n g is n o t continuous. T h i s is a n e x a m p l e of a Silva holomorphic m a p p i n g which is not c o n t i n u o u s .

R e m a r k 2. -

A s s u m e E is a q u a s i - c o m p l e t e 1. c . s .

in o r d e r t h a t

t h e closed convex balanced hull of a c o m p a c t s e t be c o m p a c t ( 0 . 2 . 2 ) and c o n s i d e r t h e bornology of t h e c o m p a c t s e t s ( 0 . 2 . 2 ) .

Then any holomorphic

mapping on h2 c E is Silva h o l o m o r p h i c for the c o m p a c t bornology of E.

2. 3.2

Silva h o l o m o r p h i c m a p p i n g s in tlx e n l a r g e d s e n s e . -

be a c o m p l e x b . v . s . dnd 1 . c . s . A mapping f :

h2 L T E - o p e n s u b s e t ,

-

U

E

F be a c o m p l e x

F i s s a i d to be Silva h o l o m o r p h i c in the

e n l a r g e d s e n s e in 0 if i t is G - a n a l y t i c in

n

and continuous in

the topology T E ( 0 . 2 . 5 ) with v a l u e s in the 1 . c . s .

0 for

F.

C l e a r l y f i s Silva h o l o m o r p h i c in t h e e n l a r g e d s e n s e in

R if and

89

Holomorphic mappings and Silva holomorphic mappings only i f f o r e v e r y c o n v e x b a l a n c e d bounded s u b s e t B of t i o n of f t o G normed space

fl E E€3’

E,

the r e s t r i c -

equipped with t h e topology induced by t h a t of t h e B’ with v a l u e s in the 1. c . s . F, is h o l o m o r p h i c .

Using t h e e q u i v a l e n c e b e t w e e n h o l o m o r p h y and d i f f e r e n t i a b i l i t y in the s e t t i n g of c o m p l e x n o r m e d s p a c e s ( s e e 2. 2 . 3 ) and u s i n g ( 1 . 4 . 2 ) , i t f o l l o w s i m m e d i a t e l y t h a t , if

F is c o m p l e t e :

f i s Silva h o l o m o r p h i c i n t h e e n l a r g e d s e n s e in R it i s Silva d i f f e r e n t i a b l e i n t h e e n l a r g e d s e n s e in

F r o m ( 2 . 2 . 3) and ( 1 . 1 . 6 ) a n y Silva h o l o m o r p h i c

Remark 1.mapping f r o m on

0 by

’,if

n

F is c o n t i n u o u s f o r t h e topology induced

t o a 1.c. s.

T E , h e n c e is Silva h o l o m o r p h i c i n t h e e n l a r g e d s e n s e i n 0 .

R e m a r k 2. 1. c . s

-

If E

i s a 1 . c . s. ,

R

a n o p e n s e t i n E and F a

,

B E d e n o t e s t h e Von N e u m a n n b o r n o l o g y of E t h e n t h e topology

7BE is c l e a r l y f i n e r than the topology of E . mapping i n

R is

TE-open s u b s e t

a m a p p i n g f r o m 0 t o F.

-

. Proof. -

R

is

BE.

THEOREM. -

and R a_ --

Therefore any holomorphic

Silva h o l o m o r p h i c i n t h e e n l a r g e d s e n s e w h e n

c o n s i d e r e d in t h e b . v . s .

2.3.3

.

i f and only if

.

Let E L e t F be -

be a c o m p l e x S c h w a r t z b. v . s.

a c o m p l e x 1.c. s.

and l e t f

be -

T h e n t h e following a r e e q u i v a l e n t :

(i)

f is Silva h o l o m o r p h i c i n

(ii)

f

n

is Silva h o l o m o r p h i c i n t h e e n l a r g e d s e n s e i n

n

T h i s is j u s t th. ( 1 . 4 . 7 ) i n t h e p a r t i c u l a r c a s e of c o m p l e x

s p a c e s but we give a d i r e c t proof which is v e r y s i m p l e R e m a r k 1 i n ( 2 . 3 . 23. S i n c e E a b a s e (Bi)iEI

i s a S c h u a r t z b.v. s . ,

of bounded s e t s which a r e

T E

. (i)

(0.5.2),

(ii) by

it aamits

c o m p a c t . If a E 0 a n d

Holomoiphic mappings, basic properties

90

i

E I there is

on

,

a t

E

B

i i'

&

.>o

s u c h t h a t a t E . B . C C-2 1

1

.

T h e r e f o r e f is bounded

h e n c e f is b - l o c a l l y bounded i n R

.m

A s in ( 1 . 4 . 7 , 8) a m o r e g e n e r a l r e s u l t is the p a r t i c u l a r c a s e of th ( 1 . 4 . 8 ) when the s p d c e s d r e c o m p l e x :

2 . 3 . 4 THEOREM.

-

Let E -

b e a c o m p l e t e c o m p l e x b. v. s . And

denote the l i n e a r s p a c e E equipped u i t h the bornology of i t s s. c be a T E - o p e n s e t and l e t F be s t r i c t l y c o m p a c t s__e t s ( 0 . 7 . 4 ) . L e t 61 let -

E

-

a c o m p l e x 1.c. s. If f

is

d

mdpping f r o m

into

F t h e follouine,

p r o p e r t i e s a r e equivalent :

E

(i)

f is Silva h o l o m o r p h i c in i-2

c o n s i d e r e d in the b.v. s .

(ii)

f is Silva h o l o m o r p h i c in t h e e n l a r g e d s e n s e i n h2

S.C.

c o n s i d e r e d in t h e b.v. s.

E.

It follows f r o m (1.4.8),but a d i r e c t proof is m u c h e a s i e r .

Proof.-

Since one c h e c k s e a s i l y t h a t TE = T(Es.c!,

is obvious,

2.3.5

m

Fundamental comments.

-

(i)

* (ii).

The converse

T h u s t h e concept of Silva

holomorphic mappings in t h e e n l a r g e d s e n s e i s e x a c t l y a p a r t i c u l a r a s p e c t of t h e c o n c e p t of Silva h o l o m o r p h i c m a p p i n g s . A s in c a s e of r e a l s p a c e s for Co3 m a p p i n g s , (1,4.9),

this basic r e m a r k strengthens

the i n t e r e s t of t h e c o n c e p t s of Silva d i f f e r e n t i a b i l i t y and h o l o m o r p h y and c o n t r i b u t e to justify t h e i r p r e e m i n e n t r o l e .

91

Holomorphic mappings and Silva holomotphic mappings

2.3.6

Example. -

In a c o m p l e x c o m p l e t e , but not S c h w a r t z , b . v . s.

E

we give a n e x a m p l e (in 2 . 5 . 3 below) of a Silva h o l o m o r p h i c m a p p i n g i n the e n l a r g e d s e n s e which is not Silva h o l o m o r p h i c i n E (but is Silva holomorphic in E

S.C.

, f r o m th. ( 2 . 3 . 4 ) ) .

A n o t h e r i m p o r t a n t r e s u l t of coincidence of t h e s e two c o n c e p t s is :

2.3.7

PROPOSITION.

Let E -

-

be a c o m p l e x b . v . s. ,

F a c o m p l e x 1.c. s. with a countable b a s e of bounded

*rE-open s e t and

-f

s e t s (for i t s Von N e u m a n n bornology). L e t

be a m a p p i n g f r o m

F which is Silva holomorphic i n t h e e n l a r g e d s e n s e in 0 Silva holomorphic in

R

.

We p r o v e that f

Proof. -

there a r e x E

fl a

is b-locally bounded i n

n

. Then

R into

f is

. Let us assume

R and a convex balanced bounded s u b s e t B of E s u c h

t h a t f ( x t a B ) i s not bounded i n F f o r any b a s e of bounded sets i n F.

1

a >o

.

Let /B', 1 F o r e v e r y n 2 1 f(x t B) Btn

-

-n

IncN

, so

Inr1

be a there

E B with f ( x t h n ) /c B',. T h e r e f o r e t h e s e t { f ( x t h n ) is not n n is M - c o n v e r g e n t to o i n E bounded in F. Since ( h ) nnEN if(x+hn)), EN is a c o n v e r g e n t s e q u e n c e in t h e 1 . c . s. F , t h e r e f o r e a is h

bounded s e t i n F and we get a c o n t r a d i c t i o n .

2.3.8

Remark.

-

Schwartz's space

r1,21 ,

An e x a m p l e of a Silva h o l o m o r p h i c function on the

&(EX) which is not continuous is g i v e n i n B o l a n d - C i n e e n

92

Holomorphic mappings, basic properties

5

2 . 4 Ca

differentiability of h o l o m o r p h i c m a p p i n g s

Let -

THEOREM. -

2.4.1

E be a c o m p l e x b . v . s ,

R a

TE

-

open s u b s e t and F a c o m p l e x p o l a r c o m p l e t e b. v . s . L e t f be a Silva holomorphic mapping f r o m for every x E CL>O

i 7 into F s u c h t h a t :

R and e v e r y bounded s u b s e t B of E t h e r e is

such t h a t f o r e v e r y bounded s u b s e t B ' of

E there i s a ' s o such

that x t a B t ~ ' B ' c0 f(x t n B t R I B ' ) (note t h a t a

and

is bounded in F,

i s a s s u m e d to be independent on Bl).

Then -

f

i s infinitely d i f f e r e n t i a b l e i n 62

.

B e f o r e the proof, which is given a t the end of t h i s

9

, we

explain a few c o r o l l a r i e s .

2 . 4 . 2 COROLLARY. s u b s e t of E and

Let E -

be a c o m p l e x 1.c. s . , R

a n open

F a c o m p l e x 1 . c . s . If f is a G-analytic mapping

from62 to F which is bounded o n a neighborhood of each point i n t h e n f i s infinitely d i f f e r e n t i a b l e in R

.

Proof.

-

W e m a y w o r k with t h e c o m p l e t i o n F of t h e 1. c . s,

o r d e r that the V o n Neumann bornology to apply th. ( 2 . 4 . 1).

I

n

F in

B F be c o m p l e t e and it s u f f i c e s

93

C a, differentiability of holomorphic mappings

2.4.3

Let E

COROLLARY. -

a n open s u b s e t of

E

be a c o m p l e x m e t r i z a b l e l . c . s ,

0

F a c o m p l e x 1.c. s. with a countable b a s e of

F i s Silva h o l o m o r p h i c i f and

R

bounded s e t s . T h e n a mapping f r o m only if it is infinitely d i f f e r e n t i a b l e .

-

Proof.

F r o m c o r . ( 2 . 4 . 2 ) and th. ( 2 . 2 . 3 ) i t s u f f i c e s t o c h e c k t h a t

any Silva h o l o m o r p h i c mapping f : R

+

F is bounded on a neighborhood

of e a c h point in fl

.

If x

is not bounded on a n y x - n e i g h b o r h o o d , t h e n f o r e v e r y

L e t u s denote by ( V ) a b a s e of n nEN o-neighborhoods i n E and by ( B ' n ) n E N a b a s e of bounded s e t s in F.

E

i-2

and i f f

p

B',. ( x ~ - x )i s~ a: null ~ ~ sequence n E IN t h e r e i s x Ext V with f(x ) n n n inE.Thetopo1ogy o f E c o i n c i d e s with TBE s i n c e E is m e t r i z a b l e , s e e c o n v e r g e s t o f(x) in F, s o T h e r e f o r e (f(x )) n nEN i s a bounded s u b s e t of F and w e get a c o n t r a d i c t i o n . I (0.6.2).

2.4.4. a

COROLLARY.

Let -

T h e n a mapping f r o m

n

of E

If x E 0

and i f

.

if

B ' a r e two given bounded s u b s e t a

that we m a y a s s u m e s t r i c t l y c o m p a c t , f o r s o m e u 9 0 1 x t a B is

n,

contained i n

h e n c e s t r i c t l y c o m p a c t in 62

s m a l l enough, x t a B t a I B ' in,, 0

B and

0

(or any complex

F is Silva h o l o m o r p h i c in R

and only if i t is infinitely d i f f e r e n t i a b l e in 0

Proof. -

In,

E be a c o m p l e x S c h w a r t z b. v. s . ,

F a c o m p l e x p o l a r c o m p l e t e b. v . s .

7 E - o p e n s e t and

1.c.s.).

-

k(xn)

.

. Therefore for

a'>o

i s contained i n h2 , h e n c e s t r i c t l y c o m p a c t

T h e r e f o r e f i s bounded on x t O B t & ' B ' and i t s u f f i c e s to

apply t h . ( 2 . 4 . 1). A

completion F

.

In c a s e F is a 1.c. s.

it s u f f i c e s t o c o n s i d e r i t s

94

Holomorphic mappings, basic properties

2.4. 5 Remark.

-

5

In

2 . 5 below we g i v e a n e x a m p l e of a Silva h o l o m o r -

phic mapping defined o n a c o m p l e t e , not S c h w a r t z , b . v . s. not

C

0 3 .

E which is

However f r o m th. ( 2 . 3 . 4 ) and c o r . ( 2 . 4 . 4 ) a n y Silva

In E .

h o l o m o r p h i c mapping i n E is C

2 . 4 . 6 COROLLARY.

-

a3

in E

S.C.

. We m a y a l s o r e m a r k :

L A E be a complex b . v . s . ,

open s e t and F a c o m p l e x 1. c . s . A m a p p i n g f r o m h o l o m o r p h i c in the e n l a r g e d s e n s e in

R

to F -

R a- T E is Silva

i f and only if it is Co3 i n t h e

e n l a r g e d s e n s e in 0 .

With c l a s s i c a l notations i t s u f f i c e s t o a p p l y c o r . ( 2 . 4 . 3 ) w i t h

Proof. E B and F

v'

B e f o r e the proof of t h . ( 2 . 4 . 1 ) w e need t h e following

2.4.7

LEMMA.-

and F -

d

Let E -

R a- T E - o p e n s u b s e t

If

a

h E E the s e t x of 0 f o r a

>

T h e n its d e r i v a t i v e f ' i s a G - a n a l y t i c

t o L ( E , F). -

mapping f r o m

-

,

c o m p l e x p o l a r c o m p l e t e b.v. s . L e t f be a Silva h o l o m o r p h i c

t o F. mapping f r o m 0 -

Proof.

be a c o m p l e x b . v . s .

.

o

b o

we s e t D

+ Dah

a

= { z E 6; with

Iz

! ~ 1. a If x

E n

and

is a (finite d i m e n s i o n a l ) s t r i c t l y c o m p a c t s u b s e t

s m a l l enough.

T h e r e f o r e i f B is a convex balanced

bounded s u b s e t of E t h e r e i s a'>o s u c h t h a t f(x t D h t a'B) i s a € I

bounded s u b s e t of F .

L e t B ' be a convex b a l a n c e d bounded B a n a c h disc,

( 0 . 2 . 10) , in F s u c h t h a t f(x t

Dab t

o'B) c B '

.

C a, differentiability of holomorphic mappings

F r o m t h . ( 2 . 2 . 3 ) one m a y c h o o s e a '

95

s m a l l enough and B ' l a r g e

enough i n o r d e r t h a t f be h o l o m o r p h i c f r o m

u

x t D h t o'( AB) c x t C h t E to F ( i n t h e s e n s e of n o r m e d B B' a o
s p a c e s ) . Applying ( 2 . 2 . 7) f o r m u l a (3) f o r we h a v e , i f

z

From ( 2 . 2 . 7 ) ,

6 Da and h ' E B

formula

(a),

one obtains

B f ( x t z h , h ' ) - & f ( x ,h ' ) =2 (2rti)

=B

:

:

f (xthh t [h')

n>1

( A 1:- a 2

( V + l

We s e t

Since in the above conditions f ( x t 1 h t [h')

E B' we obtain

:

dh dc

.

I

96

Holomotphic mappings, basic properties

Now i f B v a r i e s w e obtain a ' and

B' depending o n B a s i s d o n e

b e f o r e ( x and h r e m a i n fixed). We d e n o t e by H t h e bounded s u b s e t of

L ( E , F) m a d e of the m a p p i n g s u

E L(E, F)

such that for every convex

balanced bounded s u b s e t B of E

Therefore if d

x,h

: h'

1z

19

0 4 ,

ax,h ( h ' )

-r)

and s i n c e f r o m ( 2 . 2 . 7 ) t h e p a p p i n g

i s a n e l e m e n t of

f'(x t zh) - f'(x)

- a x , h E 14. H

L

which p r o v e s the d e s i r e d

G-analyticity result.

(2.4.8)

Proof of th. ( 2 . 4 . 1 ) .

there is

d

-

bounded Bandch d i s c

If h E a B t

B' 2 U'

and if h '

B a n a c h s p a c e F B I , ):

L ( E , F ) , w e obtain :

Using t h e n o t a t i o n s of t h . (2.4.1)

B" in F s u c h t h a t f ( x t u B t a ' B ' ) c B". a-' B ' , f o r m u l a ( 3 ) of ( 2 . 2 . 7) g i v e s ( i n t h e 4

1

f ' ( x t h ) . h ' =- 2 n 1

Assuming

B"

s

IC

f(xthtch')

I=1

(02

c l o s e d in F B I ,, w e o b t a i n f ' ( x t h ) . h ' E B"

be the bounded s u b s e t of L ( E , F ) defined by :

.

dS

*

Now let M

C

differentiability of holomorphic mappings

97

4 M = / u E L ( E , F) s u c h t h a t u ( B ' ) e - B" f o r e v e r y B ' , with a' and B" d e p e n a' d e n t on B ' )

.

Therefore f'(xtgB t that f '

a' B') c 2

is b - l o c a l l y bounded in

h o l o m o r p h i c in

n

n

,

M.

F r o m t h i s i n c l u s i o n i t follows

F r o m l e m m (2.4.7),

. But the above i n c l u s i o n p r o v e s a l s o t h a t

f ' is Silva f'

h a s the

s a m e p r o p e r t y a s t h a t a s s u m e d on f i n t h . (2.4. 1). T h e r e f o r e we c l e a r l y end the proof by induction.

98

Holomoiphic mappings, basic properties

$ 2.5 An e x a m p l e

T h e following h o l o m o r p h i c mapping is u s e d f o r s e v e r a l c o u n t e r examples. L e t K ( R ) denote t h e s p a c e of continuous functions f r o m

R

into

(Z with c o m p a c t s u p p o r t , We equip K ( R ) with i t s n a t u r a l L 5 - s p a c e topology ( 0 . 6 . 4 ) .

B(n, A ) =

A b a s e of bounded s e t s is m a d e of t h e s e t s

{ Y E K ( R ) s u c h t h a t s u p p y c [ - n , t n ) and s u p ( y ( x ) I S x t I R

r a n g e o v e r IN.

if n and

We d e n o t e by (

x

i n c r e a s i n g s e q u e n c e of points i n the r e a l interv,ll

L e t u s define a function

y

~ a s) t r i c~t l y ~

*1 ~

] 0, 1[ .

f r o m K ( R ) into C by :

taJ

L

n = O

which is a c o n v e r g e n t s e r i e s f o r a n y f function

4

f r o m K(JR) into

C by

6 K(R)

.

L e t u s define a

:

( n o t i c e that the above s u m h a s only a finite n u m b e r of non z e r o t e r m s ) . It is i m m e d i a t e to c h e c k t h a t cf and

4 a r e G - a n a l y t i c and b - l o c a l l y

99

An example

bounded in K ( R ) ,

h e n c e Silva h o l o m o r p h i c i n K ( R ) . C l e a r l y

@' , a s a mapping

W e a r e going to p r o v e t h d t

from

K ( R ) into

L(K(IR) ; C ) is not b - l o c a l l y bounded a t t h e o r i g i n of K(IR). F o r s h o r t we s e t B

B(n, 1).

n

e v e r y n E IN , of

If

9' i s b-locally bourded a t o f K(IR) t h e n , f o r

t h e r e is 6 n> o s u c h t h a t

L(K(IR) ; C )

.

B ) is a bounded s u b s e t n n q E N , Ip' ( t n B n ) . B is

(C

This means that, for every

q

bounded in 5 . We a r e going to p r o v e t h a t f o r e v e r y E > o t h e r e is a n index n E IN

s u c h that

@'(EB* ) . B n is not bounded in

C , which will

c o n t r a d i c t the a b o v e a s s u m p t i o n . L e t E > o be given and l e t r E 0 1 If f f E B 1 and g E B r , s u c h that & P

q r.

If

0

-'>

f k E & B s a t i s f i e s fk(x2p) = 1

f (x ) = o i f y 2 2 k t 2 , k q

IN be

dx 2pf 1) = E

f o r a n y p 5 k and

ther. : k

fk) =

t

(2 r

.

q = l

Let g

E

be s u c h t h a t g ( r -1)

Br 0

0

1 and

supp g c [ r o -

73

.

1 , r o - ~ )

100

Holomophic mappings, basic properties

k

q = 1

Therefore t m

and

9'

is not bounded in C

(SB1).Br

.

0

F r o m t h i s e x a m p l e w e m a y conclude

2.5.1

:

A Silva h o l o m o r p h i c m a p p i n g on K ( R ) , valued in C

n e c e s s a r i l y twice d i f f e r e n t i a b l e in K ( R ) ( s i n c e bounded in K ( R ) ) .

2.5.2 not C

@ is Cm 03

a'

is not

is not b - l o c a l l y

This example w a s used in (2.4.5).

i n the e n l a r g e d s e n s e i n K ( R ) ( f r o m 2 . 4 . 6 )

( i n S i l v a ' s u s u a l s e n s e ) in K(IR).

and is

This example was used in

(1.4.6).

2.5.3

L e t u s equip K (IR))W

= L(K(R), a

)

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 of

w i t h the topology of

K(R).

T h e n 4 ' is

Silva h o l o m o r p h i c in t h e e n l a r g e d s e n s e f r o m K ( R ) i n t o ( K ( R ) ) * ( f r o m (2. 4 . 6 ) ) and not S ilva h o l o m o r p h i c . T h i s e x a m p l e w a s u s e d in (2. 3 . 6 ) .

Series of homogeneouspolynomials

9

2.6

101

S e r i e s of hoinogeneous p o l y n o m i a l s

The s p a c e s c o n s i d e r e d in t h i s s e c t i o n m a y be r e a l o r c o m p l e x .

L e t E be a

2.6. 1 Pointwlse c o n v e r g e n c e of a f o r m a l s e r i e s . n o r m e d s p a c e and

F a 1 . c . s . F o r e v e r y n E IN l e t be given SaJ

fnE P ( " E , F ) .

We s a y t h a t the f o r m a l s e r i e s

E

r

fn converges

n = o

pointwise in a s e t A c E i f f o r e v e r y x E A

the s e r i e s

fn(X)

n = o

c o n v e r g e s in the 1 . c . s .

2.6.2

F.

Let

N o r m a l c o n v e r g e n c e of a f o r m a l s e r i e s . -

(fn)nEIN

E.F and

be a s in ( 2 . 6 . 1 ) . We s a y that the f o r m a l s e r i e s n = o

c o n v e r g e s n o r m a l l y a t s o m e point a E E if i t c o n v e r g e s pointwise i n s o m e a-neighborhood and if f o r e v e r y convex baldnced o-neighborhood V

in F t h e r e e x i s t s a a-neighborhood

z

W

in E

such t h a t

tar,

n = o

2. 6. 3

Remark.

-

(SUP X E W

Pv

(fn(X)))

<

+

.

If E i s a c o m p l e x n o r m e d s p a c e a n d F a c o m p l e x

1. c. s. it follows f r o m Cauchy i n t e g r a l f o r m u l a ( 2 . 2. 7 ) t h a t t h e T a y l o r

s e r i e s of a n y Silva h o l o m o r p h i c m a p p i n g i n t h e e n l a r g e d s e n s e is n a r m a l -

l y c o n v e r g e n t at a n y point of its domain. The c o n v e r s e i s i m m e d i a t e f r o m (2. 1. 3).

Holomorphic mappings, basic properties

102

2.6.4

THEOREM. -

-c)

L e t E be a Ba'nach s p a c e ( o v e r 1IC =

-

IR

and l e t F be a 1 . c . s . F o r e v e r y n E IN l e t f n E P ( n E , F) be

or

-

>z

-

to3

given. If t h e s e r i e s

f n c o n v e r g e s pointwise in a n a b s o r b i n g

n - o

then it c o n v e r g e s n o r m a l l y a t the o r i g i n of E .

s u b s e t of E

If t h e

series c o n v e r g e s i n a o-neighborhood of E then i t s s u m is continuous a t the o r i g i n of E .

Proof, -

Choose a fixed continuous s e m i - n o r m

absorbing s u b s e t S of E

such that for every x

on F.

y

There is an

E S t h e r e is a n

M(x) 7 o with q(f (x)) 5 M(x) f o r e v e r y n E IN. If E > o is given, n n n 1 q(fn(Ex))5 & M(x) and E M(x) ;f o r e v e r y n i f E > o is small 2 enough (dependent on x ) .

.

So q(fn(x)) 5 2 - n f o r a l l n

x in a n a b s o r b i n g s u b s e t A of E . t h e s e inequalities and, since E

r e m a i n still valid i n t h e c l o s u r e

is a B a i r e s p d c e ,

there exists a n element a in E

The mappings f

A

being continuous, Now

E=Un A n

h a s a n i n t e r i o r point. T h e r e f o r e

i f n E IN and x

E

we get :

n

s = o

-n

A,

and f o r a l l

E E and a convex baldnced o-neighborhood W

s u c h t h a t q(f (x)) 5 2-n n

q [fn(a f s x ) ) 5 2

n

E IN

when x

E 1W. n

Therefore

a t W.

F r o m (0.8.6)

Series of homogeneouspolynomials

103

n if x

1 n

E -W

, where c

n

n!

= -

n!

s = o

F i n a l l y , if x 6 W ,

n

n ;;T 5

(since

en)

.

Therefore

too This means that the s e r i e s of E . Now

if E

>o

converges normally at the origin f t n n = o

is fixed,

q[f,(x)]

5

E

if

k E N is large

n >k enough and i f x Ef(x) =

1

fn(X)

1

2e

W.

T o conclude it s u f f i c e s to o b s e r v e t h a t , i f

7

n

2.6.5

Comments. -

T h . ( 2 . 6 . 4 ) shows t h a t a r e s u l t o f

pointwise c o n v e r g e n c e is enough to i m p l y a u n i f o r m c o n v e r g e n c e . In t h e

104

Holomorphic mappings, basic properties

complex case this gives

d

convenient c h a r a c t e r i z a t i o n of Silva holomorphic

mappings i n t h e e n l a r g e d s e n s e , defined on T E - o p e n s u b s e t s of a complex complete b.v. s.

E.

In t h e r e a l c a s e ( 2 . 6 . 4 ) !

is u s e f u l t o define c o n c e p t s

E n too

of red1 analytic m a p p i n p s , a s s u m s of a s e r i e s

c o n v e r g e s pointwise in a

f (x)which

n = o T E - o p e n s u b s e t of a r e a l c o m p l e t e b . v . s .

( s e e Colombeau [ 8 ] f o r m o r e d e t a i l s ) .

E

Mappings of uniform bounded type

9

105

2 . 7 Holomorphic m a p p i n g s of u n i f o r m bounded t y p e .

2 . 7 . 1 Holomorphic mappings of u n i f o r m type. -

If E and F are -

-

coinplex 1 . c . s . a h o l o m o r p h i c m a p p i n g f : E

F i s s a i d t o be of

u n i f o r m type i f t h e r e is a convex balanced o-neighborhood V that f m a y be f a c t o r i z e d in t h e following v

L

V

N

2.7.2

such

s

with f a h o l o m o r p h i c mapping f r o m t h e n o r m e d s p a c e 1.c. s .

E

EV into the

F.

E x a m p l e showing t h a t , i n g e n e r a l , h o l o m o r p h i c m a p p i n g s a r e not

of u n i f o r m type. -

L e t E be the F r e ' c h e t s p a c e

X ( C )of

the e n t i r e

f u n c t i o n s of one c o m p l e x v a r i a b l e equipped with i t s u s u a l topology of u n i f o r m c o n v e r g e n c e nn t h e c o m p a c t s u b s e t s of mapping f r o m x(C) into

clearly 0 €K(E) , r

>

o s u c h t h a t , if

C. L e t & be t h e

C, d e f i n e d , if fc E K ( C ) , by

If 0 w a s of u n i f o r m t y p e t h e r e would e x i s t s o m e cp E E , and when n

-

t

00

,

106

Holornoiphic mappings, basic properties

v

Choose

to b e the c o n s t a n t function 3 r and

n

(2)

= 3r t (

z n -) 2 T

.

So we get a contradiction which shows t h a t 4 is not of u n i f o r m type

2. 7.3

Holomorphic m a p p i n g s of u n i f o r m bounded type. - I n ( 2 . 7 . 1) if F

is a n o r m e d s p a c e , a h o l o m o r p h i c mapping

f :E

-.

F is s d i d t o be of

u n i f o r m bounded type if it i s of u n i f o r m type ( a c c o r d i n g t o 2. 7.1)

with

N

f bounded on any bounded s u b s e t of t h e n o r m e d s p a c e E V . Z f E n o r m e d spdce we s a y m o r e s i m p l y t h a t f

Example.

-

T h e function

of

i s of bounded type

is a

.

( 3 . 2 . 4 ) is h o l o m o r p h i c on a Bandch

s p a c e ( s o it is of u n i f o r m t y p e ) and is not of u n i f o r m bounded type.

A n i m p o r t a n t r e s u l t u h i c h s h o u s the r e l e v a n c e of t h i s c o n c e p t is :

2.7.4

THEOREM.

Let E

-

-

be a c o m p l e x Silva s p a c e , F a

c o m p l e x m e t r i z a b l e 1. c . s . , f a Silva h o l o m o r p h i c mapping f r o m E into F.

T h e n f is of u n i f o r m bounded type

e x i s t a convex baldnced bounded s u b s e t B

&

following d i a g r a m :

o-neighborhood

V

i n the s e n s e t h a t t h e r e

&

E and a convex balanced

F s u c h t h a t f m a y be f a c t o r i z e d a c c o r d i n g t o t h e

107

Mappings o f uniform bounded type N

with f -

EV

a holomorphic mapping of bounded type b e t w e e n the n o r m e d spaces

-

and F

B'

L e t ( W ) be a b a s e of convex balanced o-neighborhoods in F n and l e t s : F F denote t h e c a n o n i c a l s u r j e c t i v e m a p p i n g s . We W wn n Proof. -

+

set f

n

= s w

n

o f ,

a s a mapping f r o m E

into F '

wn

.

It follows f r o m

t h a t t h e r e i s a convex balanced o - n e i g h b o r n s u c h that f o r e v e r y q F IN t h e r e is r EN wjith

l e m m a ( 1 . 6 . 4 ) applied to f hood V n in E

Since E

is a Silva s p a c e i t is the s t r o n g d u a l of a F r k c h e t s p d c e (0.6.8)

and i t follows by p o l a r i t y f r o m p r o p 2 of ( 0 . 6 . 2 ) t h a t , f o r e v e r y s e q u e n c e (Vn)

o-neighborhoods in E t h e r e is a s e q u e n c e p > o s u c h t h a t n

of

v

=

i s s t i l l a o-neighborhood i n E . r

n n

p n ~ n

Therefore for every n , q

E IN t h e r e is

E IN with

Since ( W n ) i s a b a s e of o-neighborhoods i n F , every q

E IN,

this proves that f o r

f ( q V ) is a bounded s u b s e t of F.

Now f r o m p r o p . 2 of ( 0 . 6 . 2 ) t h e r e is a s e q u e n c e

E

9o

such that

108

u

Holomoph ic mappings, basic properties

&

f(qV) is bounded i n F.

T h e r e f o r e we obtain f i n a l l y t h a t t h e r e i s a

9

convex balanced bounded s u b s e t B of F s u c h t h a t f o r e v e r y n E IN

IN with

there is p

f(n V ) c p B

.

) , i.e.

1 -

V far e v e r y n E IN, n f i s bounded on the c o m p l e x affine one d i m e n s i o n a l s u b s p a c e Now if

s (x ) = s

v

1

v

(x

2

i f x -x

1

2

E

.

+

C ( x -x ) It follows f r o m L i o u v i l l e ' s t h e o r e m applied t o y ' o f , 1 2 1 for e v e r y y ' 6 F' , t h a t f(x ) = f(x ). T h u s we obtain t h e a b o v e 1 2 factorization. I

x

2.7.5 Remark.

-

L e t i? d e n o t e t h e s p a c e & R n ) of I1

with c o m p a c t s u p p o r t in IR ,

and r a d i u s k. and i t s d u a l

Then

8'

[

2

3

functions

, and l e t Bk denote t h e

in B whose s u p p o r t is in t h e b a l l of c e n t e r

o

B is t h e inductive l i m i t of t h e s p d c e s ( B )

( t h e s p a c e of a l l d i s t r i b u t i o n s o v e r We denote by

l i m i t of the s p a c e s

,&Ik.

mapping f r o m B '

onto

k

og

a c c o r d i n g to S c h w a r t z ' s notations of

Distribution Theory, s e e Schwartz s p a c e of those functions

C

T T ~the

k kEIN Rn) is t h e projective

s u r j e c t i v e uanonical

B ' . It is proved i n Boland-Dineen [1,23 t h a t

a n y Silva holomorphic function on B ' f a c t o r s a s a h o l o m o r p h i c function on s o m e B'

k'

Since

8' is a n u c l e a r Silva s p a c e , it follows t h e n f r o m k

T h . 2 . 7 . 4 that a n y Silva h o l o m o r p h i c function on B' bounded type.

i s of u n i f o r m

109

Fock spaces of Boson Fields

6 2 . 8 - H o l o m o r p h i c r e p r e s e n t a t i o n s of F o c k s p a c e s of Boson F i e l d s .

Q u a n t u m F i e l d s a r e d e s c r i b e d by " s t a t e s p a c e s " which a r e H i l b e r t s p a c e s a n d a r e c a l l e d " F o c k s p a c e s " , a n d by l i n e a r o p e r a t o r s a c t i n g o n t h e m . In o r d i n a r y Q u a n t u m M e c h a n i c s t h e s t a t e s p a c e s a r e r e a l i z e d by function s p a c e s of f i n i t e l y m a n y r e a l v a r i a b l e s ; in t h e t h e o r y of Q u a n t u m F i e l d s the s t a t e s p a c e s a r e d i r e c t s u m s of f u n c t i o n s p a c e s of o n e , two, etc

. .. variables

a n d t h e r e e x i s t no r e a l i z a t i o n s of t h e m a s H i l b e r t s p a c e s

of f u n c t i o n s with a f i x e d n u m b e r of v a r i a b l e s ; t h e r e a r e h o w e v e r n a t u r a l r e a l i z a t i o n s of t h e s t a t e s p a c e s a s s p a c e s of h o l o m o r p h i c f u n c t i o n s ( " h o l o m o r p h i c f u n c t i o n a l s " in the P h y s i c a l l i t t e r a t u r e ) defined o n u s u a l f u n c t i o n s p a c e s , i . e . h o l o m o r p h i c f u n c t i o n s of i n f i n i t e l y m a n y v a r i a b l e s . In o r d e r t o avoid all the t e c h n i c a l i t i e s we r e s t r i c t o u r s e l v e s t o a s i m p l i f i e d c a s e of B o s o n F i e l d but all the s e q u e l m a y be v e r y e a s i l y t r a n s f e r e d t o a n y Boson F i e l d , while the s i t u a t i o n is d i f f e r e n t f o r F e r m i o n F i e l d s .

2.8.1 - T k F o c k space =simplified

Boson f i e l d . L e t

2 3n L @ )

denote

t h e u s u a l H i l b e r t s p a c e of the s q u a r e i n t e g r a b l e c o m p l e x v a l u e d f u n c t i o n s

R3"

(with r e s p e c t t o L e b e s g u e m e a s u r e t o s i m p l i f y , although t h i s 2 3 n i s not i m p o r t a n t ) . L e t L [ @ ) ] denote the c l o s e d ( H i l b e r t ) s u b s p a c e S 2 3n of L @ ) m a d e of t h e s q u a r e i n t e g r a b l e f u n c t i o n s o n @33)n= R3n

on

which a r e ( a l m o s t e v e r y w h e r e ) s y m m e t r i c f u n c t i o n s of t h e i r n a r g u m e n t s 3 in IR , i . e . f(xu , ., x U ) = f(xl, , , x n ) f o r a n y p e r m u t a t i o n u 1 n of t h e s e t [ l , . , n 1.

..

. .

..

The F o c k space

i.e.

a n element

K

of

IF is the H i l b e r t i a n d i r e c t s u m :

IF m a y be r e p r e s e n t e d by t h e c o l u m n

110

Holomophic mappings, basic properties

K =

where Kn (xl,

K € C and

. . ., x n )

in

Kn(x1,.

. . ,xn)

s t a n d s f o r the f u n c t i o n

XI,

L t [ Q3)"] a n d w h e r e

IlKIlZ = KO/'+

too

c [ I Kn(xl, . . . , xn) I2

n=l

dxl..

.dx

n

..., x n -+

< too

Of a f u n d a m e n t a l i m p o r t t a n c e a r e the c r e a t i o n a n d annihilation o p e r a t o r s a ( e p ) a n d a-(cp) r e s 2 3 pectively defined below f o r any rpE L @X ) :

2.8.2 -

C r e a t i o n and annihilation o p e r a t o r s .

I where

\

:

S y m . i s the o p e r a t o r of s y m m e t r i z a t i o n of a f u n c t i o n .

I \

111

Fock spaces of Boson Fields

T h e d o m a i n s of t h e s e o p e r a t o r s c o n t a i n obviously t h e s t a t e s

K = 0 for n a n d Kp# 0 /1\

K

K

s u c h that

K = 0 Vnf s o m e p n i s c a l l e d "a s t a t e with p p a r t i c l e s e x a c t l y " . T h e s t a t e

n

l a r g e enough. A s t a t e

with

is c a l l e d the " v a c u u m s t a t e " ( 0 p a r t i c l e ) . H e n c e we s e e f r o m (1) t h a t

(0) af(cp) t r a n s f o r m s a s t a t e with p p a r t i c l e s e x a c t l y i n t o a s t a t e w i t h

ptl

p a r t i c l e s e x a c t l y , t h u s i t s n a m e of " c r e a t i o n o p e r a t o r " . A similar r e m a r k i s in o r d e r f o r the annihilation o p e r a t o r

a-(cp) which d i m i n i s h e s by

1

the n u m b e r of p a r t i c l e s .

2 . 8 . 2 - R e p r e s e n t a t i o n of the F o c k s p a c e

?!I

by Infinite D i m e n s i o n a l

Holomorphic functions. T h e Mathematical P h y s i c i s t s had t h e idea t o i n t r o d u c e Infinite D i m e n s i o n a l H o l o m o r p h y i n t h i s s e t t i n g : i f

l e t u s c o n s i d e r t h e function

where

2 crc L

3

0

2 3 L @ )

defined o n

). It i s e a s y t o c h e c k t h a t

#

by the f o r m u l a :

is d e f i n e d o n

2

3

L @X ) :

hence the s e c o n d m e m b e r of ( 3 ) is i n a b s o l u t e v a l u e l e s s t h a n :

and it i s immediate that 2 3 bert space L @ ). I

#

is a n e n t i r e h o l o m o r p h i c f u n c t i o n on t h e H i l -

112

Holomorphic mappimgs, basic properties

Via (3) t h e F o c k s p a c e the h o l o m o r p h i c f u n c t i o n s on of

IF is c o n t a i n e d in t h e s p a c e K ( L2 (IR3 ) )

of 2 3 L @? ) : i t i s t h e s o c a l l e d " R e p r e s e n t a t i o n

IF a s a s p a c e of infinite d i m e n s i o n a l h o l o m o r p h i c f u n c t i o n s " . In t h i s

r e p r e s e n t a t i o n the r e a d e r m a y c h e c k e a s i l y that the c r e a t i o n and annihilat i o n o p e r a t o r s a r e g i v e n by t h e following f o r m u l a s :

where

Id

d e n o t e s t h e identity o p e r a t o r o n

IF

.

It is i n g e n e r a l v e r y

convenient to c o m p u t e in t h i s H o l o m o r p h i c r e p r e s e n t a t i o n a n d , a s i l l u s t r a t e d above c o m p a r i n g (1) (2)with (1') ( 2 ' ) m a n y o p e r a t o r s o n

IF h a v e

a n i c e r a s p e c t . F o r r e f e r e n c e s a n d f u r t h e r d e v e l o p m e n t s s e e the B i b l i o graphic Notes.

CHAPTER 3 CLASSICAL PROPERTIES OF HOLOMORPHIC MAPPINGS

ABSTRACT.

-

A s a continuation of chap. 2 w e p r e s e n t c l a s s i c a l p r o p e r t i e s of holom o r p h i c mappings. A l l the s p a c e s c o n s i d e r e d h e r e a r e complex. THEOREM.

-

Let E be a 1. c. s . ,

Q a n oven s u b s e t of E a n d F a

q u a s i c o m p l e t e 1. c. s . . A m a p p i n g f : 0

-t

F i s Silva h o l o m o r p h i c i n t h e

e n l a r g e d s e n s e i n fl i f and only i f f o r e v e r y y'g

n

+

(T

F' the m a p p i n g y ' o

f :

i s Silva h o l o m o r p h i c i n 0.

L e t E be a q u a s i c o m p l e t e 1. c. s. a n d l e t fl be a

Zorn's Theorem. -

connected open s u b s e t of E. L e t F be a 1. c. s . a n d l e t f be a G - a n a l y t i c mapping from

n

into F

.

If f i s b - l o c a l l y bounded a t s o m e point x

0 -

0

(i. e. for e v e r y boundet s e t B i n E theLe i s & > O s u c h t h a t f ( x t E B ) i s __- 0 bounded in F ) then f i s Silva h o l o m o r p h i c i n t h e e n l a r g e d s e n s e i n Q

.

.

-

Let E and E be two q u a s i c o m p l e t e 1. c. s. 12 respectively and let L e t n a n d R 2 be two open s u b s e t s of E a n d E .~ 1 1-2 n2 F be a s e p a r a t e l y F be a q u a s i c o m p l e t e 1. c. s . . L e t f : fll Silva h o l o m o r p h i c m a p p i n g i n t h e e n l a r g e d s e n s e . Hartoqs' theorem,

-

T h e n f i s Silva h o l o m o r p h i c i n t h e e n l a r g e d s e n s e i n

n l XO 2 '

W e p r o v e Z o r n ' s and H a r t o g s ' t h e o r e m s in t h e c a s e of n o r m e d s p a c e s ar.d we shov. how the above Z o r n ' s and H a r t o g s ' t h e o r e m s

(ill

the

c a s e of 1 . c . s ) a r e i m m e d i a t e c o n s e q u e n c e s of the n o r m e d s p a c e s c a s e results. 113

Classical properties

114

If E i s a q u a s i c o m p l e t e 1. c. s., fl a n open s u b s e t of E and F a 1. c. s.,

X

then

S, e

( a , F)

d e n o t e s the s p a c e of Silva holomorphic

mappings i n the e n l a r g e d s e n s e f r o m 62 t o F

,

equipped with t h e topology

of uniform convergence on the s t r i c t l y c o m p a c t s u b s e t s of Q .

Montel's t h e o r e m .

-

L e t E be a q u a s i c o m p l e t e 1. c. s . ,

stlbset of E and F a 1. c. s,

only i f F is.

.

Then X

S, e

(n, F) i s

n

a n open

semi-Monte1 i f and

A s usual, the t h e o r e m s obtained i n the t e x t a r e often m o r e g e n e r a l than the r e s u l t s s t a t e d i n A b s t r a c t counterexamples

.

.

W e a l s o e x p o s e several

115

Vector valued hoiomorphy

V e c t o r valued h o l o m o r p h y v e r s u s s c a l a r h o l o m o r p h y

$ . 3 . 1.

3. 1. 1. LEMMA.

-

Let E be a complex linear space and 0 a

.

finitely open s u b s e t of E Neumann bornology

L e t F be a 1. c . s.

. A mapping

with a c o m p l e t e Von

f : fi -, F i s G - a n a l y t i c i f a n d only i f

y'o f i s G - a n a l y t i c f o r e v e r y y' EF'. proof.

-

.

We m a y a s s u m e E = C

If x E

n,

a>

for a > 0 small enough,

.

s u c h t h a t 0 < r < r ' < a.

r ' > 0 be

If

t-f(xt5)definedfor

0 we

set

x t D

a

c 0

.

Let r a n d

Let u s consider the mapping

$ E D . I f h E C , U

151Srand l l \ < r l - r

a n d f r o m C a u c h y ' s i n t e g r a l f o r m u l a (see 2. 2. 7 for i n s t a n c e )

I y'o f ( x t $ t p ) I M(y', r')

,

s sup

1 zl
I y'of

I

(x t z ) < t m

.

Denoting t h i s s u p by

we have :

1 an(Y'*5 ) l Therefore i f inequalities :

11

I

G

r'-r 2

M(y', r') n (rl-r)

-,

f r o m (1) w e obtain t h e following

116

Classical properties

The set

i s a bounded s u b s e t of t h e 1. c. s. F ( B i s bounded f o r t h e weak topology CJ

( F , F ' ) ; i t s u f f i c e s to apply M a c k e y - A r e n s ' th. 0. 3. 10 a n d M a c k e y ' s

th. 0.3. 8 ) . B i s e q u a l t o i t s b i p o l a r ( f r o m the b i p o l a r t h e o r e m 0. 3 . 6 )

.

We have :

if

rl-r

1 x 1~2

and

I5 I

is continuous f r o m {

1: r

.

Therefore the mapping

5

4

f (x t

5)

1 5 I s r ) e CC to t h e n o r m e d s p a c e FB' BY

a s s u m p t i o n on F , B i s contained i n a bounded B a n a c h d i s c B1(O. 2. 10) Now i n t h e B a n a c h s p a c e F B I t h e i n t e g r a l

exists if

Ihl < 1 and

F r o m f o r m u l a s (1) a n d (2) above it s u f f i c e s now t o apply (2. 1. 3. ) . We p r o v e now a s i m i l a r r e s u l t f o r Silva h o l o m o r p h i c m a p p i c g s in t h e e n l a r g e d s e n s e

:

.

Vector valued holomorphy

3. 1. 2.

THEOREM.

-

E be a c o m p l e x

117

b. v. s . ,

n

a TE-open

set a n d F

2 1. c. s.

pilag f :

-, F i s Silva h o l o m o r p h i c i n t h e e n l a r g e d s e n s e i n 61 i f a n d

only i f f o r e v e r y y'

with a c o m p l e t e Von Neumann bornology

E F' the mapping y' o

f :

0

is

+

~

A map-

Silva h o l o m o r p h i c

.

in n proof.

-

L e t us a s s u m e y ' o f i s S i l v i h o l o m o r p h i c i n fl f o r e v e r y

y ' e F ' . F r o m (3. 1. 1 ) f i s G-analytic. L e t B be a convex :,alanced boun-

E a n d V a convex b a l a n c e d c l o s e d o-neighborhood i n F

ded s u b s e t of If x E

fl l e t u s a s s u m e (by a b s u r d ) that f ( x

n V f o r a n y n EIN a n d any s u c h t h a t f(x t h n )

y t ( f (x th,))

{

Y'(f(X

4

a> 0

.

i s not contained i n

Thereforethere i s a sequence

n V for every n c W .

By hypothesis,

y ' ( f (x)) i n C f o r e v e r y y ' E F '

.

Therefore the

hnE

1

;B

set

+ h n ) H n E r n i s bounded i n (T, f o r e v e r y y ' c F ' , i. e . t h e s e t

E f ( x + h n ) l nEw

i s bounded i n

(5

(F, F ' ) h e n c e bounded i n F ( f r o m M a c k e y

A r e n s a n d M a c k e y ' s th. 0. 3. 10 a n d 0. 3. 8 )

i fi

-i a B)

.

+hnH

.

is c o n t a i n e d i n p V for s o m e

diction. T h e r e f o r e t h e r e is a n

Therefore this

u>

0 . We get

set a contra-

a > 0 and a n n €IN with f ( x t n B ) c n V .

It i s now c l a s s i c a l , i r q m f o r m u l a s ( 2 . 2. 7 ) , t h a t we obtain t h e continuity of f

/nnEB

3. 1. 3

valued in t h e n o r m e d s p a c e FV'

Remark.

-

rn

L e t f be a Silva h o l o m o r p h i c m a p p i n g i n t h e e n l a r -

ged s e n s e w h i c h i s not Silva h o l o m o r p h i c ( s e e 2. 5 . 3)

.

Then yt o f i s

Silva h o l o m o r p h i c f o r e v e r y y ' E F' and n e v e r t h e l e s s f i s not S i l va hol om o r p hi c

3. 1. 4

.

I t follows i m m e d i a t e l y f r o m (3. 1. 2 ) t h a t , i f E is a

Remark. -

c o m p l e x b. v. s.,

n

a TE-open set and F a 1, c. s. w i t h a c o m p l e t e

Von Neumann b r m o l o g y , a n d i f

f :0

F i s a m a p p i n g , t h e n f is

Silva h o l o m o r p h i c i n t h e e n l a r g e d s e n s e i n convex b a l a n c e d bounded

n

i f and only i f f o r e v e r y

s u b s e t B of E a n d e v e r y convex b a l a n c e d

o-neighborhood V i n F the m a p p i n g

s o f/ V nY,EB

is h o l o m o r p h i c .

Classical properties

118

0 THEOREM.

3. 2. 1

connected

-

L e t E be a c o m p l e t e c o m p l e x b. v. s., -

.

g e d sense in

b. v. s.

Q 2

L e t f be a G - a n a l y t i c

If f i s continuous a t a point x E 0 f o r t h e

topology induced on 0 b~

-

.

.rE-open s e t a n d F a 1. c. s.

m a p p i n g f r o m fl to F

proof.

3.2. Z o r n ' s t h e o r e m

0

7

E then f i s Silva h o l o m o r p h i c i n t h e e n l a r -

R.

T h i s proof w i l l show how r e s u l t s i n t h e s e t t i n g of 1. c. s. o r m a y s o m e t i m e s be obtained a s a n i m m e d i a t e c o n s e q u e n c e of

r e s u l t s of t h e

s e t t i n g of n o r m e d s p a c e s

.

It is b a s e d upon Z o r n ' s

t h e o r e m in n o r m e d s p a c e s , p r o v e d in (3. 2. 6 ) below : c o m p l e x B a n a c h s p a c e and

n

"

let E be a

a connected open s u b s e t of

E ; let F

be a n o r m e d s p a c e and let f be a G - a n a l y t i c m a p p i n g f r o m 0 into F

.

If f i s continuous at a point x E 62 t h e n f i s continuous, t h u s h o l o m o r phic, i n 0"

0

.

Now i f x. E Q

one p r o v e s e a s i l y t h a t t h e r e i s a continuous s t r i c t l y a n d x,with L c R

c o m p a c t c u r v e L. of e x t r e m i t i e s x

( s i n c e 0 is

connected). L e t B be a bounded B a n a c h d i s c i n E with L c E and B For s o m e E > O , L t E B c n. Let V be a convex compact in E B' T h e n i t s u f f i c e s t o apply Z o r n ' s b a l a n c e d o-neighborhood i n F

.

t h e o r e m in n o r m e d s p a c e s t o t h e m a p p i n g s L t

E

( U h B) -. FV w h e r e o<x 4 1

L t

E

( U h B) o<x<1

Vof/L+E(UhB)' o<x<1

..

is a c o n n e c t e d open s e t i n

t h e B a n a c h s p a c e EB. When V r a n g e s o v e r a b a s e of o - n e i g h b o r h o o d s i n F, t h i s p r o v e s t h e continuity of f / L +

(u B ) O<X
3.2.2

Remark.

-

In c a s e E i s a F r C c h e t o r a Silva s p a c e , s e e

(0. 6. 2) and (0. 6. 7 ) , we r e c a l l that

of E tion

.

, thus in

7

E c o i n c i d e s with t h e given topology

t h i s p a r t i c u l a r c a s e , th (3. 2. 1) a d m i t s a s i m p l e r f o r m u l a -

119

ZomS theorem

3.2.3

THEOREM. -

c o n n e c t e d T E - o p e n s e t and F a 1. c. s.

. If

m a p p i n g f r o m fl to F then f -

n 5

L A E be a c o m p l e x c o m p l e t e b. v. s . ,

.

Let -

f be a G-analytic

f i s b - l o c a l l y bounded a t a point x

0

En

is Silva h o l o m o r p h i c i n t h e e n l a r g e d s e n s e i n Q .

In particular

i f E i s a S c h w a r t z b. v. s. o r i f F h a s

a

countable b a s e of bounded

s e t s t h e n f is Silva h o l o m o r p h i c i n

h e n c e b - l o c a l l y bounded i n

n.

n,

F i r s t l e t us q u o t e t h e following f o r m u l a t i o n of Z o r n ' s

proof. -

theorem in normed spaces :

"

let E be a

complex Banach space,

a

connected open s u b s e t of E a n d F a n o r m e d s p a c e . Let f be a

n

G-analytic m a p p i n g f r o m a point x

0

E 0 then

f is

t o F. If f i s bounded o n a neighborhood of

holomorphic in

(2. 2.7) f i s continuous at x u s e d i n proof of ( 3 . 2. 1))

.

0

( F r o m Cauchy's formulas

,If

a n d it s u f f i c e s t o apply t h e r e s u l t

Using t h e above f o r m u l a t i o n of Z o r n ' s t h e o r e m i n n o r m e d s p a c e s , t h e p r o o f of ( 3 . 2 . 3 ) i s s i m i l a r t o t h a t of ( 3 . 2. 1) , T h e last a s s e r t i o n c o m e s f r o m (2. 3. 3 ) a n d ( 2 . 3. 7)

3.2.4

Example. -

L e t E b e t h e B a n a c h s p a c e of the continuous

functions f r o m [ 0 , I ] c R L e t (x ) n ntlN L e t cp : E

-

.

into C equipped with its u s u a l s u p - n o r m .

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 of p o i n t s i n [O, 1 1 . be t h e function defined by :

We s e t I(f) =

1

f(x) d x ,

0

Let

@

be t h e m a p p i n g f r o m E i n t o t h e F r C c h e t s p a c e CIN d e f i n e d

by :

4 ( f ) = ( rp (I(f). f ) ,

n

. . . , q? (I(f 1 f ) , . . . 1

n €IN

120

Classical properties

where f

n

i s t h e n - t i m e s p r o d u c t of f, i f f c E

S i n c e t h e m a p p i n g epk : E

,

C defined by

i s G-analytic, t h e m a p p i n g @ i s G - a n a l y t i c i n E

. If

ther.

Therefore

i s bounded on a o - n e i g h b o r h o o d

6

.

f = 2 1 (i. e. f(x) = 2 i f o 6 x $; 1 ), To p r o v e t h i s , l e t o < E 4 be given 2 and let be a continuous function with d o m a i n [ O , 1 3 c IR

@

i s not bounded on a n y neighborhood

of E

of t h e point

-

Jrs,E

and range

[Z-E,

2 t E ] c I R

such that :

W e have :

Therefore i f

E

k

k -1 = (2. (1, 5 ) ) , c p k is n o t bounded on t h e

set

{ f E E such that b e c a u s e i t is not bounded o n t h e if k - t m ,

@

11 f - 2 11 5

set

i s not bounded on a neighborhood

E

k) Since of f = 2

.

E

k4

0

121

Zorn’s theorem 3. 2. 5

Remark.

-

E x a m p l e (3. 2. 4 ) s h o w s t h a t the function f i n

th (3. 2. 3 ) i s not i n g e n e r a l b - l o c a l l y bounded i n 0 (i. e. Silva holo-

n)

morphic in

e v e n i f E i s a B a n a c h s p a c e . The m a p p i n g Q

of

(3. 2. 4 ) i s a n e x a m p l e of a Silva h o l o m o r p h i c m a p p i n g i n t h e e n l a r ged s e n s e in E , e v e n of a holomorphic m a p p i n g i n E B a n a c h s p a c e , which i s not Silva h o l o m o r p h i c i n E

P r o o f of Z o r n ‘ s t h e o r e m i n n o r m e d s p a c e s

p r o o f of (3. 2 . 1) t h e point x

0’

.

We

U = { a C

,

although it is

.

i n a o-neighborhood of E

3. 2 . 6

since E is a

,

a s it i s s t a t e d in

n

f i s bounded on a n open neighborhood W c

of

set :

s u c h t h a t f i s bounded on a neighborhood of a

and w e a r e going to p r o v e t h a t U = fl

.

If x

E U l e t W be

in E such that x t W c

b a l a n c e d open o-neighborhood

n.

3

,

a convex

Then

f r o m (2. 2. 7 ) , i f h E W

a s the developrraent of t h e h o l o m o r p h i c function z .-+f (xtzh). But

since x

EU ,

f i s bounded on a neighborhood of

f

(X

th) =

with u n i f o r m c o n v e r g e n c e if h where

b n f (x,.) E P ( k , F)

fn E P ( n E , F)

.

L e t now z

.

n= o

E

E

n!

.F r o m

(2. 2. 7)

b n f (x,h)

W, for

E

> 0 s m a l l enough, and n F o r SCE a n d

T h u s b n f (x, . ) = n ! f

be given i n x t W

s m a l l e n o u g h , it follows f r o m (1) t h a t

x

.

and

/I 511

122

Classicalproperties

with u n i f o r m c o n v e r g e n c e f o r

E,

of E

,

s i n c e f is G-analytic.

i n a finite d i m e n s i o n a l c o m p a c t s u b s e t F u r t h e r m o r e , and a l s o s i n c e f i s

G-analytic,

with u n i f o r m c o n v e r g e n c e z-

for

5

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

some

neighborhood and a n y finlte d i m e n s i o n a l c o m p a c t s u b s e t of

where

f(k)

(2)

i s a k - h o m o g e n e o u s p o l y n o m i a l f r o m E into F , a

p r i o r i not n e c e s s a r i l y continuous convergence

E and

.

F r o m (2) and ( 3 ) , and for the s a m e

a s in (3),

(4) n-k k N o w , s i n c e f EP ( n E , F ) , the m a p p i n g i s i n L ( E , F) (2-x) n n Since E i s a B a n a c h s p a c e , h e n c e a B a i r e s p a c e a n d s i n c e F i s a

1

n o r m e d s p a c e , it follows f r o m (4) t h a t s i n c e it i s a pointwise

f ( k ) ( z ) i s a continuous p o l y n o m i a l

l i m i t of continuous h o m o g e n e o u s p o l y n o m i a l s of

d e g r e e k ( s e e Kathe [ I ]

$ 15. 13 f o r t h e B a n a c h - S t e i n h a u s s t h e o r e m i n

c a s e k = 1 , a n d notice t h a t a p o l y n o m i a l between n o r m e d s p a c e i s c o n t i nuous i f i t i s continuous a t s o m e point, s e e D i n e e n 1 1 1 o r Hille instance)

Now i t follows f r o m (3) a n d th. 2. 6. 4 t h a t f i s c o n t i n u o u s

a t t h e point z

. Therefore

x t W c U

We j u s t p r o v e d t h a t i f x open o-neighborhood i n E the p r o p e r t y c e i s that

L 11 f o r

I'

x t W c U

LJ = 0

. I

,

IT.

EU

.

a n d i f W i s a convex b a l a n c e d

t h e n the p r o p e r t y

If

x t W c

0 I' i m p l i e s

Since 0 i s c o n n e c t e d a n e a s y c o n s e q u e n -

123

Hartogs theorem

$ 3. 3. THEOREM.

-

Hartogs' theorem

L A E

a n d E be two comp.:x c o m p l e t e 12 b. v. s. a n d let n a n d hl b e 7 E a n d T E open s e t s r e s p e c t i v e l y Let F 1212 be a c o m p l e x 1. c . s. with a c o m p l e t e Von Neumann bornoiogy a n d l e t f 3.

be a m a p p i n g f r o m 0 in t h e e n l a r g e d s e n s

l X

.

n , i n t o F which is s e p a r a t e l y Silva h o l o m o r p h i c

l X 612. Then f i s Silva h o l o m o r p h i c i n the e n l a r g e d s e n s e i n

proof.

nP

O 2

*

-

T h e a s s u m p t i o n on f m e a n s t h a t for e v e r y fixed x E 0 t h e 1 1 f (xl,x2) i s h o l o m o r p h i c i n t h e e n l a r g e d s e n s e i n R 2 mapping x 2 and t h a t the similar p r o p e r t y holds f o r fixed x E 0 Now t h e proof 2' 2 i s a n i m m e d i a t e c o n s e q u e n c e of th. (3. 1 . 2 ) a n d of t h e c l a s s i c a l H a r t o g s ' -.I

let E a n d E be 1 2 two c o m p l e x B a n a c h s p a c e s , Q 1 a n d 0 open s e t s i n E a n d E res2 1 2 pectively, a n d let F be a B a n a c h space. If f is a s e p a r a t e l y h o l o m o r

t h e o r e m i n n o r m e d s p a c e s ( p r o v e d i n 3. 3 . 4 below) :

phic m a p p i n g f r o m

3. 3. 2

n l X2 -

If

into F, t h e n f i s h o l o m o r p h i c i n

n 1 x n2"

*

E and 1E2 a r e S c h w a r t z b. v. s. o r i f F a d m i t s a countable b a s e of bounded COROLLARY.

sets, i f --

f :

n l X n z -,

Under the a s s u m p t i o n s of (3. 3. l),if

F i s s e p a r a t e l y Silva h o l o m o r p h i c i n

then it i s Silva holomorphic proof.

-

in

n 1"

n2

I X0 2 *

Owing to (2. 3. 3. ) a n d (2. 3. 7) it i s a n i m m e d i a t e c o n s e q u e n c e

of (3. 3. 1). 3. 3. 3.

Example. -

L e t ( C ) be t h e B a n a c h s p a c e of null s e q u e n c e s 0

of c o m p l e x n u m b e r s , with its u s u a l n o r m

(Xn)n Let f be t h e m a p p i n g f r o m (C,),

\I (x ) n

EJ

= sup

( C ) into ( C ) defined by : 0

n

I ",I

Classical properties

124

L e t en

be the function f r o m (Co) to C defined by :

Let

be t h e m a p p i n g f r o m (C )

@

IN

(C ) to C

0

Q i s C-analytic in (C )

(C ) since

f and

defined b y :

a r e . C l e a r l y 6 is

sepa-

i s not l o c a l l y bounded i n ( C )

( C ).

0

r a t e l y Silva h o l o m o r p h i c a n d $Thereforem

0

i s not Silva h o l o m o r p h i c i n

( S o ) ( G o ) which p r o -

v e s t h a t H a s t o g s ' t h e o r e m i s not a l w a y s t r u e for Silva h o l o m o r p h i c

mappings

3. 3. 4

.

Proof

of H a r t o g s ' t h e o r e m i n n o r m e d s p a c e s ( a s s t a t e d i n

proof of (3. 3. I ) )

. Given

x., h. E

J

the function

z1,z2

J

E . , z.Eat if j = L 2 a n d Y' E F' , J

J

- y ' o f ( x l t z h x t z h ) 11' 2 2 2

i s separately holomorphic

.

F r o m the finite d i m e n s i o n a l H a r t o g s '

t h e o r e m t h e function t *

y'o f (xl t th

1'

x t th ) 2 2

i s holomorphic with r e s p e c t t o t r a n g i n g o v e r a s u i t a b l e open s e t i n C .

F r o m (3. 1 . 1 . )

f i s G - a n a l y t i c i n fl

".

We a r e going to p r o v e

t h a t f i s bounded on s o m e non void open s u b s e t of s i n c e without loss of g e n e r a l i t y w e m a y assume t h a t

R 2 and,

12s

Hartog's theorem

i s connected, the r e s u l t will follow f r o m Z o r n ' s t h e o r e m fll f12 i n n o r m e d s p a c e s ( s e e ( 3 . 2 . 3) and i t s proof) . Now l e t u s p r o v e t h a t if m G l N

and y e

n2 n

y

If

0

1

.

R2

R

f i s bounded on some non void open s u b s e t of

For t h i s ,

a r e given, w e s e t

(m) = { x E

is a fixed point i n 62

2'

n1 s u c h

that

1 f(x, y ) ] g

m}

.

the s e t

i s c l o s e d , s i n c e i t i s a n i n t e r s e c t i o n of c l o s e d s e t s . We c l a i m t h a t to3

u

nl= n

Indeed, f o r a fixed x E continuous

.

m , n=l

1

Therefore

1 'm, n

t h e function y 1' f(x, y ) I s m o i f

61 i s a B a i r e s p a c e t h e r e e x i s t m 1

h a s a n i n t e r i o r point

I/ y-yoI1

s E2

1 ; 1

.

As

where

1

4

\\

f(x, y) f r o m y-y,/I

and n E 1

a consequence,

I

.

4

E,

1

n

0

0 to F i s 2

.

lN guch t h a t

Since

(2' ml' n1 f(x, y ) \ s m l i f x E u) and

is s o m e open s u b s e t

of

nl.

Classical properties

126

Q 3 . 4 . Montel's theorem 3. 4. 1.

S e m i - M o n t e 1 1. c. s.

A 1. c. s.

F is said to he semi-Monte1

i f e v e r y bounded s u b s e t of F i s r e l a t i v e l y c o m p a c t .

If F i s a q u a s i - c o m p l e t e S c h w a r t z 1. c. s.

and i f B is a

c l o s e d bounded s u b s e t of F then B i s both c o m p l e t e a n d p r e c o m p a c t , therefore

compact (0.4. 1 )

. As

a consequence every quasi-complete

S c h w a r t z 1. c. s. i s s e m i - M o n t e l . So

, from results in chap

4

,

K,(n, F ) and 19 (a, F) (with t h e i r n a t u r a l topologies defined i n c h a p 4 ) a r e s e m i - M o n t e 1 i n m o s t u s u a l c a s e s . I n t h i s s e c t i o n we study d i r e c t l y t h i s p r o p e r t y a n d obtain on t h i s p r o b l e m s t r o n g e r r e s u l t s t h a n t h e a b o v e r e s u l t s t h a t follow f r o m chap. 4 case

. We limit o u r s e l v e s t o the h o l o m o r p h i c

, but s i m i l a r r e s u l t s m i g h t be obtained in the C Topologies on K s ( & F) &3C

3.4.2.

c o m p l e x c o m p l e t e b. v. s.

,

a

7

s, e

( 6 2 , F)

.

03

case.

If E i s a

E o p e n set a n d F a 1. c. s. w e

equip t h e s p a c e X (0, F ) of Silva h o l o m o r p h i c m a p p i n g s f r o m

s

n

into F

with t h e topology of u n i f o r m c o n v e r g e n c e on the s t r i c t l y c o m p a c t s u b s e t s of f)

. We

equip t h e s p a c e K

s,

e

( n , F ) of Silva h o l o m o r p h i c

m a p p i n g s i n t h e e n l a r g e d s e n s e f r o m 0 into F with t h e s a m e topology.

THEOREM.

3. 4. 3. 7

E-open

-

Let

s e t a n d F a c o m p l e x 1. c. s.

s e m i - M o n t e 1 i f a n d only F

-

proof.

Then the space

s1

a

Ks(sl, F)

k.

Since F i s a c l o s e d s u b s p a c e of K (0, F) t h e n e c e s s i t y

i s immediate Let

E a c o m p l e x S c h w a r t z b. v. s.,

. Let

S

u s now a s s u m e F is a s e m i - M o n t e 1 1. c. s.

H he a bounded s u b s e t of X (0, F) a n d let ( f . )

S

1

iEI

.

be a d i r e c t e d

s y s t e m (0. 1. 16) of e l e m e n t s of H . F o r e v e r y x E 0 w e set H(x) =

c

fi(X)1

H. H(x) i s a bounded s u b s e t of F h e n c e H(x) is rela-

tively compact in F

.

F r o m Thychonov's t h e o r e m (K6the [ 1 3

Q 3) , t h e

127

Montel's theorem

n

product set

XE

H(x) i s r e l a t i v e l y

n

c o m p a c t i n t h e topological p r o -

duct F*,T h e r e f o r e t h e r e i s a n o t h e r d i r e c t e d s y s t e m on H t h a t w e denote by (f ) s u c h t h a t f o r e v e r y x E Q t h e s y s t e m (fi(x))ic I i s i i€I convergent in F L e t f(x) d e n o t e t h e limit of (f.(x)) If y' EF' 1 ig1' and i f c E is t h e i n t e r s e c t i o n with 0 of a one d i m e n s i o n a l s u b s p a c e

.

of E then t h e function y'of

/u,

in the finite dimensional case)

is holomorphic

.

Neumann bornology i s c o m p l e t e

(from Montel's theorem

Since F is s e m i - M o n t e l i t s Von

. So f r o m

(3. 1. 1) f is G - a n a l y t i c i n n .

f i s bounded on the s t r i c t l y c o m p a c t s u b s e t s of

Furthermore

since

a r e equi-bounded t h e r e . T h e r e f o r e f E 5f ( Q ,'F ) S s i n c e t h e functions (fi) a r e cqui-bounded,

t h e functions (f.).

11EI

Now

.

f r o m (2. 2. 7 ) t h e y a r e "locally" equicontinuous, a n d a n i m m e d i a t e c o n s e q u e n c e i s t h a t (f.) 1

c o m p a c t s u b s e t of 62

.

c o n v e r g e s t o f u n i f o r m l y on e v e r y s t r i c t l y i€I T h e r e f o r e X ( Q ,F) i s s e m i - M o n t e l .

S

F r o m th (2. 3. 4) a n equivalent f o r m u l a t i o n of t h ( 3 . 4. 3 ) i s : 3.4.4 7

THEOREM.

-

Let

E (0, F) is

E be a c o m p l e t e c o m p l e x b. v. s . , R

E - o p e n set a n d F a c o m p l e x

1, c. s . ,

T h e n t h e s p a c e 5f

s e m i - M o n t e 1 i f a n d only i f F A .

s, e

F r o m t h (2. 3. 7 ) i t follows i m m e d i a t e l y : 3.4. 5 7

COROLLARY.

E-open

-

Let

E be a c o m p l e t e c o m p l e x b. v. s . , 62

s e t a n d F a c o m p l e x 1. c. s.

bounded sets.

Then the space X

S

a

with a countable b a s e of

( R ,F) i s s e m i - M o n t e 1 i f a n d o n l y

if

FL. 3.4. 6

H e r e w e show t h a t i f E i s a c o m p l e x c o m p l e t e N (but not S c h w a r t z ) b. v. s. a n d i f F = E (which d o e s not have a c o u n -

table

Example. -

b a s e of bounded s e t s )

in general.

, the s p a c e X s ( n, F )

i s not s e m i - M o n t e 1

128

Cfassical properties L e t :E

be a c o m p l e x B a n a c h s p a c e , with B its c l o s e d unit

b a l l , and Let ( f ) b e a s e q u e n c e of c o m p l e x valued h o l o m o r p h i c n n EIN i s not bounded on functions on E s u c h that, f o r e v e r y nclN , f n 1 B ( f o r i n s t a n c e define f (x) = ep(n x ) for t h e function rp defined i n n n e x a m p l e (3. 2 . 4 ) ) We c l a i m t h a t X (E, ( E r n ) i s not s e m i - M o n t e l . T o

.

prove this, if n

S

€IN,we define a m a p p i n g ‘n

m

a n d we define a m a p p i n g

: E

C

N

lim

@

n

n

(x) i n C

lN

.

ip

by:

>(ElN

K(E, C

i s a bounded s u b s e t of

“n’ntm ~p (x) =

>

: E-

{ x ~ ( E C,

lN

by

IN

) and

) since

ip

i s not

b - l o c a l l y bounded aU Ebe o r i g i n of E

3. 4. 7

Remark.

-

M o r e g e n e r a l r e s u l t s on Montel’s

theorem,

including t h e c a s e of continuous h o l o m o r p h i c m a p p i n g s a r e given i n Colombeau [ lo].

3. 4.8

Vitali’s theorem.

-

A s a n a p p l i c a t i o n of t h 3. 4. 3 o r 3. 4 . 4

one o b t a i n s i m m e d i a t e l y a V i t a l i ’ s t h e o r e m . A m o r e g e n e r a l V i t a l i ’ s t h e o r e m m a y be found i n Colombeau

-

Lazet [ 1]

and Lazet [2 1 .

ABSTRACT. -

Let E a n d F b e a r e a l o r c o m p l e x 1. c. s. a c c o r d i n g t o t h e s i t u a t i o n and 61 a n open s u b s e t of E

.

W e equip t h e s p a c e K s ( a , F) of

Silva h o l o m o r p h i c m a p p i n g s f r o m 0 into F with t h e topology of uniform c o n v e r g e n c e on t h e s t r i c t l y c o m p a c t s u b s e t s of 62 which a r e c o m p a c t in s u b s e t of E, (0.7 . 4 , ) )

an

.

(these a r e the s e t s

E B with B s o m e convex b a l a n c e d bounded

We equip t h e s p a c e 8 (Q,F) of C

m

mappings

f r o m 61 into F with t h e topology of u n i f o r m c o n v e r g e n c e o n t h e

strictly c o m p a c t s u b s e t s of

n

for the mappings and each derivative

( s e e (4. 1. 1. ) ; notice t h a t w e h a v e two n a t u r a l topologies depending on n t h e c h o s e n topology on L ( E, F))

.

L e t E be a r e a l o r c o m p l e x ( a c c o r d i n g to t h e s i t u a t i o n ) c o - S c h w a r t z 1. c. s. ((0. 5. 3. ) ; for i n s t a n c e 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 a 1. c. s . ) o r let E b e a F r C c h e t

space.

Let

s u b s e t of E a n d l e t F b e a ( r e a l o r c o m p l e x ) 1. c. THEOREM.

-

.

K ,(a,F) i s a c o m p l e t e ( r e s p e c t i v e l v a S c h w a r t z ) 1. c. s.

i f a n d onlv if F is c o m p l e t e (resD. S c h w a r t z )

- 6 (61, F )

holds f o r

S.

be a n open

. T h e same s t a t e m e n t

i n p l a c e of K s ( n , F).

Notice t h a t the s p a c e X ( n , F ) of a l l (continuous) h o l o m o r p h i c m a p p i n g s is not c o m p l e t e in g e n e r a l . 129

130

Topologies

It follows that when F i s a c o m p l e t e S c h w a r t z 1. c. s.

X ,(n, F) and Q (n, F) a r e natitrally r e f l e x i v e 1. c.

s. i n t h e s e n s e

explained i n ( 0 . 3. I ) , i. e.

. In p a r t i c u l a r t h e s e s p a c e s are (0. 3. 2 ) . T h i s r e f l e x i v i t y r e s u l t h a s a f u n d a m e n -

algebrai'cally and topologically s e m i - r e f l e x i v e 1. c. s.

t a l i m p o r t a n c e in t h e sequel. T h e r e i s a n u m b e r of w o r k s devoted to t h e s t u d y of v a r i o u s topologies on t h e s p a c e

K

(n, F) of

continuous h o l o m o r p h i c m a p p i n g s .

However we s h a l l not u s e t h e s e topologies i n t h i s book, s o t h e i n t e r e s t e d r e a d e r i s r e f e r e d t o t h e book of Dineen [ 1 1 which t r e a t s t h i s t o pic

i n d e t a i l . L e t u s a l s o m e n t i o n t h e n a t u r a l i d e a of M a z e t [ 13 t o

u s e a bornology ( o r a s i m i l a r s t r u c t u r e ) on t h e s p a c e

X (a, F ) .

131

Natural topologies

p 4. 1. 1

n

N a t u r a l topologies on @ (61, F)

4. 1

Topologies o n 61

(n, F).

K,(n. F)

Let E be a r e a l c o m p l e t e b. v. s.,

a TE-open s u b s e t a n d F a real o r c o m p l e x 1. c. s. . U ' e denote

d (61, F ) t h e s p a c e of Cco m a p p i n g s f r o m fi to F

by

.

W e equip t h e i s p a c e s L ( E, F) (0. 8. 7) with the topology of 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 of E

.

We equip n a t u r a l l y 6

(n, F)

with its n a t u r a l

topology of u n i f o r m c o n v e r g e n c e f o r m a p p i n g s a n d t h e i r d e r i v a t i v e s on t h e s t r i c t l y c o m p a c t s u b s e t s of 61 , T h e r e f o r e i f i o-neighborhood i n L( E,F),t h e sets

V(K,

E

~

.,. . , E n )= [ f

E$

such that f

denotes a

(i)( K ) c E i i f O s i l n ]

i n 8 (61, F) when K r a n g e s

a r e a b a s e of o-neighborhoods

61 , n r a n g e s o v e r IN a n d

t h e s t r i c t l y c o m p a c t s u b s e t s of over the o-neighborhoods

(n, F)

E.

i

of L ( E, F)

.

Equivalently

s t r i c t l y c o m p a c t s u b s e t s of 61

in

E.

ranges

, the s e t s yl..

a r e a b a s e of o-neighborhoods

over

. Yi)

13

4

(0, F) when K r a n g e s o v e r t h e

, L r a n g e s o v e r t h e bounded s u b s e t s

of E , p r a n g e s o v e r t h e s e t of a l l continuous s e m i - n o r m s on F and n o v e r IN. i If we e q u i p t h e s p a c e s L ( E, F) with t h e topology of u n i f o r m

c o n v e r g e n c e on t h e s t r i c t l y c o m p a c t s u b s e t s of E w e obtain a n o t h e r n a t u r a l topology o n 8 (61, F). I n t h i s c a s e L r a n g e s now o v e r t h e s t r i c t l y c o m p a c t s u b s e t s of E S c h w a r t z b. v. s. a s s o c i a t e d

. If

we denote as u s u a l by E

t o E (0. 7 . 4 )

,

s. c. t h e s e c o n d topology

the

Topologies

132

c o i x c i d e s with the first topology i f w e r e p l a c e E

(the b y Es. c. a r e i d e n t i c a l , a n d a n y bounded

s t r i c t l y c o m p a c t s u b s e t s of E a n d E s u b s e t of E

i s contained

s. c. i n a s t r i c t l y c o m p a c t s u b s e t of E). I n

s. c. any c a s e t h e s e two topologies coihcide i f E i s itself a S c h w a r t z b. v. s . ,

which will be

4. 1. 2

g e n e r a l l y t h e c a s e f r o m now on

Topology

c o m p l e t e b. v. s . ,

Ks(n, F).-

OP-

n

a

7

.

Let now E be a c o m p l e x

E - o p e n s e t a n d F a c o m p l e x 1. c. s . .

W e equip n a t u r a l l y t h e s p a c e K (Q,F) of Silva h o l o m o r p h i c m a p p i n g s S f r o m 0 into F with t h e topology of u n i f o r m c o n v e r g e n c e on t h e s t r i c t l y c o m p a c t s u b s e t s of 0

: the

f o r m a b a s e of o-neighborhoods

-

(n, F)

i n F.

inequalities that

a n d w e m a y check i m m e d i a t e l y f r o m

K s ( O , F) i s contained i n d

c e d t o p o l o g y , a s soon a s E i s a S c h w a r t z b. v. s.

4. 1. 3

strictly

C o n s i d e r i n g t h e r e a l v e c t o r s p a c e s t r u c t u r e of E we

m a y define t h u s 8 Cauchy's

when K r a n g e s o v e r t h e

and V r a n g e s o v e r t h e o-neighborhoods

c o m p a c t s u b s e t s of 62 Remark.

sets

.

(n, F)

w i t h indu-

Nachbin's p o r t e d topology i n s p a c e s of Silva h o l o m o r p h i c

mappings.-

L e t E be a c o m p l e x c o m p l e t e b. v. s . , Q a

s e t a n d F a 1. c. s.

A semi norm p on

.

S

T

E-open

Let K be a s t r i c t l y c o m p a c t s u b s e t of

Q

.

( Q ,F) i s c a l l e d K - p o r t e d i f t h e r e i s a semi-

n o r m q on F a n d a bounded B a n a c h d i s c B of E with K c o m p a c t ( in the B s e n s e of the topology induced by t h a t of E ) t h e r e is a n u m b e r C ( V ) > 0

i n EB such t h a t : for a n y open neighborhood V of K i n 0 R E with :

B

133

Natural topologies

for a n y f E K s ( 62, F)

. Now we define

c o m p a c t p o r t e d topology"

on 5c ( h, F) t h e

-

strictly

I'

a s the topology g e n e r a t e d b y t h e K - p o r t e d

s e m i - n o r m s when K r a n g e s o v e r t h e Remark.

S

When E i s a Banach

s t r i c t l y c o m p a c t s u b s e t s of

R.

s p a c e t h i s topology i s c a l l e d t h e

Nachbin's p o r t e d topology a f t e r the book Nachbin [Zl. I t i s m o r e g e n e r a l l y defined a n d s t u d i e d on X book by Dineen

(n, F )

i f E i s a 1. c. s.,

see the

1 ] and the Notes,.

W e s h a l l not u s e explicitely t h i s topology i n t h e s e q u e l f o r the following r e a s o n :

4. 1 . 4

PROPOSITION.

na TE-open

-

E be a c o m p l e x S c h w a r t z b. v. s . ,

s e t and F a c o m p l e x 1. c. s.

.

T h e n o n 3C (0, F) the S

s t r i c t l v comDact Dorted toDolor?.v c o i h c i d e s with the toDologv of unif o r m c o n v e r g e n c e on t h e

s t r i c t l y c o m p a c t s u b s e t s of

.

T h e proof i s i m m e d i a t e , s i n c e one m a y c h o o s e V i n def (4. 1. 3 ) c o n t a i n e d i n a s t r i c l y c o m p a c t s u b s e t of 0

4. 1. 5

NOTATIONS.

-

.

W e note d (E),K S ( E ) a n d X (E)when

F = (E ( o r p o s s i b l y F = R i n c a s e of & ( E ) )

.

Topologies

134

C o m p l e t e n e s s of Xs(62, F)and

9 4. 2 4.2. 1

THEOREM.

7E -open

-

E be a c o m p l e x S c h w a r t z b. v. s . ,

&J

s e t and F a c o m p l e x 1. c. s.

c o m p l e t e 1. c. s. proof.

-

6 (n, F).

.

Then Xs(

n, F )

0a

is a

i f a n d only i f F a c o m p l e t e 1. c. s.

x S (n, F ) h e n c e

F i s a c l o s e d s u b s p a c e of

the complete-

n e s s of F i s n e c e s s a r y . L e t u s a s s u m e now F i s a c o m p l e t e be a Cauchy s y s t e m i n 3c (n, F) ( s e e 0. 1.16). (fi)iE I S Since F i s a c o m p l e t e 1. c. s . , f o r e v e r y x € 0 t h e Cauchy s y s t e m

1. c. s.

and let

(fi(x))i E I is c o n v e r g e n t i n F to a n e l e m e n t t h a t we denote b y f(x) E F

. Thus

t h i s d e f i n e s a m a p p i n g f f r o m 62 i n t o F

.

F r o m the

i s a Cauchy system i t follows t h a t i f K i s a s s u m p t i o n t h a t (f ) i icI a s t r i c t l y c o m p a c t s u b s e t of 0 a n d V a c l o s e d convex b a l a n c e d o-neighborhood i n F :

(1) 3 i € 1 s u c h t h a t i , j

;1

i

0

s

F i x i n g x and i w e obtain f.(x)

fi(x)

-

-

fj(x)EV

f(x)EV since

VxEK. V is closed.

There-

f o r e f i s bounded o n K (letting V v a r y ) . F r o m t h e finite d i m e n s i o n a l results F r o m (1) it

a n d ( 3 . 1. 1) f i s G-analytic follows t h a t

.

(f ) converges to f i iC1

If E is a c o m p l e x c o m p l e t e b . v . s . ,

x S (a, F) . K S , ( n ,F) .

Therefore f c in

t-2 a T E open s e t and F a

( n , F ) t h t s p a c e of a l l S, e Silva holomorphic m a p p i n g s in t h e e n l a r g e d s e n s e f r o m R t o F , equipped

c o m p l e x 1 . c . s . we r e c a l l t h a t we denote by k

with the topology of u n i f o r m c o n v e r g e n c e of

n

01-1,the

. T h e n a n equivalent f o r m u l a t i o n of th.

4.2.2 THEOREM. the space

-

strictly compact subsets

4.2. 1 i s :

If E is a c o m p l e t e b . v . s .

and F a 1.c. s. then

S , e (n, F ) i s a c o m p l e t e 1. c. s. if and only if F is a

c o m p l e t e 1.c. s .

.

Completeness

Proof.

-

F r o m (2. 3 . 4 ) 3c

s, e (a, F) =

with t h e s t r u c t u r e of t h e b. v.

4. 2.3

COROLLARY.

-

S.

Let E

Ks(n, F)

K s ( n s. c. F) if I

E

s . c.

.

0

I

denotes

s. c.

n

be a c o m p l e x c o m p l e t e b. v. s.,

n a T E - o p e n set a n d F a L. c. s. sets. T h e n

135

with a countable b a s e of bounded

i s a c o m p l e t e 1. c. s.

F is a

i f a n d only if

c o m p l e t e 1. c. s. proof.

-

4. 2 . 4 7

THEOREM.

E-open

of

-

E be a

a

real S c h w a r t z b. v. s . ,

. Then

(B

(0, F) i s a

i f a n d only i f F a c o m p l e t e 1. c. s.

T h e condition i s n e c e s s a r y s i n c e F i s a c l o s e d s u b s p a c e

8 ( 0 ,F)

( fi)iE 1

-

s e t a n d F a real o r c o m p l e x 1. c. s .

c o m p l e t e 1. c. s. proof.

. I

it follows i m m e d i a t e l y f r o m ( 2 . 3 . 7 ) a n d (4. 2. 2 )

.

Now l e t u s a s s u m e F i s a c o m p l e t e 1. c. s.

b e a Cauchy s y s t e m i n & (0, F)

. F o r every

.

Let

n €IN it i s e a s y

t o c h e c k that t h e s p a c e L ("E, F) i s a c o m p l e t e 1. c. s. (proof similar (n) to t h a t of (4. 2. 1) a n d s i m p l e r ) T h e r e f o r e we define a m a p p i n g f n We f r o m n t o L( E , F) a s pointwise limit of t h e m a p p i n g s (). fi a r e going t o :$rove t h a t f"), denoted by f f r o m now o n , i s i n

.

8

(n, F),

.

and that t h e mappings f(n) a r e i t s s u c c e s s i v e d e r i v a t i v e s .

(f ) c o n v e r g e s t o f i n & (n, F) would t h e n i it1 i s a Cauchy s y s t e m . follow e a s i l y f r o m t h e a s s u m p t i o n t h a t ( f . ) 1 i E 1 F r o m th. (1.4. 7) it s u f f i c e s t o p r o v e t h a t f i s Cm i n t h e e n l a r g e d

T h e p r o p e r t y that

s e n s e a n d t h a t t h e m a p p i n g s f ( n ) a r e its s u c c e s s i v e T h i s follows i m m e d i a t e l y f r o m lemma (4. 2. 5) below

4. 2. 5 LEMMA.

-

.

.

L e t E be a r e a l B a n a c h s p a c e , 0 a n open

s u b s e t of E a n d F a 1. c. s. Ca,

derivatives

.

L e t (fi)icI

be a n o r d e r e d system of

m a p p i n g s f r o m 0 into F i n S i l v a ' s e n l a r g e d s e n s e s u c h t h a t

f o r e v e r y x E 0 a n d e v e r y n c l h t h e s y s t e m ( f i( " ) ( ~ ) ) ~c o~n~v e r g e s

136

Topologies

n i n L( E , F ) t o a n e l e m e n t f(”)(x) of

n L( E , F ) (we a s s u m e

L ( n E , F) i s

equipped with the topology of u n i f o r m c o n v e r g e n c e on t h e bounded n s u b s e t s of E ) L e t u s a s s u m e t h a t t h i s c o n v e r g e n c e i s l o c a l l y uni-

.

f o r m in

n

Then f = pings

f(n)

proof. -

. Let

(i. e. uniform on a neighborhood of e a c h e l e m e n t of

f(O)

i s Silva Cm

i n the e n l a r g e d s e n s e i n

).

fl a n d t h e m a p -

a r e its successive derivatives.

W e denote by

I\ I\

the norm in E a n d by x 1

0

a given e l e m e n t

z 0 be s u c h that t h e s y s t e m ( f 1. ( x ) )i E 1 c o n v e r g e s uni1 f o r m l y i n L (E, F) f o r x-xo\\ L cl If h € E w i t h h \\ S E~ w e s e t :

of

E

.

11

I\

Let V be a c l o s e d convex b a l a n c e d o-neighborhood pv denote the

in F and let

gauge of V .

r (h) = A t B t C X

with

If

-

r

denotes the c l o s e d convex b a l a n c e d hull i n F , t h e n t h e m e a n

value t h e o r e m (1. 3. 5) (which i s a l s o v a l i d f o r Silva d i f f e r e n t i a b l e mappings in the enlarged sense

f.(x t h ) J

-

f.(x) J

-

By hypothesis, f o r e v e r y

since

fi(x +h)

E

i t follows f r o m (1. 3. 2 ) ) g i v e s :

+ f . ( x ) E ?{[ f lJ. ( x t t h ) -

> 0 t h e r e i s a n index i

0

fl.(x tth)] h ]

.

0s ts 1

s u c h t h a t , i f i, j

2

i

0

137

Completeness

T h e r e f o r e , i f i, j

i

t

\\ h \ \a

and

0

Since V i s c l o s e d , letting

I'

j -.,

00

E~

I'

,

we obtain

We a l s o have

Now w e fix s o m e i

2

i

0

pV(B)

a n d we obtain

S E

\I hll

if

\Ihl\

i s small enough.

w h i c h p r o v e s t h a t f is d i f f e r e n t i a b l e i n t h e e n l a r g e d s e n s e a n d t h a t f ( l ) is i t s d e r i v a t i v e

.

T h e proof i s a n i m m e d i a t e induction on n

.

I

Topologies

138

$ 4. 3 4. 3 . 1

THEOREM.

S c h w a r t z p r o p e r t y of 3Cs(61, F)

-

Let

E be a c o m p l e x S c h w a r t z b. v. s, , 61

TE - o p e n s e t a n d F a c o m p l e x 1. c . s.

1. c. s. i f and only i f F i s a Schwartz proof.

-

of

(n, F).

S

8 (61, F)

.

Then-xS(n, 1. c. s.

a

F) is a S c h w a r - z

.

F r o m (0. 5. 4) the condition i s n e c e s s a r y s i n c e F i s a s u b s p a c e C o n v e r s e l y l e t u s a s s u m e F i s a S c h w a r t z 1. c. s.

K be a s t r i c t l y c o m p a c t s u b s e t of o-neighborhood i n F o-neighborhood in

.

Let

Let

a n d V a convex b a l a n c e d

"2/ (K, V ) , ( s e e 4. 1. Z),

xs(n,F) .

.

be t h e c o r r e s p o n d i n g

L e t V' b e a convex b a l a n c e d o - n e i g h b o r -

hood i n F with V ' c V a n d s u c h t h a t the c a n o n i c a l m a p p i n g i s p r e c o m p a c t ( V ' e x i s t s s i n c e F i s a S c h w a r t z 1. c. s. ). v ' P F V Since E i s a Schwartz b. v. s . t h e r e i s a convex b a l a n c e d s t r i c t l y

c o m p a c t s u b s e t B of E s u c h that K i s c o m p a c t i n t h e n o r m e d s p a c e T h e r e i s a > 0 s u c h t h a t K ' = K t a B i s s t i l l contained a n d s t r i B' c t l y c o m p a c t i n 0 "2/ ( K ' , V') is contained i n y (K, V) a n d w e a r e

.

-

going to p r o v e that t h e c a n o n i c a l m a p p i n g ,

i s p r e c o m p a c t . L e t C ( K , F ) denote t h e s p a c e of continuous m a p p i n g s V T h e n the following from t h e c o m p a c t s e t K i n t o t h e n o r m e d s p a c e FV* diagram is commutative :

139

Schwartz property

-

where i is the natural mapping : @

S+~;P @,K (if

rg

ExS(o,F) ,

t h e r e s t r i c t i o n of t h i s c l a s s to K c K '

and s

is the canonicalmapping from F V'V V' to FV). F r o m C a u c h y ' s i n e q u a l i t i e s (2. 2. 7 ) t h e i m a g e t h r o u g h i of t h e unit i s a n equicontinuous f a m i l y i n C ( K , F V I ) "v (K', V ' ) (Choosing V' c l o s e d i n F which is p o s s i b l e ) It s u f f i c e s now t o apply

ball of

[3Cs(n,F)1

.

[11 VII 5 o r L a n g [ 21 IX. 4) .

A s s o l i ' s t h e o r e m ( Dieudonne'

COROLLARY.

4. 3. 2.

-

L e t E be a c o m p l e x c o m p l e t e b. v. s.,

n a

_c

7 E o p e n s e t a n d F a c o m p l e x 1. c.

Then the space K

6..

Silva h o l o m o r p h i c m a p p i n g s in t h e e n l a r g e d

( 0 , F)o f S, e s e n s e i s a S c h w a r t z 1. c. s.

i f a n d only i f F i s a S c h w a r t z 1. c. s.

proof.

-

F r o m (2. 3. 4) 3€

c o n s i d e r e d in the b . v .

4. 3. 3.

E

S.

COROLLARY.

S, e

(h2, F ) = x ( G

s

S.C.

,F) if

denotes

s. c.

S.C.

-

Let E -

be a c o m p l e x q u a s i - c o m p l e t e 1. c. s.

.

s u c h t h a t its c o m p a c t bornology i s a S c h w a r t z bornoloPv L e L R k a n open s u b s e t of E a n d l e t F be a c o m p l e x denote t h e s p a c e of

. Then

x(n,F)

(continuous) h o l o m o r p h i c m a p p i n g s f r o m 0 i n t o F,

equipped with t h e topology of uniform s u b s e t s of 0

1. c. s. Let

c o n v e r g e n c e on t h e c o m p a c t

~ ( 6 1 F) , i s a S c h w a r t z 1. c. s. i f a n d only i f F &

a S c h a r w t z 1. c. s.

proof.

-

W e denote by E

nology of E a n d by 0

C

C

t h e b.

the

i s a topological s u b s p a c e of

set

obtained with t h e c o m p a c t b o r -

V.S.

fl c o n s i d e r e d i n E

3Cs(nc, F)

.

C

.

Then

x(fl,F)

140

Topologies

4. 3.4.

THEOREM.

-

Let -

s e t a n d F a 1. c. s.

7E-open

E be a r e a l S c h w a r t z b. v. s . , 0

.

T h e n 8 ( Q ,F) i s a S c h w a r t z 1. c. s.

.

a n d only i f F i s a S c h w a r t z 1. c. s. proof.

-

2

S i m i l a r l y a s i n (4. 3. 1) t h e condition i s n e c e s s a r y

.

if

Now l e t

u s a s s u m e F i s a S c h w a r t z 1. c. s .

. A proof

(4. 3. 1 ) shows t h a t f o r e v e r y n G I N

t h e s p a c e L(nE, F) i s a S c h w a r t z

1. c. s.

A l l t h e o-neighborhoods

similar to t h a t of

we s h a l l c o n s i d e r will be convex

b a l a n c e d a n d closed. A c c o r d i n g to notations of (4. 1. 1. ) let U = y ( K , in some E

B

,

If o s i s n

i. e . , if 1 E

o,.

E

. ., E n)

E

i’ assume

of &

( a , F).

K is compact

and, for s o m e a > 0, K ’ = K t a B is s t r i c t l y c o m p a c t i n 61. iitl) let E be a o-neighborhood of L( E,F) s u c h t h a t it1

-

and y E B , the mapping

E litl

X1’.

is i n

be a o-neighborhood

Since t h e

. .,xi

1 (y,

XI’.

..

Xi)

n s p a c e s L ( E , F) a r e S c h w a r t z 1. c. s.

w e may

and that h a s a precompact image in the it1 it1 for o s i s n - 1 . We set n o r m e d s p a c e [ L (itl)E, F ) I E it1 L e t j be t h e c a n o n i c a l m a p p i n g : V = y ( K ’ , E ~ E , ll,. , E ’ n t 1)C U E

c

E

..

.

We p r o v e t h a t j i s p r e c o m p a c t , i. e. s u b s e t of t h e n o r m e d s p a c e [ &

t h a t j(+) i s a p r e c o m p a c t

(n, F)]. U

Using t h e

a s i n proof of (4. 3. 1 ), l e t i be t h e m a p p i n g :

f

s a m e notations

141

Schwartz property

i w h e r e s i d e n o t e s t h e c a n o n i c a l mapping : L( E,F )

.

(L(lE, F ) ) E i From A s c o l i ' s t h e o r e m , a1 r e a d y u s e d i n proof of (4. 3 . 1) , t h e i s a p r e c o m p a c t s u b s e t of t h e n o r m e d s p a c e s e t { si o 'f (i) 1 /K f € V 4

i

C [ K, (L( E, F))E3. T h e r e f o r e t h e m a p p i n g i is p r e c o m p a c t i (($ (n, F)) is i s o m e t r i c with i t s Ratural i m a g e ir. U C(K, Ft ) 0

result.

x

C [ K , (L(E, F))e

1

1

x . .

.

x Cr(K,(L("E,F)),

n

1,

.

Since

w e get the

Topologies

142

4 4.

4.

Reflexivity of X

S

(n, F)and 8 (0,F)

We r e c a l l , (0. 3. l ) , t h a t a 1. c. s.

E i s s a i d t o be I'naturally

r e f l e x i v e " i f ElX= E a l g e b r a i c a l l y a n d topologically. F r o m t h (0. 5. 9), (4. 2 . 1) a n d (4. 3 . 1) it follows :

4.4. 1

THEOit-EM.

'r E - o p e n

x,(n,

-

s e t a n d F a c o m p l e x c o m p l e t e S c h w a r t z 1. c. s,

F ) i s a n a t u r a l l y r e f l e x i v e 1. c. s.

F r o m t h (0. 5.9), (4. 2 . 4 )

4.4.2

THEOREM.

-

Let E -

.

n a t u r a l l y r e f l e x i v e 1. c. s.

.

be a r e a l

Then

, 0a 7E-

S c h w a r t e b. v. s.

.

T h e n 8 (0, F) i s a

T h e s e r e s u l t s of reflexivity, n a m e l y X

(n, F)

.

a n d (4. 3. 4) it follows :

open s e t a n d F a c o m p l e t e S c h w a r t z 1. c. s.

d J x ( n , F) = 8

n a

L e t E be a c o m p l e x S c h w a r t z b. v. s.,

'X

S

(0, F ) = xs(n,F) a n d

w i l l be of a c o n s t a n t u s e i n t h e s e q u e l

. T h e y show

t h a t the concept of " n a t u r a l reflexivity", s e e 0 . 3 . l., i s v e r y i m p o r t a n t f o r infinite d i m e n s i o n a l a n a l y s i s , all the m o r e s o a s t h e s e s p a c e s a r e not i n g e n e r a l reflexive 1.c.s.

in the c l a s s i c a l s e n s e , s e e 0 . 3 . 2 .

.

CHAPTER 5 APPROXIMATION AND DENSITY RESULTS

A BSTRA CT If E and

F a r e 1 . c . s . we d e n o t e by

( f i n i t e type continuous p o l y n o m i a l s f r o m E mappings f r o m E

with x E E ,

XI.

THEOREM.1. c . s . ,

f f ( E , F ) = P,(E) 8 F

to F) t h e f i n i t e s u m s of

into F of t h e f o r m

E

E' if

__ Let E

1> i 5 n ,

n E IN and

y E F

.

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

0 a n y t r a n s l a t e of a balanced open s u b s e t of E a n d F

. Then

c o m p l e x 1. c . s.

6;(E, F ) , h e n c e

I t follows now f r o m is c o m p l e t e ,

5

x(n, F )

, is d e n s e in

a ",(n,

F).

4. 2 t h a t , i n t h e above conditions and i f F

K S ( R F) is the c o m p l e t i o n of

k f ( E , F ) and

x(n,F)

equipped Kith the topology of u n i f o r m c o n v e r g e n c e on t h e c o m p a c t s u b s & of 0 ( f r o m t h e a s s u m p t i o n on E

these a r e strictly compact).

If F i s c o m p l e t e and S c h u a r t z , K , ( n , F ) s e n s e (0.3.1)

is t h e i r bidual in the

:

xs(n,F) =

"(E,

F)'x = K ( n , F)'x =

143

S

( 0 , F).'x

a

144

Approximatiori and density results

Remark.

-

The above r e s u l t s r e m a i n t r u e if

0

is a f i n i t e l y Runge

E.

( 5 . 1 . 9 ) o p e n s u b s e t of

In the

Cco c a s e

THEOREM.-

Let E

R a n y open s u b s e t of E

:

be a q u a s i - c o m p l e t e d u a l n u c l e a r r e a l 1.c. s.,

and F

a 1.c. s. T h e n b f ( E , F) i s d e n s e i n

B(n,F).

It follows now a s in t h e c o m p l e x c a s e b e f o r e t h a t B(h1,F) i s t h e c o m p l e t i o n and t h e bidual ( i n the a b o v e s e n s e ) of " r e a s o n n a b l e " s p d c e of

Coo m a p p i n g s f r o m

m e a n s that t h i s s p a c e contains

hJf(E, F) and of a n y

0 into F ("reasonnable"

k f ( E , F ) and i s c o n t a i n e d i n & ( n , F ) .

It is equipped with its n a t u r a l topology which i s induced by

Important comments.

-

& ( a F, ) ) .

T h e s e r e s u l t s , both i n r e a l and c o m p l e x

c a s e show t h e r e m a r k a b l e r e l e v a n c e of t h e c o n c e p t s of Silva h o l o m o r p h i c and Silva C

a3

m a p p i n g s . T h e y s t r e n g t h e n o u r p r e v i o u s r e a s o n s to c o n s i -

d e r them a s b a s i c c o n c e p t s of h o l o m o r p h y and d i f f e r e n t i a b i l i t y i n infinite dimensions.

In

fi 5 . 3 u e p r o v e e x i s t e n c e of p a r t i t i o n s of unity which a r e

used in t h e s e q u e l . P r e c i s e l y Q e o b t a i n :

THEOREM.-

Let E

be a r e a l 1. c. s . w i t h a b a s e of p r e - H i l b e r t i a n

o - n e i g h b o r h o o d s ( f o r i n s t a n c e E m a y be a H i l b e r t s p a c e o r a nuclear_

1. c . s . ) .

Let

R

be a n o p e n s u b s e t of E which is a Lindeltlf s p a c e f o r

the induced topolopy. &t

b

be a n o p e n c o v e r i n g of

Coo p a r t i t i o n s of unity r e l a t i v e t o t h i s covering.

0. T h e n t h e i r e x i s t

145

A density result

5

5. 1

A d e n s i t y r e s u l t i n 3-C ( n , F ) S

5. 1. 1 A n a p p r o x i m a t i o n p r o p e r t y i n b . v . s. b.G.s.

. We

s a y t h a t E is a b . v . s .

a n y s t r i c t l y c o m p a c t s e t K in E

Let E

be a c o m p l e t e

with A p p r o x i m a t i o n P r o p e r t y if f o r

t h e r e is a bounded B a n a c h d i s c

B

K is a c o m p a c t s u b s e t of E

( 0 . 2 . 10) s u c h t h a t

and t h e i d e n t i t y B m a p p i n g on E B c a n be a p p r o x i m a t e d u n i f o r m l y on K by f i n i t e r a n k continuous l i n e a r o p e r a t o r s on

Example.

-

EB

'

A n y n u c l e a r b . v . s . i s a b . v . s. u i t h A p p r o x i m a t i o n

Property : from (0.5.8) the spaces E

B

m a y be c h o s e n t o be s e p a r a b l e

Hilbert spdces.

5.1.2

THEOREM.

-

Let

E be a c o m p l e t e c o m p l e x b. v. s .

with A p p r o x i m a t i o n P r o p e r t y and l e t topology on E

be a Hdusdorff l o c a l l y c o n v e x

s u c h t h a t a n y bounded s u b s e t of t h e b . v . s.

Let

in the 1 . c . s . ( E , T ) -

n

E is bounded

be a t r a n s l a t e of a b a l a n c e d

and l e t F be a c o m p l e x 1 . c . s. L e t p f ( @ , y ) F) , of f i n i t e type c o n t i n u o u s p o l y n o m i a l s f r o m Then p f ( ( E , y ) , F ) i s dense in K

d e n o t e t h e l i n e a r space

( E , %'

S, e

TE-open set

into

F.

(0,F).

B e f o r e t h e proof w e s t a t e a few c o r o l l a r i e s .

5.1.3

COROLLARY.

c o m p l e x 1. c. s.,

-

L e t E be a q u a s i - c o m p l e t e d u a l n u c l e a r

0 a n y t r a n s l a t e of a b a l a n c e d open s u b s e t of E &

a c o m p l e x 1.c. s. T h e n 6 ; ( E , F ) , h e n c e

Proof.

-

x(n, F ) ,

is d e n s e i n 3-C (0, F ) . S

Apply (5. 1. 2 ) u i t h the Von N e u m a n n b o r n o l o g y of E ( s e e

0 . 5 . 7 ) and the topology of E .

S ,e

(n,F)

F

= xS( R , F )

f r o m (2.3.3).

146

Approximation and density results

5. 1 . 4 R e m a r k . -

It follows f r o m (5. 1.3) and (4.2. 1) t h a t in t h e

a s s u m p t i o n s of (5. 1 . 3 ) on E and

n

and if F is c o m p l e t e , X ( n , F )

equipped with the topology of u n i f o r m c o n v e r g e n c e on t h e c o m p a c t

n,

s u b s e t s of

is n e v e r c o m p l e t e e x c e p t when it c o h c i d e s with

ks(n, F).

More precisely :

5.1.5 COROLLARY. c o m p l e x 1. c . s .

I

n

-

Let E

be a q u a s i c o m p l e t e d u a l n u c l e a r

any t z a n s l a t e of a balanced open s u b s e t of E

c o m p l e x 1. c . s . T h e n K S ( R , F) is the c o m p l e t i o n of

a complete

If f u r t h e r m o r e F is a S c h w a r t z 1.c. s . then

x(fi,F)

Proof.

i n the s e n s e (0.3. 1). i . e.

-

S

& F

q n , F).

( 0 , F) is the bidual of

From ( 0 . 5 . 7 ) t h e c o m p a c t s u b s e t s of 0 a r e s t r i c t l y c o m p a c t

h e n c e x ( 0 ,F) cxS(R,F) with induced topology. Now it s u f f i c e s to apply

( 5 . 1 . 3 ) , ( 4 . 2 . 1) and (4.4. 1).

5. 1 . 6 Proof of th. (5. 1 . 2 ) . -

By t r a n s l a t i o n w e m a y a s s u m e

balanced. L e t K be a given s t r i c t l y c o m p a c t s u b s e t of

R

,

R is

L e t B be

a bounded Banach d i s c in E s u c h that K is a c o m p a c t s u b s e t of E

dnd B and the identity mapping o n E B c a n be a p p r o x i m a t e d u n i f o r m l y on K by f i n i t e r a n k continuous l i n e a r o p e r a t o r s on E B . balanced

3C

S, e

L e t V be a convex

o-neighborhood i n F and l e t f be a given e l e m e n t of

(0, F)

n n EB

i s a balanced open s u b s e t of E

and B E x ( n n E B ,F). F i r s t l e t u s p r o v e the e x i s t e n c e of p ~

f/RnE s u c h tgat f o r e v e r y x € K

p(x) - f ( x ) €

-21 v.

E 6f (EB’ F)

A density result

F o r every

T P

147

o t h e r e is a continuous l i n e a r mapping

k

K

Illq ( x )

T h e r e f o r e Rq(K) c continuous on K,

-

R

x

11 5 9

for

q

therefore

/I 11

:

3 n 3 o such t h a t f ( x )

f o e

-

-

(f o R

rl

) ( x ) E-

1

4

V

bx f K

.

€ K((nmll , F). F r o m l e m m a

( 5 . 1. 7) below,

f/nn

ri s u b s e t s of 0

with d o m a i n

of E such that for B ' 7 (if d e n o t e s the n o r m i n E ). B o s m a l l enough. f i s u n i f o r m l y

E B and r a n g e a finite d i m e n s i o n a l s u b s p a c e E every x

rl

nE

c a n be a p p r o x i m a t e d u n i f o r m l y on t h e c o m p a c t

by e l e m e n t s of X ( i 2 n E

1@

rl

F.

Since

n nE

balanced a n y h o l o m o r p h i c c o m p l e x valued function on R p E a p p r o x i m a t e d u n i f o r m l y on there exists p

T

a (K) r)

by e l e m e n t s of

E b(E ) 0 F s u c h t h a t :

Therefore for every x

rl,

is

c a n be

P(E ). T h e r e f o r e Q

v

K ,

.

So ( 1 ) holds with p = p o r l r l finite d i m e n s i o n a l l i n e a r s p a c e ,

Now s i n c e p

i s a polynomial on a

p E' 6 (E ;F). Now to conclude t h e f B proof i t s u f f i c e s t o prove t h a t a n y e l e m e n t of (E ) ' c a n be approximated B u n i f o r m l y on K by r e s t r i c t i o n s to E B of e l e m e n t s of ( E X ) I . F o r t h i s l e t r denote the r e s t r i c t i o n mapping :

Approximation and density results

148

) I t h e topology on (E ) I of u n i f o r m convergence B B r on the c o m p a c t s u b s e t s oP E B F r o m the M a c k e y - A r e n s t h e o r e m

L e t u s denote by ( E

( 0 . 3 . 10) ( ( E

) I

)I

To

.

= E

algebrai’cally.

B

W e c l a i m t h a t r hcls a d e n s e r a n g e in (E x

E

((EB)lT

Since E B :E

)

’

=

EB , x

f

and s i n c e

: if not t h e r e i s

I )

To

o and s u c h t h a t u(x) = o f o r e v e r y u E (E,??)’.

(E,C)i s H a u s a o r f f t h i s i s i m p o s s i b l e ( H a h n -

Banach t h e o r e m ) .

n

5. 1.7 LEMMA. -

be a balanced open s e t i n a finite d i m e n s i o n a l

c o m p l e x Spdce and l e t F be any c o m p l e x 1.c. s . T h e n “(0)@F d e n s e in

x(n, F).

-

If f E

Proof.

x(n, F)

i t s u f f i c e s to c o n s i d e r its T a y l o r s e r i e s

e x p a n s i o n a t the o r i g i n . W

-

L e m m a ( 5 . 1 . 7 ) r e m a i n s t r u e if %2 i s a n y o p e n s u b s e t of 5. 1. 8 R e m a r k . n C A proof i s in G r o t h e n d i e c k I 1. We s h a l l j u s t u s e t h i s b e t t e r r e s u l t

.

in R e m a r k ( 5 . 1. 1 0 ) below which h a s n o f u r t h e r a p p l i c a t i o n .

5. 1.9 Definition of f i n i t e l y Runge open s e t s . l i n e a r s p a c e . A s u b s e t SIcE

-

L e t E be a c o m p l e x

is said to be finitely Runge i f f o r a n y finite

d i m e n s i o n a l l i n e a r s u b s p a c e E o c E , t h e s e t SIDE . o p e n s u b s e t of E

0

i . e . a n y h o l o m o r p h i c function on

a p p r o x i m a t e d u n i f o r m l y o n c o m p a c t s u b s e t s of on E

i s e m p t y o r a Runge

n nE

n

E o c a n be

by e n t i r e functions

. In p a r t i c u l a r a n y t r a n s l a t e of a balanced open s e t is f i n i t e l y

Runge. 5.1.10 R e m a r k .

-

F r o m (5. 1.8) (5.1.9) and proof (5.1.6),

Th.(5. 1.2)

and c o r o l l a r i e s r e m a i n t r u e if 0 is a f i n i t e l y Runge T E - o p e n s e t . S e e Colombeau-Meise-Perrot [ 1J , 5. 1.11 R e m a r k .

-

F o r o t h e r a p p r o x i m a t i o n r e s u l t s s e e chap. 14 below.

A density result

6

5.2

149

A d e n s i t y r e s u l t i n 8 ( 61, F)

We r e c a l l t h a t in (4.1.1) me defined two n a t u r a l t o p o l o g i e s o n

8 ( 0 ,F ) t h a t coi'ncide when E is a S c h w a r t z b.v. s.

5.2.1

THEOREM.

-

Let

E

.

be a r e a l S c h w a r t z b . v . s .

with

a p p r o x i m a t i o n p r o p e r t y ( 5 , l . 1) and l e t y b e a H a u s d o r f f l o c a l l y c o n v e x topology on E f o r which any bounded s u b s e t of E

(E,??).-

0 b

is bounded in t h e 1 . x

x T E - o p e n s e t and l e t F be a 1 . c . s .

. Then

6 ) ( ( E , e ) , F ) is d e n s e i n B ( R , F ) . f B e f o r e the proof l e t u s r e m a r k t h a t th ( 5 . 2 . 1) h a s c o r o l l a r i e s a n a l o g o u s t o (5. 1 . 3 ) and (5; 1.5) ; in p a r t i c u l a r :

COROLLARY.

5.2.2

real l,c.s.,

-

L e t E be a q u a s i c o m p l e t e d u a l n u c l e a r

bz a n open s u b s e t of E and F a 1 . c . s .

. Then

Ff(E,F)

is d e n s e in (s(61,F).

5 . 2 . 3 P r o o f of t h o (4.2. 1). -

F ( K , L, V , n ) = /q@(fi, where

L e t f E d(n, F) be given and l e t u s s e t F) such that v(~)(K)L'cV i f o 5 i 5 n

K i s a s t r i c t l y c o m p a c t s u b s e t of

n,

where

1

L is a bounded

s u b s e t of E , w h e r e n EN and w h e r e V is a c o n v e x b a l a n c e d o-neighborhood in F. t h a t K and on E

B

T h e r e is a bounded Bdndch d i s c B in E

such

L a r e c o m p a c t i n E B and s u c h t h a t the i d e n t i t y m a p p i n g

c a n be a p p r o x i m a t e d u n i f o r m l y on K and

continuous o p e r a t o r s on EB.

If

&

o i s given, t h e r e i s

such that sup x E K

L by f i n i t e r a n k

I l u ( x ) - x / ' B t- c

uE(EB)'@E B

Approximation and density results

150

s m a l l enough u(K) c 0 R E B .

For c > o

flnnE

o u is defined and

C 00

B on a n open neighborhood

ui

of K in E

B'

For

E > O

s m a l l enough n e

have c l e a r l y : i

(1)

If E o = f

f O

EB d e n o t e s t h e r a n g e of u , w e s e t 0

C

'*

1

Ci n E o and

E d ( n o , F ) . F r o m ( 5 . 2 . 4 ) and (5.2.5) below it follows t h a t 0

P ( E o ) b F i s d e n s e in 6(0 ,F). T h e r e f o r e t h e r e is h o e b(Eo) 0 F = P (E ) f o

ii" 0

If we s e t h = ho o u

QO

-

F such that :

hLi']

i 1 (u(K)). ( u ( L ) ) C T V

E 63f(EB)@ F

if

o 5 i 5 n.

:

F r o m ( 1 ) and ( 2 ) :

r

1

(3)

A t end of proof ( 5 . 1 . 6 ) i t is proved t h a t the r e s t r i c t i o n m a p p i n g

151

A density result

has dense range. T h e r e f o r e , since h

k 6 Pf(E,??)

E

Pf(E~)& F

8 F = P f ( ( E , Y ) ,F) s u c h t h a t

(i)

[k

(4)

-h

(i)

i (K).(L)

]

1 c7

, t h e r e exists

:

V if o S i S n

.

F r o m (3) and (4) : f- k

E 2/ ( K , L, V , n).

5 . 2 . 4 A review on N a c h b i n ' s A p p r o x i m a t i o n T h e o r e m . L e t 0 be a n m n open s u b s e t of IR and l e t C (0)denote t h e s p a c e of m - t i m e s r e a l valued continuously d i f f e r e n t i a b l e functions on with t h e c o m p a c t open topology of o r d e r m . Let A

be a s u b - a l g e b r a of

R ,

equipped a s u s u a l

W e assume m 2 1 ,

C m ( n ) . Then A

is d e n s e in

C

m

(C?)

if and only i f the following conditions hold :

f o r e v e r y xE

(2)

for every x , y

that g ( 4

E 0

with x

#

s u c h t h a t g(x)

y there is g

for every x E R

s u c h that g ' ( x ) . v

#

and v EIRn with v

#

A

o such

LEMMA.

-

Let Cf

o t h e r e is

0.

T h e proof of t h i s c l a s s i c a l t h e o r e m i s in Nachbin n c o n s e q u e n c e , P ( R ,IR) is d e n s e in d ( n , I R ) ,

5.2.5

E

#

f dY).

(3) g E A

t h e r e is g E A

R

(1)

be a n open s u b s e t of

o r c o m p l e x 1. c . s . Then d(n)

[6]. A s a

IRn and l e t F be a r e a l

@F i s d e n s e i n d ( n , F).

152

Approximation and density results

Proof. -

We m a y a s s u m e without loss of g e n e r a l i t y t h a t F is a r e a l

F u e m a y a l s o a s s u m e without loss of

1.c. s . .Using t h e c o m p l e t r o n of

g e n e r a l i t y that

. With the u s u a l n o t a t i o n s of t h e

F is a complete 1.c.s.

c l a s s i c a l book of S c h w a r t z [ 2 , on D i s t r i b u t i o n T h e o r y i t s u f f i c e s t o p r o v e

z

t h a t b ( R )p F i s d e n s e i n E(R, c o m p a c t s u p p o r t in (P1(x) 2

0,

0). Let

til€ J(n),

F) ( i . e . t o c o n s i d e r f u n c t i o n s w i t h

(F1)lEI

bt a

p a r t i t i o n of unity in

Cm

B,(x) = 1 ) and l e t

( s 1)l E I

h

be e l e m e n t s of

1EI

n

s u c h that

.&( n)

pi({l) #

o , Let J

be the m a p p i n g f r o m

8(64,F) into

F defined by

(Jdx

If EN is a s e q u e n c e of s u c h m a p p i n g s s u c h t h a t the d i a m e t e r of t h e subdivision of 0 a s s o c i a t e d to J x t e n d s to o when t 00,

x v ) (x)

t h e n one p r o v e s e a s i l y t h a t ( J in a c o m p a c t s u b s e t of

-

m € N ) , if

t e n d s to + ( x ) u n i f o r m l y i f x i s

G and cp in a bounded s u b s e t of L ( n , F ) . T h u s

introducing the s p a c e s B to order

-

m

( 0 , F) ( w h e r e d e r i v a t i v e s i i r e c o n s i d e r e d u p

Jhv

6 B ( 0 , F ) then

-

C&

in D o ( n , F ) when

tco.

Now l e t

p

be a f u n c t i o n in

n

&(R ) with P

p o and

[

p(x)dx=l.

Rn If

E B o ( n , F ) and x E IRn we m a y c o n s i d e r t h e i n t e g r a l

T h e mapping

R defined by t h i s f o r m u l a is l i n e a r continuous f r o m

D o ( R n , F ) into B ( R n ; F ) . We c o n s i d e r a s e q u e n c e (C )

P PGN

Of

above

A density result f u n c t i o n s s u c h thdt t o o if p

-

t

.

03

P

( 0 ) f o and t h e d i a m e t e r of s u p p o r t of P tends /J P T h e n one p r o v e s e x a c t l y a s i n t h e s c a l a r c a s e ,

Schwartz [ 1 3 , that if

Nou l e t

B o ( R , F) if flxed

-

and f o r

i-(

153

E B ( i 2 , F ) then R

-

P

in B ( n , F ) .

be a g i v e n e l e m e n t of B ( R , F ) . J h q

CT

t

A

00

. Therefore t

M.

. f + ( n , F )T.h e r e f o r e s i n c e R

R (Ja v )

R rp i n d ( 0 , F ) f o r

3

1

I-r

Furthermore if 1

(Jhv) E IJ

.&(a)Qp

y in

+cul

RPQ

F we o b t a i n t h a t

-

ec

h ( n )& F i s d e n s e i n & ( n , F ) .

5.2.6

THEOREM.

Let -

-

a p p r o x i m a t i o n p r o p e r t y . L& complex 1 . c . s .

.I

~ =Y i

0

X J

T E - o p e n s e t and l e t F be a

eiqj

finite

d e n s e in

be a r e a l S c h w a r t z b. v . s . u i t h

E

1 C.EC J Tj'

, t&Y@F& x

E

a(n ,F).

T h e proof i s s i m i l a r t o t h a t of

t h . ( 5 . 2 . 1).

in

154

Approximation and density results

8

5 . 3 . 1 THEOREM.

cm

5.3

-

p a r t i t i o n s of unity

__ Let E

be a r e a l 1. c . s. which h a s a b a s e

of p r e - H i l b e r t i a n o - n e i g h b o r h o o d s and l e t U c E

be a n oDen s u b s e t

Let %

which is a Lindelbf s p a c e f o r t h e induced topology.

T h e n t h e r e e x i s t s a s e q u e n c e ( ' Y ) of C functions n which have r a n g e i n the r e a l i n t e r v a l [ 0 , 1 ] a n d s u c h t h a t :

c o v e r i n g of- U .

on

E,

be a n open_ 03

every x E U admits

1)

f i n i t e n u m b e r of the s e t s

/ x €U

t h e c l o s u r e in E

2) e l e m e n t of

d

neighborhood which i n t e r s e c t s onlv a

such that

Y

n

(x)

of e v e r y s e t s u p p \Y

zi

>o 1 n

i s contained in a n

n

5.3.2 Remark.

-

F r o m t h e proof below it follows thdt the f u n c t i o n s

obtained h e r e a r e C

m

of u n i f o r m bounded type o n E , s e e ( 1 . 6 . 2 ) .

5 . 3 . 3 P r o o f of t h . 5 . 3 . 1. -

Since E a d m i t s a b a s e of c l o s e d

o - n e i g h b o r h o o d s t h e r e e x i s t s a n open c o v e r i n g every

U' E 24' i t s c l o s u r e

e v e r y open s e t V in U function f on E , and

supp f c V

n

'lit

of

U

such that for

i s contained in s o m e e l e m e n t of

and f o r e v e r y x E V

there exists a

%. F o r

Cm

of u n i f o r m bounded t y p e , s u c h t h a t f 5 0 , f ( x ) > o

( b e c a u s e t h i s i s t r u e in p r e - H i l b e r t s p a c e s , u s i n g

exactly t h e s a m e function a s in Bn). T h e s e t

( y EE such that f ( y ) h

is contained i n V and o p e n i n E .

V

a union of o p e n s e t s of the type

Therefore

{y EE

\

m a y be considered a s

I

s u c h t h a t fi(y) > o

when i

r a n g e s o v e r a c e r t a i n s e t of i n d i c e s . T h e r e f a r e t h e r e e x i s t s a n open covering finer than fi(y) 7 0 Since U

I

and m a d e of s e t s of t h e t y p e

w h e r e the functions f . a r e

C

W

IyEE

such that

of u n i f o r m bounded t y p e .

is Lindel6f t h e r e e x i s t s a c o u n t a b l e c o v e r i n g ( W . )

J

jEm

where

c a, partitions ofunity W . = Iy E E such that f.(y) 3 o J J

V

a n d , if

1

1 .

W e set :

= { y eE s u c h t h a t f ( y ) 1

o

I

r 6 IN

= / y E E such that f

‘rt

Therefore V

r+ 1

c Wr+

.

Since

rt

( y ) > o and f . ( y ) C i i f i r

(W,)

J j EN

e v e r y x E U t h e r e is a n index n(x)

is a c o v e r i n g of

U,

I.

for

IN s u c h t h a t in(.)(”)3 o and

f . ( x ) = o if j <: n(x). T h e r e f o r e x E V J is a c o v e r i n g of U .

1s i 2 r

’

and thus t h e f a m i l y ( V )

r rEN

L e t x 6 U be given and l e t a ( x ) 3 o be s u c h t h a t

o < g ( x ) < fn(x)(x). We s e t :

’V = { y E E s u c h that f X

P ’

X

is a n open neighborhood of x i n U .

’r+ 1

4x1(y)

1.

3 &(x)

Since

c { y E E such that f

( Y ) <1 ~ ) 4x1

, Vx n V r fl f o r r l a r g e enough. L e t 1 be a Cco function f r o m IR to IR s u c h t h a t b ( x ) P o if x < o and P ( x ) = i f n(x) < r

o if

x 2 0 . We s e t

i s a Cco function f r o m E to Rs = cs , t a),of u n i f o r m bounded ‘rt1 1 ( x ) = o o r f . ( x ) > - f o r a t l e a s t one index i type. If x f V r t l then f rt1

Approximation anddensity results

156

s u p p y n c Vn

and t h u s ( x ) = o if x y V r t l , rtl f o r e v e r y n 6 IN , and t h e r e f o r e s u p p tp is contained in

a n e l e m e n t of

Q.

with

15 i

5 r.

-

Therefore

cp

n ey (x)9 0 . If x $ V then 4x1 n Q (x) = 0 . Since 6’ i n t e r s e c t s only a f i n i t e n u m b e r of t h e sets X n ’n’ t h e n only a finite n u m b e r of the functions a r e d i f f e r e n t f r o m o on ‘n ?Jx. T h e n the functions For every x 6 U,

h a v e a l l the r e q u e s t e d p r o p e r t i e s .

5.3.4 Remark. s p a c e and i f U

-

If E i s a n u c l e a r Silva s p a c e o r a n u c l e a r F r C c h e t is a n y open s u b s e t of

E t h e n E and U h a v e a l l p r o p e r -

t i e s needed in (5. 3. 1) ( s e e P i e t s c h [ 1 ) 4.4. 10).

E

CHAPTER 6 -PRODUCT AND KERNEL THEOREMS

ABSTRACT. If E and F a r e 1 . c . s . we r e c a l l t h a t the S c h w a r t z

E

-product

of E and F i s defined by

where E '

C

i s the d u a l of E

equipped with i t s c o m p a c t open topology and

w h e r e t h e s u b s c r i p t e s t a n d s f o r the topology of u n i f o r m c o n v e r g e n c e on the equicontinuous s u b s e t s of E l . We obtain : THEOREM.

-

Let

E be a c o m p l e x q u a s i - c o m p l e t e 1. c . s . ,

s u b s e t of E a n d F a c o m p l e t e 1 . c . s .

a lg e b r a fc a 11y and to polo g i c a 11y

THEOREM. -

fi

Let

.

62 a n open

Then

.

E be a r e a l q u a s i - c o m p l e t e c o - S c h w a r t z 1. c . s.,

a n open s u b s e t of E

and

F a complete 1.c.s.

a l g e b r a i c a l l y and topologically. 157

. Then

158

E

- product and kernel theorems

A s i m m e d i a t e c o n s e q u e n c e s w e obtain the following k e r n e l theorems : COROLLARY.

-

E and F be c o m p l e x ( r e s p e c t i v e l y r e a l ) c o -

S c h w a r t z 1.c. s . , R

R' open s u b s e t s of E

a&

F respectively.

T h e n the a l g e b r a i c and topological i s o m o r p h i s m s ( i n the c o m p l e x and r e a l c a s e r e s p e c t i v e l y ) hold :

T h e y a r e given by t h e f o r m u l a

O t h e r c o n s e q u e n c e s a r e s u f f i c i e n t . conditions on t h e a p p r o x i m a t i o n p r o p e r t y :for

xs(0)

and c$(fZ).

Holornorphic functions

5 6. 1

159

The S c h w a r t z & - p r o d u c t in s p a c e s o f h o l o m o r p h i c functions

E be a c o m p l e t e c o m p l e x b.v. s . ,

6 . 1 . 1 THEOREM. a T E - o p e n s e t and

a c o m p l e x c o m p l e t e 1.c. s.

F

algebrai'callv and topolgeicallv, w h e r e 3-C

s, e

n

. Then

(f'j,F)

denotes a s usual the

s p a c e of Silva h o l o m o r p h i c m a p p i n g s i n the e n l a r g- e d s e n s e .

F r o m th (2.3.4) an

equivalent f o r m u l a t i o n is :

Let

6 . 1 . 2 THEOREM. a T E - o p e n s e t and

E be a c o m p l e x S c h w a r t z b. v . s.,

F a complex complete 1.c.s.

.TI'

n

algebrai'callv and topologically.

6. 1.3 P r o o f of 6 . 1 . 2 . r e f l e x i v e 1. c . s .

xs(h2) ( s e e

x s(h2)

is a n a t u r a l l y

. T h e r e f o r e f r o m ( 0 . 3 . 11) t h e s t r o n g topology on

x'S (0)c o i n c i d e s with of

F r o m (4.4. I )

the topology T K

I

S

(0).F r o m the Monte1 p r o p e r t y

$ 3 . 4 ) t h i s topology coi'ncides a l s o with t h e topology

(Xs(n))lc of u n i f o r m c o n v e r g e n c e on t h e c o m p a c t s u b s e t s of u s c o n s i d e r the mapping A EL(T3(C'S(S2);F),

G defined on C ( T K

then w e s e t

I

S

(n),F)

xS (0). L e t .

by : if

160

f

"Is(n)

(6

i s the D i r a c m e a s u r e at t h e point a E

p r o v e t h a t the mapping and r a n g e

S

0

S

(0) =

XIx

S

En , thus f

=

a

= (TX'

(n)

0.

S

by (4.4.l ) ,

f

E

for every a E 0 ,

= o

xs(n) and

f(a)

G is i n j e c t i v e .

L E

(f2) S

sc

c x ' S (n)

A E V(K,&)

sc

from K

€

K

and if F

0

( t h e p o l a r of V ( K , E ) in

0

'

>

o a r e such that

K Is(n)),

*

I

Icp ( k ) ( y ' ) 5 1.

Therefore

* cp

F u r t h e r m o r e cp

maps 3t-

X

S 0

(1)E ( E K ' )

0

1

E

If

3C ( ( 2 ) S

&

F.

sc

&

S

(n)

* c

(A)EF.

(a) E

1 -K' E

if

T o prove

i n t o F it s u f f i c e s t o p r o v e t h a t 0

TX'

L E V ( K , E ).

S

(n) ,

G i s a bijection f r o m

jc

S

and t h i s follows

Therefore

I t is i m m e d i a t e t o c h e c k t h a t G ( q ) =

surjective, therefore

on F' ,

and t h u s

(bipolar t h e o r e m 0.3.6).

= -K'

is continuous f r o m

f r o m the fact t h a t cp cp

K'.

( 0 ) into F and q y c i s c l e a r l y l i n e a r .

it is bounded o n a n y equicon tinuous s u b s e t of

sc

C

w e s e t K' t o

*( 1 ) is continuous

Therefore

F r o m t h e M a c k e y - A r e n s t h e o r e m (0.3. 10) ( F ' ) ' = F

that

to

( K ' c F i s the p o l a r of K ' ) t h e n y ' o cp E V(K, E ) and

therefore

* cp

(n)

( A ) : F'

be a convex balanced c o m p a c t s u b s e t of F with y ( K )

E

S

,

If K i s a s t r i c t l y c o m p a c t s u b s e t of

y'

(n).

S

jr

cg

0

= o for every

is d e n s e in Tx'

T h e r e f o r e the s e t

the algebrai'c d u a l of F ' . We define a m a p p i n g y

0

I t is e a s y t o

be a g i v e n e l e m e n t of k s ( n , F ) . We d e n o t e by (F')

Nom l e t b y , if

.

i s contained in 3-L ( 0 , F ) . S

G

(0))'and i f f ( 6 ) cc

A s a c o n s e q u e n c e the m a p p i n g

(F')'

gl)

is Silva h o l o m o r p h i c with d o m a i n 0

-+

T h e r e f o r e t h e r a n g e of

(0).

E K:(n)

If f since

- product and kernel theorems

(n) E

v.

Thus

F onto

G is

KS (62,F).

Holornovphie fu nciions

161

L e t K be a s t r i c t l y c o m p a c t s u b s e t of balanced o - n e i g h b o r h o o d in F. 0

?J =

T(T(K, 1),V)

=

and

V a c l o s e d convex

With the n o t a t i o n s of (4.1 . 2 ) we s e t

1 .& eg(X(S(fl),F) s u c h

that

a(?(,,

I))c V

I.

If A E 1/ then c l e a r l y G ( A ) E a ( K , V ) which p r o v e s t h e continuity of the m a p p i n g

lA(yl o

G. 0

Now if cp

E ? J ( K , V ) , A E ?f ( K , I ) , y ' E

c,c)I

thcrefore

2 1,

Tx(A)

8 =V

v" c F' t h e n

(bipolar theorem),

T h e r e f o r e C i - ' ( v ( K , V ) ) c %' which p r o v e s t h e continuity of t h e m a p p i n g -1 G . I

6.1.4

COROLLARY ( K e r n e l t h e o r e m ) . -

c o m p l e x b.v. s., 0

& 0' k ' T E

Let E

and TF-open

T h e n the following topological i s o m o r p h i s m h o l d s

and t h i s i s o m o r p h i s m is given by

:

F be c o m p l e t e s e t s respectively.

I62

E

If we r e p l a c e E and

Prooi. -

E

and F

S.C.

- product and kernel theorems

S.C.

F by the a s s o c i a t e d S c h w a r t z b . v . s .

r e s p e c t i v e l y (0.7.4),

the s p a c e s of h o l o m o r p h i c

functions and t h e i r topologies d o not c h a n g e , ( f r o m 2. 3 . 4 and 2. 3. 7 ) s o we m a y a s s u m e f r o m the beginning t h a t E and

F a r e Schwartz b.v.s.

.

N o w it s u f f i c e s t o apply th. ( 5 . 1 . 2 ) and l e m m a ( 6 . 1.5) below.

6. 1 . 5 LEMMA. - Let E - 7F-open s e t s R' be T E and

F be c o m p l e x S c h w a r t z b . v . s . ,

R a&

r e s p e c t i v e l y . T h e n t h e following topologi-

c a l i s o m o r p h i s m s hold :

KS(R

x

n') =

3(

S

(ill,

"$2))

T h e proof is quite i m m e d i a t e .

6. 1. 6 R e m a r k . -

A d i r e c t proof of (6. 1.4) i s in C o l o m b e a u - P e r r o t [ 4

1.

T h e following c o u n t e r e x a m p l e s show t h a t t h e s e r e s u l t s a r e not valid f o r continuous h o l o m o r p h i c f u n c t i o n s i n p l a c e of Silva h o l o m o r p h i c functions.

6. 1.7 C o u n t e r e x a m p l e . m a y h a v e , f o r c e r t a i n 1.c. s .

XS(E)

E

F

If E

is a c o m p l e x c o m p l e t e b . v . s .

F , "is, ( E , F ) 3 K S ( E , F ) ( s e e 2 . 5 . 3 ) .

f

# XS(E, F).

6.1.8 Counterexample. general K ( E )

E

F

# K ( E , F)

If E and

F a r e complex l . c . s . ,

[ 1]

.

Then

in

( t h e s e s p a c e s of h o l o m o r p h i c f u n c t i o n s a r e

equipped with the c o m p a c t open topology). An e x a m p l e i s given i n Dineen

ue

163

Holomorphic functions

6 . 1. 9 C o u n t e r e x a m p l e on c o r o l l a r v ( 6 . 1. 4). 1. c . s.

Let E

be a c o m p l e x

and l e t E ' denote i t s s t r o n g d u a l . W e a s s u m e t h a t both E

E' a r e dual n u c l e a r q u a s i - c o m p l e t e 1. c. s . duality b r a c k e t between E and

El.

< ,

>

the

We denote by L the mapping

w h e r e < ~,a 9

E Ks(E1)

t h i s mdpping

L we h a v e , f r o m (6.1.4),

is the mapping T

E ) and N L f ' J C ( E ' x E )

. We denote by

and

-

< T , a 3 if T

e

El. F o r

N (a,T)= .

L

a s i t h a s been a l r e a d y noticed

(El NLf S s e v e r a l t i m e s . One m a y c h e c k e a s i l y t h a t

L 6 S((K(E'))' , X ( E ) ) = K ( E ' ) E K ( E ) . On t h i s e x a m p l e s e e Colombeau - P e r r o t [ 3 ] appendix 2.

:

164

E

9

6.2

- product and kernel theorems

The Schwartz

E

- p r o d u c t i n s p a c e s of

C

03

functions

T h e proof of t h i s s e c t i o n a r e e x a c t l y s i m i l a r to t h e p r o o f s i n

$ 6. 1 , although t h e y a r e m o r e t e c h n i c a l to w r i t e i n d e t a i l . So w e j u s t s t d t e the r e s u l t s and give r e f e r e n c e s f o r p r o o f s .

6.2.1

Let E

THEOREM. -

' i E - o p e n s e t and

F a complete 1.c.s.

n

be a r e a l S c h w a r t z b.v. s .

.T

a_

U

a l g e b r a r c a l l y and topologically.

A m o r e g e n e r a l r e s u l t valid f o r Meise [ 1 6.2.2

1.

See a l s o Meise

i 1 1.

COROLLARY. -

n &

Cn m a p p i n g s is i n C o l o m b e a u -

L A E &a

F be r e a l S c h w a r t z b.v. s.

L T E a n d T F - o p e n s e t s r e s p e c t i v e l y . T h e n the following

topological i s o m o r p h i s m holds.

T h e p r o o f i s obviously a d i r e c t c o n s e q u e n c e of ( 6 . 2 . 1 ) w i t h ( 6 . 2 . 3 ) b e l o u . A d i r e c t proof is in C o l o m b e a u 11 1).

M e i s e [ 13.

See aIso Colombeau-Meise

&

1 ] and

c a,functions 6 . 2 . 3 LEMMA. -

be

T E

Pnd

Let

E a n d F be r e a l S c h w a r t z b . v . s . ,

165

0

and

0'

T F - o p e n s e t s r e s p e c t i v e l y . T h e n t h e following t o p o l o g i c a l

i s o m o r p h i s m holds

:

T h e p r o o f is s t r a i g h t f o r w a r d but c u m b e r s o m e t o w r i t e in d e t a i l . It i s in Colombeau

[

1

1,

see also Meise [ 1 1

.

166

E

- product and kernel theorems

§ 6 . 3 A p p r o x i m a t i o n p r o p e r t y in ks(n)

6. 3 . 1 R e c a l l , -

We need t h e following r e s u l t of Si-hwartz r 3 1

L e t E be a q u a s i - c o m p l e t e 1.c.s. property (i.e.

d(n)

:

Then E has the approximation

E ' 8 E is d e n s e in g ( E , E ) f o r t h e c o m p a c t open topolo-

g y ) if and only i f E

@

F is d e n s e i n E

E

F f o r all B a n a c h s p a c e s F.

N m f r o m (4. 2. 1) ( 5 . 1 . 2 ) and ( 6 . 1. 1) i t follows : 6.3.2 THEOREM. -

3 E

be a c o m p l e x c o m p l e t e b. v. s.

with a p p r o x i m a t i o n p r o p e r t y (defined in 5. 1. 1) and a balanced

R a n y t r a n s l a t e of

' r E - o p e n s e t ( m o r e p e n e r a l y 0 m a y be any finitely Runge

'TE-open s e t , s e e 5.1.10).

T h e n the 1.c. s .

xs(n) h a s

the a p p r o x i m a -

tion property.

In the r e a l c a s e it follows f r o m ( 4 . 2 . 4 ) , (5.2. 1) and ( 6 . 2 . 1 ) that : 6.3.3

THEOREM.

-

L e t E be a r e a l S c h w a r t z b . v . s .

a p p r o x i m a t i o n p r o p e r t y ( 5 . 1. 1) and 1. c . s .

n any

with

TE-open s e t . T h e n the

d(62) 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 .

CHAPTER 7 THE FOURIER-BOREL AND FOURIER TRANSFORMS

A BSTRA C T . If E transform &

if

i s a c o m p l e x 1 . c . s . we r e c a l l t h a t t h e F o u r i e r B o r e l i s the mapping f r o m 3C'

, t E k ' ( E ) and S

THEOREM.-

7

s (E) t o K

(El)

defined by :

EE'.

If E i s a d u a l n u c l e a r q u a s i c o m p l e t e 1. c. s . t h e n t h e

F o u r i e r B o r e l t r a n s f o r m i s a topological and b o r n o l o g i c a l i s o m o r p h i s m

of 3i

IS(E) onto t h e s p a c e E x p ( E ' ) of e n t i r e f u n c t i o n s of e x p o n e n t i a l type

on E ' ( i . e . h o l o m o r p h i c functions where 17

IB=

c

>

o i s constant, where

SUP x E B

I+)

$

o n

El

such that

B i s a bounded s u b s e t of E

exp

171

and

B

.)I

In f a c t we p r o v e m o r e g e n e r a l r e s u l t s in c a s e E n e c e s s a r i l y d u a l n u c l e a r and f o r open s u b s e t in E .

I$(T)15 c

Xls(n)

All these results a r e

where

is n o t

R is a c o n v e x b a l a n c e d

f u n d a m e n t a l t o o l s f o r the s e q u e l .

The F o u r i e r isomorphism (in the r e a l c a s e ) i s m o r e complicated than the F o u r i e r - B o r e 1 i s o m o r p h i s m : if E h a s a n infinite d i m e i i s i o n a l bounded s e t t h e u s u a l f o r m u l a t i o n i s no l o n g e r valid. 5 8 ' ( ~ )the i m a g e through the F o u r i e r t r a n s f o r m functions on E l t i E '

5 of t h e s p a c e & ' ( E ) ,is a s p a c e of e n t i r e

( E ' d e n o t e s t h e r e a l d u a l of E ) t h a t , b e s i d e s the 167

168

Fourier-Bore1 and Fourier transfomis

u s u a l inequality, s a t i s f y a s u p p l e m e n t a r y t e c h n i c a l condition (which holds a l w a y s if E

i s finite d i m e n s i o n a l ) . T h i s condition m a k e s i t difficult t h e

u s e df t h i s F o u r i e r i s o m o r p h i s m a s a tool. So in o r d e r t o c i r c u m v e n t t h i s difficulty we define and study a d e n s e s u b s p a c e IE(E) of

B(E),

equipped with a p r o p e r topology, s o t h a t 5 lE'(E) is m a d e of t h e e n t i r e functions on E l t i E l

that s a t i s f y only the u s u a l inequality. Since

IE (Rn) = 8(IRn) if n 6 IN t h i s s p a c e E ( E ) is a n a t u r a l c a n d i d a t e a s s p a c e of

C

transform.

CD

functions on E when one n e e d s t o u s e t h e F o u r i e r

169

Preliminaly results

5

7. 1

P r e l i m i n a r y r e s u l t s on the F o u r i e r - B o r e 1 t r a n s f o r m

Let E

be a c o m p l e x S c h w a r t z b . v . s .

s t r i c t l y c o m p a c t s u b s e t of E .

dnd

K a convex b a l a n c e d

T h e r e is s o m e c o n v e x b a l a n c e d bounded

a B a n a c h s p a c e and K c o m p a c t i n E We B B' m a y a s s u m e f r o m ( 0 . 7 . 1) t h a t K = /xn = fA Ixn f o r s o m e n u l l 1 s u b s e t B of E

with E

1

in E s e q u e n c e (x ) n nE IN B'

For n

E IN

the s p a c e L("E)

i s equipped w i t h its n a t u r a l topology

of u n i f o r m c o n v e r g e n c e on t h e s t r i c t l y c o m p a c t ( o r equivalently bounded n I t s d u a l L'("E) is s i n c e E i s a S c h w a r t z b.v. s . ) s u b s e t s of ( E )

.

equipped w i t h the equicontinuous bornology. We s e t V(B)= { u E L ( ~ E )s u c h t h a t

For every n E

IN

Iu(x

,

1

..., x n ) 15 1

if x i E B

1

.

we c o n s i d e r t h e m a p p i n g i

i(kl,.

. ., kn)

defined by

if u t L ( n E )

.

We n o t e k w...@kn 1

(L'(nE))o

for i(k

l,. . ., kn).

is a B a n a c h s p a c e dnd t h e m a p p i n g i

YB) (EB)n to (L'(nE))o

YB)

from

n is continuous. So i(K ) is c o m p a c t in t h e B a n a c h

170

Fourier-Bore1 and Fourier transforms

n

, t h e r e f o r e i ( K ) is s t r i c t l y c o m p a c t i n the b.v. s .

space (L'(nE))o

1JB) L f ( n E ) . So f r o m (0.7. 1) f ( i ( K n ) ) is a convex balanced s t r i c t l y I1 c o m p a c t s u b s e t of L'("E).

We note Kmn f o r i(Kn).

The bornological d u a l E x of E is equipped with its n a t u r a l n x topology ( 0 . 3 . 3 ) and we denote by s( E ) the l i n e a r s p a c e of n - l i n e a r continuous s c a l a r valued m a p p i n g s on E X . $ ( n E x ) i s equipped with t h e bornology of the equicontinuous s e t s and is thus a c o m p l e t e b.v. s.

. We

define a mapping I :

defined by I(A) ( T I , .

if

T i c E X and if

T

I

@...@

T

n

.. T n ) = A ( T 1 .

E L(nE)

.

@.

I

19 i 5 n.

Tn)

is defined by

(T1@. . 8 T n ) ( x l , .

if xi E E ,

.@

.. , x n ) = T 1(x1) . . . Tn(xn)

It is i m m e d i a t e t o c h e c k t h a t I i s l i n e a r bounded

and injective i f (Ex)@" i s d e n s e in L("E)

(i.e

if a n y u E L("E)

may

be a p p r o x i m a t e d u n i f o r m l y on a n y s t r i c t l y c o m p a c t s u b s e t of E by elements

T1

... 0 T n

with T i €Ex). F r o m a s i m p l e r c a s e

finite of (5. 1 . 2 ) t h i s l a s t p r o p e r t y holds if E i s a b.v. p r o p e r t y (5.1.1).

S.

with a p p r o x i m a t i o n

For m o r e s i m p l i c i t y w e a s s u m e f r o m now on t h a t E

is a b.v. s. with A p p r o x i m a t i o n P r o p e r t y and we d o

not d e n o t e

Preliminary results

e x p l i c i t e l y the m a p p i n g I , in

171

t h u s c o n s i d e r i n g t h a t L'(nE) is contained

EX). If

fl

.

in C("EY)

is a continuous h o l o m o r p h i c function on E x

We s e t , i f K is a n y s t r i c t l y c o m p a c t

f K]

=

{fl E5C(Ex)

s u c h t h a t f o r e v e r y n EN

r

N o t i c e t h a t t h i s definition m a k e s s e n s e s i n c e

e

T

If T F E : x

-

x

n 21

KBnC L ' ( n E ) ~ C ( n E x ) .

it is i m m e d i a t e t o c h e c k t h a t the function

e x p ( T ( x ) ) ( w h e r e x E E ) is i n 3c (E).

S

Let

7 . 1 . 2 DEFINITION. -

n

is

So w e m a y define :

7.1.1 DEFINITION. s u b s e t of E ,

t h e n fl("'(0)

L T E - o p e n s e t . If_ L

if T E E x .

E xL(0)

E

be a c o m p l e x S c h w a r t z b . v . s . and

w e define a function 3 L = E X

It is i m m e d i a t e t o c h e c k t h a t 5L

continuous on E x ,

G a n a l y t i c and

3L E3C(E').

The mapping

3

: x g?) L

is c a l l e d the F o u r i e r - B o r e 1 t r a n s f o r m .

X(EX)

5L

by_:

Fourier-Bore1 and Fourier transforms

172

n

If K is a s t r i c t l y c o m p a c t s u b s e t of by

T ( K ) the following o-neighborhood of

0

(h2)

I f(x) I

%'(K)= { f C X s ( n ) s u c h t h a t s u p xfK We denote by

S

we r e c a l l t h a t w e denote :

5 1

1

.

x'S(n).

V ( K ) i t s p o l a r in

Now w e a s s u m e t h a t R is convex balanced. In p r o p . ( 7 . 1 . 3 ) we choose K of the type K = in

n

n

rL

(x ) w h e r e (x ) is a null s e q u e n c e n n nEIN E B ( w h e r e B is s o m e bounded Banach d i s c ) . F r o m ( 0 . 7 . 1)

1

a n y s t r i c t l y c o m p a c t s u b s e t of

0 is contained in s o m e s e t of t h e a b o v e

t Y Pe.

7 . 1 . 3 PROPOSITION.

-

T h e i m a g e through the F o u r i e r - B o r e 1

( K ) i s contained i n t h e s e t [ K ]

t r a n s f o r m of t h e s e t

.

C o n s i d e r t h e duality between L(nE) and L'(nE). Since Proof. On n ra ( K ) is a convex balanced s t r i c t l y c o m p a c t s u b s e t of L'( E ) it 1 cgn follows f r o m t h e bipolar t h e o r e m ( 0 . 3 . 6 ) t h a t I; (K a ( K ) Jo0 = 1 1

r

rA

We define a mapping J f r o m L ( n E ) to K S ( E ) by

(notations of 0.8.3) continuous,

If ep

if

Err,

cp E L ( n E ) and x E E .

(KBn)JO c L ( ~ E )then JY, c v ( K ) . 1

LE; (K) c X t S ( E ) then / L ( J ' p ) ) then

I En(cp) 151.

Therefore

n

Clearly J i s linear

6:

5

1.

If

If we s e t .X = L o J E L ' ( n E ) n ( K @ n ) ] O O = r , (K@ n).

5

1

Now i f

173

Preliminary results Therefore

n

S i n c e L'(nE) C S ( " E x )

~ L ( oJ )= j L ( i )

1

s

1

n

( s e e above),

( 3 L )( n )(0)= En.

Furthermore

.

L e t e d e n o t e t h e r e a l n u m b e r s u c h t h a t L o g e = 1. a s s u m e now t h a t eK c 0. L e t k

>1

be s u c h t h a t ke K c 62.

If # EK(E*)

7 . 1 . 4 PROPOSITION. -

Let us

i s in [ K ]

and i f

cp

E Ks(n)

then t h e n u m e r i c a l s e r i e s

n

is c o n v e r g e n t (V(n)(o).$( n )(0) h a s a m e a n i n g s i n c e p(4(0) E L(nE) #(")(o)

c

L I ( ~ E ) ) We . set :

L n V

3

$

i s a n e l e m e n t of X l s ( 0 )

and

n!

Fourier-Bore1 and Fourier transforms

114 Proof.

-

F r o m Cauchy's i n t e g r a l formula

-

ke choose r . = , a i E K and c p E q k eK ). i n

lli151

with

and

a

.E K.

i, J

Then

Therefore

i

It is i m m e d i a t e t o c h e c k t h a t 3 and T h e r e f o r e in c a s e 0 = E we obtain :

V

3 a r e i n v e r s e mappings.

Preliminary results

Let

7 . 1 . 5 COROLLARY. -

175

E be a c o m p l e x S c h w a r t z b.v. s.

with a p p r o x i m a t i o n p r o p e r t y . The i m a p e 33C' ( E ) CK{EX) of 3c IS(E) S through the F o u r i e r - B o r e 1 t r a n s f o r m 5 is t h e l i n e a r s p a n of the s e t

CK] & ( E X ) when K r a n g e s o v e r a b a s e of bounded s e t s in E . If we equip t h i s s p a c e 3 K ' (E) with t h e bornoloEy defined by t h e S h o m o t h e t i c s of t h e s e t s [ K ] a s a b a s e of bounded s e t s , then 5 b o r n o l o g i c a l i s o m o r p h i s m from x f S ( E ) , equipped with its equicontinuous bornology, onto t h i s s p a c e W ' ( E ) . S If w e equip x t S ( E )with i t s s t r o n g d u a l topology and & F S ( E ) with the bornological topology a s s o c i a t e d t o its above bornology ( s e e

0 . 2 . 7 ) t h e n 5 is a l s o a topological i s o m o r p h i s m .

Proof.-

U s e ( 7 . 1.3) and ( 7 . 1.4) . F o r t h e topological i s o m o r p h i s m it

s u f f i c e s to r e m a r k t h a t T 3C.l ( E ) coi'ncides with t h e s t r o n g d u a l topology S of 3CtS(E) which follows f r o m (4.4.1)and ( 0 . 3 . 1 1 ) . 1

7.1.6 Remark. -

If

fl E[K]

then

I$(T)

I

e x p IT

IK

for any T E E Y

( i m m e d i a t e computation). T h e r e f o r e 3g ( E ) is m a d e of e n t i r e S

functions of exponential type o n E X . T h e c o n v e r s e i s p r o v e d in the n e x t s e c t i o n if

E is a n u c l e a r b.v. s. (th. 7 . 2 . 1).

Fourier-Bore1 and Fourier transfonns

116

7.2

!$

The Fourier-Bore1 isomorphism

7 . 2 . 1 THEOREM. -

be a c o m p l e x n u c l e a r b . v . s.

356' ( E ) of3C'

( E ) t h r o u g h the F o u r i e r - B o r e 1 t r a n s f o r m S is the l i n e a r s p d c e of t h e h o l o m o r p h i c functions fl EX s u c h that :

T h e n the i m a g e

3

LA E S

t h e r e e x i s t a convex balanced bounded s u b s e t B c

2

of

E and a n u m b e r

o such that :

for every T E

{fli li,

E'.

If we d e f i n e the bounded s u b s e t s of

3F ( E ) a s the f a m i l i e s

S s u c h t h a t the a b o v e inequality i s valid with the s a m e c d

for a l l i

E I,

B

then 3 is a b o r n o l o g i c a l i s o m o r p h i s m f o r K ' ( E ) S

equipped with its equicontinuous bornology onto 3 K ' ( E ) . S

5 is a l s o a topological i s o m o r p h i s m f r o m

x'S (E)

equipped w i t h

its s t r o n g d u a l topology onto the 1. c . s. T33C IS(E) l i . e. the b o r n o l o c i c a l topologv of 3 3 dS(E)). ~ B e f o r e t h e proof w e need :

7.2.2

LEMMA.

-

U

X be a c o m p l e x s e m i - n o r m e d s p a c e and l e t h

be a n e n t i r e function of exponential t y p e o n X

and I h ( x ) 1 T c

e x p I. lixll X c o n s t a n t s and w h e r e

//

]Ix

for every x E X ,

e

with c , p s o m e p o s i t i v e

d e n o t e s t h e s e m i - n o r m of X ) .

e v e r y n E IN :

where

h is G-analytic

i s defined a s u s u a l by L o g e = 1.

Then for

177

Fourier-Bore1 isomorphism

Proof.

-

F r o m Cauchy’s integral formula

f r o m which w e d e d u c e t h a t if

Ih (n)( o ) x l . . . x n l

R =

n P

-

\)xi

5 n

1 1 51

c exp(’R) Rn

for every

R >o,

g i v e s the r e s u l t .

7 . 2 . 3 P r o o f of t h 7 . 2 . 1. -

h [ K ] ( with

>

‘sE T € E X a n i m m e d i a t e

If $ E 33C ( ) then $ is in s o m e

o ) f r o m (7.1.5).

T h u s if

computation gives :

F o r t h e c o n v e r s e w e u s e the n u c l e a r i t y a s s u m p t i o n on E. a s s u m e that

Let us

$ E X ( E ” ) and t h a t

for every T E E X

.

We m a y c h o o s e B

such that E

s p c e ( s t r u c t u r e of n u c l e a r b . v . s. s e e 0.5.8). c a n o n i c a l injection f r o m E B i n t o E .

is a H i l b e r t B L e t u s d e n o t e by i t h e

T h e n t i ( E X ) is d e n s e in (E

B

)I

(if not t h e r e is x E (E ) I ’ = E with x # o and i(x) n u l l on E X w h i c h B B c o n t r a d i c t s t h e a s s u m p t i o n t h a t E is s e p a r a t e d by its d u a l ) . F r o m l e m m a ( 7 . 2 . 2 ) applied with X =

Ex s e m i - n o r m e d with t h e s e m i - n o r m

Fourier-Bore1 and Fourier transforms

178

a s s o c i a t e d t o t h e p o l a r of

B,

i t follows f r o m ( 1 ) t h a t :

f o r e v e r y n 6 IN, with c ' = c and N o w if X , Y

x = (xl ' . . .

1 Xi-Yi 1

If

p ' = ep.

E ( E X ) n and if (

t. n 1)

t n (X) =( i) ( Y ) in ( ( E B ) t ) n with

i 1 n , then Y,), X i E E x , Y i E E X , 1 I 1""' 1 5 i 5 n . L e t L be the c o m p l e x e line l X t z ( Y - X )

, X ) and Y = ( Y n

= o

15 i 2 n

if

1 Xitz(Yi-Xi) I

bounded on L.

5

I Xi

1 B.

LEc.

T h e r e f o r e f r o m (2),@( n )( 0 ) is

F r o m L i o u v i l l e ' s t h e o r e m @(")(o) is c o n s t a n t o n L and

t h e r e f o r e @(")(o) X = @(")(o) Y .

T h u s t h e quotient m a p p i n g U

in d i a g r a m

below e x i s t s . Now f r o m t h e d e n s i t y of ti(EY) in (E ) ' and f r o m (2), U B m a y be extended continuously a s a m a p p i n g 0 ) such that the following d i a g r a m is c o m m u t a t i v e :

179

Fourier-Bore1isomorphism

If

llYill(E

B

)I

for every n

< 1 for -

E N

1 5 i I n then f r o m ( 2 ) :

.

Let B

denote a convex balanced bounded s u b s e t of E s u c h 1 is n u c l e a r and s u c h t h a t E that t h e n a t u r a l injection j : E B EB1 B1

-.

i s a H i l b e r t s p a c e ( t h i s is p o s s i b l e s i n c e E

0.5.8). nuclear

F r o m (0.4.2)

. If

i s a nuclear b.v.s,

the t r m s p o s e d mapping

Lj : (E

)I

Bl

-

see

( E B ) l is

x' E (EB )' 1

lpq 1 < t co ,

where

yq

E (EB)l with n o r m

5

1 for every q,

ci f

4 (:j)"

E B1 ( s i n c e E f r o m ((E

B1

B1

is a r e f l e x i v e Bandch s p a c e ) . T h e p r o d u c t mapping t o ( ( E B ) l ) n is given by t h e f o r m u l a :

into E

If we denote by k the c a n o n i c a l injection mapping f r o m E B we have the following c o m m u t a t i v e d i a g r a m

:

1

180

Fourier-Bore1arid Fourier transforms

If xliE E X , 15 i,' n , t h e n

fl(4(

.

0 ) ~ ~ . x~' .=

n

i-I

1'

Since f

qi

E B

1

-

fl(n)(p) o (tj)n o (tk)n xtl... X I

. . . p 'n

c E ,

Therefore from (3) :

f

'1

.

(tk(x'l)).. f

n

=

qn

(tk(x'n))&"'(o)

y

'1

. .. yn' .

Fourier-Bore1isomorphism i. e.

T h e r e f o r e f r o m ( 7 . 1 . 5 ) the algebrai'c i s o m o r p h i s m is p r o v e d . T h e b o r n o l o g i c a l i s o m o r p h i s m follows i m m e d i a t e l y f r o m t h e a b o v e proof and ( 7 . 1 . 5 )

. T h e topological i s o m o r p h i s m

i s proved in (7. 1. 5).

181

Fourier-Bore1 and Fourier transfoms

182

5

7.3

Holomorphic g e r m s

7 . 3 . 1 Definition of h o l o m o r p h i c g e r m s . -

K a c o m p a c t s u b s e t of E and F neighborhood of

complex 1.c. s.

. If

into F.

On t h e s e t

u

VSK

r a n g e s o v e r a b a s e of open neighborhoods of

t h e equivalence relAtiOn : f such that f

/w

V

,

8.

is a n open

K we denote by 3 € ( V , F ) t h e l i n e a r s p a c e of continuous

h o l o m o r p h i c functions f r o m V where V

d

L e t E be a c o m p l e x 1 . c .

= g/w

.

N

3((V,F),

K,

we define

g i f t h e r e e x i s t s a neighborhood

W

of

K

T h e s e t of equivalence c l a s s e s is c l e a r l y a

l i n e a r s p a c e t h a t we denote by X ( K , F ) (K(K) if F = C ) and t h a t we c a l l t h e s p a c e of F - v a l u e d h o l o m o r p h i c g e r m s on K.

7. 3.2 Bornology and topology on above we denote by K f

on V

Kc"(V)

cn

K(K).-

In t h e s e q u e l F = C .

In the s a m e conditions a s

( V ) the l i n e a r s p a c e of the h o l o m o r p h i c functions

such t h d t

is a Banach s p a c e with the n o r m

If cpFX(K) , t h e r e i s s o m e V

s u c h t h a t cp c3C

03

(V). 3C ( K ) is a l g e b r a r -

c a l l y the inductive l i m i t ( s e e f o r i n s t a n c e Ktithe [ 13

5

19 f o r p r e c i s e

definitions of inductive limits) of t h e l i n e a r s p a c e s X "(V) r a n g e s o v e r a b a s e of open neighborhoods of

Ifi

liEI

when V

K.

We equip 3C (K) with t h e following n a t u r a l bornology : a s u b s e t

C K ( K ) is said to be bounded if and only if t h e r e is s o m e V such

t h a t a l l the functions f i m a y be extended a s e l e m e n t s of 3C c"(V)

and

183

Holomolphic g e m s

c o m p o s e a bounded s e t in the B a n a c h s p d c e X

00

(V).

W e shall also

c o n s i d e r on 51 ( K ) the bornological topology TX(K).

7 . 3 . 3 PROPOSITION.

x(0

-

( I s o m o r p h i s m between E x p E

X

and -

be a c o m p l e x n u c l e a r b . v . s. dnd l e t 0 _denote t h e EX

)). Let E EX

of E . T h e n the mapping G defined below is a o r i g i n of t h e d u a l E X bornological i s o m o r p h i s m f r o m E x p E x onto X ( 0 ) EX

#

G#

T-

x + 1

n

n

where x E Ex

.

Proof. -

#

If

#(")(o) xn

E E x p E X , then by ( 7 . 2 . 1 )

#

E c[K]

for some c 9 0

and f o r s o m e convex balanced s t r i c t l y c o m p a c t s u b s e t K of E .

.

c E" b e the #(")(o) E c (Pa KDn) f o r e v e r y n E IN Let 1 p o l a r of K. If p > o and x E p ( i ) t h e n [ #( 4( 0 ) x n I Cc(M)". Therefore

u

T h e r e f o r e if we s e t 0 = o< p q M0<

I Gfll * I c

z n 2 o

n

(Po)

.

p ( g ) , 'i$EX(0). F u r t h e r m o r e 1

T h i s p r o v e s t h a t the mapping

G is w e l l

Fourier-Bore1and Fourier transfoms

184

defined and bounded. C o n v e r s e l y if

0

\Y

ex ( K )

l'fl

and

0

f r o m C a u c h y ' s i n t e g r a l f o r m u l a ( 2 . 2 . 7 ) t h a t if x f K

1 '?(")(o)x n 1

then

5a n

1

!

.

f#(x) 5 a e x p ix

F o r every x f Ex ,

IK '

p r o v e s t h a t the m a p p i n g

7.3.4 Remark.

-

Therefore

G- 1

is


K

i t follows

, then

setting

6 E Exp E x .

(i = Y .

This

w e l l defined and bounded.

W e defined t h e F o u r i e r - B o r e 1 t r a n s f o r m 5 by t h e

formula

if 1 EkiS(E), T € E x and

A J(T*")

= (T"),

see 0.8.3.

Nowif w e s e t

w e h a v e obviously t h e following c o m m u t a t i v e d i a g r a m ( w h e n E complex nuclear b.v.5.)

:

EX

is a

Holomoiphic germs

T h e r e f o r e f r o m ( 7 . 3 . 3 ) and ( 7 . 2 . 1) t h e topology T K ( 0

185

) is i s o m o r p h i c Ern

with the s t r o n g topology on 3C' ( E ) . S

7.3.5

Remark.

-

S p a c e s of g e r m s of h o l o m o r p h i c f u n c t i o n s h a v e b e e n

intensively s t u d i e d - s e e the Notes.

186

Fourier-Bore1and Fourier transfoniis

4

T h e F o u r i e r t r a n s f o r m and the

7.4

Paley -Wiener-Schwartz i s o m o r p h i s m s .

T h e proofs(in A n s e m i l - C o b m b e a u a r e r a t h e r long ,

11 and C o l o m b e a u - P o n t e [ 11

s o w e j u s t explain t h e r e s u l t s without p r o o f s . L e t E

be a r e a l n u c l e a r b. v. s . and l e t E X = E x t i E x be t h e c o m p l e x i f i c a t i o n of i t s r e a l d u a l E x , w e d e n o t e by the s e t

ieicjvEEX

c

Y c B(E)

and , f r o m ( 5 . 2 . 6 ) ,

= d(E,IC) the l i n e a r s p a n of

Y is

d

d e n s e s u b a l g e b r a of

B(E). We define a s u s u a l t h e F o u r i e r t r a n s f o r m

We equip a s u s u a l B'(E) w i t h i t s s t r o n g d u a l topology and with i t s equicontinuous bornology.

( P a l e y - W i e n e r S c h w a r t z i s o m o r p h i s m ) . fI

7.4. 1 THEOREM. -

E i s a r e a l n u c l e a r b. v. s . , then the F o u r i e r t r a n s f o r m i s a topological and bornological i s o m o r p h i s m f r o m d ' ( E ) o n t o the s u b s p a c e defined below and denoted by 5B'(E).

3&'(E) is the l i n e s r s u b s p a c e of K { E X ) m a d e of t h e functions f s u c h t h a t both

1) and

1)

\€IN

and

for e v e r y

below hold :

c

t h e r e a r e a convex balanced bounded s u b s e t B of E ,

M >o

6

2)

X

EEc

such that

.

The Fourier transfonn

2)

The sequence

(5 )

n nEN

187

6 Y ' (which is defined below) is

equicontinuous.

Now we define a n a t u r a l bornology on 56'(E) i n the following way : a subset (f.).

1 1 E I

is bounded if t h e functions f . s a t i s f y inequality

t h e s a m e B , v, M and i f t h e f a m i l y

(5

1) f o r

) is equicontinuous. We equip n, i n € N

iEI

36 ' ( E ) with t h e b o r n o l o g i c a l topology a s s o c i a t e d t o t h i s bornology (see 0.2.7).

B is such t h a t E ble s i n c e E

(5

) of 2 ) . F o r t h i s we a s s u m e n nEN is a s e p a r a b l e H i l b e r t s p a c e (which is a l w a y s p o s s i -

Now we d e f i n e t h e s e q u e n c e

B is a n u c l e a r b.v. s. ( s e e 0 . 5 . 8 ) .

o r t h o n o r m a l b a s i s of E

B

Let (e ) be a n n nEIN and l e t Pn denote t h e o r t h o g o n a l p r o j e c t i o n

on t h e s u b s p a c e spanned by t h e s e t

/el,

.. . , en I.

we notice that t h e r e a r e h o l o m o r p h i c functions folowing d i a g r a m

f

If f i s given with N

and f s u c h t h a t t h e

is c o m m u t a t i v e :

f

EX canonical surjection

canonical injection

' N

( T h e proof of e x i s t e n c e of f and f proof f o r & n ) ( o ) i n ( 7 . 2 . 3 ) ) .

is quite s i m i l a r t o t h e analoguous

1)

Fourier-Bore1 and Fourier transforms

188

We equip the d e n s e s u b a l g e b r a Y c &(E) w i t h topology and we define a s e q u e n c e

('n'n

E

t h e induced

of e l e m e n t s of Y '

L

by :

R J

j = l

j = l

'Fj a r e l i n e a r l y independent if 8 . E C , y.E E x , ( n o t e t h a t t h e f u n c t i o n s e J J f o r d i s t i n c t (r.). Not& t h a t t h e s e q u e n c e ( 5 ) d e p e n d s upon B J n nEN

.

If E

7 . 4 . 2 T h e n e c e s s i t y of condition 2 ) . -

is a r e a l v e c t o r space

eqoipped with the finite d i m e n s b n a l b o r n o l o g y , condition 2 ) is a u t o m a t i c a l l y s a t i s f i e d f o r a n y f with p r o p e r t y

1). So t h e P a l e y - W i e n e r

-

S c h w a r t z t h e o r e m b e c o m e s m u c h s i m p l e r and s i m i l a r t o its f i n i t e d i m e n s i o n a l f o r m u l a t i o n . It is p r o v e d in A n s e m i l - C o l o m b e a u [ 13 t h a t in a l l o t h e r c a s e s condition 2) is i n d i s p e n s i b l e .

7.4.3

DEFINITION of E ( E ) c d ( E ) .

-

In o r d e r t o g e t r i d of t h e

t e c h n i c a l condition 2) in ( 7 . 4 . 1 ) u e d e f i n e a new s p a c e E ( E ) of

Co3 functions on E . T h e p r o o f s a r e in C o l o m b e a u - P o n t e

and we j u s t give the r e s u l t s . E

c &(E) 1]

d e n o t e s a r e a l n u c l e a r b. v. s. a s b e f o r e .

We s e t e

i'ch

with

c

h

E C and 5ohEEY )

c

finite

and we n o t i c e t h a t the functions distinct

e

icPh

a r e l i n e a r l y independent f o r

(~1

h'

F o r e v e r y convex balanced bounded s u b s e t B of E , and e v e r y

V

E N we s e t

:

every m > o

189

The Fourier transform

for every

c 1

LEE'

.

We a l s o s e t :

?J(B, m , v) =

i

X finite

c e h

icrh

€Z such that

Ix c h $ ( y h ) Ig

1 for every

When B r a n g e s o v e r t h e c o n v e x balanced bounded s u b s e t s of E , w h e n t m and V r a n g e r e s p e c t i v e l y o v e r R and IN, t h e f a m i l y of sets

V(B, m , V) d e f i n e s a

f u n d a m e n t a l s y s t e m of o - n e i g h b o r h o o d s for a

, and which

H a u s d o r f f l o c a l l y c o n v e x topology on Z , t h a t w e d e n o t e by?? i s f i n e r t h a n t h e topology induced on

Z by d ( E ) ,

We p r o v e t h a t t h e c o m p l e t i o n E ( E ) of

( Z , r ) i s contained i n

B(E) and w e w'rite i n the s e q u e l IE(E) c B(E).

7 . 4 . 4 T h e s i t u a t i o n of IE(E).l i n e a r s u b s p a c e of

-

f u n c t i o n s 'p : E

f o r a~ n

E IN, a

B(E) m a d e of t a m e

'f E 8 ( R n ) , x I i E E Y i f

c=

C

CYl

( E ) t o be t h e

functions o n E : i . e .

t h a t a r e of t h e type

We f u r t h e r m o r e c o n s i d e r t h a t t h e s p a c e f u n c t i o n s on E

We s e t d m

1 5 i 9 n and w h e r e x F E .

s( E c )

(of Silva h o l o m o r p h i c

E t i E ) i s c o n t a i n e d i n B(E) v i a t h e r e s t r i c t i o n t o E

of f u n c t i o n s defined on E

c

( t h i s r e s t r i c t i o n is i n j e c t i v e on 3c (E

s

c )).

Fourier-Bore1and Fourier transforms

190

PROPOSITION 1. -

PROPOSITION 2.

T h e followinp inclusions hold

-

The (algebrai'c and topological) equality lE(E)=G(E)

holds if and only if E is a v e c t o r s p a c e equipped with the finite, d i m e n s i o n a l bornology.

Remark. -

T h i s i s exactly the unique c a s e when t h e P a l e y - W i e n e r -

Schwartz t h e o r e m h a s the u s u a l f o r m u l a t i o n

7 . 4 . 5 PROPOSITION.

-

-

s e e 7.4.2.

E ( E ) is a c o m p l e t e S c h w a r t z 1 . c . s .

( h e n c e i t is n a t u r a l l y r e f l e x i v e : 0.3. 1 and 0 . 5 . 9 ) . is a n u c l e a r 1. c , s.

7.4.6

.

The F o u r i e r t r a n s f o r m .

-

It is unknown i f E ( E )

We have the following situation

w h e r e the symbol c d e n o t e s obvious n a t u r a l i n c l u s i o n s

s(E c ) > U

E'(E)

3

XK'S(E C )

( s e e 7 . 2 . 1)

U

3E'cE)

( s e e 7 . 4 . 1)

The Fourier transform

6

w h e r e 3a(C) = L(e 1 'c) if d e s c r i p t i o n of 5 ~ (E '

s c

THEOREM,

( t h i s obviously d o e s n o t change t h e

) in 7 . 2 . 1 ) .

Now w e d e s c r i b e 7.4.7

EEi

191

3IE'(E):

-

l s e c o n d v e r s i o n of t h e P a l e y - W i e n e r -

S c h w a r t z i s o m o r p h i s m ) . If E is a r e a l n u c l e a r b . v . s . the i m a g e

5E'(E)

of

IE'(E), is d e s c r i b e d in the following way :

3 E ' ( E ) = { $ E X ( E $ ) s u c h that t h e r e e x i s t s a convex balanced bounded subset B o f E ,

m T o

d '

EN u i t h

If w e equip t h i s l i n e a r s p a c e

='(E) u i t h t h e bornology defined

by t h e s e t s & ( B , m ,V) (defined in 7 . 4 . 3 ) and with t h e a s s o c i a t e d b o r n o l o g i c a l topology ( 0 . 2 . 7 ) , then the F o u r i e r t r a n s f o r m 3 is a bornological and a topological i s o m o r p h i s m f r o m , E ' ( E ) (equipped with i t s equicontinuous bornology and i t s s t r o n g d u a l topology ) onto 3EYE). 7.4.8 Remark.

-

In c a s e E

is a Banach s p a c e and if w e r e p l a c e

by a s p a c e d N b c ( E ) (defined i n D i n e e n - N a c h b i n [ 1 1 ) of s o called functions of n u c l e a r bounded type

OR

t h e o r e m w a s obtained i n A b u a b a r a

8(E) Cco

E , a Paley-Wiener-Schwartz

[ 1 , 2 1. In c a s e E is a B a n a c h s p a c e ,

a s p a c e analogous to E ( E ) w a s c o n s t r u c t e d i n C o l o m b e a u - P a q u e s [ I ] .

S e e t h e Bibliographic Notes.

CHAPTER 8 NUCLEARlTY OF SPACES OF HOLOMORPHIC OR Cm MAPPINGS

ABSTRACT. -

If E is a 1. c. s .

we denote by E ' i t s s t r o n g d u a l equipped C

with t h e c o m p a c t open topology. THEOREM.

-

L e t E be a c o m p l e x q u a s i c o m p l e t e 1. c. s. f o r which

.

Elc i s a n u c l e a r 1. c. s.

Let Q

be a c o m p l e x n u c l e a r 1.c. s .

'I

by

"

strongly nuclear

T h e c a s e of C THEOREM.

T h e n 3c

a n open s e t i n E S

( n , F ) i s a n u c l e a r 1. c. s.

-

Let E -

a3

I'

mappings is completely different :

be a r e a l q u a s i c o m p l e t e 1. c . s.

with a n infinite

.

T h e n 8 (0)

nuclear.

Remark. of

.

.

d i m e n s i o n a l bounded s e t a n d l e t 61 be a n open set i n E i s not

and let F

Same statement a s before replacing everywhere

THEOREM. nuclear

.

be

-

In the above a s s u m p t i o n s for E

, K ( 0 ,F) i s a s u b s p a c e

' 3 ~( n , F ) , t h e r e f o r e K ( ( n , F) i s n u c l e a r (and a l s o

S

when K s ( n , F ) i s .

192

strongly nuclear )

193

Nuclearity

$ 8. 1 N u c l e a r i t y of K s ( n ) 8. 1. 1 0

-

THEOREM.

a TE-open

s e t . Then

L e t E be xs(n) i s a

b. v. s.

a complex nuclear nuclear

.

1. c . s.

and

An e q u i v a l e n t m o r e g e n e r a l f o r m u l a t i o n i s :

8.1.2.

n

a

7

THEOREM.

-

Let -

E

be a c o m p l e x S c h w a r t z b. v. s . ,

.

E - o p e n s e t a n d F a c o m p l e t e c o m p l e x 1. c. s.

a nuclear nuclear

I. c.

i f a n d only i f E

s.

1. c. s.

nuclear

(n, F )

is a nuclear

F is a l s o a n u c l e a r

s u b s p a c e of t h e c o n s t a n t m a p p i n g s

.

EX i s a l s o a

X (n, F) of l i n e a r S

.

valued i n a one d i m e n s i o n a l s u b s p a c e of F)

(0. 5. 6 ) E X ' i s a n u c l e a r b. v. s. a r e f l e x i v e b. v. s.

n u c l e a r b. v. s. a r e nuclear nuclear

1. c. s . ,

1. c. s. ( 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 of

mappings

S

n u c l e a r b. v. s. a n d F a

is a

,

If Xs 1. c. s., a s t h e

. .

.

Therefore from

S i n c e E is a S c h w a r t z b. v. s. it i s

(0.3. 3. ) a n d (0.5. 10)

: Ex' = E

Now f r o m (6. 1. 2 ) K s ( n )

1. c. s.

1. c. s.

-

T h e n K (Q,F ) i s

E

F =

.

Therefore

Ks(n, F) .

i t i s quite e a s y to check that E

T h e r e f o r e we proved the equivalence

E

If E

E

is

a

and F

F is also a

(8. 1 . 1.) a n d

of

( 8 . I . 2.) .

T h e end of t h i s

section is devoted to the proof

a n d E d e n o t e s now a c o m p l e x n u c l e a r b. v. s.

.

I n l e m m a s (8. 1. 3 . )

( 8 . 1. 4. ) a n d ( 8 . 1. 5. ) 0 w i l l d e n o t e a c o n v e x b a l a n c e d while i n t h e proof 8. 1. 6 (of t h . 8 . 1. 1 . )

w i l l be

X S b ) by

~ ( ( n )a n d

X',(0)

A s u s u a l we

of

n, bJ-L'(c));(B)

7

E-open

set

a n y T E - o p e n set.

In o r d e r t o s i m p l i f y t h e n o t a t i o n s we denote m o r e s i m p l y (no c o n f u s i o n with the F o u r i e r

of t h (8. 1. 1 . )

Z K i ( E ) by 5:

Bore1 t r a n s f o r m will be p o s s i b l e )

,

by X ' ( n ) .

r e c a l l that i f B i s a s t r i c t l y c o m p a c t s u b s e t

denotes the Banach s n a c e which is the l i n e a r

Nucleariv of spaces

194

s p a n of

)'J

, (8.

L e t u s a s s u m e i n l e m m a s (8. I . 3 ) B

0

is a

?J(B).

(B) in X I ( Q ) , n o r m e d with t h e Minkowski functional of

strictly compact

s u b s e t of 62

1. 4) a n d (8. I. 5) t h a t

of the type

r

n , that n with

(x ) f o r

some

el

B 1 a n d B2 a r e c o n v e x b a l a n c e d R c z B 6 B with E a Hilbert 0 1 2 ' B, s p a c e and s u c h t h a t t h e i n c l u s i o n m a p p i n g i : E E i s 'nuclear. B1 B2 We note, i f x E E B , 1 M a c k e y n u l l s e q u e n c e ( xn ) i n s t r i c t l y c o m p a c t s u b s e t s of

4

with

c I),n \

Yn E B2

c r

(0)

T h e i n j e c t i o n m a p p i n g ZLBl,

=

Iun,

1s 1

---+

E

z Z. r

pl,.

and

x

2

L 1, 1

-5

is -

tBZl

. E B 1 . F r o m the f o r m u l a

ns J

, it follows

:

B2

,

un,r pn

x p1 ... x Pn f PI (xn* r I) .... f p n ( x n , a

y

If

\

n B

If @ E I B l ] and n €IN t h e n

Proof. -

(n)

and I f

.

nuclear

with

)I

*

LEMMA :

8. 1. 3

@

:: q c l ( f o r s o m e q > o ), f n E (EB

c EX i s the p o l a r of

B2 we

set

Q... P1

. @

yPn

n

)

195

Nuclearity

n x T h e n B ( B ) i s a bounded s u b s e t of , C ( E ) 2

is a . .

Banach s p a c e in which t h e a b o v e s e r i e s r e p r e s e n t i n g If ( p l , . E X into C

.., pn) €INn

E B2, 1 si s n , Jr P1'. 'i space the restriction mapping X

r : E T

E . . ' prl

d

(

E

[B

EX and we always a s s u m e t h a t E X

Therefore (r(E )I.

B1

X

))o

B

If 6 E [Bl]

= (E

1

)o

then

B1

X

2

1.

Since E

1

from

is a H i l b e r t

1

)I'

B1

= E

B

separates E,

which i s n u l l o n 1 hence a contradic-

i s a d e n s e subspace of the B a n a c h 6(n)(0)EL(n((EX), )) = L ( n ( E B f i ~ , C ("Ex) 1 B 1

A s a c o n s e q u e n c e , t h e following m a p p i n g F 3 [: 5

B1

. . 'n

B ) -1

h a s a d e n s e r a n g e ( i f not t h e r e i s x E ( E

space (E

P1'.

by :

Since y

tion).

Jr

we define a m a p p i n g

convergent.

(0) i s

@

PI'.

..

3

'n

from

into C :

i s w e l l defined

by 1 on [B,].

, . . . ,f

E ( E B ) a n d bounded i n a b s o l u t e value pn 1 Now we m a y w r i t e : if f

PI

.

196

Nuclearity of spaces

Therefore

m=

n, pl,

. . . , P,

1

. . . 1P,

F

P1

PI'* * . ' P n

(m)

f

1*

'n

n Since

n

5

IF

1 and

4

P1'".,

P,

E[BZ]

We a s s u m e f u r t h e r m o r e that k e B c 2 (recall Log e = 1 )

LEMMA :

8 . 1.4

we

get t h e r e s u l t .

m

for some k > 1

. The canonical mapping

is nuclear.

Proof.

-

I t follows from the d i a g r a m :

continuou (7. 1. 3 )

continuous

s i n c e the c o m p o s e d of a n u c l e a r mapping and a continuous l i n e a r mappir.g i s obviously a n u c l e a r mapping.

197

Nuclearity

8 . 1. 5

LEMMA. -

The c a n o n i c a l m a p p i n g :

is quasi-nuclear.

Proof.

-

We have t h e following c o m m u t a t i v e d i a g r a m

denote the canonical surjection mappings. The 2 t r a n s p o s e d d i a g r a m is :

where

0

and

yr

(x where t

I

is nuclear

from

(8. 1. 4)

.

(k e B2)

Nuclear@ of spaces

198

We have the c o m m u t a t i v e d i a g r a m :

I

I

tt

I

tt a r e i s o m e t r i e s (of a n o r m e d s p a c e into its bidual) I is 2 t n u c l e a r s i n c e I i s n u c l e a r . T h e r e f o r e I i s q u a s i n u c l e a r (see 0.4. 3).

j

0

.

and i

I Now we p r o v e th. 8. 1. 1. a n d fl

i s no l o n g e r convex b a -

lanced. 8. 1. 6

P r o o f of th. 8. 1. 1.

-

L e t K be a s t r i c t l y c o m p a c t s u b s e t of

0 and l e t B be a convex b a l a n c e d s t r i c t l y c o m p a c t s u b s e t of 0

rk i x and s u c h that K i s c o m p a c t i n t h e 1 " . There exist2zonvex balanced strictly compact

o f t h e type s p a c e EB 0

E

normed s u b s e t s of

such that E

is a B1 H i l b e r t s p a c e and s u c h t h a t t h e i n c l u s i o n m a p p i n g f r o m E into E B1 B2 is a n u c l e a r m a p p i n g with t h e p r o p e r t i e s l i s t e d b e f o r e lemma (8. 1. 3). E denoted by B

1

and B

B c B1c B 0

2 '

g i v e n . F o r e v e r y x CK t h e r e e x i s t s a n

L e t k > 1 be E X

with

2

> 0 s u c h that x

t

~

~ Bk 2 ec

n.

199

Nuclearity

Since K i s c o m p a c t i n E

B '

t h e r e is a n & E IN

such that

0

If BIZ i s obtained f r o m B 2 in the s a m e way a s B 2 from B 0 , we s e t I

K =

U (xi t lsise

K ' is a s t r i c t l y c o m p a c t s u b s e t of Q We c h o o s e a 7 E - o p e n

k i

E

2 2 e BIZ).

.

set f l ' c n ,

0''is a convex b a l a n c e d 7 E - o p e n s e t

-

.

n'= x

t fl"

where

W e apply (8. 1. 5) with

(and a t r a n s l a t i o n ) a n d c o n s i d e r the following d i a g r a m :

x(nl)

quasi nuclear

V(XjfEX. k e B 2 )

injective continuous

1

sc(nl) 2/

Bo)

(Xi+ EX.

1

i n j e c ti ve with same n o r m

It follows t h a t the c a n o n i c a l m a p p i n g

i s q u a s i - n u c l e a r . F o r t h e s a m e r e a s o n t h e c a n o n i c a l mapping

n'

Nuclearity ojspaces

200

i s quasi-nuclear

.

T h e r e f o r e the product mapping :

i s n u c l e a r . T h u s from

'K, K'

t h e following d i a g r a m , t h e c a n o n i c a l m a p p i n g

i s q u a s i - n u c l e a r which, f r o m (0. 4. 3), e n d s t h e proof.

1

w h e r e I 1 ( f ) =( /f ( x i t E X

k2e2BL) i lds8

kis

e

20 1

Strong nucleanty

0 8. 2. 1

8. 2 . S t r o n g n u c l e a r i t y of

Strongly n u c l e a r m a p p i n g s

.-

‘jCc,(n)

L e t E and F be two n o r m e d

s p a c e s and u a l i n e a r m a p p i n g f r o m E into F

.

u i s s a i d t o be

s t r o n g l y n u c l e a r ( s - n u c l e a r f o r s h o r t ) i f i t m a y be w r i t t e n

where

(xd€s (the s p a c e of 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 of /I n \I s 1 . Since c

E E ’ with n s 1 for e v e r y n

n u m b e r s , s e e 0. 7. 5 ), x’

//

yn € F with

yn\!

mapping is nuclear

8. 2. 2

.

s - n u c l e a r 1. c. s. a n d b. v. s .

XI

El

n

A 1. c. s.

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

convex b a l a n c e d o-neighborhood

FVPietsch

CJ

F

11.

U

is

A b. v. s. bornologically

Vc U

for every n and

11n I < t

00

, a s-nuclear

F i s called s-nuclear

U there exists another

such that the canonical mapping

s - n u c l e a r . For stafidard examples s e e th. 8.2.5 a c d

E is called s-nuclear

E = lim

if

i t m a y be w r i t t e n

E. (0. 2. 4) w h e r e t h e s p a c e s E

i €I s p a c e s s u c h t h a t f o r e v e r y index i E 1 is

complex

i

are Banach

the i n c l u s i o n m a p p i n g E .

-

E

1 . i

s-nuclear. We a d m i t t h e following r e s u l t w h i c h m a y be found i n

P i e t s c h [ 11 , 8 . 2. 3

A c h a r a c t e r i z a t i o n of

s

-nuclear mappinvs.

-

Let E

and F

be two n o r m e d s p a c e s a n d l e t u be a l i n e a r m a p p i n g f r o m E to F Then u

is s-nuclear

iff f o r e v e r y nEIN

the c o m p o s i t i o n p r o d u c t u = u

o u o n n-1 a s shown ir. the d i a g r a m below :

.. .o

it c a n be u

1

represented

.

as

of n n u c l e a r m a p p i n g s ,

Nucleanfy of’spaces

202

U

E

F r o m t h i s c h a r a c t e r i z a t i o n i t follows from t h e c l a s s i c a l r e s u l t s (0. 4. 2 ) and ( 0 . 4. 3) on n u c l e a r m a p p i n g s t h a t the t r a n s p o s e d of a s - n u c l e a r mapping i s a l s o s - n u c l e a r , and, i f we define a q u a s i s - n u c l e a r mapping in t h e s i m i l a r way a s a q u a s i n u c l e a r m a p p i n g (0.4. 3 ) , that any q u a s i s - n u c l e a r m a p p i n g is i n f a c t 8. 3. 4 E’

C

A s a consequence i f E i s a q u a s i c o m p l e t e 1. c. s. I. c. s.

is a s - n u c l e a r

s-nuclear s-nuclear

.

bornology

i s a s-nuclear

bornology

7

f o r which

t h e n t h e c o m p a c t bornology of E

If E i s a s - n u c l e a r b. v.s

is a

. t h e n itrs d u a l

E

X

1. c. s.. A n y s u b s p a c e of a s - n u c l e a r 1. c. s. is a

1. c. s.

under t h e induced topology and a n y Mackey c l o s e d slibs

p a c e of a s - n u c l e a r

8. 2 . 5

s-nuclear.

b. v. s. is a s - n u c l e a r b. v.

R

.

f o r the induced

,

THEOREM,

-

Let -

E - o p e n s e t . T h e n K,(n)

E be a c o m p l e x s - n u c l e a r

i s a s-nuclear

b. v. s.

andn

1. c. s.

B e f o r e the proof we m a y r e m a r k a s w a s done i n s e c t i o n 8. I t h a t th 8. 2. 5 i s equivalent t o :

8. 2 . 6

THEOREM.

-

E

E be a c o m p l e x S c h w a r t z b. v. s . ,

TE-open s e t and F a c o m p l e t e c o m p l e x 1.c. s. s-nuclear

1. c. s.

s-nuclear

1. c. s.

i f a n d only i f E i s a

.

T h e n 3Cs(

s-nuclear

n, F)

b. v. s.

n

2

is a

and F a

-

203

Strong nucleanty

8.2. 7

P r o o f of th. 8. 2. 5.

of E contained i n

R

.

-

Let K be a strictly compact subset

Since E i s a s - n u c l e a r

two convex b a l a n c e d bounded

sets

the injection i : EA

there exist

A c B of E s u c h t h a t

1) K i s c o m p a c t i n the n o r m e d 2)

b. v. s.

*

E

B

space

:

EA

is s-nuclear

3) E B i s a B a n a c h s p a c e . L e t s ( E ), defined i n (0. 7. 6), denote the n u c l e a r bornology of the B B a n a c h s p a c e EB. T h e n t h e following o b s e r v a t i o n s h o w s that K i s a s t r i c t l y c o m p a c t s u b s e t of the b. v. s.

s(E ). F r o m def. 8. 2. 1 : B

1 E s a n d i t follows t h a t (In/Z ) n E m E

(1n)n

i s a bounded

compact in E

C

s

.

Therefore the

set of

.

s(E ) a n d K i s B

Now we c h o o s e a n o p e n neighborhood Q of K i n flE such B' lim that ( i f we w r i t e a s u s u a l E = -+ E bornologically) u€A a the c l o s u r e of Q in s o m e E is compact i n E a n d c o n t a i n e d i n n.

a

Let

a

X (0) denote t h e s p a c e of Silva h o l o m o r p h i c functions on

S c o n s i d e r e d a s a n open s u b s e t of the b. v. s.

Q

,

s ( E B ) . Since s(E ) is a B n u c l e a r b. v. s. (0. 7. 6), it follows f r o m th. ( 8 . 1 . 1) t h a t 3C is a nuclear

of

1. c. s.

(a)

. T h e r e f o r e t h e r e e x i s t s a n i n c r e a s i n g s e q u e n c e (K n ) n G I N S

s t r i c l y c o m p a c t s u b s e t s of Qc s(EB) s u c h t h a t :

204

Nuclearity of spaces

l ) K = K 1

2 ) t h e c a n o n i c a l mapping

is n u c l e a r for e a c h n G l N , Now f o r e v e r y n E l N

we have the following c o m m u t a t i v e

diagram :

(with obvious definitions of I and J )

.I

is continuous a n d

J is

a n isometry of n o r m e d s p a c e s . T h e r e f o r e J o r is t h e c o m p o s i t i o n p r o d u c t of n n u c l e a r m a p p i n g s a n d t h i s holds

f o r e v e r y n ElN

.

F r o m (8. 2. 3) J o r i s s - n u c l e a r . Since J i s a n i s o m e t r y it follows from (8. 2. 3) t h a t r i s s - n u c l e a r .

8.2.8

Remark.

-

F o r f u r t h e r n u c l e a r i t y r e s u l t s , including t h e p r e s e n t

ones a s particular cases, see Borgens-Meise-Vogt Bibliographic Notes.

c 1,21

and t h e

Non nucleanty

Q 8 . 3 Non

(n)

n u c l e a r i t y of 8

r e c a l l t h a t i f 0 i s a n open s u b s e t of a finite n dimensional r e a l space l R t h e n the s p a c e (n) i s a n u c l e a r I. c. s.

8 . 3. 1

Recall. -

205

we

( s e e P i e t s c h [ 13 6. 2 ). F r o m

8 . 3. 1

PROPOSITION.

-

this

result

one d e d u c e s e a s i l y :

L e t E be a r e a l v e c t o r s p a c e equipped with

i t s finite d i m e n s i o n a l bornology (0. 2. 2 ) a n d let The & ( 0 )i s a pucLe5r

1. c.

1. c. s.

(6. 2. 1)

we

( a )E

F =

An i m p o r t a n t r e s u l t

6 (n, F)

if F i s a c o m p l e t e

1. c. s.

iff F is a nuclear

due t o M e i s e [ 3 ]

Let

-

.

the hornology of

E i s the finite d i m e n s i o n a l one

The proof

( a )is

nuclear :

E be a r e a l S c h w a r t z b.v. s.

open s u b s e t of E

T hen $ -

(n)

i s a nuclear

1. c. s .

shows that the c a s e

c o n s i d e r e d in 8 . 3. 1 is t h e unique one i n which $ THEOREM.

E-oDen s e t .

deduce i m m e d i a t e l y that, i n t h e a b o v e conditions f o r

E a n d F, then d ( 0 , F) i s a n u c l e a r

8.3.2

T

.

S.

Since i n t h i s c a s e 6

be a

1. c. s.

.

and 0

an

i f and only i f

i s t e c h n i c a l and, s i n c e we s h a l l not u s e t h i s r e s u l t

i n t h e sequel, we d o not give i t . It m a y b e found i n M e i s e [ 3 ]

and

Colombeau-Meise [ 13 .

8.3.3 the

Comments. -

Th. 8 . 3 . 2

shows a n i m p o r t a n t d i f f e r e n c e b e t w e e c

Cco and the h o l o m o r p h i c c a s e f r o m on hand, and f r o m t h e o t h e r hand,

in the

C

OD

c a s e , between t h e finite and t h e infinite d i m e c s i o c . T h i s

r e s u l t m a y be c o m p a r e d with s i m i l a r d i f f i c u l t i e s f o r the P a l e y - W i e n e r Schwartz theorem in 7 . 4 .

.

.

PART I1 CONVOLUTION AND

a

EQUATIONS

INTRODUCTION

T h e p u r p o s e of Part I1 is t o study s o m e m a i n b a s i c p a r t i a l differ e n t i a l e q u a t i o n s , whose s o l u t i o n s w e r e a l r e a d y known i n t h e f i n i t e d i m e n s i o n a l c a s e . In the infinite d i m e n s i o n a l c a s e , t h e i r s o l u t i o n s a r e m a i n l y obtained by extending c l a s s i c a l m e t h o d s , but t h e s e e x t e n s i o n s a r e in g e n e ral m u c h m o r e t e c h n i c a l t h a n t h e p r o o f s i n t h e f i n i t e d i m e n s i o n a l c a s e a n d

r e q u i r e t o u s e new t o o l s s u c h a s f o r i n s t a n c e n u c l e a r i t y of t h e s p a c e s , t o pological t e n s o r p r o d u c t s , i n t e g r a t i o n t h e o r y i n infinite d i m e n s i o n , t h e t h e o r y of m a p p i n g s of type

QP

...

a s well a s new m e t h o d s t h a t have no

analogues o r a r e t r i v i a l i n t h e finite dimensional c a s e a n d that a r e added t o the c l a s s i c a l m e t h o d s t o y i e l d a s o l u t i o n . In c h a p t e r 9 a n d 10 we obtain

v e r y g e n e r a l r e s u l t s of e x i s t e n c e

of s o l u t i o n s of convolution e q u a t i o n s i n s p a c e s of p o l y n o m i a l s a n d of e n t i r e functions of exponential t y p e . I n c h a p t e r 11, f r o m a v e r s i o n of t h e W e i e r s t r a s s p r e p a r a t i o n t h e o r e m , we obtain a g e n e r a l r e s u l t of d i v i s i o n of a d i s t r i b u t i o n by infinite d i m e n s i o n a l non z e r o h o l o m o r p h i c f u n c t i o n s , which g i v e s a new proof of a n e x i s t e n c e r e s u l t of c h a p t e r 10 a n d t h e n we s t u d y t h e d i v i s i o n by r e a l a n a l y t i c f u n c t i o n s . I n c h a p t e r 12 we study t h e convolution e q u a t i o n s in v a r i o u s s p a c e s of h o l o m o r p h i c f u n c t i o n s on n o r m e d a n d l o c a l l y convex s p a c e s , and in c h a p t e r 13 we obtain e x i s t e n c e a n d a p p r o x i m a t i o n of s o l u t i o n s f o r f i n i t e - d i f f e r e n c e p a r t i a l d i f f e r e n t i a l e q u a t i o n s i n s p a c e s of

Coo

func-

t i o n s on n o r m e d a n d l o c a l l y convex s p a c e s . T h e e n d of P a r t I1 t r e a t s of t h e s o m e w h a t d i f f e r e n t , but a l s o basic,, p r o b l e m of t h e r e s o l u t i o n of t h e

6

e q u a t i o n , i n pseudo-convex open s u b -

s e t s of l o c a l l y convex s p a c e s . In c h a p t e r 14 we e x p o s e m a i n b a s i c r e s u l t s o n pseudo-convexity in infinite d i m e n s i o n , a n d s o m e a p p r o x i m a t i o n r e s u l t s 206

Convolutionand a equations

201

in pseudo-convex d o m a i n s . In c h a p t e r 15 we p r o v e t h e r e s o l u t i o n of t h e b

e q u a t i o n in D F N s p a c e s , a n d m o r e g e n e r a l l y i n n u c l e a r s p a c e s , (with

s o m e a s s u m p t i o n on the given s e c o n d m e m b e r in t h i s l a t t e r c a s e ) . C h a p t e r 16 is c o n c e r n e d with a p p l i c a t i o n s t o t h e f i r s t C o u s i n p r o b l e m a n d t o s o l u t i o n s of s o m e homogeneous convolution e q u a t i o n s in s p a c e s of e n t i r e f u n c t i o n s of exponential t y p e . T h e r e s u l t s i n t h i s p a r t show t h e d i f f i c u l t i e s , but a l s o t h e r i c h n e s s , of the t h e o r y of p a r t i a l d i f f e r e n t i a l e q u a t i o n s i n i n f i n i t e d i m e n s i o n .

CHAPTER 9 CONVOLUTION EQUATIONS IN P(E)

ABSTRACT.

-

If E i s a r e a l o r c o m p l e x l o c a l l y s p a c e a n d i f n €IN, we

n denote b y p ( E ) the v e c t o r s p a c e of continuous homogeneous polyno-

m i a l s of d e g r e e n on E , equipped with t h e bornology of the

equiconti-

nuous s e t s . The s p a c e p ( E ) of e l l continuous polynomials on E i s tm n n a t u r a l l y the d i r e c t sum @ p ( I?) equipped with t h e d i r e c t s u m b o r n o n =o logy We define a convolution o p e r a t o r on P ( E ) a s a bounded l i n e a r

.

m a p p i n g f r o m 63 ( E ) into i t s e l f which c o m m u t e s with t h e t r a n s l a t i o n s .

THEOREM.

-

L e t E be a r e a l o r c o m p l e x n u c l e a r

1. c. s.

e v e r y non z e r o convolution o p e r a t o r on 65 ( E ) i s s u r j e c t i v e .

208

.

Then

209

Formal power series

5 9. 9. 1 . 1.

1

F o r m a l power s e r i e s a n d d u a l i t y

Uefinition of 6 ( E ) . -

c o m p l e x 1. c. s .

.

t f nclN

we

In this section E denotes a r e a l o r n denote by p ( E) t h e s p a c e o f c o n t i n u o u s

homogeneous polynomials of d e g r e e n on E

,

equipped with the e q u i n continuous bornology (i. e. a bounded s u b s e t of 6) ( F) i s a f a m i l y of

homogeneous polynomials of d e g r e e n t h a t o-neighborhood

in E ;

a r e equibounded on the

same

this property implies that these polynomials a r e

equicontinuous s e e (0.8-2). We denote by 6 ( E ) the s p a c e of continuous polynomials on E , i. e. the algebrai'c d i r e c t s u m

(with F ( O E ) =

lR o r

a ) equipped with

t h e d i r e c t s u m bornology, i. e. a

6 (E) is bounded i f i t i s contained i n s o m e produut

s u b s e t of

for some n

and w h e r e

set

i a r e bounded i n p ( E ) f o r o 5 i I n i i 1 0 ) d e n o t e s t h e o - s u b s e t of t h e v a r i o u s s p a c e s p ( E ) f o r

€IN w h e r e the s e t s B

,

i > n.

9. 1 . 2.

Formal p o w e r s e r i e s on a b. v. s. F. I n t h i s s e c t i o n F d e n o t e s

a real o r c o m p l e x b. v. s.

. We d e n o t e

by

P("F) t h e s p a c e of t h e

bounded homogeneous p o l y n o m i a l s of d e g r e e n on F , equipped with t h e topology of 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 of F fa3 n P( F), equipped with the p r o d u c t (P(OF) = lR o r C ). T h e 1. c. s. n=o topology, i s c a l l e d t h e s p a c e of the f o r m a l power series on F a n d i s

n

denoted by S(F)

.

power s e r i e s , i. e.

In S ( F )

w e c o n s i d e r t h e u s u a l p r o d u c t of t h e f o r m a l

210

Convolution equations

if (a

n

1

I

(bp)

nEN

9. 1. 3.

EN

with

qEN

c

for every q

a r e e l e m e n t s of S ( F )

and ( c )

PEN =

c

a

n t p =q

b

n'

P

.

The Fourier-Bore1 transform

.-

If T E FX a c d i f n €lN#

a n e l e m e n t T B n of P("F) i s defined by :

f o r e v e r y x EF

e E (P("F))'

Now i f into lR ( o r

a ) by

-. e

define a m a p p i n g

f r o m FX

:

1

i(T) = i

we

TE FX. C l e a r l y

n d e d from (P( F ) ) ' to

@n

e(T

I

E b (nFx) and t h e m a p p i n g

is linear b i n -

e-

p (nFx) ( w h e r e (PnF))I i s n a t u r a l l y equipped with

i t s equicontinuous bornology).

9. 1. 4.

PROPOSITION.

T h e n the m a p p i n g to p -

e

+

t?

-

Let F -

be a r e a l o r c o m p l e x n u c l e a r b. v. s.. n i s a b o r n o l o g i c a l i s o m o r p h i s m f r o m ( P ( F))'

(nFx).

Proof:

If K i s a s t r i c t l y

?J

(K) = ( f E P("F) s u c h that

(K) d e n o t e s t h e we s e t :

c o m p a c t s u b s e t of F we

p o l a r of

?J (K)

sup xEK

i n (P("F))'

I f(x)l b .

1)

set

.

Using notations of $ 7. 1

Fomal power series

21 1

[ K ] = { Q E P (nFx) s u c h t h a t

{

Then t h e proof of ( 7 . 1. 3 ) s h o w s t h a t if @ E n Now i f @ E [ K ] a n d cp E P( F ) we s e t :

which,

Now i t is i m m e d i a t e in a n equicontinuous

A s in (7. of the

1.

--

t o check that i f n ' s u b s e t of ( P ( F))

sets [K]

the s e t s [ K ] .

8

A

.

@

r a n g e s in [ K

Furthermore

5) w e denote by (P( F))' the when K

v

then

E =

r a n'g e s e E (P(n F )).

linear span in

@

n ( FX )

T h e n the p r e v i o u s

(P( F))' w i t h t h e b o r n o l o g y of

results

show t h a t t h e m a p p i n g

is a b o r n o l o g i c a l i s o m o r p h i s m f r o m (P(nF))' o n t o I-( P ( F ) ) .

Now a p a r t of p r o o f (7. 2. 3) shows t h a t c a l l y and b o r n o l o g i c a l l y

9. 1. 5

i'

l

r a n g e s o v e r a b a s e of s t r i c t l y c o m p a c t

s u b s e t s of F and we equip t h i s s p a c e

-e

re (KBn).

i n (7. 1. 4) i s w e l l d e f i n e d s i n c e @ ( n ) ( o ) E

like

.

e (P( F ) ) ' = P (nFx)

algebrai-

I

Duality of s p a c e s of f o r m a l power s e r i e s a n d p o l y n o m i a l s

F is a S c h w a r t z b.

, Ks(+-) i s a S c h w a r t z 1. c. s. (4. 3. l ) , h e n c e

V.S.

P("F) i s a S c h w a r t z 1. c. s. a l s o a complete

1. c. s.

a naturally reflexive

.

. It i s i m m e d i a t e t o check

t h a t P(nF) i s n T h e r e f o r e f r o m (0. 5. 9 ) a n d (0. 3. 1) P( F ) i s

1. c. s . , i. e . (P( nF))'X= P(nF)

T h e r e f o r e , f r o m (9. 1.4), r e , i f w e d e n o t e by

S(F) =

+m

n

. Since

n=o

(p (nFx))"

is i s o m o r p h i c t o P("F)

the i s o m o r p h i s m

P("F)

tw

fl

n=o

. .

Therefo-

of (9. 1. 4 ) :

(P (nFx))" = ( @ p(nFx))" =(p(FX))"

212

Convolution equations

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 . W e have a l s o : too

63 (FX) = 8

n=o

63 (nFx)

-

too @

n= o

algebraically and bornologically

I

(P("F)) =

too

(n

n=o

P(nF'))l=(S(F))'

213

A division result

p 9. 2 . A 9. 2. 1.

division -

result. -

R e c a l l of a o n e - d i m e n s i o n a l r e s u l t .

-

Let P :

-, C

L

a

u n i t a r y polvnomial of d e g r e e n and l e t f b e a h o l o m o r p h i c fupction.of one c o m p l e x variable. T h e n :

-

Proof.

W e a r e going to p r o v e t h a t f o r e v e r y r > o a n d e v e r y L ~ C

one has :

p(z) = (z-

cl). . . . ( z - C n ) .

fl(z) = ( z - c 1) f ( z )

If

,

then

( ~ - 5 z~r 1 t h i s

9.2. 2.

-

let m , n

By induction it s u f f i c e s to p r o v e t h a t i f

LEMMA. -

i s o b v i o u s . If I X - C , I

Let F

6r

, then

be a r e a l o r c o m p l e x b. v. s. a n d

n

€IN be g i v e n . Let p be a given n o n - z e r o e l e m e n t of P( F).

T h e n the mapping :

P(ntm)F)

2

p.

P(mF)

(9

aP(mF)

214

Convolution equations

m

is w e l l defined and continuous when p. P( F) = { p.g } g E p(mF)

is equipped with the topology induced b y that of

-

Proof.

T h i s mapping (

1 )

P p r o v e that the mapping g x EF be such that p(xo) 0

s u c h t h a t p(x) thus if

E

#

i s w e l l defined b e c a u s e i t is i m m e d i a t e p. g i s injective

to

( a s s u m e pg = o ; l e t

f 0 , thus t h e r e i s a b o r n i v o r o u s s u b s e t P of F

0 for e v e r y x E x

0

t P ; therefore

g = o on x

0

t P and

g = o on F f r o m (2. 1. 6 ) ) . If K ibs a bounded s u b s e t of F and

> 0 we s e t : Vm(K,

L e t K and p(xo)

.-)

). P((“+dF

#

0

.

E

>o

be

= { g E P ( ~ Fsuch ) that

E

given and l e t

I

sup g(x)l xEK

G E

1.

x E F be s u c h that 0

Without loss of g e n e r a l i t y we m a y a s s u m e p(x )= 1

.

F i r s t we a s s u m e F is a c o m p l e x l i n e a r s p a c e . F o r e v e r y fixed

y EF we c o n s i d e r t h e complex polynomial

m P is a unitary polynomial of d e g r e e n . If g E P ( F) s i m i l a r l y C(x)

g ( y t xxo)

.

we define

F r o m (9. 2. 1) we obtain :

i. e.

T h e s e t K’ = ( y t Ax

3

o YEK

\X l g 2 n

i s a bounded

s u b s e t of F

21s

A division result &’ - (K’, E ) is a o - n e i g h b o r h o o d i n F‘((ntm)F). If n+m

hence

p. g

-

.

g E 2/ ( K , E )

the a b o v e i n e q u a l i t y p r o v e s t h a t

m

g i s continuous

i n t h e corri,>lex c a s e .

Now we a s s u m e

F i s a r e a l linear

Fa: = F t i F i t s c o m p l e x i f i c a t i o n

.

p. g. E

Therefore

. . y)

m

where g EL(

s e e ( 0 . 8. 7 )

.

) ,

the mapping

s p a c e a n d we

F o r any polynomial

d e n o t e by m g E P( F ) we

2 E P(mFa: ) i t s e x t e n s i o n t o F a * If x , y E F g ( x t i y , . . . , x t i y ) is i m m e d i a t e l y e x p r e s s i b l e i n t e r m s denote by

E

-

g(xtiy) =

.

of g ( x , , , x, y , .

F ) i s t h e m - l i n e a r s y m m e t r i c f o r m a s s o c i a t e d to g ;

W e d e n o t e by P ( m F )c P(lnFC) t h e

g] g E P ( m F ) and

set {

we h a v e t h e m a p p i n g s :

P((ntm)F)

T h i s m a p p i n g i s continuous from t h e above

s t u d y of t h e

complex c a s e

a 9 . 2 . 3.

LFMMA. -

L et -

F b e a r e a l o r c o m p l e x b. v. s.

and let

f 6 S ( F ) b e a lion z e r o f o r m a l p o w e___ r s e r i e s or. F. T h e n t h e _”r-n a p p 2 I _

( S ( F ) 3 f. S ( F ) f .

i s w e l l d e f i n e d a n d continuous

g

1

7) +

S(F) g

when f. S ( F ) is e a u i w e d with t h e t o p o -

logy i n d u c e d by S ( F ) If f = (f ) a n d s i n c e f # 0 t h e r e is r €IN s u c h t h a t n nElN 0 and f = 0 i f p < r If g E S ( F ) a n d i f g # 0 t h e r e i s q elN

proof. -

fr

#

P

.

.

.

216

Convolution equations

f 0 a n d g = 0 if p < q . T h e f i r s t n o n - z e r o c o m p o n e n t i n 9 P t h e p r o d u c t f. g. is f ( w h i c h i s m n - z e r o f r o m t h e beginning of r * gq T h e r e f o r e fg = 0 i m p l i e s g = 0 a n d t h u s t h e proof of (9. 2. 2 . ) 1 mapping i s well d e f i n e d . We s e t 9 J O ( c ) a s the c l o s e d b a l l of with g

.

(7)

center

o and r a d i u s

~

>

in 0

IR o r a . L e t

be a given o-neighborhood i n S ( F )

.

to f )

.

:

L e t r c l N b e a s above

r e lati ve 1y

B y definition

tS 0

hence

P F r o m (9, 2. 2) , g

P

t>O

will be i n 9/ (K, P

c e r t a i n o-neighbcrhood

E

) a s soon a s f,gp

and, f r o m t h e above e q u a l i t y , t h i s

of P((’”)F)

l a s t condition will hold t r u e i f (fg) Ptr s m a l l enough in P(( r t p ) F ) . Now

and e v e r y

t > We want to have frtt

g

( t f s = p,

gp-l

E Vp-l(K,

t > o )

.

E

w i l l be i n a.

frtt.

g (tts=p,t>O) a r e s

0

) a s w e l l a s t h e above r e q u i r e m e n t s on

Both r e q u i r e m e n t s

on gp-l a n d f

r t l ’ gp-1

is small enough, t h e r e f o r e f r o m (9. 2 . 2. ) i f P-1 i s s m a l l e n o u g h , h e n c e from t h e above f o r m u l a i f (fg) and f r gp-1 ptr-l

will hold t r u e if g

g ( t t s = p-1, t > 0 ) a r e s m a l l e n o u g h . The p r o c e s s of proof rtt’ s i s obvious a n d e n d s a f t e r a finite n u m b e r of s t e p s

each f

.

217

Convolution operators

tj 9. 3 Convolution o p e r a t o r s on p (E)

-

In t h i s s e c t i o n E is a r e a l or c o m p l e x n u c l e a r c o m p l e t i o n of

1. c . s.. If I? is t h e

E , (0. 1. 16), we h a v e c l e a r l y 63 ( E ) = p ( E ) a l g e b r a i ' c a l l y

a n d b o r n o l o g i c a l l y , s o we m a y a s s u m e without l o s s of g e n e r a l i t y t h a t E i s a c o m p l e t e 1. c. s.

.

F = E ' i s a n u c l e a r b. v. s. a n d FX = ElX= E

(0. 3. 1 a n d 0. 5. 9 ) . We r e c a l l t h a t , f r o m (9. 1 . 4), t h e d u a l i t y b e t w e e n n (%) a n d P ( El) i s given b y t h e f o r m u l a

if + n p~("E) a n d

c i

Iu i \

n

I 1 a n d T i , J.

a s s o c i a t e d to

n

EP(%I).

1 I f ip = n n !

CKc E ,

i f rp d e n o t e s t h e s y m m e t r i c -n

- c ui

1 ( h e n c e cg =-n n !

qn

i

. .'Ti,

T i l ~. .

,

( n ) ( o ) ) we h a v e

T h e d u a l i t y (9. 1. 5) b e t w e e n 63 ( E ) and S f E ' ) is

n

with

n-linear f o r m

:

given b y t h e f o r m u l a

(finitesum)

9. 3.

1.

Convolution p r o d u c t .

-

F o r e v e r y fixed Q E S ( E ' ) t h e m a p p i n g

R -, OR f r o m S ( E ' ) i n t o i t s e l f i s c o n t i n u o u s . T h e r e f o r e , f o r a n y f i x e d ip

€ P (E),t h e m a p p i n g

Convolution equations

218

i s a continuous l i n e a r f o r m on S ( E ' ) ; h e n c e it d e f i n e s a n e l e m e n t

Q

*

of 6 (E) by the f o r m u l a :

p

I n t h i s way we define a l i n e a r o p e r a t o r into p ( E )

9. 3. 2.

. This

translation operator

E

@

0

4

o p e r a t o r i s c l e a r l y l i n e a r bounded

Convolution o p e r a t o r s ,

for e v e r y

0* :

-

If a E E

,

@

from p (E)

we denote by

7

f r o m P (E) into i t s e l f defined by :

I t i s i m m e d i a t e t o c h e c k that

,

* .

m a p p i n g f r o m p ( E ) into i t s e l f .

7

U

a

the

i s a bounded l i n e a r

We define a convolution o p e r a t o r on p ( E ) a s a bounded l i n e a r m a p p i n g f r o m p ( E ) into i t s e l f which c o m m u t e s with all t r a n s l a t i o n s 7

a, a E E .

9.3. 3 .

LEMMA, -

t

If Q E S(E ) , t h e o p e r a t o r 0

.

operator

on p ( E )

proof. -

If a E E we have t o check that , f o r e v e r y

*

i s a convolution

m ~p ( E ) ,

If suffices c l e a r l y to check i t i n the c a s e OE P ( m E ' ) and

@

E P ("E) f o r

s o m e a r b i t r a r y m , n E IN. Since it h a s b e e n p r o v e d i n (9. I . 4 ) t h a t i n s o m e set [K] , i t s u f f i c e s t o c o n s i d e r the c a s e iP = T B n f o r T CE'.

ip

is

219

Convolution operators

N o w , f r o m (5. 1. 3), t h e finite type h o m o g e n e o u s p o l y n o m i a l s a r e d e n s e i n 64m , for P ( m E 1 )s o i t s u f f i c e s to c h e c k t h e a b o v e e q u a l i t y i n c a s e 0 = 0 s o m e a € E l X= E a n d s o m e m EIN . If R e S (

BBi .

B E E and i c l N , then Q R =

and if i

If

f n - m we have < 0

P = T@

with T c

El

* @R, >

=

0 .

El)

a n d R = pdFifor s o m e

Therefore i f i = n-m

Hence

and r ElN , and i f a EE then Newton's binomial

Hence

On t h e o t h e r hand ( f r o m f o r m u l a ( 2 ) ) : Ta

@ =

n

c

p =0 F r o m (1) a n d (4) we o b t a i n :

that is ,

o

:

f o r m u l a gives :

(4)

2

n ! P I (n-p)

( - l ) n - P ( T ( a ) ) n - P T@

220

Convolution equations

9. 3.4

obtain (3)

i n (5), we

Setting q = p - m

PROPOSITION. -

.

The mapping @

Q

4

*

i s a bijection

f r o m S(EI) onto t h e s p a c e of the convolution o p e r a t o r s o n' 6 ( E )

proof.

-

L e t G denote t h e l i n e a r s p a c e of t h e convolution o p e r a t o r s o n

6 (E) and l e t

if 0

E

Q

Q

iy

-

.

(=

*

y

be the m a p p i n g f r o m C; into 63 (E)' = S ( E ' ) defined by:

a n d @ E 63 ( E )

.

L e t u s denote b y

f r o m S ( E 1 ) into G

.

If

$ = TBn

7

the mapping

and Q =

Q m ( with T

E E , m , n €IN),then f r o m f o r m u l a (1) i n (9. 3. 3. ) ,

(CI

*

+)

(Q

*

@ )

(0)

(0)

m ,

=

o

=

m ! ( ~ ( 1)" a

if

n f

i f n= m

Therefor e

hence

yo

7

i s the identity m a p p i n g on S ( E ' )

and (4) i n 9. 3. 3 it follows t h a t :

i f Q E S ( E 1 ),

+

E p (E) and a E E

.

Therefore

.

F r o m formulas (1)

EEl,

221

Convolutionoperators Hence

7

v and

9. 3. 5. If 0

y

oy

is the i d e n t i t y m a p p i n g

a r e i n v e r s e mappings

on C a n d thus the mappings

.

G e n e r a l f o r m of the convolution o p e r a t o r s on p (E)

E G ,

+ €63 ( E ) a n d

a E E t h e n f r o m (9. 3 . 4 ) t h e r e i s a

Q = (QnInEW E S ( E ' ) such that

0- 0

*.

Therefore from formula

( 1 ) of (9. 3 . 4 ) :

hence

and t h u s :

We m a y r e m a r k t h a t the above s u m i s finite s i n c e

with

C

Ivi

I 51

and

E 63 (F) ,

T . . E K w h i c h i s a bounded 'J

( s e e 7 . 1. 1 f o r m o r e d e t a i l s on the f a c t t h a t

ip

s u b s e t of E '

("'(a) E L' ( n E ' ) )

.

Convolution equations

222

8 9.4 9.4. 1

THEOREb

E x i s t e n c e of solutions

L e t E be a r e a l o r c o m F . 3 ~ n u c l e a r

-

T h e n e v e r y non z e r o convolution o p e r a t o r on 63 ( E ) is s u r j e c t i v e proof.

-

9. 3 we may

A s a l r e a d y r e m a r k e d at the beginning of

assume without l o s s of g e n e r a l i t y t h a t E i s c o m p l e t e non z e r o convolution o p e r a t o r O€S(E'), Q

#

.

Let 0 be a

on 63 (E). F r o m (9. 3. 4 ) t h e r e i s a .

s u c h that 0 = Q

0 ,

.

1. c. s.,

*

.

Now l e t 4

P ( E ) . We s e e k f o r a n X EP ( E ) s u c h t h a t Q

be a given e l e m e n t of

*X=

4

,

Let u s conai-

d e r the m a p p i n g s :

+

F r o m (9. 2. 3) the c o m p o s e d m a p p i n g l i n e a r f o r m on Q.S

(El)

o(

61 )

i s a continuous

equipped with t h e topology induced by (S(E') I

.

1

F r o m t h e Hahn-Banach t h e o r e m l e t X E (S( E )) be a continuous l i n e a r 1 T h e n f o r e v e r y R ES(E') : e x t e n s i o n of @ o ( - )

Q

.

1

=<X,QR> = ( $ o(-))(QR)=$(R) Q hence Q

9.4.2

*

.

X = 4

Remark.

-

I

If E i s a n u c l e a r

Frdchet space o r a nuclear

Silva s p a c e one m a y r e m a r k t h a t a l i n e a r o p e r a t o r f r o m 63 (E) ink, 63 (E) i s bounded i f it i s continuous f o r the d i r e c t sum topology of the n s t r o n g topologies on the s p a c e 63 ( E ) , S e e C o l o m b e a u - P e r r o t [ 8 ] for m o r e details. 9.4, 3

t 81.

Remark.

-

A p h y s i c a l m o t i v a t i o n i s explained

Colombeau-Perrot

CHAPTER 10 CONVOLUTION EQUATIONS IN SPACES OF ENTIRE FUNCTIONS OF EXPONENTIAL TYPE

ABSTRACT.

-

If E is a c o m p l e x 1. c. s.

we denote b y E x p E t h e

s p a c e of e n t i r e functions of exponential type on E

,

i. e. , t h o s e e n t i r e

on E s u c h t h a t t h e r e a r e c > o and a continuous

functions

semi

n o r m p on E with

f o r a l l xE E

. A s u b s e t of E x p E i s

s a i d to be bounded i f i t i s a family

t h a t s a t i s f y the above inequality f o r all x e E with t h e s a m e c ('i)ic 1 T h u s E x p E: is a b. v. s. a n d we equip i t with t h e b o r and p f o r all i

.

nological topology a s s o c i a t e d t o t h i s bornology. Remark.

-

If F i s a c o m p l e x q u a s i c o m p l e t e dual n u c l e a r 1, c. s.

we

know f r o m chap 7 that E x p (F') is a l g e b r a i c a l l y a n d topologically i s o -

B

m o r p h i c , through t h e F o u r i e r B o r e 1 t r a n s f o r m 3 , w i t h (3C (F))'

s

THEOREM 1.

-

Let E -

be a c o m p l e x n u c l e a r 1. c, s.

.

8'

T h e n a n y non

z e r o convolution o p e r a t o r on E x p E i s s u r j e c t i v e . THEOREM 2 . -

-

Let E -

be a c o m p l e x n u c l e a r

1. c. s

a n d 6 a convolu--

tion o p e r a t o r on E x p E. T h e n a n y 6 E E x p E solution of 6 @ = 0 i s limit

Ti, finite which are a l s o solutions. i n E x p E of functions

QD..

. .@ T i ,

e x p (Ti, n+l ), (n

€rn,

T h e s e r e s u l t s a r e proved ir. m o r e g e n e r a l c a s e s . 223

T i € El),

224

Convolution equations in spaces

9

Convolution o p e r a t o r s on 5x1(n) S

10.1

We a s s u m e t h a t F is a c o m p l e x S c h w a r t z b.v. s. with a p p r o x i m a tion p r o p e r t y ( 5 . 1. 1) and t h a t

10. 1. 1.

n

is a convex, balanced,

Topologies and b o r n o l o g i e s on 3Kt,(n).

-

T F

open s e t .

F r o m 9 7. 1 we

X

denote by 35(' ( n ) c3c ( F ) the i m a g e u n d e r the F o u r i e r - B o r e 1 S S t r a n s f o r m 5 of the s p a c e 3C' (n) ( s e e 7. 1.5). We r e c a l l t h a t 3 i s a S bornological and a topological i s o m o r p h i s m .

(defined by : 7

5x1,(n)

Proof.

F o r each a

LEMMA. -

10. 1 . 2

-

0

$(p)

=

E FX, t h e t r a n s l a t i o n o p e r a t o r

$ ( p -a)) is

7

a

a continuous l i n e a r mapping f r o m

into i t s e l f .

It suffices to check that 7

is a bounded l i n e a r mapping a f r o m 33c' (0) into i t s e l f . If aEF , A E Xts(62), w e define S X

(exp(-a)).k f

EX

,(n)

.

x's(n)

by ( ( e x p ( - a ) ) . A ) ( q ) = A((exp(-CI)).y) f o r a n y

It i s i m m e d i a t e to c h e c k t h a t

T h u s the mapping 1

Kls(n) into i t s e l f .

m

-

(exp(-a}).& is a bounded l i n e a r mapping f r o m

225

Convolution operators

1 0 . 1. 3 Convolution product and convolution o p e r a t o r s . l e t T Y be the mapping f r o m 3X' S

if

@ E

S

(0)and a E F

X

(n) to x s(F

X

-

If

TE(355k(n))l,

) defined b y :

( i t i s e a s y to check that T

+

E

We r e c a l l t h a t , by definition, a convolution o p e r a t o r on

S

(FX)).

3(K's(n)

is a continuous l i n e a r mapping (equivalently a bounded l i n e a r m a p p i n g s e e 7.1.5) f r o m XK' lations

'r

a

if

a'

(n) sX

6 F

.

into i t s e l f which c o m m u t e s with a l l the t r a n s We denote by

convolution o p e r a t o r s on ZK

10. 1 . 4 LEMMA. onto the s p a c e

Proof.-

since t

3(T) E

If

S

(a)

ci

t

3

.

Is(n)

.

The mappinp T

*

G the l i n e a r s p a c e of t h e

T3c

d e c o t e s the t r a n s p o s e of

i s bijective f r o m ( 3 Y t s ( O ) ) '

5 , we have

i s n a t u r a l l y r e f l e x i v e (4.4. 1). So, i f

xs(n).

TF(5SCC!S(n))', t h e n

L e t us define a mapping U f r o m 3 ( '

S

(n)

into i t s e l f by :

Convolution equations in spaces

226

(T =

* 3 a ) ( x ) = T(T

-X

ZJ?)= T i y

-

J? ( e x p ( x t y))]

T i S ( ( e x p x ) . a ) ] = ( T 0 3 ) ( ( e x p x ) . a ) = [ ( e x p x).

=

a ] (T

o 3) =

= A I ( e x p x ) . ( T o 311 E

( w h e r e exp x and ( T o 5 ) F X (n)). On the o t h e r hand S

Therefore :

( T 3103

5 o U

and a s a consequence the following d i a g r a m is c o m m u t a t i v e .

c

Convolution operators

It is i m m e d i a t e to c h e c k t h a t U s u b s e t s of X l S ( n ) ) ,

so U

is bounded (on t h e equicontinuous

is continuous ( s e e proof of 7. 1.5). T h e r e f o r e

is conticuous s i n c e t h e mapping

T SC

3 is a topological i s o m o r p h i s m .

It i s i m m e d i a t e t o c h e c k t h a t the mapping translations. Therefore T Let

if

221

T

*

c o m m u t e s with t h e

.

*

is a convolution o p e r a t o r on 3'Hts(n)

.

One c h e c k s i m m e d i a t e l y ( t h e d e t a i l s of

Y be the mapping

(d F 3 H t s ( n ) and

6 6 G

c o m p u t a t i o n s a r e done in 1 2 . 1 . 2 ) t h a t which p r o v e s the bijection.

10. 1.5

Y o ( T % ) = T and

(Y6)t

= h,

I

G e n e r a l f o r m of t h e convolution o p e r a t o r s on E x p E.

-

-

If E

is

a c o m p l e x n u c l e a r 1.c. s . dnd i f E is t h e completior. of E , E x p E = E x p E a l g e b r a i c a l l y and topologically, s o t h a t we m a y a s s u m e without l o s s of g e n e r a l i t y that E is c o m p l e t e . T h e r e f o r e f r o m ( 0 . 3 . 1) and ( 0 . 5 . 9 ) ,

E i s a n a t u r a l l y r e f l e x i v e 1 . c . s.,

F = E ' . Now i f 0 g G ,

f r o m (10. 1.4), t h e r e is a

i. e.

ElX

= E. We s e t

TF(35(IS(F))'

such that 6 = T* , i.e.

for every -1

R = 5

$

f

3 K ' (F) = E x p E and e v e r y 5 6 F X = E . Now i f S

6 KIS(F),

it follows f r o m the a b o v e d i a g r a m t h a t

(ExpE)'

228

Convolution equations in spaces

t w h e r e we s e t p = 5 T gxS(F) ( p is c a l l e d t h e c h a r a c t e r i s t i c function of the convolution o p e r a t o r

6 ). T h e r e f o r e f r o m 7 , 1 . 4 ,

r . = o

229

Approximation of the solutions

9

10.2

L e t F and

A p p r o x i m a t i o n of t h e s o l u t i o n s

62 be a s i n

5

10. 1. Since the F o u r i e r - B o r e 1

t r a n s f o r m 3 i s a n i s o m o r p h i s m f r o m XI (0) onto XU,' (n) , i t s S s = ( 3 K ' (n))x onto t r a n s p o s e t3 is a bijection f r o m ( 3 X ' (0))' S S

K

'762) = (n) . S

S

10.2. 1 LEMMA. 3C

S

(n)

IfS

( X K 's(0))',

t h e n t3(S) is t h e e l e m e n t of

defined by : t

( 3 ( S ) ) (x) = S(exp x)

f o r e v e r y x E 62

.

~ y ' ~ ( ,nt h) e n is c o n s i d e r e d ir. K ( 6 2 ) S S t t f o r m u l a d x ) = y ( bX). So ( 3(S))(x) = ( 3 ( S ) ) ( E x ) = S ( 3 6,) Proof,-

If

Q

by the

= S ( e x p x).

We s e t E x p R c 3 K 1 (n) be t h e l i n e a r s p a n of t h e f u n c t i o n s e x p x S if x e n

.

10.2. 2 LEMMA,

-

E x p 0 is d e n s e ir? 3 H ' (0) ( f o r its l o c a l l y convex S

topology t h a t is t h e i m a g e of t h e topology of

Proof. x E 0

-

Let S E

(ax'S(0))'be

. F r o m (10.2.1)

hjective.

I

t

XI

S

(62)).

such that S(exp x) = o for e v e r y t 3(S) = o i n xs(n), s o S = o s i n c e 5 i s

Convolution equations in spaces

230

10.2.3.

-

Convolution product.

define a n e l e m e n t X

*Y t

3(X

If x , Y E ( 3 ~ ~ ~ ( n we ))l,

of (3x’ ( 0 ) ) l by t h e f o r m u l a : S

-)t

Y ) = t5(X).t3(Y)

w h e r e the r i g h t hand s i d e i s the o r d i n a r y p r o d u c t ir,

ws(n) . F r o m

and the above definitior. it follows t h a t

(X

(1)

for e v e r y x

c 0. t

andif x c R ,

* Y ) ( e x p x ) = x ( e x p x) . Y

If S , T 6 ( 5 K K ’ S ( R ) ) ’ , we have

:

5 [ ( t ( T * ) ) ( S ) ] = t3[S o ( T * ) ] = [S o ( T * ) ] o 3

5

E F”:

so t h a t

(21

(exp x)

(t(T * ) ) ( S ) = T

f o r e v e r y S, T C 3 3(

‘S( 0 ) ) l .

.)(

S

( 1 0 . 2 . 1)

Approximation of the solutions

231

10.2.4 P r o p o s i t i o n on d i v i s i o n of h o l o m o r p h i c f u n c t i o n s . complex b.v.6.

g fo,

and

7 F - o p e n set. L g f , g

a connected

t F

cxs(n) ,

be s u c h t h a t f o r e v e r y open s e t S of e v e r y affine s u b s p a c e of

d i m e n s i o n 1 of F , with S

C

G , and in which g is not i d e n t i c a l l y is d i v i s i b l e by t h e r e s t r i c t i o n g

z e r o , t h e n t h e r e s t r i c t i o n f/s

the quotient a s a h o l o m o r p h i c functior. in S.

, with /S Ther, f is d i v i s i b l e by g

with the quotient a s a Silva h o l o m o r p h i c function in fl

Proof.

k

-

.

It c l e a r l y s u f f i c e s t o p r o v e the r e s u l t in the c a s e F i s a

c o m p l e x n o r m e d s p a c e , u h i c h we a s s u m e f r o m now on in t h i s proof. Now i t is enough to p r o v e the r e s u l t l o c a l l y (in 0 which is a n open s e t of t h e n o r m e d s p a c e F). If x x

t hy F

F

n

1x1 5

1 F C,

for a l l

t h e r e is y 1.

F such that g(x t y )

#

0 and

Since the z e r o e s of a h o l o m o r p h i c

functior. of one c o m p l e x v a r i a b l e a r e i s o l a t e d , t h e r e is o < r < 1 s u c h that

if

1 e C,

IA 1

= r

{ x t Ay , A 6 C ,

.

IA I

V of x , V t f l y , all x

V acd h

f o r a l l x E V.

.$

S i n c e g is coctinuous in = r

lx

C,

I=

1AI

1

i s c o m p a c t , t h e r e is a n open neighborhood

r

] cn, = r

such that

~y

X

in

Ig(xtAy)l 5 6 P o f o r

. Now we define

h i s locally bounded in V .

a holomorphic f u n c t i o n

and the s e t

:

By o u r a s s u m p t i o n , t h e r e is

1 h F C , IX 1 e 1 1

such that

Convolution equations in spaces

232

f o r a l l t E Q:

, I t I < 1.

Hence, for e v e r y x E V

T h u s f(xf = g(x) h(x) f o r a l l x C V . bounded in V ,

on F

P f F

@n

and x C 0 , then t h e function

:

is in 35{' (0)( i f S

R

Since h is G-analytic and l o c a l l y

h is h o l o m o r p h i c in V .

N o w l e t u s r e m a r k t h a t , if X

:

P = x

~ 3 C ' ~ ( nis) defined

1

~1

... @ x n , x i E F ,

then P e x p x = 3 A

where

by

d ( 3 K ' (0))' with T # 0 . L e t u s S a s s u m e that f o r a l l P and x F 0, ther. the equality T * ( P e x p x ) = o t t i m p l y the equality X ( P e x p x) = 0. T h e n 3 ( X ) i s d i v i s i b l e by 5 ( T )

10.2.5 LEMMA.-

x X , T b F @n

with the quotient a s a Silva h o l o m o r p h i c function on

h2

.

If x , x E 0 we s e t S = { x .f w x when w r a n g e s o v e r 1 2 1 2 If w o is a z e r o of a suitable open s u b s e t of C s u c h t h a t S c 0 Proof.-

.

233

Approximation of the solutions

o r d e r m of the function

( s e e 10. 2. I ) , then a3k T ( x2

exp ( x l - t w o x 2 ) ) = o

for a l l k < m ,

k

@k

T+(x2

k

exp(x + a x ))= 1 0 2 i = o

!

1 i ! (k-i).

x

gk-i exp(x t w x ) 2 1 0 2

Therefore , 8k T 31 ( x 2 e x p ( x l t w o x 2 ) ) = o

f o r a l l k < m.

By assumptior. ,

x

( x y k e x p ( x l t w 0 x 2)

= o

.

for all k e m T h e r e f o r e w is a z e r o of o r d e r 2 m f o r t h e r e s t r i c 0 t t t tion t o S of 5 ( X ) . T h u s ( ~ ( X ) ) , I ~i s d i v i s i b l e by ( 3(T)) with t h e /S quotient a s a h o l o m o r p h i c function in S.

Now i t s u f f i c e s t o a p p l y (10.2.4).

I

234

Convolution equations in spaces

n

Now we denote by P . 6 x p

t h e l i n e a r s p a n i n 35cIs(n) of t h e

Qn

functions P . e x p x w h e r e P E F x

for s o m e v a r i a b l e n and w h e r e

en.

10.2.6 A p p r o x i m a t i o n t h e o r e m . b.v. s . and

-

L e t F be a c o m p l e x n u c l e a r

61 a convex balanced open s u b s e t of F.

convolution o p e r a t o r on 3 X ' (0). T h e n e v e r y solution t h e homogeneous equation 6

.

of solutions in P.Gxp 61

Proof.

-

fl

fl ( g 3 K ' S(n)) of

= o is a l i m i t , i n the topology of

If 6 = o t h e r e s u l t is t r u e s i n c e , by ( 1 0 . 2 . 2 ) ,

d e n s e i n 3X'

S

( f r o m 10.1.4).

P. d x p 0

s

6 be a

n

(n).

#

If 6

.

Q = (t3)"(h)

t

n),

P . & x p 0 is

be s u c h t h a t X i s null on

* (P. exp x)

o imply X ( P exp x)=

0.

t h e r e e x i s t s a n h E 5( ( 0 ) s u c h t h a t : S t

If we s e t

(a))'

S Therefore T

T h e r e f o r e , by ( 1 0 . 2 . 5 ) ,

Is(

o l e t T F(3Xts(n))'be s u c h t h a t Q = T j c

Let X 6 (SKI

Ker 8

3K

3(X) = h ( t 3 ( T ) ) .

then

E (3XtS(61) I ,

3(X)

= t3(Q)

t

3 ( Q ) = h and , f b m ( l 0 . 2 . 3 ) ,

%(T) = t3(Q

* T).

Therefore

Hence, f r o m f o r m u l a ( 2 ) i n ( 1 0 . 2 . 3 ) , t = Q 0 B , = ( (T+))(Q) = and x($)=Q(Q$)=o if !dg K e r 6 . T h e r e f o r e X is null on Ker 6 and i t

x

s u f f i c e s t o a p p l y the Hahn-Bandch t h e o r e m t o c o m p l e t e the proof.

a

235

Existence of solutions

9 lo. 3

A s before,

n

p r o p e r t y and

10.3. 1 LEMMA.

a Is(n) . T h e n

E x i s t e n c e of s o l u t i o n s

F is a c o m p l e x S c h w a r t z b.v. s. with a p p r o x i m a t i o n

a convex balanced

-

t

TF-open set.

be a nor, z e r o convolution o p e r a t o r on

@

S ( 3 , ~ s ( ~=) )( K I er

t h e p o l a r of K e r @, contained

@)O,

L ( 3 K I s ( 0))l.

T h e ir.clusion

Proof. if

t

& ( 3 X t ,(n))' c ( K e r 6)'

S 6 t@(3K's(n))1. t h e r e is a U E ( 3 K '

S

$

= o

i m p l i e s S($) =

0.

t h a t X($) = o w h e n e v e r G ( $ ) = T t h e r e is a 0

with

.

t

S = $(U)=UoO

Hence 6

(n))l

i s immediate :

Conversely let X F

*

$

=

(3KtS(n))' such that X =

such

F r o m the proof of (lO.Z.6),

0.

t

(3x'S(0))'be

b

(a).

10.3. 2 LEMMA { E x i s t e n c e of solutions i n a p a r t i c u l a r c a s e ) .

- If

F

is a Silva s p a c e , then e v e r y non z e r o convolution o p e r a t o r on 33CI (n) S is sur,jective.

Proof. -

In t h i s c a s e

",(n)

is a F r 6 c h e t - S c h w a r t z

s p a c e (4.2. 1 and

4. 3. 1) , h e n c e sKls(n) is a Silva s p a c e (0.6.8). F r o m (10. 3.11, t &(m1s(C2))' if X

i s w e a k l y c l o s e d in (35CtS(0))'

E (axl,(n))'

.

t @ is i n j e c t i v e

is s u c h t h a t t&(X) = o and if

:

8 = T * , it follows f r o m

Convolution equations in spaces

236

t t h e f o r m u l a (2) in ( 1 0 . 2 . 3 ) that T + X = 0 , h e n c e 3 ( T -X X ) = o a n d , t t t ( 1 0 . 2 . 3 ) , 3(T). % ( X ) = 0 . Since 3 ( T ) # o we h a v e 3 ( X ) = o t ( t Z ( T ) and 5 ( X ) E K S ( o ) and n is c o n n e c t e d ) , h e n c e X = 0 . Now it s u f f i c e s to apply the following c l a s s i c a l r e s u l t t h a t m a y be found in Hogbe Nlend

[

1

:

<< L e t E be a Silva s p a c e and u a continuous l i n e a r mapping f r o m E i n t o E. T h e n u is s u r j e c t i v e i f t~ is i n j e c t i v e a n d c l o s e d in E '

>>

.

t

u ( E ' ) is weakly

I

be a n i n c r e a s i n g s e q u e n c e of convex balanced Let ( C n ) n c m s t r i c t l y c o m p a c t s u b s e t s of F s u c h t h a t , f o r e v e r y n E IN , Cn is c o m p a c t and that t h e identity m a p p i n g in Fc may nt1 'n+ 1 be a p p r o x i m a t e d u n i f o r m l y on c by continuous l i n e a r finite r a n k n o p e r a t o r s (given a n y C , one m a y obtain the s e q u e n c e ( C ) icduc 0 c nEN in the Banach s p a c e F

tively, s i n c e F is a s s u m e d t o be a S c h w a r t z b.v. s . with a p p r o x i m a t i o n p r o p e r t y ) . If

8 d e n o t e s t h e b o r n o l o g i c a l inductive limit of t h e B a n a c h

c n ) n € N , 81

s p a c e s (F

i s a Silva space,(O. 6.7),

p r o p e r t y . We c o n s i d e r

R n 8 a s a 7 8 - o p e n s e t and denote by K ( n n & )

t h e s p a c e of t h e holomorphic functions on f2 fl 8 ,

6 is a Silva s p a c e ) . W e denote by r t h e r e s t r i c t i o n mapping

with a p p r o x i m a t i o n

(3CS(h2n&)=3((nnd)s i w e

Existence of solutions

Proof.

r(3cs(n)) is d e n s e in K ( n n d ) .

LEMMA. -

10.3.3

-

Since

237

8 is a Silva s p a c e u i t h a p p r o x i m a t i o n p r o p e r t y , i t

( a ) is d e n s e in 3( ( 6 1 n d ) . Now r(b;(F)) f is d e n s e in P f ( 8 ) s i n c e r ( F ” ) i s d e n s e ir, d * ( i f not t h e r e is

follows f r o m th (5. 1 . 2 ) that ‘6 x F 6’

=

8, x

#

0,

such that x’(x)= o for e v e r y

X I

i m p o s s i b l e s i n c e we a s s u m e t h a t F Y s e p a r a t e s F ,

10.3.4 Existence t h e o r e m . a p p r o x i m a t i o n p r o p e r t y and

K

6 FX

.

T h i s is

s e e (0.2.3)).

F be a c o m p l e x S c h w a r t z b.v. s . with

61 a convex balanced ‘TF-open s e t . T h e n

a n y n o n - z e r o convolution o p e r a t o r on Z X 1 (0) is s u r . j e c t i v e . S

B e f o r e t h e proof we s t a t e the m a i n c o r o l l a r y ,

COROLLARY. -

10.3.5

Let E

be a c o m p l e x n u l c e a r 1.c. s . T&

a n y non z e r o convolution o p e r a t o r on E x p E is s u r j e c t i v e .

Proof.

-

A s explained in ( 1 0 . 1 . 5 ) we m a y a s s u m e without l o s s of

g e n e r a l i t y thkt E

i s c o m p l e t e and we s e t F = E ’ equipped with the

equicontinuous bornology. T h e n , f r o m ( 7 . 2 . I ) , i t s u f f i c e s t o apply ( 1 0 . 3 . 4 ) .

I

10.3.6

Proof of t h e e x i s t e n c e t h e o r e m ( 1 0 . 3 . 4 ) .

z e r o convolution o p e r a t o r on 3 K t S ( R ) . commutative d i a g r a m

-

Let

6 be a non

We c o n s i d e r the following

238

Convolution equations in spaces

w h e r e , s e e t h e proof of (10. 1 , 4 ) , U

is given by

U(1) = ( t 3 ( T ) ) . a if 1 6

xfS(n).

Obviously, i t s u f f i c e s t o p r o v e t h a t U

If K i s a s t r i c t l y c o m p a c t s u b s e t of F contained i n

n

is s u r j e c t i v e .

and if

6 3 o

we r e c a l l that we s e t

V(K, E

EKS(h2) s u c h t h a t s u p

) =

xEK

and we denote by V o ( K , E )

i t s p o l a r in

WS(n)

.

I dx)15c I

Let u s also recall that

if K is a s t r i c t l y c o m p a c t s u b s e t of F contained i n

convex balanced hull of K is s t i l l contained in

-

n

%2 ,

L e t Y be a given e l e m e n t of t h e r e e x i s t s a n e l e m e n t X of

S

(n)

t h e r e i s a Z F X ' (0) s u c h that UZ S

K

x'S (0). We a r e g0ir.g to p r o v e

#

s u c h t h a t UX = Y.

1'

{Y,Z

U

that

#

is contained in s o m e Yo(K

E )) i s a n equicontinuous s u b s e t of k

1

Since

so

o

0.

1 bounded f o r t h e equicontinuous bornology on K

U(Vo(K

R n F B and

, E ), w h e r e w e m a y 1 1 convex and balanced ( a s a l r e a d y explained above). U is

The s e t assume

1

then t h e closed

( s i n c e K is c o m p a c t

r K i n F B is c o m p a c t in FB ' "closed" i s understood above in t h i s s e n s e ) ,

in s o m e Banach s p a c e

,

S

(n).

ls(O),

Therefore hence there i s a n

239

Existence of solutions

f

2

>

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

K'

2

c

n

such that

a convex balanced s t r i c t l y c o m p a c t s u b s e t of F s u c h 1 We denote by d t h a t K i s c o m p a c t in the B a n a c h s p a c e 1 FT 1 ( r e s p e c t i v e l y 6 ) t h e d i s t a c c e in FK1 ( r e s p : F T 1 ) obtained with the We denote by

T

.

n o r m s of t h e s e Banach s p a c e s . We s e t

KY =

if

IX

F 0

nF

K1

IN aiid w h e r e

p

s u c h that

CF

FT1

,

K1

,

which is contained in

s o i t is contained in Kp 1

Kp in F

1 contained in R .

T1

, denoted by K p

T1

1

FT 2

The

, is c o m p a c t in

n

1

u

K'

and is

2

and c o m p c t in s o m e Bdnach s p a c e F

2 in F c l o s e d convex balanced hull of K U K li K ' 1 1 2

u

is also

f o r s o m e p F IN.

i s a convex and balanced s t r i c t l y c o m p a c t s u b s e t of

r(K1

0,

Therefore

K1 U K

is contained in

.

in F

T1

-F c l o s u r e of

n

R d e n o t e s the c o m p l e m e n t of

E v e r y c o m p a c t s u b s e t of F c o m p a c t in

b(x,

Kf. U Klz),

F. c1

,

, where c 1 c1 T h e r e f o r e the

denoted by

is s t i l l contained in 0 . T h e r e f o r e t h e r e is

g 7 o 1

240

Convolution equations in spaces

s u c h that

We s e t

K

2

=

r

(K

1

u ~ f u. ~

1 t a~ l c) l ,

i s a convex balanced s t r i c t l y c o m p a c t s u b s e t of F contained 2 in h2 Define the s e q u e n c e of s u b s e t s ( KP) ir, n n F in the 2 pcIN K2 s a m e way a s it w a s done b e f o r e with K in p l a c e of K 2 , ( u s e s o m e 1 T2 in place of T ). T h e r e f o r e , e v e r y c o m p a c t s u b s e t of F , which is 1 K2 contair.ed ir, n, is cnntained in Kp f o r s o m e p F IN 2

and thus K

.

.

Now, s i n c e t h e mapping

U is l i n e a r bounded on

e x i s t a s t r i c t l y c o m p a c t s u b s e t K'

3

c

n

and a n

E

3

4. o

X'

S

( 0 ) , there

such that

;

3 K3 U K U K r 3 t h e w o r k Blready d o c e with 1 2 3 3 K u K2 U K ' b e f o r e : it follows t h a t r ( K U K u K U K' ) is c o m p a c t 1 1 2 2 1 2 3 in s o m e Banach s p a c e Fc (with c a convex balanced s t r i c t l y c o m p a c t 2 2 s u b s e t of F) and contained in n T h e r e f o r e t h e r e is a n a 3 o s u c h 2 that N o w w e d o with K

2

u

.

We s e t

K

-

3

= r(K

2

3

U KI

u

3 K 2 U KI3) t g

2 c2

241

Existence of solutions

and t h u s K

3

i s a convex balanced s t r i c t l y c o m p a c t s u b s e t of F

.

contained i n hl

nn

F

in A s b e f o r e we define now a s e q u e n c e of s u b s e t s ( KP ) 3 p c N

.

K3 subset K’

F u r t h e r m o r e , a s before, t h e r e exist

4

c fl and a n

E

4

>

o s u c h that

a strictly compact

:

B y a n obvious inductior. we obtain s e q u e n c e s (Cn)nc N (O

of s t r i c t l y c o m p a c t s u b s e t s of F and s e q u e n c e s (& ) nn@N ’ of s t r i c t l y p o s i t i v e r e a l n u m b e r s s u c h t h a t :

c)nF N

Kn+ 1

(Kn)I?FN’ ( K ’ n ) n E N ’

c R a n d F

compact in F

C

n

K

.

C F r*

C F

with cnntinuous i n c l u s i o n s , and

Kn+ 1

r .

K

n

F u r t h e r m o r e , s i n c e F i s a S c h w a r t z b.v. s . with a p p r o x i m a t i o n p r o p e r t y (5. 1. l ) , then c

nt1

,

a t e a c h s t e p of t h e induction, w e m a y c h o o s e

l a r g e enough s u c h that the identity o p e r a t o r on F

a p p r o x i m a t e d u n i f o r m l y on c on F

.

n

m i g h t be C n t1 by finite r a n k s continuous l i n e a r o p e r a t o r s

C n t1

We denote by ( F c ) n € IN n

6 t h e inductive limit of t h e B a n a c h s p a c e s

, which is a l s o t h e inductive l i m i t of t h e B a n a c h s p a c e s

Convolution equations in spaces

242 : clearly

n

8 is a Silva s p a c e with a p p r o x i m a t i o n p r o p e r t y

(in the s e n s e of 5 . 1. 1).

Now , f r o m (10.3. 3 ) , the r e s t r i c t i o n mapping

h a s a d e n s e r a n g e ; thus its t r a n s p o s e

x

is injective. So K l s ( n n & ) m a y be identified a s a s u b s p a c e of

S( 0 ) .

6 which

Now l e t u s o b s e r v e t h a t a n y s t r i c t l y c o m p a c t s u b s e t of

i s c o n t a i n e d i n Q , i s contained i n K f o r s o m e n ( t h i s is d u e to the n for every n F N). F o r convenience u s e of t h e s e q u e n c e s (Kp ) n p F N one m a y a s s u m e without l o s s of g e n e r a l i t y t h a t lim E = o Therefore n I? a3 the sets

.

4

a r e a b a s e of o - n e i g h b o r h o o d s of 51

Wo(K,'C

n

) in X

IS(n n

S

(n n 6 )

Therefore their polars

6 ) a r e a b a s e of equicnntinuous s e t s .

F r o m t h e d e n s i t y of r(5( ( 0 ) ) i n that

.

S

Ks(n n 8 ) (10.3.3),

it follows

243

Existence of solutions

Since, by c o n s t r u c t i o n ,

then the s u b s p a c e 5 ( ' (0 n 8 ) of S restriction U

Z F3CiS(nn&) ( b e c a u s e Z e V o ( K

i t s e l f . S i n c e , by c o n s t r u c t i o n ,

U

Now

6

is mapped i n t o i t s e l f by U

and

is a bounded l i n e a r mapping f r o m K t (npd) into S

/X i s ( n ns)

and UZ # o ,

xis(n)

/~i,;nn

E )) ,

1' 1

is non z e r o .

8)

is the o p e r a t o r which c o r r e s p o c d s to U

dvS(nn 6

/33cfs( 0 na)

i. e. s u c h that the following d i a g r a m i s c o m m u t a t i v e

(since

6 = T* ,

/ ~ 3 cp

n a

=

i s the m u l t i p l i c a t i o n of e l e m e n t s of t

3(T)/

t

nn5

= 5(T

' ~ x i ~ ns) jn

).

)c

Is(

Thus @

convolution o p e r a t o r on 3%' (rind) S

(7%n ns)) x'S( n n 5 ) by

.

' 5 3 ~is(n n s )

.and U

/xis(n n8 1

is a non-zero

'

244

Convolution equations in spaces

From (10.3.2), is s u r j e c t i v e on

x

ssris(61ns).

I s ( n n 4 ) . Since Y F

C o n s i d e r i n g X in

10.3.7

Remark.

and s i n c e B

-

XI

S

x ’S(n)

Thus

is a Silva s p a c e ,

U/k~,(n”s)

6

i s s u r j e c t i v e on

(nn8) t h e r e e x i s t s a n X F K’ ( f i n d ) s u c h t h a t S

we have UX = Y

.

A n o t h e r d i f f e r e n t proof of th. ( 1 0 . 3 . 4 ) w i l l be given

i n the next c h a p t e r . N e v e t h e l e s s the c o n s t r u c t i o n ir. the above proof

(10.3.6) w i l l b e used i n chap. 15 f o r a d e e p e r s t u d y of t h e k e r n e l s of convolution o p e r a t o r s on Exp

E.

CHAPTER 1 1 DIVISION OF DISTRIBUTIONS

ABSTRACT. THEOREM

-

L e t E be a c o m p l e x 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

( i n p a r t i c u l a r d u a l n u c l e a r ) 1. c. s. and l e t h2 be a connected s e t . L e t T be a given e l e m e n t of h o l o m o r p h i c function on

TE-open

d ' ( 0 ) and p be a g i v e n non z e r o

. T h e n t h e r e is a n

S E d ' ( n ) s u c h that

p . S = T .

A n i m m e d i a t e c o n s e q u e n c e is a new proof of the e x i s t e n c e of s o l u t i o n s of convolution e q u a t i o n s i n s p a c e s of e n t i r e functions of exponential type ( a l r e a d y obtained with a d i f f e r e n t proof i n the p r e c e d i n g c h a p t e r 3. T h e n we show t h a t division by r e a l polynomials and a n a l y t i c f u n c t i o n s is i m p o s s i b l e in g e n e r a l e x c e p t f o r finite type polynomials and f o r finite type analytic functions.

245

246

Division ofdistributions

6

The Wei'erstrass preparation theorem.

11.1

11. 1. 1 R e c a l l s on h o l o m o r p h i c functions of one c o m p l e x v a r i a b l e .

-

If

r 9 o we s e t :

1.1

E C such that

A

= /z

C

= {z € C s u c h t h a t

12;

L r = {z C C s u c h t h a t

< r] ]=r

1z15r

1

A

=

u cr.

We denote by 3C(b ) the B a n a c h a l g e b r a of t h e c o m p l e x valued continuous functions on 11 with the s u p - R o r m on a r e in

x(A

br.

which a r e h o l o m o r p h i c in If U r

hr,

equipped

d e n o t e s t h e set of t h e functions which

U r is a n open 'r' t h e s u b s e t of U

) and which d o not take t h e value o on

s u b s e t of K ( A r )

.

If n E N ,

we denote by U

r,n m a d e of the functions t h a t h a v e e x a c t l y n z e r o e s i n A r

( e a c h z e r o is

counted with i t s o r d e r of multiplicity). C l a s s i c a l l y (Dieudonnk 9.17.4)

U

r,n

is a n open s u b s e t of 3

(2

)

.

[ 1

3

1 1 , 1 . 2 The W e i ' e r s t r a s s d i v i s i o n t h e o r e m i n the one d i m e n s i o n a l c a s e .

(1)

For every

e x i s t unique y E X (

L ~ and )

'

g 'r,n (ao, a l , .

and e v e r y f

. . , an - 1 F a?

f ( z ) = g(z) q ( z ) t a. t a z t 1 for e v e r y z E

Lr

.

cK

(ar)

such that

... t a n - 1 z n - 1

there

Weierstrass preparation theorem

(2)

T h e mapping

247

8

n - 1 a . z.

1

i = o is a n h o l o m o r p h i c d i f f e o m o r p h i s m .

Proof.

-

( 1 ) i s t h e c l a s s i c a l W e i e r s t r a s s division t h e o r e m

(Hormander Clearly

[

1

]

c h a p , VI) and a s a c o n s e q u e n c e

8 is continuously d i f f e r e n t i a b l e . F o r a n y

the d e r i v a t i v e 6 ‘ ( a ) is bijective

8

is a bijection.

g

6 Ur,n

x

“(~,)XC”

( u s e (1)). F r o m t h e c l a s s i c a l i m p l i c i t

function t h e o r e m ir. Banach s p a c e s (DieudocnC [ 1 ] c h a p 10) t h e i n v e r s e @-l . is d i f f e r e n t i a b l e , h e n c e h o l o m o r p h i c . mapping

11.1.3

The Weierstrass division theorem in Banach spaces.

a c o m p l e x Banach s p a c e and

n

If g F X ( % 2 ) , g f o N g ( 0 ) =

(1)

a convex balanced 0

- Let

E

o-neighborhood in E .

ther,:

t h e r e e x i s t s a d e c o m p o s i t i o n of E i n a topological

d i r e c t sum

E = F & c ~e ,f o

N

s u c h t h a t , i f w e denote by g ( 5 ) the function z a convex balanced

o-neighborhood

W

&

-, g ( 5 t

F and a n r

ze), there exists

>o

s m a l l enouph

Division of distributions

248

s u c h that, if

5 E

f o r some p E

IN

(2)

01

C

UJ,

;

if f

( 0 ) t h e r e i s a cocvex balanced o-neighborhood

h2 and t h e r e e x i s t unique

(1

E 3(

( n ' ) and a,,'.

s u c h that

- . , a P- 1c h ' ( f a l n F )

i = o for every x E R I

Proof.

-

(1)

. Since g

f

o t h e r e is a n e 6 k-2

, e # o , such that

E=F$(I:e x = ! , t z e N

(topological d i r e c t s u m ) . T h e function defined i f

z

r, g ( 5 t z e ) = (g(5))

5 t ze E n. L e t r > o be s u c h t h a t g/

t a k e the value o on C e c 0 hood in F such that

. Let

W

Ce

n

g!

(2)

is

does not

be a c o n v e x balacced o - n e i g h b o r -

249

Weierstrass preparation theorem N

T h u s g(6) c X ( i r ) i f

5

F W

and t h e r e f o r e we m a y c h o o s e

5 E N

g(5)

ui

.

r-d

W

s m a l l enough s o t h a t g ( s ) E U

. T h e c u m b e r of z e r o e s (with t h e i r o r d e r of m u l t i p l i c i t y ) of

in b r

is a continuous function of

w

hence g ( 5 ) F U

r *P

(2)

for some

5 ( f r o m Dieudonne' [ 1 ] 9.17.4),

p E N and a l l

5 6

W

.

We set

N

Thus f ( 5 ) E X ( & r ) if N

q ( $ ) € 3 r ( a r ) and

if

ai($)

5

W

.

F o r a n y fixed

5

UJ

l e t u s define

C ( o > i s p - 1 ) by ( s e e 1 1 . 1 . 2 )

Therefore

-r

i = o

:

250

Division of distributions

--

Clearly g, f 6 c . 4

q

c

X (W,X ( A r ) ) .

A ) and a i E X ( w , Q .

,3((br)) ,K(W

3((w

It follows t h a t , via

and it s u f f i c e s t h a t R ' c U J t b r e and

0' n F c

e-', T h e r e f o r t . qF3C("+Are)

UJ

. T h e u n i q u e n e s s of

and

follow s i m m e d i a t e l y f r o m the uniqueness in ( 1 1.. 1. 2) p a r t ( I ) ,

( a 1'02 is p- 1 applied f o r any fixed

11.1.4

5.

The W e i e r s t r a s s p r e p a r a t i o n t h e o r e m in B a n a c h s p a c e s .

Let E

U =open

be a c o m p l e x Banach s p a c e ,

o-neighborhood

i n E and l e t be given a g

3((U) , g $

e x i s t s a decomposition of E

in a topological d i r e c t sum

E = F G C e

g(o) =

0 , &w

Then there

0.

( e f o )

x = 5 t z e , a n i n v e r t i b l e holomorphic functior, J i n a

o-neighborhood

V

E

a polynomial

with ai f X (V p F) , ai(o)

0,

for all x

V.

Proof.-

We a p p l y (11.1.3)

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

g to U

g(o)=o, gl(o).e=o,.

(Ir 1 5 i 5 p),

: let p f

c@e , i . e .

such that

IN d e n o t e the o r d e r of the

:

.. , g ( P - l ) ( o ) . e p - l = o ,

g(P)(o) -eP #

0.

ar.d

25 1

Weierstrass preparation theorem

We c h o o s e 62

C

U s m a l l enough s o t h a t g

than the origin in C e

F r o m (11.1.3)

n

R

.

/Ce n R Define f FX(62) by :

there a r e V c

n,

h a s no o t h e r z e r o

q F K ( V ) and a . c X ( V f l F ) , o 5 i 5 p - 1 ,

such that :

i = o

for all x =

5

t ze

a.(o) = o since g/ q(o)

#

V.

Ce

n

Therefore

a d m i t s t h e origir, a s a z e r o of o r d e r p and

o f o r t h c s a m e r e a s o n , T h e r e f o r e q is i n v e r t i b l e in a o - n e i g h b o r -

hood i n E .

m

252

Division of distributions

11.2

Division b y a c o m p l e x polynomial

E 6 ( W ” ) , m F IN

If

and i E INn

i s a m u l t i - i n d e x of d e r i v a t i m

we s e t

(possibly M

m

1 1 . 2 . 1 LEMMA. degree p 6 N ,

I’ = P y .

t a).

(6)

- Let P s u c h that

P(x) = o implies

x =

J f - Cp F

r .

Ther, the followiing inequality h o l d s :

w h e r e the c o n s t a n t A ( m , P) d e p e n d s only on m

Proof. -

0.

p.

,of 8(R ) we

be a h o m o g e n e o u s p o l y n o m i a l on JR

& P,

but not on

W e u s e the c l a s s i c a l m u l t i - i n d e x n o t a t i o n s (Schwartz

c2]

f o r i p s t a n c e ) . L e t u s d e v e l o p p \Y a c c o r d i n g to T a y l o r ’ s f o r m u l a u p t o

o r d e r m t p - 1 , G i t h the r e m a i n d e r w r i t t e n in i n t e g r a l form ( s e e Hoffman

[1 ]

5.4. f o r i n s t a n c e ) :

I

Rmtp-1

1 q I =mtp-1 i=l,2,.

. ., n

.

25 3

Complex polynomial

T h i s i n t e g r a l i s c o m p u t e d on a n y r e c t i f i a b l e c u r v e of o r i g i n o and endpoint x in

5

1

s i n c e f o r a n y fixed x t h e d i f f e r e n t i a l f o r m of d e g r e e

, under the integral sign , i s closed. F r o m ( 2 ) me h a v e :

(3)

S i n c e P is h o m o g e n e o u s and

'Y d i v i s i b l e by P,

p o l y n o m i a l s of v a r i o u s d e g r e e s t h a t a r e i n

by

P,

and t h u s

Qmtp- 1

P

a l l its d e r i v a t i v e s of o r d e r m

the homogeneous

Qmtp- 1

a r e a l l divisibke and s o

is a p o l y n o m i a l of d e g r e e 5 m - 1

a r e z e r o . T h e r e f o r e , if

I r 1 =m

:

s lr

By i n d u c t i o n on s ,

where

6

S

one p r o v e s e a s i l y t h a t D S ( 1/P) is a q u o t i e n t

i s a h o m o g e n e o u s p o l y n o m i a l of d e g r e e

w h e r e c ( s , P) is a c o n s t a n t t h a t d e p e n d s o n l y on s and hand,

1 I.

(p-I) s

P.

e

S ,

So

On a n o t h e r

Division of distributions

254

i f a is the i n f i m u m of

1 P(x) 1

if

Ix I

= 1.

F i n a l l y f r o m ( 5 ) and ( 6 ) :

w h e r e c ( s , P) is a n o t h e r c o n s t a n t which d e p e n d s only on s and L e t u s now c o n s i d e r the d e r i v a t i v e s of o r d e r S m

Of

P.

Rm+p-l

*

If m 2

5 i s a s u m of 2 t e r m s : d e r i v a t i o n of 1, a derivative bx k R m + p - l

(x-5)'

in the i n t e g r a l and d e r i v a t i o n ic x which is the endpoint of the

path of integration. F o r t h i s l a s t d e r i v a t i o n we m a y a s s u m e t h e path of i n t e g r a t i o n e n d s by a segment p a r a l l e l t o the x - a x i s and finally one

k

obtains :

T h e s u c c e s s i v e d e r i v a t i v e s m a y be computed in the s a m e way ( t h e expocents of (x-5 ) r e m a i r , I m ) . F i n a l l y one obtains

70

s i n c e w e c o m p u t e d e r i v a t i v e s of o r d e r

:

i = l,Z,...,n

We compute t h e i n t e g r a l i n ( 9 ) on t h e s t r a i g h t l i n e s e g m e n t

[ o,x]

of

25 5

Complex polynomial

length

I XI

where

c ( m , p ) is a c o n s t a n t t h a t d e p e n d s only on m a n d

and it follows t h a t

p.

F r o m (4) ( 7 ) and (10) it f o l l o w s :

(since r with

- 1 s l - p t m+p-lr-sl = m - I r I = o ) . Ir

I

i m one o b t a i n s (1).

11.2. 2 P a r t i c u l a r c a s e . -

From(I1)writtenforall

I

If n = 2 , IR2

a

C by s e t t i n g x t i y

= z

and if P ( x , y ) = z w e obtain

w h e r e t h e c o n s t a n t A ( m ) d e p e n d s only upon m F

IN

.

S i n c e M ( 8 ) is t r a n s l a t i o n i n v a r i a n t , t h e s o m e bound h o l d s if i s r e p l a c e d by d e n t of

z

) :

z

- z0'

z

0

z

C , with t h e s a m e c o n s t a n t A ( m ) ( i n d e p e n -

256

Division of distributions

LEMMA.

11.2.3

complex variable

-

k

C

z

2 If y E d(IR )

t P be a u n i t a r v polynomial of d e p r e e p in t h e

\Y

:

Pq then the following i n e q u a l i t v h o l d s :

w h e r e t h e c o n s t a n t A ( m , p ) d e p e n d s only upon m and

p, and n o t on cp

and the coefficients of P.

P(z) = (z-z ) ( z - z ) 1 2 w i t h z - z . , i = 1, ,p. Proof. -

...

... ( z - z P)

and it s u f f i c e s t o a p p l y (12l)

If 52 is a n o p e n s u b s e t of IR2? i f m E

11.2.4

C,i

LEMMA.

fn'

1 1 . 2 . 3 , if

-

( l o c a l f o r m of 1 1 . 2 . 3 ) :

is a bounded open s u b s e t of

cp

IN

and cped(n), w e s e t

If n

C with

i s a n open s u b s e t of c

R , if P

i s a s in

d ( n ) a n d \Y = Pcp, t h e n

w h e r e t h e c o n s t a n t A ( m , p, 61,

a')

d o e s not depend on cp a n d P.

Complex polynomial

a E J(n), a

-

1 in a neighborhood of 6'' and a 2 , t h e n o w and a'Y = P a f p F 6 ( R ). F r o m (13)

Proof.

Cn

-

251

If

E

E

o in

F r o m Leibnitz's formula

since a

d e p e n d s only on

N o-

and

n'. I

w e a r e going t o a p p l y thes:

r e s u l t s in the infinite d i m e n s i o n a l

c a s e , by i c t r o d u c i n g a p a r a m e t e r t v a r y i n g in a n o r m e d s p a c e . Let A

be a o-neighborhood in a r e a l n o r m e d s p a c e F (of n o r m

11 /I F )

and l e t t d e n o t e a v a r i a b l e point r a n g i n g o v e r p 2 Let fl be a o-neighborhood i n C = IR and l e t z d e n o t e a v a r i a b l e

denoted by

point rar.ging o v e r

R.

We c o n s i d e r t h e p o l y n o m i a l s in z

w h e r e the f u n c t i o n s a . a r e in in

A

.

W e set

.

&(A) ,

with a l l t h e i r d e r i v a t i v e s bounded

258

Division of distributions

be a function i n d(n x 1\)

Let

and

'f = P.7.

0' d e n o t e s a n o p e n

n'

is contained in 0. bounded o-neighborhood in C such t h a t i t s c l o s u r e 2 If @ E d ( R A ) , h and k E IN, r = ( r l , r 2 ) F IN , we s e t :

11.2.5

LEMMA. -

With t h e s e notations

w h e r e t h e c o n s t a n t B d o e s not depend on CF: only on h , k , p. a k ,

Proof.

-

n, n '

.

T h e proof i s by induction on k.

E d ( n x I\) and on P, but

For

consequence of (14). L e t u s a s s u m e (15) holds f o r k i t holds f o r k. W e fix t

1'""

t

k

.c F with Ilt.11

I F

15) is a

k= o

-

5 1

1 and l e t us prove

.

F r o m Leibni.tz s

formula :

S'

(16)

( bk k

P) t l . .

bt

w h e r e 8 d e n o t e s t h e s y m m e t r i c p r o d u c t of a n e l e m e n t of S

e l e m e n t of

L( ( k - s ' ) F ) .T h e r e f o r e :

S'

.tk

L( F ) and an

259

Complex polynomial

L e t 0 ' ' be a boucded open s u b s e t of

C , such that

bk F o r m u l a (14) applied t o (q). tl.. k at

.tk

121' C

$2

and E ' C R ' ' .

f o r a n y fixed t 6 A

gives

w h e r e A ( h , p, rill, n') i s independent on t E i\ , Now w e s e t r = (r

r )

1' 2

em2

with

r t r = h t p. F r o m (17) it s t i l l follows from 1 2 r tr

L e i b i t z ' s f o r m u l a t h a t , if D r =

b

r

dx

k

r

( s l ) < r l ) (D

1

r'

2

1

's'cp

by

Q S

r tr tk

(if

1

Dr'k =

2

r

bx

r by

1btk

r

with z = x t i y , 2

Dr-r'' k-s'

P)tl..

.t k

:

260

a)

Division of distnbffrjoffs

L e t u s f i r s t notice t h a t by definition :

SUP xtiy

b)

R"

IDr'

.k

y(x, y , t). t l . . t 15 M

htp,k

(YY,Oxh)

.

Now we c o n s i d e r the o t h e r t e r m s i n the ri&t hand s i d e of ( 1 9 ) :

f r o m the f o r m u l a of

P,

( w h e r e the above bound only d e p e n d s upon t h e q u a n t i t e s between p a r e n t h e s e s ) , On a c o t h e r h a n d , s i n c e s ' 5 k - 1 , f r o m f o r m u l a ( 1 5 ) f o r k-1 w e have :

sup

05 j I k - 1

M h t p ( j t l), k- 1- j ( y , n

F i n a l l y , f r o m a ) and

X

A).

b) and s e t t i n g j ' = j t 1, h ' = h t p , we obtair. f r o m (19) :

Complex polynomial

N o w i t s u f f i c e s to a p p l y (14) t o obtain, f r o m (20), i n e q u a l i t y (15) of order

k.

261

262

Division of distributions

0

11.3

D i v i s i o n of d i s t r i b u t i o n s by h o l o m o r p h i c f u n c t i o n s

-

11.3.1 T H E O R E M

and fl

a connected

p g",(n)

with p

#

Let

E

be a c o m p l e x Schwjartz b. v. s .

7 E - o p e n s e t . Let be given a n y TEd'(h2) and a n y 0.

Then t h e r e exists a n S E d'(n) such that p . S = T.

B e f o r e t h e proof w e need a l e m m a .

1 1 . 3 . 2 LEMMA. -

The mapping

is well defined a c d c o n t i n u o u s if p. d ( n ) = ip.cp[

with the topology induced by t h a t of

Proof.

-

p. y = o i m p l i e s

= o

balanced bounded s u b s e t B of E , point in p.(r

-

R

n EB,

so

VmE,

d(h2).

since p

#

cFEd(fi)

is equipped

o ( f o r e v e r y l a r g e enough

the s e t of z e r o e s of p h a s n o i n t e r i o r

o by continuity). T h e r e f o r e the m a p p i n g

is w e l l d e f i n e d , Now we p r o v e its continuity. L e t U'

o-neighborhood in d ( n ) , t h a t we m a y a s s u m e of t h e f o r m :

be a

263

Distributions by holomorphic functions

n,

K is a s t r i c t l y c o m p a c t s u b s e t of

where

balanced s t r i c t l y c o m p a c t s u b s e t of E , Let K'

L is a c o n v e x

where

where

E

>o

and n E IN.

E such that K

be a c n n v e x balanced s t r i c t l y c o m p a c t s u b s e t of

is c o m p a c t in t h e B d n a c h s p a c e E K , a n d t h a t K a n d L a r e c o n t a i n e d i n K ' ( K ' i s not necessarily contained i n 0). L e t xo be ap. e l e m e n t of

K s u c h t h a t p ( x )=o. In o r d e r t o u s e (11. 1.4) l e t u s a s s u m e 0

p r o v i s i o n a l y (by t r a n s l a t i o n ) t h a t x

0

=

0.

T h e n if

g = p/

(Ep)'

it

f o l l o w s f r o m ( 1 1 . 1.4) t h a t E K l = F C€ C e

and

g = P.J

R' of E K l ,

in a o-neighborhood

P and J h a v e t h e p r o p e r t i e s

where

l i s t e d i n ( 1 1. 1.4). F is equipped with t h e topology induced by E Let

E

0

9 o be s u c h t h a t E o K '

i n F and If

U

K" be a n o p e n o - n e i g h b o r h o o d

c nl. L e t

a n open o-rieighborhood i n

Ce

such that

~ d ( n we ) a p p l y l e m m a ( 1 1 . 2 . 5 ) . t o P.(Jv)/

A y W

A

x

and

c EoK'.

(Jv)/A

( ( J C ; ) / , ui~ is denoted by y i n 11.2. 5). T h i s g i v e s u s a l o c a l ( i n a ) m a j o r i e a t i o n of Jv f r o m a m a j o r i z a t i o n of K' W r i t i n g now e x p l i c i t e l y t h e point x for c o n v e n i e n c e of t h e

o-neighborhood of E

P. J.

.

0'

end of t h e p r o o f , we o b t a i n : f o r e v e r y n" an n'

N and a n

& I >

pcp

implies

o such that

V(xot

C

K',

E

', n ' )

N and

& "

> o there exist

264

Division of distributions

where

W'

comes f r o m lemma (11.2.4), i.e.

W' C VJ

.

(indeed (I)

implies that

w h i c h , b y l e m m a ( 1 1 . 2 . 5 ) i m p l i e s (11)). But J i s i n v e r t i b l e on x SE K ' , 0

if

F

0

>o

is s m a l l enough. H e n c e , f r o m (II), f o r e v e r y n C N and

0

E>o

t h e r e e x i s t a n n ' Fm and a n 0 ' 3 o s u c h t h a t , if

ther.

for some

3 o

s m a l l enough.

F o r a n x o F K s u c h t h a t p(x )

-

0

#

o , the implication (I)

(111)

is t r i v i a l l y valid. Since K i s c o m p a c t i n E K l , w e obtain i m m e d i a t e l y t h e continuity of the mapping

p ( ~

q

.I

265

Distributions by holornotphicfitnctions

11.3.3

We define a l i n e a r f o r m

Proof o f t h . (11.3.1).-

R

on p . & ( n ) c d ( n ) by t h e f o r m u l a

F r o m ( 11.3.2)

k

is continuous and s o from t h e Hdhn-Banach t h e o r e m

it m a y be continued a s a n S E 6 ' ( n ) . We h a v e

pS = T.

I

Division of distributions

266

9 11.4

Application to e x i s t e n c e of s o l u t i o n s

We r e c o v e r th ( 1 0 . 3 . 4 ) a s a n i m m e d i a t e c o n s e q u e n c e of ( 1 1.3. 1). L e t u s r e c a l l th. (10.3.4) : l e t E be a c o m p l e x S c h w a r t z b . v . s .

R

a p p r o x i m a t i o n p r o p e r t y and

a convex, balanced

with

TE-open set. Then

a n y Eon-zero convolution o p e r a t o r on 3KVS(n)i s s u r j e c t i v e .

11.4. 1.

-

T h e new proof of t h i s r e s u l t , u s i n g ( 1 1.3. 1) is t h e following. 8 = T3e f o r s o m e T 6

F r o m ( 1 0 . 1.4).

. Let

33(' (n) be giver. and l e t u s s e e k f o r ap, f c 3 X ' (0) s u c h t h a t

g T

(3K lS(cL))' , T # o

S

3t

f = g.

S

T h i s eyuatior, i s equivalent to :

&(T

-)c

f) = 3-lg.

F r o m the c o m m u t a t i v e d i a g r a m in proof of (10. 1.4), we have 3-'(T+f)

t

3(T).3-'f,

s o that o u r equation b e c o m e s :

where

t

3(T) E

xs(n)

and

t

3(T) f o .

3

-1

g F ~ ' ~ ( and 0 ) K s ( n ) is a

topological s u b s p a c e of d ( n ) , s o t h a t , f r o m Hahn-Banach,

5

-1

g may

be continued a s a n e l e m e n t h 6 d ' ( n ) . F r o m (11.3. 1) t h e r e i s an t A d V ( n ) s u c h that 5 ( T ) . A = h in d ' ( 0 ) . A

h

= 3-l g

, s o t h a t it s u f f i c e s to choose 3 - ' f

= 1

. I

261

Existence of solutions

11.4. 2.

-

If E is a r e a l n u c l e a r b.v. s . me c h a r a c t e r i z e d i n ( 7 . 4 . 1)

the i m a g e 3 d ' ( E ) of

d ' ( E ) , u n d e r the F o u r i e r t r a n s f o r m 3 , a s a

s p a c e of e n t i r e functions on E'

+

i

E '

(Paley-Wiener-Schwartz

t h e o r e m ) . Now, a s i n t h e d i a g r a m in proof of (10. 1.4), i f

6 is a

convolution o p e r a t o r on 3 d ' ( E ) then, via 3 , @ b e c o m e s in d ' ( E ) the t multiplication o p e r a t o r by 5 ( T ) g 8 ( E ) , f o r s o m e T C ( 3 d ' ( E ) ) ' with

8 = TW.

t

Z ( T ) is c a l l e d , a s ir, c h a p t e r 10, t h e c h a r a c t e r i s t i c f u m t i o n

of @ . P r o o f (11.4. 1) g i v e s :

PROPOSITION.

-

Let E

be a c o m p l e x n u c l e a r b . v . s. and l e t 8

be

a n o n - z e r o convolution o p e r a t o r on 5 d ' ( E ) s u c h t h a t its c h a r a c t e r i s t i c function is Silva h o l o m o r p h i c . T

d 14 is s u r j e c t i v e .

Division of distributions

268

$

i 1.5

Division by r e a l a n a l y t i c f u n c t i o n s of finite type

We prove a division r e s u l t which is a c o n s e q u e n c e of t h e c l a s s i c a l finite d i m e n s i o n a l c a s e r e s u l t s ,

11.5. 1 Definitions of r e a l analytic m a p p i n g s and r e a l a n a l v t i c m a p p i n g s of finite type.

-

If E is a r e a l b . v . s . ,

n o r m e d s p a c e , we s a y t h a t a mapping f : convex balanced bounded s u b s e t B of E , locally (in fi

0 a -+

T E - o p e n s e t and F a

F is a n a l y t i c i f f o r e v e r y

the r e s t r i c t i o n f

nEB

is

nEB ) the s u m of a n o r m a l l y c o n v e r g e n t s e r i e s of E B -

continuous polynomials ( s e e 2 . 6 . 2 ) .

-. F is a finite type a n a l y t i c

We s a y that a mapping f :

mapping in 0 i f f o r e v e r y B a s a b o v e t h e r e s t r i c t i o n

f/n

n E B is

l o c a l l y a n a n a l y t i c mapping of finite t y p e , i. e. f o r e v e r y x E OnE 0

B

t h e r e a r e a n E 3 0 , a d e c o m p o s i t i o n of the n o r m e d s p a c e E B i n a 1 2 1 topological d i r e c t s u m EB = EB @ E B , ( x = x 1 t x 2 ) , with d i m E
1

/x0tC B

2

(x t x ) = fl(X1)

.

In p a r t i c u l a r a n y finite type continuous polynomial ( s e e A b s t r a c t of chap. 5 ) i s a finite type a n a l y t i c mapping. If E is a l i n e a r s p a c e equipped with the finite d i m e n s i o n a l bornology, a n y a n a l y t i c m a p p i n g is c l e a r l y of finite type.

269

Real analytic functions of finite type

1 1 . 5 . 2 PROPOSITION.

-

Let E

h2

be a r e a l S c h w a r t z b . v . s . ,

a

connected 'TE-open s e t and p a n o n - z e r o f i n i t e tvpe r e a l a n a l v t i c mapping in 0

.

T h e n f o r e v e r y S 6 6 ' ( 0 ) t h e r e is a T F d I ( 0 ) s u c h

= S.

=PT

Proof.

-

F r o m proof 1 1 . 3 . 3 it s u f f i c e s t o show t h a t t h e mapping

is w e i l defined and cnntinuous. T h e continuity w i l l be obtained a s a

c o n s e q u e n c e of t h e s a m e r e s u l t in the finite d i m e c s i o n a l c a s e which follows f r o m L o j a s i e w i c z

[

1, 2

1

). L e t V ( K , L, E , n ) be a given

o-neighborhood in d(n) ( s e e proof of 11.3.2) and l e t B be a cnnvex balanced bounded s u b s e t of E L i s bounded in E x

E

n n EB,

s u c h t h a t K is c o m p a c t in E B and t h a t

F r o m t h e a s s u m p t i o n on p , f o r e v e r y B' m a y be w r i t t e n a s a topological d i r e c t s u m

EB x E E ) with d i m E < t m and E B = E l 6 E 2 (x = x t x x E E 1 2 ' 1 1 ' 2 2 1 0

p f x ) = p(x ) for e v e r y x in a s u i t a b l e x -neighborhood in E 1 0 B

m a y a s s u m e B h a s the f o r m B

1

f€

B2,

where B

1

= El. *I notations we w r i t e E x E i n s t e a d of E 6 E 2 . 1 2 1

convex balanced and bounded and w h e r e E

We s e e k f o r a o-neighborhood would i m p l y f

V(K,L,C ,n).

W

C

E

1'

B2 c E

(1)

are

i n d ( 0 ) s u c h t h a t pfFW

K in 0 n E B ,

F r o m the c o m p a c t n e s s of

tx EK 1 0,2 4. o and a o-neighborhood W(x ) in d ( n ) s u c h t h a t : 0

E

2

.

F o r convenience of

it s u f f i c e s , a s in proof 1 1 . 3 . 2 , to p r o v e t h a t f o r e v e r y x = x

there a r e an

.We

0

pf 6 W ( x o )

3

f E V ( x t& B , B , 0

0

E

,n),

0,

Division of distributions

270

f € V(x t & oB, B,E , n ) m e a n s t h a t i f 0

i ti

I(

(2)

b bx

1

i ) with

1' 2

2

i

< n and

1 -

i

i

1

i = (i

.bx

i

+E

B ,x

f ) ( x ~1, o 1

2

0,2

t E

i < n,

2i

o B 2 )(B1) '(BJ212-2.

2

i

If

i sn, 2

2 x 2 E x o , 2tEoB2, y2E(B2) , w e s e t

T h e r e f o r e ( 2 ) i s equivalent to :

f o r e v e r y i 2-

i 2 n, x 2 F x 0, 2+coB2' Y2E(B2)

i s finite d i m e n s i o n a l and if n c n is a neighborhood 1 1 of x te B in E f r o m L o j a s i e w i c z ' s r e s u l t i n the finite 0,1 0 1 1 ' d i m e n s i o n a l c a s e ( c h o o s e B l a r g e enough in o r d e r t h a t p /n n E Bf 0 ) Sicce E

t h e r e is a o-neighborhood

W

1

in

d ( n l ) s u c h that

271

Real analytic functions of finite type

T h e r e f o r e , ir. o r d e r t o have f E V ( x , t EoE3,B,e,n) it s u f f i c e s that

(4)

P. gi

2’ x 2 ’ y2

E W

1

i

tEoB2,y2E(B2) f o r every i 5 n, x E x 2 2 0,2 and K

1

c o m p a c t ir.

0

1’

‘.

W 1 = V ( K I , B l , E I , n l ) , (4) is e q u i v a l e n t to :

T h e r e f o r e , i n o r d e r to have f F V ( x

t E B, 0 0

B,

T h i s l a s t inequality d e f i n e s a o-neighborhood

1 1 . 5 . 3 PROPOSITION.

-

If, with u s u a l n o t a t i o n s

Let

E,

n),

it s u f f i c e s t h a t :

W ( x ) ir. d ( 0 ) .

E be a r e a l n u c l e a r b.v. s . and

@

n o n - z e r o convolution o p e r a t o r on 5 6 ‘(E) s u c h t h a t its c h a r a c t e r i s t i c function i s r e a l a n a l y t i c of finite type. T h e n

8 is sur,jective.

The proof i s s i m i l a r to 11.4.1 and 11.4.2.

,

212

Division of distributions Y 11.6 Impossibility of the division by r e a l polynomials

If E is a r e a l 1 . c . s . and 0 a n open s u b s e t we c o n s i d e r on d ( 0 ) e i t h e r topologies defined in (4.1. 1) and ?he. 11.6.2 below holds f o r both.

-

11.6. 1 DEFINITION.

If E

i s a r e a l 1 . c . s . we s a y t h a t E

p r o p e r t y (P) if t h e r e is a convex balanced o-neighborhood V L

-

and a convex balanced bounded s u b s e t B ( w h e r e sv: E

E

v

E

& E such t h a t d i m

s (E )=+a V B d e n o t e s a s u s u a l the canonical s u r j e c t i o n map).

Any infinite dimensional n o r m e d s p a c e has c l e a r l y p r o p e r t y (P), C l e a r l y i f E h a s a continuous n o r m , then i t h a s p r o p e r t y ( P ) i f and only if it h a s a n infinite d i m e n s i o n a l bounded s e t . F r o m ( 0 . 6 . 2 prop. 2 ) i t i s e a s y to prove that if E i s a s t r o n g dual of F r k c h e t s p a c e , always p r o p e r t y (P) except

if E =

i R(m)o r i f E

E has

is finite d i m e n s i o n a l .

F r o m ( 0 . 6 . 2 prop. 2) one a l s o p r o v e s e a s i l y that a F r e ' c h e t s p a c e E h a s always p r o p e r t y (P) except i f E = JRN

11.6.2 a&

THEOREM.

-

o r i s finite dimensional.

L e t E be a r e a l 1. c . s. with p r o p e r t y

(P)

R a n open s u b s e t of E . Then the division by a n y nor. z e r o

continuous polynomial i s i m p o s s i b l e ir. 8

If e l

Proof.-

vl(el)

#

0.

#

0 and

e

cs

1

'(n).

(B), t h e r e exists a v

v

E(E ) ' with

1

v

then E = R e & E a s a topological If we s e t E = k e r v 1 1' v 1 1

d i r e c t s u m . If e f 0 and e 2 C s V ( B ) flE 2 1'

there exists a u F(E ) ' 2 1

( E l is equipped with the topology induced f r o m E V ) with u ( e )#O. 2 2 If we s e t E = k e r u then E = R e 2 f€ E2 a s a topological d i r e c t s u m . 2 2' 1

273

Division by real polynomials

Thus

EV = R e

1

E2

%Re2

x = $ e t s e t x 1 1 2 2 2

a s a topological d i r e c t s u m . If w e s e t v (x) = 2 v 2 6 ( E V ) I , v ( e ) = 0 and v ( e ) 2 1 2 2

. .,en - 1 F

there exist e v ( e ) = o k . t

k#A

#

0.

5

2

u (e ) 2 2

~

then

Now, by induction, l e t u s a s s u m e

sV(B), v l , .

.., vn - 1

E (EV)I s u c h t h a t

and

EV = IR e l CE R e

2

C€.

. . CE IR e n - l

6€ E

n- 1

a s a topological d i r e c t . s u m . Since d i m s (E ) = t a, t h e r e is an e V B n e n 6 sV(B) n E n - 1 and u n F ( E n - l) I with u ( e n ) # 0 . If w e s e t

#

l?

= k e r u , then E = R e t€ E a s a topological d i r e c t s u m . n n- 1 n r. Therefore

E

n

EV

= Re

1

6

x = $ e 1 1

If we s e t v (x) = n

6

n

u (e ) , n n

f

... & R en CL E n

... + E n e n + xn .

t h e n v E (E ) I and v ( e ) = O n V n . t

n

#

1 .

-W T h e r e f o r e we proved by induction t h a t t h e r e is a s y s t e m ( e ) n nEN

of continuous of icdependent v e c t o r s i n EV and a s e q u e n c e (v ) n nElN We m a y c l e a r l y a s s u m e with v ( e d = 0 n l i n e a r f o r m s on E V n 1Iv 51 acd 00 (en) 5 I. We s e t n V n

11

&I.

0,

Division of distributions

214

a

If x gE

n

=

vn(en) n

1 n

5 -

we set

V'

and thus P is a continuous polynomial on E V V' continuous polynomial on E.

.

So P = P

v

o s

v

is a

Smce e n c s V ( B ) , t h e r e is an e ' F B s u c h n

We m a y a s s u m e without loss of g e n e r a l i t y t h a t oFR. n 1 T h e r e f o r e t h e r e is n G I N s u c h t h a t - e ' C f i i f n > n ( s i n c e e ' 6 B). 0 n n 0 r. L e t u s define an S C d ' ( a ) by : that s ( e ' ) = e

v

n

n S n

if f E d(n).

L e t u s a s s u m e by a b s u r d t h a t t h e r e is a

TCd'(n) s u c h

that P T = S i . e . T ( P f ) = S(f)

f o r e v e r y f e d ( n ) . Since T E d ' ( n ) t h e r e is a c o m p a c t s u b s e t K of a bounded s u b s e t L of E , 0

an

E

>o

and a n a

TEV ( K , L , & , a ) . Now we define a functior?

E IN s u c h t h a t

f r o m IR into IR by

:

0,

Division by real polynomials cp

E &IR)

and s o t h e r e is a c 2 o s u c h t h a t

k =

F o r e v e r y n E IN

*

0,

...,b

and x F E we s e t

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


0

Since L is a boucded s u b s e t of E t h e r e is a b 2 0 s u c h t h a t

Therefore

sup x F E

h.€ J

L

I f E ( k ) ( x ) h l .. .hk 15 ~ ( a , ) ~ - ~ ( 2.b ) ~

275

216

Division of distributions

Since o g a such t h a t

n

5

1 , n

( a n ) n - k = 0 , and t h e r e f o r e t h e r e is an N E N lim 0 n -t m

v

F r o m definitions of

and f

(fn(X)

n

, a

#

3a

11

( 25-vn o s v ( x )

0)

5

-+.

I?

F r o m definition of P, n

E N.

P(x) = o is equivalent to v n 0 sV ( X ) = o f o r e v e r y T h e r e f o r e , if P(x0) = 0 , e a c h f n is z e r o on a neighborhood of

x , A s a consequence, 0

T h e r e f o r e , if N 1 n

0

,

fn P

is w e l l defined a s a n e l e m e n t of

d(n).

F r o m t h e definitions w e h a v e :

1

f N ( v e l N ) = (a,) TUN)

>-

2

N

N

~ ( 0=)2 ( a N )

.

2 2

(’ N ( e N ) )

which, f o r n l a r g e enough, is c o n t r a d i c t o r y with the f a c t s that f E V(K , L , N & , & ) and T i n its p o l a r .

CHAPTER 12 CONVOLUTION EQUATIONS IN SPACES OF HOLOMORPHIC FUNCTIONS

If E is a c o m p l e x q u a s i - c o m p l e t e d u a l i i u c l e a r 1. c.

ABSTRA CT.

S.

we r e c a l l t h a t we e q u i p the s p a c e 5c (E) with the c o m p a c t open topology. S If E is a c o m p l e x 1.c. s. we denote by 3( ( E ) the l i n e a r s p a c e of t h e u, b h o l o m o r p h i c functions on E of u n i f o r m bounded type. X u , b ( E ) is t h e inductive l i m i t , when V

r a n g e s o v e r a b a s e of convex balanced

o-neighborhoods in E ,

of the s p a c e s 3c ( E ) of t h e h o l o m o r p h i c b V t h a t a r e bounded on e v e r y bounded s u b s e t of E functions on E V V' i s a F r e ' c h e t s p a c e u n d e r the topology of u n i f o r m c o n v e r g e n c e on (E ) b V the bounded s u b s e t s of E and ( E ) i s equipped with t h e l o c a l l y V u, b convex inductive limit topology.

x

Existence theorem. -

L e t E be a c o m p l e x n u c l e a r 1. c . s. T h e n

a n y c o n z e r o convolution o p e r a t o r on X u , b ( E ) is s u r j e c t i v e .

If E

is a n u c l e a r Silva s p a c e X

a l g e b r a i c a l l y and topologically, s o :

COROLLARY.

- 3

-

when E =

(E)

=x ( E ) = 31u , b (E)

E be a c o m p l e x n u c l e a r Silva s p a c e . T h e n a c y

non z e r o convolution o p e r a t o r on 3(

Remark.

S

S

( E ) is s u r j e c t i v e .

T h e s a m e e q u a l i t y of t h e a b o v e t h r e e s p a c e s hold a l s o

&'(n),

s o t h e c o r o l l a r y still h o l d s in t h i s c a s e . 277

Convolution equations in spaces

278

Approximation t h e o r e m A . n u c l e a r 1.c. solution f

L e t E be a c o m p l e x q u a s i - c o m p l e t e d u a l

and l e t 8 be a convolution o p e r a t o r or,

S.

FX

-

S

( E ) of t h e homogeneous equation 8 f

x S(E).

o is l i m i t (in

of exponential-polynomial solutions (i. e . solutions of the type a,(x)

. .. a n ( x ) exp (B(x)) with

Then a n y

n FIN, x F

3; ( E ) ) S

E, a i and P F E l ) .

finite

A p p r o x i m a t i o n t h e o r e m B.

-

Let E

be a c o m p l e x n u c l e a r 1.c. s . and

X

(E). Ther, any solution u, b f C Y U , b ( E ) of the homogeneous equation 8f = o is l i m i t , in (E), of u, b exponential polynomial s o l u t i o n s ,

let

be a convolution o p e r a t o r on

We a l s o define in

12.8 the convolution o p e r a t o r s of finite type

and obtain f o r t h e s e o p e r a t o r s s t r o n g e r e x i s t e n c e r e s u l t s .

219

Convolution operators

4

Convolution o p e r a t o r s on

12.1

12.1. 1 PROPOSITION. T g x K s ( E )& f

-

X S(E)

.

scS(E)

L e t E be a c o m p l e x c o m p l e t e b. v . s., T h e n the function T

*f

m E into C

f

defined by :

(T

-h

f ) (x) = T ( 7

-X

f)

i s a n e l e m e n t of Y ( E ) and t h e mapping T 9 :f S

into XS(E),

Proof.

-

T

*f

f r o m xs(E)

is a convolution o p e r a t o r on ?( (E). S

It is e a s y to c h e c k t h a t the mapping x

to c h e c k t h a t the mapping

-h

f

tions. Now we p r o v e T* i s continuous. If

1>

s u b s e t of E and if

E) and

x13 o

0,

let

YB,A)

Since T F X i ( E )

Xs(E) ( s e e 4. 1.2).

7

f from E

into

-X

(E). It i s s t i l l i m m e d i a t e S T-X i s l i n e a r and c o m m u t e s with t h e t r a n s l a -

Ys(E) is Silva h o l o m o r p h i c , h e n c e T

of

-

be s u c h t h a t

c

3(

B is a s t r i c t l y c o m p a c t

be a given o-neighborhood i n

let B ' (a strictly compact subset

1 T(f)l

Y

v ( B ' , 1'). We m a y

if f

a s s u m e that B and B ' a r e convex and b a l a n c e d , T h e n if f g Y B t B ' , A ' ) and x E B, 7 -xf F Y B 1 , X I ) , whenever f E v(BtB',

I(T*f)(x) I S

1. T h u s T * f c v ( B ,

a)

I).

Now l e t u s denote by o p e r a t o r s on K s ( E ) ,

hence

G

-

t h e l i n e a r s p a c e of t h e convoIution

L e t u s define the l i c e a r m a p p i r g Y

from

S( E )

into G by :

if

T 6 KlS(E).

L e t u s define t h e l i n e a r m a p p i n g

Y

f r o m 12 i c t o X ' S (E)

280

by

12.1.2

PROPOSITION.

-

y and

N

Y

N

a r e i n v e r s e mappings.&

is a n a l g e b r a i c i s o m o r p h i s m f r o m K IS(E) onto

Proof. -

*

((7o Y)

= (Y(Y(Q)) ) ( f ) = ( Y ( 8 ) )

(of) (x).

* f ) (x) = (Y(Q))

Hence

.

*f .

N

(9) ( f )

( ~ ( 8 ) ) f = @ I s i n c e (Y(Q) =((T - x @ ) f )(0)=

G

Y

Furthermore

( ~ - ~ =f ( )Q ( T - ~ ~ )(0) ) =

N

y o Y is t h e identity mapping on G

. On

t h e o t h e r hand :

Hence

Y

N

o 'f

is t h e ider.tity mapping On

x'S(El.

(E). Since G is a n a l g e b r a u n d e r S composition w e m a y define t h e "convolution product" on X ' (E) by : S 12.1.3 Convolution p r o d u c t on

if

T

1'

(T1

T2 FXIS(E).

+

T2)

Now, if f EKs(E)

N

and

* f = ( @ 1o 8 2)(f) = 81( 8 2(f)) = T 1

@ . = Y ( T . ) , (i = 1 , 2 ) , t h e n :

.K

( 8 f ) = T I * ( T 2 * f) 2

.

28 1

Convolution operators

LEMMA. -

12.1.4

&(T 1 F o u r i e r -Bore1 t r a n s f o r m .

Proof.

-

* T2) =

If T F XIS(E),

5

F E

(d

X

T 1 ) . ( 5 T 2 ) , when 5 d e n o t e s the

and x E

E then

:

hence :

(1)

T*e'=T(e

If T 1 , T 2 E X I s ( E )

and

5

by definition ( 7 . 1. 2 ) of 5 .

s ) .' e 6

.

6 E X we have

:

F r o m f o r m u l a ( 1 ) above

:

Hence f r o m ( 1 2 . 1 . 3 )

T h u s we obtain :

c

c:

( 3 ( T l * T2))(C) = T 1 ( T 2 * e 5 ) = T ( T ( e 6 ) . e 5 ) = T 2 ( e ).T ( e )= 1 2 1

282

Convolution equations in spaces

1 2 . 1 . 5 G e n e r a l f o r m of t h e convolution o p e r a t o r s on X (E). S

We

a s s u m e E is a S c h w a r t z b . v . s . with a p p r o x i m a t i o n p r o p e r t y in o r d e r to use

5

7.1

on the F o u r i e r Bore1 t r a n s f o r m . If

B E G, then it follows

f r o m ( 1 2 . 1 . 2 ) that t h e r e i s s o m e T 6 K l S ( E ) s u c h t h a t 8 f = T every f F

*f

for

x S(E), i.e. ( @ f ) (x) = ( T

(1)

for every x 6 E

. Now,

if

y

* f ) (x) =

T(7

-X

f)

E

t m

n = o hence

We s e t

fl

=

3 T C 3('XIS(E)). T h e n f r o m (7.1.4),

( 1 ) and ( 2 ) a b o v e , we

have :

I

n = o

C o n v e r s e l y the s a m e computation p r o v e s t h a t f o r a n y flg 5X' ( E ) the above f o r m u l a d e f i n e s a convolution o p e r a t o r

8 on

S

(E). The

S convolution o p e r a t o r s on TS(E) a r e t h u s c h a r a c t e r i z e d by the a b o v e

f o r m u l a , with

@

€ 53(IS(E).

283

Entire functions of nuclear bounded type

5

12.2

E n t i r e functions of n u c l e a r bounded type on a Banach s p a c e .

In this s e c t i o n E is a n o r m e d s p a c e .

1 2 . 2 . 1 Nuclear m u l t i l i n e a r mappings. If n E N w e denote by n xf( E ) c L(nE) the l i n e a r s p a c e of the mappings (XI,.

. . xn) - $ 1(x 1) .. . $n( xn ) t

if x.

E and Pli€EI.

the s y m m e t r i z a t i o n (0.8. 1) of any e l e m e n t of W e write

2,

f, s

("E) = s A n E )

n

L

( s e e 0.8.3), and the mapping A

s

-

*

A

It is c l e a r t h a t

Cf(nE) is a l s o in

Ef(nE).

If A E L k n E ) , then i E p f ( k ) , is a l i n e a r bijection f r o m

onto P f ( n E ) .

If T 6 L f ( = E ) we d e f i n e the n u c l e a r n o r m of T by :

w h e r e m r a n g e s in IN and w h e r e

]Iffi,

] / = sup x (1

/I It

1 gi, j ( x ) 1 .

xE E

n i s the n a t u r a l n o r m in L( E ) , then c l e a r l y , i f T E L f ( n E ) ,

If

Lf(nE)

Convolution equations in spares

284

// I/

then the Lf(nE) is equipped with the n u c l e a r n o r m i " n a t u r a l inclusion mapping @ f ( n E ) L ( n E ) is continuous. If we denote Thus if

-

( n E ) the completion of Q ( n E ) , ( 0 . 1. 16), then the m a p p i n g i is N extended a s a continuous mapping If r o m L N ( n E ) i n t o L("E), s i n c e

by

C$

L ( n E ) is a Banach s p a c e . It is known ( c f S c h w a r t z

L 5 3 , e x p o s e 14 p 8)

t h a t i f E i s a Banach s p a c e s u c h t n a t its d u a l E ' i s a Banach s p a c e with A p p r o x i m a t i o n P r o p e r t y , t h e n t h e mapping T is n injective ( i n t h e notation of t e n s o r p r o d u c t , L f ( E ) is denoted by &In (El) ). In t h e s e q u e l we s h a l l u s e t h e c a s e E is a s e p a r a b l e H i l b e r t s p a c e , s o we m a y a s s u m e t h a t E l h a s the A p p r o x i m a t i o n P r o p e r t y , n and s o LN(rlE) is identified a s a l i n e a r s u b s p a c e of L( E). XN(nE) is called the s p a c e of t h e n u c l e a r n - l i n e a r m a p p i n g s on E and i s a l w a y s equipped with the n u c l e a r n o r m , which m a k e s i t a Banach s p a c e .

1 2 . 2 . 2 N u c l e a r polynomials.

-

T h e s e t XN(nE) n L ("E)

(nE). We N, s S N ( n E ) , which

of n u c l e a r s y m m e t r i c n l i n e a r m a p p i n g s is denoted by

2

( n E ) with the n u c l e a r n o r m induced by N, s m a k e s i t a Banach s p a c e . L ("E) is d e n s e i n ("E). f, s N, s endow

T

P

-

g

-

The i m a g e of

T,

(0.8.3),

6'N(nE)

L

N,s

n ( n E ) in P( E ) u n d e r the n a t u r a l i s o m o r p h i s m

i s denoted by

b N ( n E ) . An n - h o m o g e n e o u s polynomhl

-

is called a n u c l e a r n - h o m o g e n e o u s polynomial.

6'N(nE) is

equipped with the topology making this mapping T T a topological n i s o m o r p h i s m , It follows t h a t pN( E ) i s a Banach s p a c e and t h a t n b f ( n E ) i s d e n s e i n ' 6 ( E ) However we a r e going to c o n s i d e r on N b N ( n E ) a n o r m which d o e s not m a k e the mapping T T an isometry.

.

-

We define the n u c l e a r n o r m on 6'N(nE) in the following w a y : if

P

F

6'f(nE)

we define t h e n u c l e a r - p o l y n o m i a l n o r m

m

m

j = 1

j = l

IIPllN b y :

285

Entire functions of nuclear bounded type

n If T E- g f , s ( E ) ,

then c l e a r l y by definitior, ,

N o w if T E 2

(nE), a n e a s y computation based on t h e p o l a r h a t i o n f, s formula (0.8.4) gives :

( s e e Gupta [1,21 ). T h u s 63N(nE) m a y be c o n s i d e r e d a s t h e c o m p l e t i o n of

11 /IN

f o r t h e n u c l e a r polynomial n o r m , s t i l l denoted by

T E g N , s ( n E ) , we have :

12.2. 3 E n t i r e functions of n u c l e a r bounded tvpe. Banach s p a c e with

-

b;("E)

, and if

If E

is a c o m p l e x

A p p r o x i m a t i o n P r o p e r t y we s a y t h a t a n e n t i r e

function f f r o m E into C is "of n u c l e a r bounded type" i f , f o r e v e r y

m F N , $m)(o) E PN(mE) and if

We denote by 3C

N,b

( E ) the l i n e a r s p a c e of t h e e n t i r e functions

of n u c l e a r bounded type on E.

3C

N, b

(E) is a m e t r i z a b l e 1. c. s. f o r the

topology defined by a l l the s e m i - n o r m s

286

Convolution equations in spaces

t O 3

m = o Remark.

-

If E is a n o r m e d s p a c e s u c h t h a t its d u a l

El

is a

Banach s p a c e with A p p r o x i m a t i o n P r o p e r t y ( f o r i n s t a n c e w e s h a l l u s e l a t e r on the c a s e €3 is a p r e - H i l b e r t s p a c e ) , t h e n one d e f i n e s P N ( m E ) and

xN, b (E)

e x a c t l y a s b e f o r e and one h a s

P ( m E ) = PN(mk ) and N

If E ' h a s not the A p p r o x i m a t i o n P r o p e r t y the N , b (k). definitions of P ( E ) acd 3c (E) a r e still p o s s i b l e ( s e e Gupta [1,2] N N,b and Matos-Nachbin [ 2 ] )

K N , b(E) =

1 2 . 2 . 4 PROPOSITION.

Proof.-

xN, b ( E )

-

Let (fk)k

is a Fre'chet space.

(E). T h i s gives N, b e v e r y I? and s o t h e r e

be a Cauchy s e q u e c c e ic 3c

t h a t k ) ( o ) i s a Cauchy s e q u e n c e exists a p

n

every p

E

> 0,

f o r e v e r y k.

in '6

PN(nE) such that 03

-,

Now,

Letting k

tcr,

we have ~

(k). Also,

s u c h t h a t Ilf 11 I- M k N9P P M 1 \Ifk - ( n )( o ) l l N 5 f o r e v e r y k and n. n n. P

there exists o 5 M < t P Therefore

N

~

M p5 ~ 2\ l f o ~ r every n, Pn

for

287

Entire functions of nuclear bounded type

which i m p l i e s t h a t

for every p

>

o

. Letting

p

-

t m we h a v e

:

T h i s shows t h a t too

n - o

-

d e f i n e s a n e n t i r e function of n u c l e a r bounded type on E. T h u s i t r e m a i n s only t o p r o v e that (f ) t e n d s t o f i n k p > 0,

N,b

( E ) when k

t

CO.

m

n = o too

n =mtl

tco

t~

= mtl

F a r any

288

Convolution equations in spaces

Since

n = o and

too

too

n = o

n = o

when o < p < p such that

1'

we have that, f o r a n y C 3 0 , t h e r e e x i s t s m

>o

too

n = mtl

and

too

n = mtl Using t h i s and the above m a j o r i z a t i o n of /If -fII

k

If

5 E E',

N,p

then c l e a r l y t h e function expc

, we obtain that

is in x N , &E).

So

289

Entire functions of nuclear bounded type

we define a s u s u a l , if T E ( K N , b(E))'

, t h e F o u r i e r - B o r e 1 t r a n s f o r m of

T by the f o r m u l a :

1 2 . 2 . 5 PROPOSITION. -

The Fourier-Bore1 transform 3 i

algebrait isomorphism from

X'

N,b

(E) onto the s p a c e E x p ( E ' ) of e n t i r e

functions of exponential type on the B a n a c h s p a c e E ' ( i . e . e n t i r e fumtions

fl

=El

1 fl ( 6 ) I

such that

c o n s t a n t s a , b 7 o and f o r a n y

Proof.-

For every

5 E

5

= a e x p b1)s

for some

E El).

El,

n = o in the topology of X

N, b

(E).

Thus

n = o

(where

Can

is t h e function x

there exist c , p 9 o such that

for every f

EP

N. b

(E).

*

m

(C(x))")

.

Since T is continuous

290

Convolution equations in spaces

hence

and

Since i t is e a s y t o prove t h a t the mapping 3(

N, b

( E ) is G - a n a l y t i c , C o n v e r s e l y , if

5

exp

+

6

f r o m E ' into

3 T is G - a n a l y t i c and t h e r e f o r e 5 T E E x p E ' .

E Exp

El,

then f r o m ( 7 . 2 . 2 ) , t h e r e a r e c , ,I%

s u c h t h a t , for e v e r y n E N ,

m Let us r e m a r k that i f

PEP("E')

and i f p =

(@i)n E 6;("-E) , i = 1

m ( f l i E E 1 ) ,then P(p) is well defined a s

P(fli),

i.e.

d o e s not depend

i = l m ( @ i ) n (using the language of

upon the way p is x r i t t e n a s i =1

topological t e n s o r p r o d u c t s ( G r o t h e n d i e c k [ 2 3 , S c h a e f e r [ 1 3 , Tr'eves [ 1 1 ) : b("Et)

-

J? ( n E f )c S("E')

Pf(nE) = E s

f, s

a

( n E ) C Lf("E)

w")' E'V) .

(El

a

29 1

Entire functions of nuclear bounded type

and

Therefore

m

m

m

i =1

i = l

i = l

, b y continuity, p E (P,("E))'.

f o r e v e r y n. T h e r e f o r e : I$")(o). ;'"'(o)

Therefore taJ

I

I n! c.

x

(- )n M(p) P

.

292

Convolution equations in spaces

If w e s e t

n = o V

then

T EX

$ 6%’ (E) and N,b

’

V

S $ is the i d e n t i t y on Exp E ‘ . F u r t h e r m o r e if

V

n

( E ) , 3(T)

N,b

T.

So the m a p p i n g s $

Bore1 t r a n s f o r m 3 a r e i n v e r s e mappings.

I

-+

3 and t h e F o u r i e r -

293

Convolution operators

Q

12.3

Convolution o p e r a t o r s on 3c (E) u, b

(E). - F i r s t w e r e c a l l , u, b i f G is a c o m p l e x n o r m e d s p a c e , w e d e n o t e by Kb(G)

12.3. 1 Topology and bornology on t h e s p a c e (2.7.3),that,

the l i n e a r s p a c e of the h o l o m o r p h i c f u c c t i o n s of bounded type on G , i.e. which a r e bounded on e v e r y bounded s u b s e t of G.

The s p a c e Kb(G)

is naturalby equipped with the topology of 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 of G.

It is i m m e d i a t e t o c h e c k t h a t xb(C)is a F r e c h e t

s p a c e and t h a t its bounded s e t s a r e the f a m i l i e s of functions t h a t a r e equibounded on e v e r y bounded s u b s e t of G.

( E ) t h e s p a c e of t h e h o l o m o r p h i c f u n c t i o n s of u, b u n i f o r m bounded type on E , i. e. (2.7.3) t h o s e h o l o m o r p h i c functions f W e denote by %

on E f o r which t h e r e is a convex balanced o-neighborhood

V

in E

s u c h that f f a c t o r s in the following way

with

? E Xb(EV)

.

is s u r j e c t i v e , t h e mapping g € X b ( E V ) gosv€Ku!f) V is i n j e c t i v e , s o t h a t we m a y c o n s i d e r (E ) a s a l i n e a r s u b s p a c e of b V (E) A l g e b r a i c a l l y 3c (E) is t h e inductive limit of t h e s p a c e s u, b u, b 3’? (E ) , when V r a n g e s o v e r a b a s e of convex b a l a n c e d o - n e i g h b o r h o o d s b V ( E ) i s n a t u r a l l y equipped with t h e b o r n o l o g i c a l inductive in E. 3C u, b (E 1 (0.2.4). Then, f r o m (0.2.7), a convex limit of t h e s p a c e s b V balanced s u b s e t Q of K ( E ) is a o-neighborhood for t h e a s s o c i a t e d u, b

Since s

.

-t

294

Convolution equations in spaces

n

( E ) if Q 3( ( E ) is a b o r n i v o r o u s s u b s e t u, b b V ( E ) is m e t r i s a b l e , f r o m ( 0 . 6 . 2 ) , t h i s a m o u n t s t o :

bornological topology T3c of

X (E ). Since X

b V b V Q OK b(EV) is a o-neighborhood in 5( (E ). T h e r e f o r e t h i s topology on b V ( E ) coi'ncides with the l o c a l l y convex inductive l i m i t of the F r e ' c h e t u, b s p a c e s X (E ) We s h a l l only c o n s i d e r t h e above bornology and b V topology o n 5(u, b ( E ) *

.

12.3.2 PROPOSITION.

-

If E

i s a c o m p l e x n u c l e a r 1. C . s . ,

then

( E ) i s the algebrai'c, bornolopical and topological ( l o c a l l y c o n v e x ) u, b inductive l i m i t of the s p a c e s 3C when V r a n p e s o v e r a b a s e of

3c

N ,J E V )

c o ~ v e xbalanced o - n e i g h b o r h o o d s i n E .

-

T h e proof 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 a s s u m p t i o n t h a t E i s a n u c l e a r 1. c. s . and f r o m l e m m a 1 2 . 3 . 3 below.

12.3.3 LEMMA.

-

El-

E

xN,b(E1).

be two n o r m e d s p a c e s w i t h a

into E If f 12' M o r e o v e r the mapping

l i n e a r n u c l e a r mapping j f r o m E: f o j

2

EK (E ), t h e n b

2-

f o j

f

is continuous.

Proof.-

A f t e r a s u i t a b l e choice of n o r m s in E

a s sume

n = l

1

and E

2

w e may

Convolution operators

295

ta3

n = l

Heme

If r

>

o we set

M(r) =

1

sup

1I4& s 2

f(z)/

.

r

F r o m C a u c h y ' s i n t e g r a l f o r m u l a , a l r e a d y u s e d in (7. 1.4),

F r o m (1) we get

and thus,

hence

:

we have :

296

Convolution equations in spaces

and s o f r o m (12.2.2)

For a n y fixed M fixed r

> 0 , (M) l'm

-.

1 if m

>o ,

-

too. T h e r e f o r e , f o r e v e r y

Since t h i s inequality holds f o r e v e r y r 3 o ,

T h i s p r o v e s that f o j

E

N,b

(E ), 1

Now,from

inequality (2), w e g e t

too

m = o

Choosing p

0

too

too

m = o

m = o

= 2 p e

2

, we have

which p r o v e s t h e continuity of o u r mapping f

-

f o j

.

8

29 I

Convolution operators

3G'

(E). If E is a n y c o m p l e x u, b 1. c. s . u e define obviously a convolution o p e r a t o r on (E) a s a u, b continuous l i n e a r mapping f r o m 3C ( E ) i n t o i t s e l f which c o m m u t e s u, b with the t r a n s l a t i o n s . 12.3.4

Convolution o p e r a t o r s on

12.3.5

PROPOSITION.

and

f

xu , b(E).

E

T h e n t h e function T

(T if x -

'u,b

E E,

(E)

* f) (x)= T ( T

be a c o m p l e x 1. c. s , T E +?

-X

f)

( E ) , and t h e m a p p i n g T * , f r o m u, b (E), is a convolution operator on (E). u, b u, b

into

b(E),

then, f o r e v e r y V, T

+

EX' (E ). Now

b V f E Xb(EV), a n d ,

* *f

(E ) into X b ( E V ) , t h u s TW is continuous f r o m V (E). I u, b b

If w e denote by

xu, b ( E )

&E)

f f r o m E i n t o C defined bv

i m m e d i a t e to c h e c k t h a t i f f €3C (E ), t h e n T b V a s in (12.1. I ) , w e p r o v e that the mapping f T $(

x:,

is a n e l e m e n t of

If T EX:,

Proof.-

Let E

-

it i s

is continuous f r o m u, b

( E ) into

G the l i n e a r s p a c e of a l l convolution o p e r a t o r

w e d e f i n e a s u s u a l t h e two m a p p i n g s

Y and Y :

298

Convolution equations in spaces

T h e computations in 1 2 . 1 . 2 p r o v e t h a t N

Y

Therefore

12.3.6

N

Y

and

Y

a r e inverse mappings.

i s a n a l g e b r a i c i s o m o r p h i s m f r o m 5(

'

u, b

E n t i r e functions of exponential type on a b.v. s.

(E) onto G

-

.

If F i s a

we denote by E x p ( F ) t h e l i n e a r s u b s p a c e of X (F) S S m a d e of t h e functions $ on F such t h a t f o r e v e r y convex balanced complex b.v.s.

bounded s u b s e t B of F t h e r e a r e n u m b e r s

for every x EE

B '

12.3. 7 F o u r i e r - B o r e 1 t r a n s f o r m . if T

c , p 3 o such that

-

If E is a c o m p l e x 1. C . s . and

b(E) we define a s u s u a l the F o u r i e r - B o r e 1 t r a n s f o r m by t h e

E%jU,

f o rmula

+

for every

E

If E ' i s equipped, as u s u a l , with i t s equicontinuous

El.

bornology, then w e c h e c k t h a t 5 T E K S ( E ' ) .

1 2 . 3 . 8 PROPOSITION.

-

If E

i s a c o m p l e x n u c l e a r 1. c. s . , then I

(E) t h e F o u r i e r - B o r e 1 t r a n s f o r m 5 is a n a l g e b r a i c i s o m o r p h i s m of 3C u, b onto E x p ( E l ) . S Proof.-

If

We r e c a l l t h a t , f r o m ( 1 2 . 3 . 2 ) ,

T C3c

if CF E

(

IU,

~

b ( E ) , then T ~

=1

( 1~

1

)

8

'&N,b (EV

c EI,

xu , b ( E ) =

l i m 3t 4

V

N,b

(E ). V

€ ( x N , b(EV))I. T h u s f r o m (12.2.5),

I s~(cp)l5

c e x p ( p l l c c ~ ~ , ) i, . e .

V

Gmvolution operators

(3T',

and, a s a consequence,

E Exp

v

C o n v e r s e l y ,if

i.e.

5 T E ExpS(E').

V

(EIIo

borhood

299

$ E E x p S E l , t h e n , f o r e v e r y convex balanced o - n e i g h t h e r e is T

V in E ,

for e v e r y p E ( E l )

(E ) s u c h that V

V '"N,b

, T V ( e x p P) = $ (b)

3(TV),

/(Wo v

.

If V

V

1

cV

l e t r denote

2'

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

f

where i

f o i

v1'v2

into E

is the c a n o n i c a l mapping f r o m y1'v2

T

s i n c e the s e t

v2

l e x p cp

(f) = T V ( r ( f ) ) : 1 is dense in

I

V (12.2.5)

it s u f f i c e s t o prove t h a t T

v2

N,b

(E

V2

(exp

y) = $

TV ((exp ~ p o ) i 2 v11v2 ( T ) when cp is

and l e t E = G x GIP

. Then

E l = GI

B

x G

bornologically. If f is t h e duality b i l i n e a r function on G' x G , then

f E Exp

S

where < ,

(El)

1

L e t G be aninfinite d i m e n s i o n a l n u c l e a r , d u a l

n u c l e a r , r e f l e x i v e 1.c.s. f(x',x) = <xl,x>,

)

2

E ( E ) ' , and both a r e equal t o v2 c o n s i d e r e d a s a n e l e m e n t of El.

f o r any

12.3. 9 E x a m u l e . -

. One h a s

v2

>

d e n o t e s the duality between GI

and f $ E x p E l ,

B

s i m e f is not continuous.

and G ,

:

300

Convolution equations in spaces

T h e r e f o r e , f r o m (7.2.1) and (12.3.8),

XIS(E)c KfU, b(E).

f

12.3. 10 G e n e r a l f o r m of the convolution o p e r a t o r s on 3 (12.3.8) they a r e given by the foamula

u, b

(E).

-

From

n = o

if f

E X U ,b(E),

that of (12.1.5).

E ExpS(E') and x E E.

The proof is quite s i m i l a r to

301

Convolutionoperators

9

12.4

Convolution o p e r a t o r s on '3c

N, b

(E)

Owing to t h e t h r e e l e m m a s below, t h e study of t h e convolution

(E) is s i m i l a r t o t h a t in X (E) or 3( ( E ) of N, b S u, b and 9 12.3. The m a i n r e s u l t is t h a t a n y convolution o p e r a t o r on 3C

$ 12.1

o p e r a t o r s on X

is an o p e r a t o r (T-k) f o r s o m e T

N, b

(E)

A l s o w e have ( s e e 1 2 . 1 . 3

E "INib(E).

and 12. 1 . 4 )

(T

1

.)(

T ) # f = T l * (T2-k f ) 2

N o w , w e s t a t e and p r o v e the l e m m a s .

12.4. 1 LEMMA.

Let E

be a c o m p l e x n o r m e d s p a c e , a C E

(E). T h e n the t r a n s l a t e d function

"N,b

(T

-

l *f

-af) (x) =

7

and

is in vN,$E) a& -a f -

(x) ( a ) in the topoloev of

xN,b ( E ) ,

n = o

Proof,

-

For every x E E ,

t w

tco (T

:-

-af)(x) = f ( x t a ) =

f

(x) a

n = o

n = o

too f

n i (o).x . a

(i) i (x) a =

n = o

.

1fix) (a).

n!

=

Convolution equations in spaces

302

I t is e a s y t o p r o v e t h a t

n

Jnti)

(0)

ai

E q E )

and that

It follows t h a t , when x i s v a r i a b l e in E , V

C m

n = o

n =Vtl

n =vtl

n = V t l

which t e n d s t o o a s

V

-

.

t oo

n = o

a s functions of x and i n the s e n s e of 3C

n = o

N, b

(E). Now i f p

>o

is g i v e n ,

Convolution operators

n = V t l

i = o

too

t w

n= v t l

i = o

Choosing E > o C(E)

3-0

303

s u c h t h a t 2Ep < 1, 2 E I/a

(IE<

I,

we

s u c h that

tw

L

i = o

n = o too

i = o

tw

n=Vtl

tm

n=Vt1

have that t h e r e e x i s t s

Convolution equations in spaces

304

which tends t o o when

V

tends t o t m

.

Thus

I

n = o

in the s e n s e of 3C

1 2 . 4 . 2 LEMMA.

N, b

-

(E)

.

If_ T

EX(”, b ( E )

SO

that

f E3CN,b(E) and s o m e c , p 7 o and if n

A E E N , s ( E),

I T(f)j

ifllfIIN,pf o r e v e r y

€ PN(nE) with

then the polynomial or: E

-

y

k n-k T~(A.x.y )

E

c

A

k

denoted by Tx@x ) belongs t o PN(n-kE) f o r e v e r y

ks

n . ( w h e r e Tx

denotes T acting on functions of the v a r i a b l e x). F u r t h e r m o r e

m

F i r s t we a s s u m e A E

Proof.we have

f,s

(“E).

If p =

j = l :

m

j = l

305

Convolution operators

s o that

j = l

A l s o we have

:

m

A

j = 1 n

which g i v e s the d e s i r e d inequality. The r e s u l t f o r a r b i t r a r y A E b ( E ) N now follows f r o m t h e d e n s i t y of ' b ("E) in PN(nE). f

12.4.3 LEMMA.-

Let

d e f i n e s a function

T+ f

the mapping f

-

T

+

T

E

ExNN, b(E)

f from

(E).

N, b continuous and t r a n s l a t i o n i n v a r i a n t . Proof.

-

T h e n ( T + f ) ( x ) = T(T

for every f ( E ) into

E xN,b(E).

xN, b ( E )

-X

.n!!-

f) =

n = o tw

tw

p.=o

i = o

4

f), x 6 E ,

Furthermore

is linear,

By (12.4. l ) , f o r e v e r y x E E ,

( T + f) (x) = T ( T

-X

T z ( f t n ) ( z ) (x)) =

Convolutiorr equations in spaces

306

Now l e t c and p (depending on T ) be a s in ( 1 2 . 4 . 2) and l e t p' Z p. Then

tco

too

n

i = o

i = o

i = o

s o that the s e r i e s

i = o c o n v e r g e s in' 6

N

( n E ) to s o m e p

n

such that :

i = o and

which i m p l i e s that

1

307

Convolution operators

for every p'

> p.

Thus if p'

-

t

we g e t

00

Accordingly

tw (T* f) =

'

1 n: 'n

n = o

-

is in 3C (E). T o p r o v e that the mapping f T*f i s continuous l e t N, b p 1 > 0 . Taking p ' = p t p1 > p i t follows f r o m ( 1 ) t h a t

Therefore

n = o

'1

n = o

'IN, 2 ( p t p 1 ) .

T h e l i n e a r i t y and t r a n s l a t i o n i n v a r i a n c e a r e i m m e d i a t e t o prove.

Convolution equations in spaces

308

9

12.5.1

s u c h that if f , f , f 1 2 3

E

v(C) I

1 ii(z)l

if e

5

fl=f2.f3,

Qil

0,

there exist g

iff1(0)

3’

A 5 o 3

f2(0) = 1 and if

ZI

1, 2 and a l l z E C , then

for a l l z E C

Proof.

A division result

Given a l , a 2 , A , A 2 > 1

LEMMA.-

for i -

12.5

-

.

It is a n i m m e d i a t e c o n s e q u e n c e of L e m m a s 12.5.5 and

1 2 . 5 . 6 proved below, which a r e e x t r a c t e d f r o m LindelBf [ 1 ] and Malgrange

i 1] .

The d i v i s i o n r e s u l t (12.5. 1) is i m m e d i a t e l y extended to t h e infinite d i m e n s i o n a l c a s e i n t h e following way 12.5.2 PROPOSITION,

$1,$2 E E x p ( E Y )

Proof.

-

$1

If E

:

is a c o m p l e x b . v . s . ,

if

T-

a G-analytic function on Ex, t h e n

We m a y a s s u m e $ . ( o )

f o if

i = 1 , 2 since we m a y do any

t r a n s l a t i o n in E and s i n c e w e m a y a s s u m e $. convex balanced bounded sets in E and l e t c

f

.

Let B , B

be 1 2 c 2 > o be s u c h t h a t o

A division result

f o r i = 1 , 2 arid a l l B =

r ( B 1u

5

for all of $,

and

6

309

€ E x . If we l e t c = m a x ( c , , c 2 ) and

B ) ( t h e convex balanced hull of B u B ), 2 1 2

Ex

8,

.

Then i t suffices to apply (12.5. 1)

for the r e s t r i c t i o n s

to all complex l i n e s C x , x € E X ,x

1 2 . 5 . 3 LEMMA. -

T h e r e is a constant

A

complex number

K

then

f

0.

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

,

I I-AI

5 exp K

I X[

l/2

The proof is obvious. 1 2 . 5 . 4 LEMMA. -

and

I ~ ( x ) 5I J

the c i r c l e

I

XI

f

f(o)I = r

,

exp

be a n e n t i r e function on

( ~ ( 1 11

f o r m a closed s e t of m e a s u r e

Proof.

-

If

i

1)

the points

- . ., 5,

m I

a

T

C , w i t h f(o)#o,

a positive function. O L

such that

2TTr ( 0 2o I t 0

denote the z e r o e s of

f

is given).

of absolute

Convolution equations in spaces

310 value

5 r

we r e c a l l J e n s e n ' s f o r m u l a ( T i t s c h m a r k

[

13

T h i s f o r m u l a shows t h a t the functior.

is a n i c c r e a s i n g functionof

From t h e a s s u m p t i o n on

o

5

r

>

o

. F(o) =

0,

thus

F(r) 2 o

.

f

F(r) 5

m

- -

G

m T(r) t ( 2 n - - ) T ( r )

.

Therefore

2 nr lta

m s

*

m

12.5.5

LEMMA, -

If

f

Exp C

f(o)

#

o

we s e t ( d e v e l o p m e n t

I-

~

and we a s s u m e

Then t h e r e exist t w o constants

C

and - D

which depend only upon

A division result

A

and

B

31 1

s u c h that

f o r any

n 2 1

P la

2")

- 11 s

+

'i

1

for any

I f(o)I

p 2 o

.

F r o m Jensen's formula ,

Proof.-

So, f o r e v e r y definition of

n E N,

ISn If we c h o o s e

we have ( s i n c e

a s a function of

p

r

Since

D

r .

5

I"

r =

. r

B

rn

I 6 ..6, I we obtain

I f ( o ) I 5 A , we obtain :

I (i I

5

1

if

izptl

by

r)

P 5

lcl.. . I p I

5 -

A If(0)

I

exp (Br).

Convolution equations in spaces

312

1")

which proves

. Now

let 'us prove

2"). We set

P

i = 1 with

F r o m l e m m a (12.5.3)

From

1")

w e have

:

:

P

1/

P 1 -

since i = 1

5

-

2 Vp

.

AS

a consequence

1/

A division result

313

In the s a m e w a y one p r o v e s

I 1 =p ,

T h e r e f o r e if

Therefore, since

If(o)

w h e r e the cor,stant a)

If

p 2 1

in a closed s e t

El

I

5 A,

and if

IhI

= p

,

A

and

B.

K' d e p e n d s only on

we a p p l y l e m m a 1 2 . 5 . 4 t o

of

-

m l 1 4n p ,

measure

ql.q2.

We obtain

If w e s e t

P

aP = a t

Z T

one p r o v e s e a s i l y ( g e o m e t r i c a l l y ) t h a t t h e

i = l set

R e ( a h ) 5 - p1 l a p l and P 2 2rr m e a s u r e m 2 = ~ pT h.e r e f o r e

1x1 El

= p

n

is

E2

# $

a closed set

.

If

of

E2

10 E E l

n

E2,

Convolution equations in spaces

314

and then

Since

If(o)

a number

b)

I

5 A

, t h i s l a s t m a j o r i z a t i o n i m p l i e s the e x i s t e n c e

D'

>

If

p = o

and, from 1")

o

which d e p e n d s only on

we h a v e , f r o m above

A

and

,

B, s u c h t h a t

of

A division result 12.5.6 LEMMA. n u m b e r s , with

315

If a and ( 6 i ) i E N (or 1 5

-

lcitl I

L

1Ci1

f o r e v e r y i,

i 5 q) a r e complex

such that, for every n a n d

P E N

P

i = 1

lfor some

1")

defines a n 2 a)

C,3

>

0)

then :

t h e p r o d u c t (finite o r infinite)

e n t i r e function there a r e A , B

> 0,

t h a t depend only

011

C and D ,

such that

Proof. -

F r o m t h e a s s u m p t i o n s and l e m m a 1 2 . 5 . 3

P T h e r e f o r e the infinite p r o d u c t

+-

i =1

h

[ ( 1 -7)

c o n v e r g e n t on e a c h bounded s u b s e t of

'i

exp

x

7;-

'i

)

is u n i f o r m l y

C , and f i s a n e n t i r e function.

316

Convolution equations in spaces

T o p r o v e the m a j o r i z a t i o n 2 ) it s u f f i c e s t o p r o v e it f o r

1x1 EN .

A s in

1 2 . 5 . 5 we w r i t e P

with

t$

1

and

a s in 1 2 . 5 . 5 . F r o m 1 2 . 5 . 3 and t h e a s s u m p t i o n s :

2

i =1

In the s a m e w a y one o b t a i n s :

Therefore, if

11

1

= p ,

we have

:

T h e r e f o r e we obtain

-1

_3

I f ( 1 ) ) 5 exp(pL2 K ( C 2 t C2) t D ]

)

.

317

Existence and approximation results

9

12.6

E x i s t e n c e and A p p r o x i m a t i o n r e s u l t s i n

In t h i s s e c t i o n E

N,b

(E)

s t i l l d e n o t e s a c o m p l e x n o r m e d s p a c e with

Approximation P r o p e r t y .

12.6. 1 THEOREM. -

lf

N,b

Let 8 -

be a convolution o p e r a t o r on

( E ) . Then the v e c t o r s u b s p a c e of

p.e5

(with p

E

Pf(E)

t h e topology of

Proof. -

N,b

s

c

N,b

( E ) s p a n n e d by t h e functior,s

C E l ) s u c h t h a t @ ( pe 5 ) = o is d e n s e f o r

( E ) ir, t h e k e r n e l of

@

.

F i r s t we r e m a r k e a s i l y t h a t a n y p . e

s i n c e t h e l i n e a r s p a n of t h e s e t

[el ,

t El

1

c I. S ir. . xN,b ( E ) .

Then

is d e n s e i n

N , b(E) ( t h i s follows f r o m khe bijectivity of the F o u r i e r - B o r e 1 t r a n s f o r m : 1 2 . 2 . 5 ) , we m a y a s s u m e without loss of g e n e r a l i t y t h a t

T E X I N , b(E),

"' I

N, b

T

f

o , such t h a t

8 = TY.

8

# 0.

So t h e r e is s o m e

L e t u s a s s u m e now

5

( E ) is s u c h t h a t Q ( p e ) = T-X ( p e ' )

= o imply S(p e

5)

= o

,

f o r e v e r y p € bJ ( E ) and e v e r y 5 E E l . We a r e going to p r o v e that f S(f) = o f o r a l l f in the k e r n e l of 8 and t h e n the d e s i r e d r e s u l t w i l l be a n i m m e d i a t e a p p l i c a t i o n fo the Hahn-Banach t h e o r e m . that

3s E 3T

Exp

El.

F o r this,we s e

If w o is a z e r o of o r d e r m

@k then T(C2 e x p

F i r s t we p r o v e

(61tw,62))

of the function

= o f o r a l l k .< m

.

If k < m ,

Convolution equations in spaces

318

k

i = o Therefore

f o r a l l k < m , and thus wo to L .

Thus by (12. 2 . 4 )

T h e r e f o r e , by ( 1 2 . 5 . 2 ) ,

is z e r o of o r d e r

2 is a 3'T

''

-E 3T

3s r e s t r i c t e d

G - a n a l y t i c functior. on E ' .

Exp E'.

i s o m o r p h i s m (12.3.8) t h e r e i s ar. R

2 m for

So f r o m t h e F o u r i e r - B o r e 1

Ex'N . b (E)

such that

Hence

S = R * T . Now, if

Bf =

0,

then

S* f = ( R * T ) + f = R+(T r f )

Therefore S(f) = which c o m p l e t e s the proof.

(s r f ) (0)=

0

R * ( @ f ) .=

0 .

319

Existence and approxima!ion results

1 2 . 6 . 2 PROPOSITION. -

Let Q -

on

(E)

t

K N , b(E) and l e t t @ : K ;,

4

be a non z e r o convolution o p e r a t o r

x ; . ~ , ~ (bE e its ) transpose. T :-

@(XI;,b(E)) = [Sf XIN, b(E) s u c h that S(f) = o f o r a l l f E K e r (9 t If S E Q(K;\l. b ( E ) ) t h e r e is a U E X '

Proof. -

s =t Q Hence

.

( E ) with

( U ) = U o Q .

C o n v e r s e l y l e t SEX' ( E ) be such N, b @f= o = T r f . F r o m the proof of ( 1 2 . 6 . 1)

of = o i m p l i e s S(f) =

that S(f) = o w h e n e v e r

N, b

1

0.

t h e r e is a n R E K I N , b(E) s u c h t h a t

= 3 T . 5R

3s aiid thus

S = R + T .

T h e r e f o r e , if f

ExN,b(E) ,

and thus S E t @ ( x k , b ( E ) ) .

12.6.3

THEOREM.

on 3C (E). N, b

& T

-

Let

@ be a R o n - z e r o convolution o p e r a t o r

@(KN,b(E))=XN,b(E).

Convolution equations in spaces

320

Proof. -

By (12.6.2)

t 6 (KCIN,b(E)) is a w e a k l y c l o s e d s u b s p a c e of

F u r t h e r m o r e l e t u s p r o v e t h a t t & is i n j e c t i v e . A s s u m e t is s u c h t h a t qR) = 0 . If @ = T r , t @(R) = ~r T = o

(E). 5?, b R E 3C' (E)

N, b ( c o m p u t a t i o n a t the end of proof 1 2 . 6 . 2 ) . T h u s

5(R* T ) Since on

6'

El).

#o

=

(3R)( 3 T )

= o

, (5T) # o hence 3 R = o (they a r e holomorphic functions

T h u s R = o ( 1 2 . 2 . 5 ) and t @ is i n j e c t i v e . Now l e t u s r e c a l l the

following c l a s s i c a l r e s u l t ( B o u r b a k i [ 1 1 , D i e u d o n n 6 - S c h w a r t z [ I ] ) : "Let 6 from B

be a F r e c h e t s p a c e and u a continuous l i i i e a r m a p p i n g

into 6 , T h e n u is s u r j e c t i v e if

is w e a k l y c l o s e d in B '

'I.

Since 3C

N,b

Lu is i n j e c t i v e and

'u ( 8 ' )

(E) is a Fre'chet s p a c e (12. 2.4),

the a b o v e r e s u l t p r o v e s t h e s u r j e c t i v i t y of

@

.

m

32 1

Existence and approximation results

6

12.7

E x i s t e n c e and A p p r o x i m a t i o n r e s u l t s i n 3C

12.7. 1 Existence theorem. -

k

t E

be a c o m p l e x n u c l e a r 1 . c . s .

a n y c o n - z e r o convolution o p e r a t o r on K

Proof.-

By ( 1 2 . 3 . 2 ) ,

3C

u, b

u. b

( E ) = lim 3C

9

N,b

(E ) w h e n V r a n g e s o v e r V S i n c e E is a n u c l e a r

is a p r e - H i l b e r t s p a c e ( 0 . 5 . 8 ) ,

V

t h e r e f o r e it 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 , We p r o v e d i n if T

from

If

X

8 i s a convolution o p e r a t o r on

E 3C'

8

u, b

#

o ,

u,

(E). Given V a s b e f o r e ,

$ 12.4, ( T T

#o

)

i3

T K '

( E ) , then

(E )

8

=

&

12.3 that

T % for some

E K C I N , b ( E V ) , so,

N,b V

is a convolution o p e r a t o r on 3CN,b(EV 1.

and w e m a y c h o o s e a l l t h e

consider such that ( T

Then

( E ) is s u r j e c t i v e .

a b a s e of c o n v e x balanced o - n e i g h b o r h o o d s in E . 1 . c . s . we m a y a s s u m e each E

aEd 3C (E) S

u,

)

#

0 .

o-neighborhoods V that we

Since

=( T

'KN, then

i s a n o n - z e r o convolution o p e r a t o r on 3 C N , b(EV 1 ,

(9

12.7.2 COROLLARY. -

L A E be a c o m p l e x n u c l e a r Silva s p a c e .

T h e n a n y n o n - z e r o convolution o p e r a t o r on

x S (E)

is s u r j e c t i v e .

(E) a l g e b r a i ' c a l l y . S u, b F u r t h e r m o r e i t f o l l o w s f r o m t h e proof of ( 2 . 7 . 4 ) t h a t a n y bounded Proof. -

F r o m th. ( 2 . 7 . 4 ) ,

3f ( E ) = 3C

s u b s e t i n 3C ( E ) i s in f a c t a l s o bounded i n 3C

( E ) ( i . e . bounded i n u, b S i n c e 3-C ( E ) is m e t r i z a b l e , its topology c o i n c i d e s b V S TKS(E) ( 0 . 2 . 7 and 0 . 6 . 2). F r o m (12. 3. 1) t h e n a t u r a l topology

S s o m e X (E )).

with

Convolution equations in spaces

322

(E) c o i n c i d e s a l s o w i t h T5( (E). So XS(E) = 3c (E) u, b u, b u, b topologically and t h u s (12. 7. 2) follows a t once f r o m (12.7. 1). I of

12.7.3 Remark.

-

A n y non z e r o convolution o p e r a t o r on X ( 8 ' )is S

surjective. Proof.

-

Dineen [ 7

In ( 2 . 7 . 5 ) w e r e m a r k t h a t , f r o m B o l a n d - D i n e e n [1,2]

3

, w e h a v e 3C (2')= 3C S

d e t a i l s and o t h e r r e s u l t s .

u, b

(f'). S e e B e r n e r [ 1 3

and

for

( E ) . - L e t E be a c o m p l e x u, b 8 be a convolution o p e r a t o r on Xu, &E). ' T h s

1 2 . 7 . 4 A p p r o x i m a t i o n t h e o r e m i n 3C n u c l e a r 1. c . s. and l e t a n y solution f E 3C

(E)

of solutions of t h e t y p e u'b

of

5-

( E ) of a s e q u e n c e u, b . @ u i n e x p ( p i ) w i t h a,. ., FiEE'. J

G f = o is l i m i t i n X

I

ai

finite

Proof, -

It 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 proof

Df

(12. 7. 1) a n d f r o m

(12.6.1).

-

1 2 . 7 . 5 A p p r o x i m a t i o n t h e o r e m in X (E)AL e t E be a c o m p l e x S E). T h e n n u c l e a r b . v . s . and l e t 8 b e a convolution o p e r a t o r on a n y solution f

E K S (E)

of@f

%(

o i s limit i n X (E) of s o l u t i o n s of t h e S

finite Proof.

-

T h e proof is e x a c t l y s i m i l a r t o t h a t of ( 1 2 . 6 . 1).

323

Existence and approximation results

-

If E is a q u a s i c o m p l e t e d u a l n u c l e a r 1 . c . s . then

X ( E ) is d e n s e in

and h a s the induced topology (5. 1 . 5 ) h e n c e

12.7.6 R e m a r k .

xS( E )

the two s p a c e s h a v e the s a m e convolution o p e r a t o r s . F u r t h e r m o r e i t is i m m e d i a t e to c h e c k that E ' is d e n s e ir. E X ( s i n c e E

( 0 . 3 . 3) and ( 0 . 5 . lo)), h e n c e E x p E ' works if Abstract.

0.

. and p i E E ' and

we

a

1JJ

12.7.7 Remark.

-

12.7.8 Problem. -

= E

. So t h e proof

from of ( 1 2 . 6 . 1)

obtain the e x i s t e n c e th. A of t h e

A m o r e g e n e r a l f o r m u l a t i o n of t h e e x i s t e n c e t h e o r e m

i n 3C ( E ) is in C o l o m b e a u - P e r r o t [

S

= Exp E X

X'

6

1 .

L e t E be a c o m p l e x b . v . s . ( r e s p . I.c. s . ) and

@

a non z e r o convolution o p e r a t o r on Jc ( E ) ( r e s p . 3C(E)). D o w e h a v e S @(3(,(E)) = 3CS(E) ( r e s p . @((X(E))= k ( E ) ) ? The p a r t i c u l a r c a s e E is a

n u c l e a r F r k c h e t s p a c e is p a r t i c u l a r l y i n t e r e s t i n g .

324

Convolution equations in spaces

9

1 2 . 8 Convolution o p e r a t o r s of f i n i t e type

1 2 . 8 . 1 DEFINITION, -

If E

we s a y that a convolution o p e r a t o r

is a c o m p l e x 1.c. s. (or b . v . s . 1 ,

6 02K ( E ) h K s ( E ) ) i s of f i n i t e

type i f t h e r e i s s o m e f i n i t e d i m e n s i o n a l c o m p a c t ( o r s t r i c t l v c o m p a c t ) subset K o f E

and

/ < T ,v >

( f o r some c

.L

such that

8 = T % , w i t h TE K ’ ( E )

1 5 1 for a n y

>

o),

i.e. T E

E K ( E ) (or XS(E))

% (K,&).

b ~ xS (E)) ’

ICJ

. w w

T h e s e convolution o p e r a t o r s a r e w r i t t e n

I

n = o

K6 ’I f o r s o m e p 3. o and s o m e ( c o n v e x balanced) 1 finite d i m e n s i o n a l c o m p a c t ( o r s t r i c t i y c o m p a c t ) s u b s e t K of E (apply

w h e r e $‘“’(o)

E iJ- l a

7 . 1. 5 and 12.1.5

12.8. 2

in a finite d i m e n s i o n a l s p a c e containing

K).

If E i s a c o m p l e x v e c t o r s p a c e we e q u i p E with its f i n i t e

d i m e n s i o n a l bornology 0 . 2 . 2 .

B y a proof of t r a n s f i n i t e inductior. based

upon the e x i s t e n c e t h e o r e m in the f i n i t e d i m e n s i o n a l case, Boland-Dineen

[ 3 ] obtain : Existence Theorem.

-

L e t E be any complex vector space

equipped with t h e finite d i m e n s i o n a l bornology. If convolution o p e r a t o r on 3c ( E ) , then S

1B.8. 3

8 i s a n y non z e r o

8 K ( E ) = Ks(E) S

.

A n o t h e r d i f f e r e n t r e s u l t is obtained by a proof e x a c t l y

s i m i l a r to t h a t ir, c h a p t e r 13 ( s o we j u s t s t a t e the r e s u l t ) .

325

Convolution operators of finite type

Existence Theorem. -

E be any c o m p l e x 1. c . s . and

a f i n i t e type convolution o p e r a t o r . Ther,

G(Ku , b ( E ) =

f9

xu,&E) .

We note t h a t in t h i s c a s e we d o not need a n y a s s u m p t i o n of n u c l e a r i t y on E ,

a s needed i n Th. 1 2 . 7 . 1 w h e r e g e n e r a l convolution

o p e r a t o r s w e r e c o n s i d e r e d . In p a r t i c u l a r i f E is a n o r m e d s p a c e

6(Kb(E)) = X b ( E ) and i f E is a Silva s p a c e i t f o l l o w s f r o m t h . 2 . 4 . 7 that

Q(K(E)) = K(E).

#

o

CHAPTER 13 LINEAR FINITE DIFFERENCE PARTIAL DIFFERENTIAL EQUATIONS IN %I (E)

A BSTRA CT.

If E is a r e a l 1.c.s. we d e f i n e a l i n e a r p a r t i a l

d i f f e r e n t i a l - d i f f e r e n c e o p e r a t o r with c o n s t a n t c o e f f i c i e n t s (1. p . d . d .

0.

f o r s h o r t ) by the f o r m u l a m

j = l

if m

E N , c E C , hj. and y . E E a r e fixed and w h e r e x E E and

v E &(El.

j

J

We r e c a l l that we denote by 8 ( E ) the l i n e a r s p a c e of t h e u, b c o m p l e x valued Cco functions on E t h a t a r e of " u n i f o r m bounded type" (defined in 1 . 6 . 2 ) .

Existence Theorem. 1.p.d.d.o.

on E

is a Silva s p a c e

.

-

If E -

then D &u, b ( E ) =

is a r e a l 1 . c . s . and D a nor. z e r o

a

d u , b(E). In p a r t i c u l a r C & ( E ) = & ( E )

In the p a r t i c u l a r c a s e E is a n o r m e d s p a c e , we have obviously a3

(E) = 8 ( E ) , t h e s p a c e of the C functions on E that a r e bounded 'u, b b a s w e l l a s a l l d e r i v a t i v e s on any bounded s u b s e t of E , and t h e r e f o r e D6b(E) = db(E). We a l s o obtain a p p r o x i m a t i o n of t h e s o l u t i o n s of t h e homogeneous equation by exponential polynomial solutions. 326

321

Imaginaty-exponentiabpoly nomials

5

13.1

A d i v i s i o n r e s u l t by i m a g i n a r y - e x p o n e n t i a l - p o l y n o m i a l s

By a n i m a g i n a r y exponential polynomial ( I . e. p. f o r s h o r t ) we m e a n a l i n e a r combination of p r o d u c t s of polynomials by p u r e i m a g i n a r y exponentials. L e t Q be a non z e r o i. e. p. on one c o m p l e x v a r i a b l e and write

... t Q k e x p ( i a k' ) Q ( z ) = Q , ( z ) exp(i a z ) t. . exp (i a z), 1 k a r e non z e r o polynomials and w h e r e a < a <. . . < a 1 2 k' Q

i. e.

1

.+Q,(z)

where the Q

L = ak

Q 1 e x p ( i a .) t

-

j a l and l e t

Q1 = b

( i t is allowed t h a t b

13. 1. 1 L E M M A , -

0

=

0

X

m

t bl Xm-'

t

Call

... t bm

0).

In t h e above notation, t h e r e e x i s t s a K > o which

.

and on the d e g r e e s of Q such that e a c h j k i n t e r v a l I in the c o m p l e x plane which is p a r a l l e l to t h e r e a l a x i s and

depends only on a l , a 2 , . . , a

h a s ler.gth 2 2 ( L t 1) contains a point y a t which

w h e r e a is t h e d i s t a n c e of I f r o m t h e r e a l a x i s .

328

Linear finite difference

1 3 . 1 . 2 LEMMA.

-

With Q a s above s u p p o s e t h a t

f o r a l l c o m p l e x n u m b e r s x, z ,

r

where

M i s a p o s i t i v e function and

7o

.Lety

>o

we c a n d e s c r i b e a c i r c l e a b o u t y of r a d i u s

be a n y c o m p l e x n u m b e r which s a t i s f i e s ( 1 ) . T h e n f o r a n v

r ' such that

r i r' 5 2r

where B , c d

d

a r e c o n s t a n t s and w h e r e d is a p o s i t i v e i n t e g e r .

We m a y obviously a s s u m e c < o ( s i n c e A we c h o o s e M(y) 2 1 we m a y a s s u m e B < o

>

o and

r'

lhence

IF((C) I

s

m(1

+ 1161

fixed non z e r o i . e . p . on C n

KP E

(A d i v i s i o n r e s u l t ) .

1 ) ~ e blllm

.

'I1

V

5

Let us assume

3 d ' ( l R n ) , i . e . t h e r e e x i s t m ' , V ' and

where m ' , v ' and b ' d o not depend on

9

If

. We d o not give the p r o o f s

of t h e s e two l e m m a s which m a y be found in E h r e n p r e i s [ 1 ]

1 3 . 1 . 3 PROPOSITION.

> 0).

E

8)and

fl E

and [ 2

58'(1Rn)

let P

a' E3C(Cn). P

Then

b' with

but only on m,

U '

a n d b.

3.

3 29

Imaginary-exponential-polynomials

n F o r c o n v e n i e n c e we s e t F = IR and l e t

Proof.-

(2)

P. e

P=

lYj

J

j € J

(ficite)

w h e r e y . E F and y if j , fy. J jl J2

# j2

, and w h e r e

(finite)

whith

Cj

the s e t s

Q

6 C , hj F F and P. a,P J

#o

We cl aim t h a t t h e r e is a t l e a s t one

n

hence F' =

A. J

. For each

# fl ,

j E J let us consider

s i n c e i f not

Q

j € J

,fj =

fl ,

, and F' would be c o n t a i c e d in a finite ucion of

j € J c l o s e d h y p e r p l a c e s (the h y p e r p l a n e s that

# fl

. L e t u s d e n o t e by

t h a t P,(q) by :

#

0.

For

U

-y

P

) = o i f er

# p).

We a s s u m e

m the d e g r e e of the polynomial

# $ , and P1 # 0 , t h e r e i s a n q we c o n s i d e r the function each 5 E F'

is open,

(2). Since

i$y

i\l

c

E

P

1

in

A1 s u c h

Q defined

Linear finite difference

330

Let ai = q(yi)

E

Since q E h l

JR.

the n u m b e r s a . a r e a l l d i s t i n c t . We

( k = c a r d J), and w e h a v e numerote them a < a < . . . < a 1 2 k '

where

where b

0

=

Pl(q e

it

(Y,)

and w h e r e b 1,

.. , bm a

a r e the o t h e r

- a 1 ' By l e m m a 13.1. 1 t h e r e is a K > o k and on the d e g r e e s of t h e which depends only on a l , a 2 , a k polynomials Q such that t h e r e a l i n t e r v a l [ - ( L t 2 ) , L t 2 3 c o n t a i n s a j point Y a t which coefficients. L e t L = a

R e m a r k that a 1,

... a k

...,

a r e fixed independently of

5 , and t h a t t h e r e i s

only a finite n u m b e r of p o s s i b l e d e g r e e s of O., 1 I j 5 k, when 5 J r a n g e s i n FIa; , hence i t i s p o s s i b l e to find K > o independently of Now, ir. o r d e r t o a p p l y t h e l e m m a 1 3 . 1 . 2 ,

w e a r e going t o p r o v e

that

IQ(h+

where

R

>o

u)1

5.

~ M ( Pe )

and w h e r e M is a positive function t h a t we a r e going t o

Irnagimrpexponential-polynomials

compute

. Since

we a r e only doing t h e c o m p u t a t i o n s f o r one index j ,

Hence t h e p r o b l e m is t o obtain a m a j o r i z a t i o n of a t e r m of t h e f o r m :

Such a t e r m is a s u m of 2

We have

:

n

t e r m s of the f o r m :

331

Linear finite difference

332

where

I/h

= max

19 ( h i , k) I } far t h e hJ , k i n ( 2 ' ) *

If N = s u p n . , j E J , ( s e e ( 2 ' ) ) , then e a c h J

H e n c e t h e r e is a A 2 1 ,

We h a v e

where

W e have a l s o :

w h e r e w e set

independect

OR

IT

5 ,

I

i s m a j o r i z e d by

such that :

Imaginnlyexponenrial-polynomials

333

T h e r e f o r e , f r o m (4)t o (8) ,

Therefore

i. e.

Where

and

k = 2.4'.

Now we apply the l e m m a 13. 1.2 with the point y = Y with the choice r = L t 3 .

obtained i n ( 3 )

T h e n we obtain that t h e r e is a

r'

,

r< r

such that

inf

( 9)

where B <

1 A I =r' 0,

fQ(1 t

c < o and d

>

\)I

2

B d d (M(L)) e x p ( c I r') K l b o l

o a r e constants.

, and ' 2 r~

334

Linear finite difference

CoRsider now t h e following e n t i r e f u n c t i o n F on

g

F(X) =

(5

t

1 q) , A

C :

C , q fixed a s a b o v e and

F r o m the maximum principle

because

o is i n s i d e t h e c i r c l e of c e n t e r

F r o m a s s u m p t i o n on

Since

and

g,

I X t Y l i 3Lt8 ,

we have

Y

and r a d i u s

r ' . Hence

Ec F'

c*

Imaginary-exponen tial-poly nornials

hence

and f r o m (9)

S i n c e , f r o m t h e definition of b ' ,

then

We s e t 2 A 1 ( L t 2 ) - B .-c.l(2Lt6) 1 C1 = ( k A e C2 = m ( 1 t ( 3 L t 8 ) IlqII) e b ( 3 w

I I 11 ~

335

Linear finite difference

336

which a r e independent on

5.

T h e n f r o m (10),(11) and (121,

and we obtain t h e d e s i r e d inequality.

Vg E F i C

337

A Paley-Wiener-Schwartz theorem

5

space

13.2

A-Paley-Wiener-Schwartz

t h e o r e m and a divisior, r e s u l t

If E is a r e a l Banach s p a c e w e r e c a l l that 8 ( E ) d e n o t e s t h e b of t h e c o m p l e x valued Cco functions on E t h a t a r e bounded a s

w e l l a s t h e i r s u c c e s s i v e d e r i v a t i v e s on t h e bounded s u b s e t s of E .

db(E)

is n a t u r a l l y equipped with the topology of 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 of E f o r t h e functions a n d the d e r i v a t i v e s . It is e a s y t o c h e c k t h a t d b ( E ) is a F r e ' c h e t s p a c e , L e t E be a r e a l Banach s p a c e and F be a f i n i t e d i m e n s i o n a l s u b s p a c e of E . We s e t E = F t a r y t o F.

R(F;T: , dIb(G))

G,

where G is a subspace complemen-

d e n o t e s t h e l i n e a r s p a c e of all m a p p i n g s f r o m

F i e into dlb(G).

13.2. 1 DEFINITION.

-

We define a F o u r i e r t r a n s f o r m z i n t h e

following way :

13.2.2 PROPOSITION. -

T h e F o u r i e r t r a n a f o r m 13.2. 1 is i n j e c t i v e .

Linear finite difference

338

yi.eFi) =

T h e r e f o r e k. (

0

for

Y i E B ( F ) , p i € Bb(G). But

finite

6 ( F ) Q" S , ( G ) is d e n s e in d ( F ) E 6,(G) , s i n c e d ( F ) h a s the approximation property (Schwartz (I)).

B(F)

6 . 2 . 1 and

a($)

=

6

d b ( G ) = G ( F , Bb(G)) from

d ( F , db(C)) = d b ( F x G) = Bb(E), see 6.2.3 0, f o r a l l $ E 6 ( E ) , t h a t is ,t = 0

.

b

13.2.3

. Therefore

The Fourier

PROPOSITION ( P a l e y - W i e n e r - S c h w a r t z t h e o r e m ) . -

t r a n s f o r m is a n a l g e b r a i c i s o m o r p h i s m f r o m

8 ' (E) onto a l i n e a r s p a c e

b i n t o 6 ' (G) t h a t we denote by 3 d l b ( E ) and w e cb d e s c r i b e now : 3 S r b ( E ) is the l i n e a r s p a c e of the m a p p i n g s of mappings f r o m F '

-

$ :Frc (1)

B1,(G)

such that :

f o r e v e r y tgEQb(C) , the function 5

-,

a' ( s ) . y

is holomorphic

on F ' c (2)

f o r e v e r y cq such that

(3)

E 6

b

(G) , t h e r e a r e c o n s t a n t s m

t h e r e i s a 0-neighborhood

'V

cp'

v

so-

and b

i . 8 (G) s u c h t h a t i f

b c'lf t h e n w e m a y c h o o s e t h e c o n s t a n t s m

b

va

independent of

uEn.

cc

,v 9,

a&

339

A Paley- Wiener-Schwortz theorem

5

where

d e n o t e s the F o u r i e r t r a n s f o r m on 6 ' ( F ) . T h e n by the P a l e y F and b W i e n e r - S c h w a r t z t h e o r e m i n finite d i m e n s i o n , t h e r e a r e m

CF

such that

for e v e r y

GF'

0

. Now

c

B E

, where

?f

lVCy

v

is a o-neighborhood

1

(convex balanced bounded and K in d b ( E ) = Bb(FUG). T h e r e a r e K F G s u b s e t s of F and G) s u c h t h a t we m a y a s s u m e

such t h a t

?fl = / f € g t ( E ) ,

If

(i)

(KFxK

G

).(KFxK

G

)

i

I@,

i f 0 5 is n

.

We s e t

?f= { v C Q b ( G ) ; s u c h t h a t l(i(i)(KG). (KG )i 15 p i f

Then on rp

(iP E

E

E

0

in &'(F),i f

k

?f.T h e r e f o r e

e p G ? f and for s o m e

05 i 5 n

E

>o

1

independent

ICQP ] c ; c l r i s a bounded s u b s e t of d ' ( F ) . By

the P a l e y - W i e n e r - S c h w a r t z t h e o r e m s in f i n i t e d i m e n s i o n , we have (3). Conversely let

a'

i n t o 6' (G), t h a t v e r i f i e s ( 1 ) , b in Bb(G) be fixed and define fl : F' C by

be a function f r o m F '

( 2 ) and (3). L e t cc,

Idcp(!)

=

$(c) .y

. By (1) and ( 2 ) ,

IdCF E

B'(F) with ( 3 .4 ) ( { ) = A & e i 5 ) = Fcp w e define a m a p

2

:

8 (F)

db(G)

-

c

v

c

-

Z F ( S ' ( F ) ) , s o t h a t t h e r e is

cp

(5),

for every

G by i ( y , @) =

T € F I C .

CF

A

CF.

in

Now

( y ) , if @Edb(G)

340

and

Linear finite difference

2

'f E B(F), and i t is i m m e d i a t e t o check that (given by (3)) then

cp E ?f

is bilinear. F o r

{ A q 1 is a n equikontinuous s u b s e t of .$'(F),

f r o m the Paley-Wiener -Schwartz t h e o r e m in finite d i m e n s i o n , t h e r e f o r e is bounded on

v,

k

0

k in B ( F ) s u c h t h a t k?

is a 0-neighborhood

that is

3

E II;

cp is continuous. So

continuous l i n e a r mapping ,f, f r o m d ( F )

QP

Schaefer [ 13, cor. 2,p. 172, that

&(F)@,

1

2 comes from a

8 (C) into C. Since d ( F )

n b

is a nuclear s p a c e , i t follows f r o m T r i v e s [ I ] ,

-

. T h i s i m p l i e s that

thCor'eme 50. 1,

db(G) = d(F) @c Bb(C),

( W h e r e n and

E

&(F)Q~ S,(C)

is a topological s u b s p a c e of B(F) E db(G) acd is d e n s e in

it

denote the topologies on the topological

product.).

, since &(F) h a s the approximation p r o p e r t y (Schwartz [I 1).

T h e r e f o r e B(F) @E Bb(G) = 8(F) & db(G). By 6.2.1, B(F) E d b ( G ) = B(F, B b ( G ) ) = db(E). T h e r e f o r e & E 8' ( E ) acd b

13.2.4 Remark.

-

fl

= 378.

T h e Paley-Wiener-Schwartz t h e o r e m 13.2.3 r e m a i n s

t r u e i f E i s any Schwartz b.v.s.

and if w e r e p l a c e d b ( E ) by &(E) :

the proof is the s a m e a s above.

+- -.

- 2 fl E

13.2.5 PROPOSITION ( a division r e s u l t ) . a non z e r o i.e. p

on

F and if

is a mapping f r o m F ' ,f

s u c h that f o r e v e r y v,€Sb(C) , then P P

then

P

(2-

if

P

into tjIb(G)

cp is holomorphic on F '

E 5 8lb(E).

Proof.

-

If cp

#,(C)

=

#(C).q ,

that

38',(E),

F

.$,(C)

is fixed a n d i f

$

CF

C '

E K ( F I C ) is defined by

then by hypothesis t h e r e a r e m

cp' veF,

and b

cp

such

341

A Paiey-Wiener-Schwartz theorem

F r o m prop.

mcC'

"CP

13.1.3

and b

CP

there a r e m'

such that

cp'

v'

T h e r e f o r e it follows f r o m ( 1 3 . 2 . 3 ) t h a t ty ( 3 ) 13.2. 3 since

#

h a s it).

(P

and

b'

cp

w h i c h depend only on

$ E 3 8' (E),(pry P

b

has proper-

Linear finite difference

342

9

E x i s t e n c e and A p p r o x i m a t i o n of s o l u t i o n s in a B a n a c h s p a c e .

13.3

In t h i s s e c t i o n E s t i l l d e n o t e s a r e a l Banach s p a c e . If h l , and y a r e in E we d e n o t e by

Dh l , . . . , h n ;

y

. .,, h n

the l i n e a r o p e r a t o r f r o m

& ( E ) i c t o d ( E ) defined b y :

(Oh , , . . . ,

(x)

h r? ; y

if q E d ( E ) and x E E .

Clearly D

rC(n)(x+y).hl...hl,

hl,...shn;

y

i s also a linear

o p e r a t o r f r o m B'&E) i n t o BIb(E). We define a l i n e a r p a r t i a l d i f f e r e n t i a l d i f f e r e n c e o p e r a t o r with c o n s t a n t c o e f f i c i e n t s (1.p.d.d.

0.

for short) a s

finite

with

c.E

C. T o the l . p . d . d . o .

J defined by

D we a s s o c i a t e t h e i . e . p .

PD on E'

j C J finite

We c h o o s e F and G w i t h E = F @ G s u c h t h a t t h e s e t is contained i n F.

{hq/ u /yj jj P PIq

343

Existence and approximation

1 3 . 3 . 1 PROPOSITION ( A p p r o x i m a t i o n of s o l u t i o n s ) .

-

Each u E & ~ ( E )

solution of D u = o is l i m i t i n B (E) of so1utior.s of t h e f o r m b

E J P.e

a.

.1. with J

P . finite type continuous p o l y n o m i a l s on E , J

finite

Proof.

-

It s u f f i c e s to c o n s i d e r the c a s e D

#o

. We a r e going t o

p r o v e that the s e t of t h e s e p a r t i c u l a r s o l u t i o n s is d e n s e i n K e r C f o r t h e topology induced by 8 ( E ) . F r o m the H a h n - B a n a c h t h e o r e m it b s u f f i c e s t o p r o v e t h a t a n y A? E 8Ib(E) which is null on t h i s s e t is null on K e r D . F i r s t we show t h a t i f

A is a s a b o v e then

''

- is

a well

pD

defined mapping f r o m F r C into BIb(G) a fixed qr E Sb(C) , fixed

6

. F o r t h i s we f i r s t c o n s i d e r

and r) in FIC and the e n t i r e functions

on C defined by :

if h E

C

.

in (12.6.1)

f o r all j s n

If

1 E C is a z e r o of o r d e r n of Q, the c o m p u t a t i o n s 0

give t h a t

. T h e n for a l l

E db(G)

I

344

Linear finite difjerence

j i ( S -t 1 .q) q .e .Y

T h e r e f o r e the function

is i n the a b o v e s e t of

p a r t i c u l a r s o l u t i o n s and t h u s by h y p o t h e s i s

T h e r e f o r e F ( j ) ( A o ) = o i f j < n , t h a t is g r e a t e r t h a n n o r e q u a l to n. H e n c e function on C function g

-

. T h e n by ( 1 0 . 2 . 4 ) ,

n r

o

Q

i s w e l l defined a s a n e n t i r e

f o r e v e r y fixed

-t

q 6 6b ( G ) the

is w e l l defined and h o l o m o r p h i c on

FIc.

PD(S)

5 E F'

Now we a r e going to p r o v e that f o r a n y fixed q-

i s a z e r o of F of o r d e r

3.l (0.y

is a n e l e m e n t of

pD

c'

the m a p p i n g

8 ' b (G). If PD(S) # o t h i s i s

. If

P ( 5 ) = o t h e r e i s a null s e q u e n c e ( A ) of c o m p l e x n D n u m b e r s , and a 1-I c F t C s u c h t h a t P ( s t 1 p ) # o f o r all n. D r . 31 5a S i n c e -(5tinp).cF 3 ( 5 ) . y if n t a o , w e obtain t h a t -(g) P D pD pD obvious

-

4

is a l i n e a r f o r m on db(G)

. Now if

{cp

1

i s a bounded s e t i n db(G),

a &A

f r o m 1 3 . 2 . 3 and 13.1.3,

w h e r e m ' , v 1 and b' a r e independent of

0

is a bounded s u b s e t of

e

p

. Therefore

C and t h e r e f o r e

- (3Rz ) pD

is a bounded l i n e a r f o r m o n 8 (G) , i . e . a continuous l i n e a r f o r m s i n c e b 3.t db(G) i s a F r e ' c h e t s p a c e . F r o m 1 3 . 2 . 3 , E 3dIb(E), that i s ,

-

t h e r e is a n R E 6Ib(E) s u c h t h a t 51 = P 3R. If D' E 6b(') 1

5 C

FIc

and

Existence and approximation

ie ( ei'

.

q)=3,4( 8 ) . q = PD(g). ( 3 R (5 ) .

cp)=x

n c j ( i ) Jh$

345

5)

iSYj ...h h ( 5 )e .R ( e.'

v)

j

j€J finite

By d e n s i t y in &F) of the l i n e a r s p a n of t h e exponentials, we have R(f.cp)

\YE NF).

= (R o D ) ( f . ~ )f o r any

Since, f r o m 13.2.2,

Bb(G) i s d e n s e i n 8 (E), i = R o D i n dIb(E). T h e r e f o r e b if f E K e r D.

a(f)=o

d(F)

Remark.

13.3.2

-

The A p p r o x i m a t i o n R e s u l t 13.3. 1 m a y be adapted to

o t h e r s p a c e s and i m p r o v e d . For i n s t a n c e if E i s a n u c l e a r b . v . s . c o n s i d e r d(E) ar.d We know, ( 5 . 2 . h ) ,

we

B(G) i n p l a c e of db(E) and &b(G) r e s p e c t i v e l y . that the l i n e a r s p a n of t h e exponentials i s d e n s e i n

S(G) , s i n c e G is a l s o a n u c l e a r b.v.

s, and the proof of

in t h i s c a s e t h a t t h e s e t of s o l u t i o n s of t h e f o r m P. f i n i t e type continuous polynomials on E and J the s e t of a l l solutions.

fiEite

13. 3. 1 g i v e s

P . e a j with J a.EE' a r e dense in

J

C

N o w the following r e s u l t s a r e analogous t o (12.6.2) and (12.6.3).

13.3.3 LEMMA. E and C be a s u s u a l i n t h i s s e c t i o n and l e t t D : dfb(E) 6' ( E ) be t h e t r a n s p o s e d of D. T h e n b

-

t

13.3.4

E ( d l b ( E ) ) = /SEd',(E)

s u c h t h a t S(f) = o f o r a l l f C K e r D

E x i s t e x e Theorem. -

l e t D__ be a non z e r o l.ptd,dlo.

.I__.

]

.

L e t E be a r e a l B a n a c h s i _ a c_ e_ and -on -__I_. E . T h e n D b b ( E ) = B b( E ) .

Linear finite difference

346

9

13.4

E x i s t e n c e of solutions in locally convex s p a c e s

L e t E be a r e a l 1.c. s. and C a non

13.4.1 THEOREM. z e r o l.p.d.d.o.

Proof.

-

on E.

Then D . d ( E ) = d U , b(E). u, b

By definition (1.6.2), 8

u, b

( E ) = l i m 8 ( E ) when V r a n g e s b V

v"

o v e r a b a s e of convex balanced o-neighborhoods in E . Th. 13.4.1. follows immediately f r o m the s a m e r e s u l t in the p a r t i c u l a r c a s e E is a normed space and since $ ( E ) = &b(E) algebrarcally i f E i s a normed

.

space acd E its completion it suffices to apply 1 3 . 3 . 4 .

13.4.2 COROLLARY. z e r o l.p.d.d.o.

-

Let -

.

E be a r e a l Silva s p a c e and D a non

on E. Then Dd(E) = d(E).

The r e s u l t follows immediately f r o m 13.4. 1 and 1.6.3. ,

CHAPTER 14 PSEUDO-CONVEX DOMAINS AND APPROXIMATION RESULTS

A BS TRA CT

.

We s t a t e c l a s s i c a l definitions and p r o v e v a r i o u s m a i n

r e s u l t s on pseudo -convex d o m a i n s , d o m a i n s of h o l o m o r p h y , d o m a i n s of

. S o m e of t h e m w i l l be used

e x i s t e c c e , h o l o m o r p h i c convexity..

in the

next c h a p t e r .

THEOREM (Solution of the L e v i p r o b l e m ) . Banach s p a c e with b a s i s and

-

L e t E be a complex

0 a pseudo-convex open s u b s e t of E .

Then fl is a d o m a i n of e x i s t e n c e of a h o l o m o r p h i c function ( t h e r e f o r e 0

R

is a d o m a i n of h o l o m o r p h y , h e n c e

is h o l o m o r p h i c a l l y convex). L e t u s

p o i n t out that the s a m e r e s u l t holds i f E i s a DFN s p a c e , a Silva s p a c e with b a s i s , a F r 6 c h e t s p a c e with b a s i s and s e v e r a l o t h e r s p a c e s .

THEOREM (Runge A p p r o x i m a t i o n t h e o r e m ) . Banach s p a c e with b a s i s and

n

Let E

-

be a c o m p l e x

a pseudo convex open s u b s e t of E .

K is a c o m p a c t s u b s e t of 0 s u c h t h a t

Go,

= K,

If

then ahy holomorphic

function in a neighborhood of K m a y be a p p r o x i m a t e d u n i f o r m l y on K by h o l o m o r p h i c functions on 0 .

THEOREM Ian A p p r o x i m a t i o n r e s u l t ) .

-

W

E &F

be two

s e p a r a b l e H i l b e r t s p a c e s with a c o m p a c t i n c l u s i o n m a p p i n g f r o m F L e t R be a pseudo-convex open s u b s e t of E with 0 n F # $ . Then E* the r e s t r i c t i o n mapping,:

xc(n)

+

qfl n F)

h a s dense range. 341

Pseudo-convex domains

348

$ 14.1

G l i m p s e a t pseudo-convexity and d o m a i n s of h o l o m o r p h y

L e t E be a l i n e a r s p a c e o v e r -

1 4 . 1 . 1 DEFINITION. -

C,

“rf: a

0 a n open s e t f o r (E,?f).Let v be a

Hausdorff topology on E

function defined on %1 and with r a n g e i n ’

[-00,

f a [ , with v

,d -00. The

function v is called p l u r i s u b h a r m o n i c if a) v(z) < c

1

b)

v is upper s e m i - c o n t i n u o u s (i.e. the s e t is open f o r a n y c

E

if(a,b)

E R)

~ (E x - 101)

is s u b h a r m o n i c o r i d e n t i c a l to w h e r e it is defined

.

I z 6 R such that

the function

-OD

6

-v(atcb)

(5 E C)

on e a c h connected component of

C

We r e c a l l t h a t if a ) is s a t i s f i e d , b) is e q u i v a l e n t t o

When E i s a l c s and we do not m e n t i o n t h e t o p o l o g y c , w e assume that

b e i s t h e topology of

1 4 . 1 . 2 DEFINITION. s u b s e t of E .

-

E . F o r o t h e r topologies s e e K i s e l m a n [43.

L e t E be a c o m p l e x 1 . c . s . and

We denote bv d n t h e function :

n

a n open

Glimpse at pseudo-convexity

349

We s+y t h a t %2 is pseudo-convex i f the function -Log d m o n i c on 0 dn

x

(E- l o

1)

( f o r e v e r y fixed z I ir, ( E - l o ] ) ,

is t h e d i s t a n c e f r o m z to t h e c o m p l e m e n t of

14.1.3 and

R

is a h u b s e t of E

if %1

n

is p l u r i s u b h a r -

the function

0 in t h e d i r e c t i o n zl).

If E is a c o m p l e x l i n e a r s p a c e

P s e u d o - c o n v e x open s e t s . -

dimensional subspace

R

such that 0

n

F i s open f o r a n y finite

F of E , we s a y t h a t R is f i n i t e l y pseudo convex

F i s p s e u d o convex f o r a n y finite d i m e n s i o n a l s u b s p a c e F of E. T h e n i t is proved ir, N o v e r r a z [ 1 ] 2 . 1.5 that i f E is a c o m p l e x

1 . c . s . and R an open s e t in E , then

R

is p s e u d o - c o n v e x i f and only i f

s1 i s finitely pseudo-convex.

1 4 . 1 . 4 DEFINITION.

Let E

-

be a c o m p l e x l i n e a r s p a c e with be a n open s u b s e t of E. L e t

s o m e topology o r bornology and l e t

A ( n ) denote a f a m i l y of h o l o m o r p h i c o r Silva h o l o m o r p h i c functinns on L e t K be a s u b s e t of R s u c h t h t

f EA(R)

. Then w e call

If

1 K=

s u p I f ( z ) 1
zE K

A(R)-enveloppe of K the s e t

={zEh2 s u c h t h a t I f ( z ) 1 5 1 f / K f o re v e r y f c A ( R ) ] K A(R)

If K'(n) -

.

is a f a m i l y of p l u r i s u b h a r m o n i c functions on R we s e t :

-

K p ( n ) = { Z E D s u c h t h a t f ( z ) S s u p f ( x ) f o r .every f g q n ) 1 xE K

14.1.5

n.

A( n) convexity.

-

.

If E i s a c o m p l e t e 1 . c . s. and R ar, open

then R i s said to be A ( n ) - c o n v e x i f K is c o m p a c t i n R A(0) f o r a n y c o m p a c t s u b s e t K of R set in E,

.

Pseudo-convex domains

350

14.1.6

Domains of Holornorphy

.-

If E i s a complex 1. c. s .

0 a n open s u b s e t of E , 0 is said t o be a domain of holomorphy

and

if t h e r e d o cot e x i s t two non void connected open s u b s e t s R1 and

of E

14. 1.7

fi

such t h a t

n h2 1

kh2

a)

n,cR

b)

for every f E X ( n ) there exists f

and

R

1

Domains of E x i s t e n c e .

R ar, open s u b s e t of E , 0

2

-

1

€ q R 1)

such that

If E is a complex 1 . c . s . and

is said to be a domain of e x i s t e n c e (of a

holomorphic function) i f t h e r e i s a n f

Eqn)

e x i s t two non void connected open s u b s e t s Rl

such t h a t t h e r e do not and

hl

2

of E ,

m e a n s that f cannot be continued to a connected open s e t l a r g e r than 0).

14. 1.8 R e m a r k .

-

The l i t t e r a t u r e

OR

t h i s topic d e a l s with holomorphir

functions although one m i g h t c o n s i d e r Silva -holomorphic functions. In a n y c a s e , in a l l this c h a p t e r , we s h a l l w o r k in s p a c e s w h e r e t h e s e two co

~ e pts c

c oihc id e

.

14. 1 . 9 V a r i o u s p r o p e r t i e s of open s e t s .

-

L e t E be a complex 1. c. s.

and h2 a n open s u b s e t of E. We c o n s i d e r the following p r o p e r t i e s : (PI)

0 is domain of existence of a holomorphic function.

(P,)

f o r e v e r y sequence

( 5n ) nEJN of e l e m e n t s of h2 which c o n v e r g e s t o 5 E on ( b n is the boujidary of 0) t h e r e e x i s t s a n fEx(C(h)

351

Glimpse at pseudo-convexity

such t h a t

R i s a d o m a i n of h o l o m o r p h y R i s h o l o m o r p h i c a l l y convex ( i . e .

n

14. 1.5 with A ( 0 )

a"(n)

is p l u r i s u b h a r m o n i c a l l y convex (14. 1.5 with

x(6-2)) the s e t of

t h e p l u r i s u b h a r m o n i c functions in 0)

R is pseudo-convex R is f i n i t e l y pseudo-convex. Then it i s proved i n N o v e r r a z

('5)

1 3 that

('6 P1)

F u r t h e r m o r e the i m p l i c a t i o n s

* (P3)

In 14. 1. 10 below we p r o v e t h a t (P,) not (P,)

(P,)

and ( P 4 )

3

(P,) a r e obvious.

. T h e p r o b l e m w h e t h e r or

(P,) is called t h e L e v i p r o b l e m and it w i l l be studied i n the

next section. If (P,)

(P,)

, thele'vi problem

is a f f i r m a t i v e l y solved

in 6-2 , and then p r o p e r t i e s ( P ) ( P ) ( P ) ( P ) ( P ) and (P,) a r e 1 3 4 5 6 equivalent.

14.1.10 THEOREM.

-

E v e r y d o m a i n of h o l o m o r p h y in a c o m p l e x

c o m p l e t e 1. c. s . is h o l o m o r p h i c a l l y convex.

Proof.

-

L e t U be a d o m a i n of h o l o m o r p h y i n t h e c o m p l e t e 1. c.

and l e t K be a g i v e n c o m p a c t s u b s e t of U.

E,

L e t u s a s s u m e , by a b s u r d ,

that f o r e v e r y continuous s e m i - n o r m p on E (and i f d a s s o c i a t e d d i s t a n c e ) , w e have :

S.

P

denotes the

352

Pseudo-convex domains

T h e r e is s o m e p with

dp(K,

By a s s u m p t i o n , t h e r e is a

IU) = E 30.

5 E

. If

Fix any y E E with p(y) 4 6 let

%

(U)

such that

A d e n o t e s the open unit b a l l i n C ,

b > 1 be s u c h that 1

We s e t

K

If f

ex(U)

Y

= K t b1 b y .

t h e r e is a convex balanced o-neighborhood

W

in E

such

t h a t K t 6 W c U and Y 1

If w E W

Since 5 C

,

f? A and n EN we c o n s i d e r t h e h o l o m o r p h i c function on U

%

(U)

we have :

353

Glimpse at pseudo-convexity

from Cauchy's integral formula, since x t T h e r e f o r e the function

(hytw) 1

eK

Y

+€I

1

W

.

t m

L

n = o

c o n v e r g e s u n i f o r m l y in a neighborhood of we l e t y be v a r i a b l e , with p(y)

e

6

5 t xy,

(then W

for every

A€A

. Now

becomes variable). Thus

we define a h o l o m o r p h i c fLhnction in the ball

B

P

( 5 , 6)

= / z CE

s u c h that p ( z - s ) . c 6

1.

E ) i s not contained i n U s i n c e d P ( 5 ,[U) . f and f coi'ncide 2 This on e v e r y neighborhood of 5 contained in B ( 6 , 6 ) n U


Bp(E,

N

.

P c o n t r a d i c t s the a s s u m p t i o n that U i s a d o m a i n of holomoxphy ,

T h e r e f o r e , we proved by a b s u r d the e x i s t e n c e of a o-neighborhood - c Vt i)n uE (s u c. h that , , & V c U. L e t K denote the closed convex hull of K

ir,

E.

gc

is c o m p a c t f r o m Kdthe

[l ] $ 20.6. C l e a r l y

%(uf :?I

in U and E a r e i d e n t i c a l . Since &(U) T h e c l o s u r e s of K is K(U) closed in U i t is c l o s e d in E , h e n c e c l o s e d in k , h e n c e c o m p a c t .

Pseudo-convex domains

354

5

14.2

-

The Levi p r o b l e m

Schauder b a s e s ir. Banach s p a c e s .

14.2.1

-

A Schauder b a s i s in a

Banach space E is a sequence ( e )'O0 of e l e m e n t s of E s u c h t h a t n n = l every x

E

E may be writter. in a ur.ique way

x

n

e

n

n = l

(xn IK = R or C and the s e r i e s c o n v e r g e s in E). We denote by E n the l i n e a r span of the s e t the mapping f r o m E , ,e and by u n n into E defined by :

lei,. .

1

11

n u (x) = n

7-

I

xj ej

j = l

T h e n i t follows f r o m the c l a s s i c a l fact that a Banach space has the B a i r e p r o p e r t y that

( s e e Ktithe [ I 1

9

15. 13).

The Hilbertian b a s i s of a s e p a r a b l e H i l b e r t s p a c e is the m o s t c l a s s i c a l example of a Schauder basis. F o r o t h e r e x a m p l e s and a study of this concept s e e L i n d e n s t r a u s s - T z a f r i r i

[,1 ] ,

Singer [ l ]

.

355

The Levi problem

14.2.2

LEMMA. -

b a s i s and l e t U

E be a c o m p l e x Banach s p a c e with a S c h a u d e r

be a pseudo-convex open s u b s e t o f , E.

A s usual w e

denote by d the d i s t a n c e in E _given by d ( x , y) = /Ix-yII i f x , y e E . We s e t :

Ib

Ism

d e n o t e s a s e q u e n c e of points of

n n=l

T h e n t h e r e e x i s t s a sequence

Ifn

function f

Proof, -

= fn-l

E U n n K n n L n - l , u ~ - ~ ( E'x U) n - ] )

m a y be choser. s u c h that

n

U

p.

'

. Furthermore each

is i t s d o m a i n of e x i s t e n c e ,

We u s e s e v e r a l r e s u l t s of f i n i t e d i m e n s i o n a l c o m p l e x

a n a l y s i s t h a t m a y be found in H o r m a n d e r 'n

IN

, fn E x(Un), such that

'O0

n =1

n- 1

(note t h a t , i f x

U s u c h t h a t f o r e v e r y nE

Kn "n-1

€ 13

. F i r s t we r e m a r k t h a t

i s a Runge c o m p a c t s u b s e t of U

n

s i n c e t h i s l a s t s e t is

Pseudo-convex domains

356

E

!x

Un

n Kn

s u c h t h a t -log d(x,CU) t log 2 lxn ( 5 - l o g ( 2 lien

s e e Hormander [ 1 Hormander [ 1

fl,

1

1

-

th. 4.3.2

I} 1

t h i s s e t i s equal t o i t s P ( U ) - h u l l , 11

def. 2.6.6.

Now we prove the l e m m a by induction on n. L e t u s a s s u m e -1 f 1 e x i s t . If x C Un n u n q l ( U n - l ) we s e t n-

...,

g(xl,.

Let (Un

..,

xn) = f n - 1 ( x I

n

n

Kn

unn(ur,- I 1(

Ln-l)

r

u Un-l

1) '

E 1 i n a neighborhood of

i n 6(U ) he s u c h t h a t I?

J * * * Jn ~-

and (p

5

0 i n a neighborhood of

bLr,-U n-1 ) ) ( t h e s e a r e two disjoint closed s u b s e t s of U 11).We

1

set

h is holomorphic ir, U n

if and only i f

bh =

0 ,

i.e.

8v = x

L e t v E S(U ) be a solution of t h i s equation ( H o r m a n d e r n

-1

n

g

. :(+ .

[ 1 ] 4.2.6).

Since v is holomorphic in a n open neighborhood of the Runge c o m p a c t set Un

Kn

n

Ln-l

( 5 ~o

t h e r e ) , t h e r e exists a w c K ( U ) such that n

( f r o m H o r m a n d e r [ 1 ] th 4.3.2).

F=h t x

n

NOWi f

w = yg

-

, xn

(V

-

W)

357

The Levi problem

then

We c h o o s e : f

n

= F t x GtPxnH n

where G E x ( U n ) is such that

IG(b n

)I

is l a r g e enough and

IG

1

'n nKn i s s m a l l enough ( H o r m a n d e r [ 1 ] th. 4.3.4) : we c h o o s e G s u c h t h a t

and

We c h o o s e f o r H a h o l o m o r p h i c function s u c h that U e x i s t e n c e (solution of the Ldvi p r o b l e m i n C th. 2.5.5 and 4.2.8)

. We c h o o s e F t x

has

lso U

r .

n

E

is its d o m a i n of n see Hormander [ 1 1

> o such that

G t E x

n

H

a s its d o m a i n of e x i s t e n c e and

SUP x E :UnnKn)U ibn

T h e n f,

n

1

[E

xnH(x)

has all the requested properties.

I c- 2 - n - 2

Pseudo-convex domains

358

Let E -

THEOREM. -

14.2.3

be a complex Banach s p a c e with c

a b a s i s and l e t U be a pseudo-convex open s u b s e t of E. Then is a - Udomain of existence of a holomorphic function.

Since E i s s e p a r a b l e t h e r e e x i s t s a sequence ( b ) of n e l e m e n t s of U such that a n y e l e m e n t of b U , i s a l i m i t of s o m e Proof.-

subsequence of ( b ). We m a y a s s u m e that, f o r e v e r y n , bnEUn"K$[En-,. n We consider the sequence (f n ) obtained in l e m m a 14.2.2. We a r e going to prove that f o r e v e r y a t? U t h e r e i s a n r > o s u c h that f o r e v e r y E>O there is a

6 > o such that

a s soonas x

U

n

,y E

Urn,

[ I x - y l l ~ 6and

(i.e. that the function obtained on

Ilx-aI!s r

, Ily-aII< r

U n by patching together t h e

u

functions ( f ) is 13cally uniformly continuous on U f o r the topology n n n of E). T h i s will prove the existence of a holomorphic function f on U defined by f(x) = l i m f ( x . ) w h e r e ( x . ) is a sequence of points in J j + t m "j J

K Un

which c o n v e r g e s to x 6 U ( x is such that x . E U ) j J "j

above a s s e r t i o n we m a y a s s u m e that

/I

1lu 5 1 for every n n L(E)

( s e e 14.2. 1). T h e r e f o r e un(B(a, r)) c B(un(a), r ) the open ball of c e n t e r a and r a d i u s r in E.

and l a r g e enough so that

no t 1

. T o prove the

n En

if

B(a, r ) denotes

L e t u s choose

359

The Leviproblein

and

Iju ( a ) - a l l s g 1 d ( a , [ U ) if

"

n 2 n -1 0

1

If

r 5 .(a) = m i n ( - d(a,[u) , l ) , 8

in

Kn

if n 2 n

Ln-l

(recall u (a) n

B ( a , r ) and

E

( t h e r e f o r e if x

0

u n i f o r m l y continuous in U

n

n Kn 0

a).

u ( B ( a , r ) ) a r e contained n

B(a, r ) p E

. Choose a fixed

p p . ( x ) - f n - l ( u n - l ( x ) )15 2 - " )

I?

, and

r < .(a).

p.

>n

0

f

'

is "0

, h e n c e t h e r e is a

b

o with

0

6 5 r ( a ) - r such t h a t

If,.

0

('n

( y ) ) 15

(x))-fh (u, 0

0

T h e r e f o r e if n , m 5 n

E

.

if

x, y

E B ( a , r t 6 ) and 1Ix-y

I/ < 6 .

0

:

j=n + I

j = n t l 0

n

j = n tl

a s soon a s x

E U , y E Urn, n

Ilx-aIIS r

,

l/y-aII< r

and

I'x-yI'< 6

.

360

Pseudo-convex domains

Now it r e m a i n s to p r o v e t h a t U is the d o m a i n of e x i s t e n c e of f . T h i s c o m e s e a s i l y by a b s u r d f r o m the p r o p e r t i e s of the functions

14.2.4

Remark.

-

A c o u n t e r e x a m p l e d u e to J o s e f s o n [ 3

3

( f r? ).

shows that

t h e Lgvi p r o b l e m c a n n o t be solved in g e n e r a l in non s e p a r a b l e B a n a c h s p a c e s . In s e p a r a b l e B a n a c h s p a c e s the a n s w e r is unknown.

1 4 . 2 . 5 THEOREM. -

Let E

be a c o m p l e x nuclear Silva s p a c e

and l e t U be a pseudo convex open s u b s e t of E. T h e n U i s a d o m a i n of e x i s t e m e .

Proof.

-

F r o m the s t r u c t u r e of E (0.6. 10 and 0.5.8)

E =

lim +

n €IN

E

, then

n

w h e r e the s p a c e s E

a r e s e p a r a b l e H i l b e r t s p a c e s with c o m p a c t n and w h e r e e v e r y c o m p a c t s u b s e t of inclusion m a p s f r o m E n i n t o Z nt 1 So we p r o v e i m m e d i a t e l y the E i s contained and c o m p a c t in s o m e En* of c o m p a c t s u b s e t s of U e x i s t e n c e of a n i n c r e a s i n g s e q u e n c e (Kn)nE IN K F o r e v e r y n EN , t h e r e is a o-neighborhood s u c h that U = r . n E N

u

V

n

.

s u c h that K n + Vn C U

.

F r o m ( 0 . 6 . 2 , p r o p , 2 ) i t follows by p o l a r i t y t h a t : f o r a n y s e q u e n c e (V n )

361

The Levi problem of o-neighborhoods in E t h e r e is a s e q u e n c e p

v =

is s t i l l a o-neighborhood.

( E , p V ) if pv

pn

n

n

Po

s u c h that

Vn

So U is oper, ir. the s e m i - n o r m e d s p a c e

d e n o t e s the Minkowski functional of V .

N o w , f r o m (0.6.11)

E X= El is a n u c l e a r F r e ' c h e t s p a c e and it i s e a s y t o p r o v e that a n u c l e a r F r e ' c h e t s p a c e i s s e p a r a b l e . F r o m ( 0 . 6 . 2 p r o p , 2) t h e r e is a bounded s u b s e t B of El s u c h that

(ElB)

is d e n s e in E'. Its p o l a r in E

is a

zero-neighborhood which c n n t a i n s no s t r a i g h t line. So E a d m i t s a continuous n o r m . So, siiice E t h e r e is a o-neighborhood

i s a n u c l e a r 1 . c . s . , f r o m 0.5.8 ,

W of E , W c V , with E

s p a c e and s u c h t h a t the gauge of W EW

pre-Hilbert W is a n o r m . T h e r e f o r e U is o p e n in

. Now it s u f f i c e s to a d a p t t h e proof of (14.2.3)

in t h e p r e - H i l b e r t

s p a c e E W ( t h e s e q u e n c e (b ) of (14.2.3) is obtained a s i n (14.2.3)). n

14.2.6 R e m a r k . for E

Th. 14.2.5 r e m a i n s valid in m o r e g e n e r a l s i t u a t i o n s

s e e Colombeau-Mujica

[ 3 3 . The L&i p r o b l e m w a s a l s o solved

in F r g c h e t s p a c e s with b a s i s o r in Silva s p a c e s with b a s i s : s e e the Notes of t h i s c h a p t e r .

14.2.7

T h e proof of (14. 2. 3) and (14.2. 5) g i v e s a l s o : L e t E be a

Banach s p a c e with a b a s i s o r a n u c l e a r Silva s p a c e , l e t

n

be a p s e u d o

convex open s u b s e t of E and l e t F be a finite d i m e n s i o n a l s u b s p a c e of

E. T h e n the r e s t r i c t i o n mapping

is surjective.

362

Pseudo-convex domains

9

14.3

T h e Runge a p p r o x i m a t i o n t h e o r e m

In a l l t h i s s e c t i o n E is a c o m p l e x B a n a c h s p a c e with a S c h a u d e r b a s i s and U i s a pseudo convex open s u b s e t of E . W e u s e the n o t a t i o n s of

8

14.2 and w e s e t

14.3. 1 LEMMA. n (K) E IN

1

F o r every compact subset K o f U

s u c h that c A

if n -

2 n (K)

Proof.

1

-

n

.

Since u

r,

c o n v e r g e s t o the identity mapping uniformly on

e v e r y c o m p a c t s u b s e t of U t h e r e is a n l ( K ) C

if n p n (K).

1

t h e r e is a

IN s u c h t h a t

Therefore

and w e know f r o m t h e proof of 13.2.2 t h a t A n is a Runge c o m p a c t s u b s e t of U , hence t h e r e s u l t . n

The Runge approximation theorem

1 4 . 3 . 2 LEMMA. -

L e t n E IN,

t h e r e is a g c K I U n t l )

such that

f c"(U

3.

n -

aiid

E > o

363

be given. T h e n

T h i s r e s u l t follows f r o m t h e proof of 14.2. 2 .

14.3. 3 LEMMA. n (K) 2

If K is a c o m p a c t s u b s e t of U , t h e r e e x i s t s a

-

I

IN such t h a t if n > n ( K ) , i f& 3 0 and i f f

E

exists a g E x ( U ) such that a) b)

g

/un

2

Ex ( U n )

there

= f

Ig-f o u

1

n K

5 e

.

-

L e t n ( K ) be the n u m b e r obtained in 14.3. 1 and i f n>n ( K ) , 1 1 apply ( 1 4 . 3 . 2 ) w i t h E 2 - n and f E x ( U ) : t h e r e is a g C x ( U n t l ) s u c h n 1 5 e 2-n Apply a g a i n (14.3.2) = f and g -f o u I that g n A 1 nt1 '/U n -(&I) and g 1c q U n t l ) . C l e a r l y w e obtain by icduction a s e q u e n c e with E 2 Proof.

I

.

of functions (g,). such that : J JZl

A s i n t h e proof of (14. 2 . 3 ) w e obtain g fx(U) with a l l t h e r e q u e s t e d p r o p e r t i e s , and w e m a y c h o o s e n ( K ) = n l ( K ) 2

.

Pseudo-convex domains

364

1 4 . 3 . 4 LEMMA.

n3(K)

E

-

F o r every compact subset K

U t h e r e is a

IN s u c h t h a t , i f n 5 n 3 ( K ) , t h e s e q u e n c e of s u b s e t s of E

is d e c r e a s i n g , w h e r e E

Proof.

of

-

n

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

Itm

( e1. i - n t l

W e a r e going to p r o v e t h a t , if n 2 n ( K ) , 3

F o r t h i s , if n (K) i s t h e n u m b e r o b t a i n e d in (14.3, I), a e c o n s i d e r 1 a n n > n (K) and a n x E E s u c h t h a t 1 0

W e set

365

The Runge approximation theorem

F r o m (14. 3 . 2 ) applied with f and with a g EX(Untl)

such that

g/u

= f

and

E

=

-a3

Ig-f o u

n

2 a . 3

Furthermore

F o r n large enough, u

(K) c A n t l

n+ 1

F r o m ( 2 ) and ( 3 ) we obtain :

Therefore

we obtain t h a t t h e r e is

, hence

I

n A

U

c-

-3 nt 1

. Therefore

Pseudo-convex domains

366

14.3.5

LEMMA.

-

of U

F o r e v e r y c o m p a c t s u b s e t -K

t h e r e exists .

a a n 4 ( K ) E IN s u c h that

Proof.

-

F i r s t we p r o v e that f o r a n y n E N :

u (K) c U n

If x ( K3C(U)t h e r e is a n f E x(U) with l a r g e enough, un(K) c U n and

f(‘,k))

f/u E

I > 1 1 un(K) K(Un)

,

(f(x) ( >( f

1,

it f o l l o w s t h a t un(x)

Now let u s p r o v e that, f o r s o m e n ( K ) E 4

p.

for n

( b y continuity of f ) . Since u I2( x ) E Un ,

f

n

If

-

. Therefore,

IN ,

> n l ( K ) , ( s e e 14.3, l ) , l e t xo E E be s u c h t h a t

( u (K))qu n n

, i.e.

The Runge approximation theorem

T h e r e is a n N if

> n 1( K )

s u c h t h a t xo

n > N

We s h a l l apply ( 1 4 . 3 . 3 ) with K

u

fi

$ ( u h(K))K(U N

~

/xo

1

367

G3 E N

.

F r o m (14.3.4),

and we c o n s i d e r n 2 n ( K u /x

2

F r o m (1) t h e r e i s a n f E x ( U ) s u c h that

1)

n

We s e t

We a p p l y (14.3.3) with K

u {xo] , n r n 2 ( K

T h e r e is a g EK(U) such t h a t g/

Un

= f

u

/x

I),

E =

a 3

aad f.

and

(a detailed proof of t h i s l a s t inequality is given in proof of (14. 3 . 4 ) ) .

Therefore x

0

k%

(U)

'

I

-C

If K is a c o m p a c t s u b s e t of E w e r e c a l l t h a t w e denote by K the c l o s e d convex hull of K , a Banach s p a c e (Kbthe [ I ] and i f n

E JN (with

II

which i s still c o m p a c t in E ,

s i n c e E is

2 0 . 6 ) . If K is a c o m p a c t s u b s e t of U

l a r g e enough s o that un(K) c U ) ,

we set

:

.

368

Pseudo-convex domains

LEMM4. -

14.3.6

Proof,

-

F o r e v e r y n,

Since T n c S n , un(Tn) c u n ( S n ) .

c o m p a c t convex s u b s e t of E n . t h e r e f o r e u (K) i n En)

n

Since

-

-c c u n ( K ), (un(K)'

. O n t h e o t h e r hand

K c

k'

-C

C o n v e r s e l y u n (K ) is a -C

, un(K) c u n ( K ) , and

is the closed convex hull of

u I? (K)

,

and t h e r e f o r e ,

T h e r e f o r e , i f x E un(Sn),

0

u n ( y ) = x 6 ( u , ( K ) ) ~ (u

n

s u c h t h a t x = u n (y)

.

, and s o y E Sn. T h e r e f o r e x E u ( T ) which

p r o v e s that u (S ) C u,(Tn) n n

14.3.7 LEMMA. -

-C

t h e r e is a n y E K

Let K

n

.

n

be a c o m p a c t s u b s e t of U

K. T h e n f o r e v e r y open neighborhood V K V U ) n u m b e r n(K, V ) E IN s u c h t h a t , i f n 2 n ( K , V ) ,

I\ (Un(K))qu ) n

=v

*

of

K t h e r e is a

369

The Kunge approximation theorem L e t u s a s s u m e by a b s u r d t h a t t h e r e is a n infinite s e q u e n c e

Proof. -

of i n d i c e s n f o r which

T h e r e is a s e q u e n c e (v

tco of e l e m e r k s of U nk)k = 1

such that, for every

k E N ,

So,

v

1.e.

=k

v

ELI

n

nk

(S

nk

= u nk(Ynk)

from t h e s e q u e n c e by

{y nk

(yn ) ''k k

{u

From

) ar.d t h e r e f o r e , f r o m (14.3.6), W'ith YE {y,

k

Ik

ET k

*

E KC ,

Yn

nk

c o n v e r g e s to y

is closed

u

nk

(T

nk

)

,

so we may extract

dn infinite s e q u e n c e - t h a t we still d e n o t e

0

since

,

E kc

. The sequence

:

{T,. k

nk

E

~

(14.3.4) the s e q u e n c e of s e t s

every s e t T

nk

k

- which c o n v e r g e s t o s o m e y 0

ltco k=l

v

I Sk a=J

i s d e c r e a s i n g , and s i n c e

370

Pseudo-convex domains

F r o m (14.3.5),

f r o m the a s s u m p t i o n s on K

u

(yn )

k

14.3.8

3

y

.

But u

nk

(y

nk

) $V

for every k,

E K C V and V is a n open s e t , s o w e g e t a c o n t r a d i c t i o n .

I

Runge a p p r o x i m a t i o n t h e o r e i n . -

s p a c e with b a s i s ,

U a p s e u d o convex o p e n s u b s e t of E and K a Runge

c o m p a c t s u b s e t of U ( i . e . neighborhood of K ,

a g €x(U) __

L e t E be a c o m p l e x Bandch

ifE

K =

% ( u ) ) .If V c U

6

.

> o &f

is a n open

I

x(V) a r e g i v e n , t h e n t h e r e e x i s t s

s u c h that

-

mqu

E

and , For n l a r g e enough, un(K) c V , If o uE-f Ks 7 A CV ) is a Runge c o m p a c t f r o m 14.3.7, (UP)K(U,) n , t h e n by the Runge a p p r o x i m a t i o n s u b s e t of U ; if f = f / v n n n t h e o r e m in finite d i m e n s i o n ( H o r m a n d e r 1 1 ] th. 4.3.2 and th. 4.3.4) Proof.

t h e r e is a g

.

n

6 x(U ) s u c h t h a t n

F r o m ( 1 4 . 3 . 3 ) (and f o r n l a r g e e n o u g h ), t h e r e is a g E Y U ) s u c h t h a t

The Runge approximation theorem

lg

Therefore

,

-

gn

UnlK

E

'7

'

37 1

372

Pseudo-convex domains

6

14.4. I

14.4

An approximation theorem

L e t E and F be two s e r i a r a b l e H i l b e r t

THEOREM. -

with a compact inclusion mapping f r o m F into E. convex open s u b s e t of E w i t h

h a s d e n s e r a n g e , i.e. of R n F -

and if

Let

Proof. -

E

R

pF

f $

if h 6x(n fl F) ,

-

Let 9

be a pseudo--

. T h e n the r e s t r i c t i o n mapDinP

if K

is a compact subset

N

2 0 is given, t h e r e is a n h

Eyl?)

such that

b e a n o r t h o n o r m a l b a s i s of F which is a n

{en :=: \

orthogonal s y s t e m in E ( s u c h a b a s i s e x i s t s f r o m P i e t s c h Cl] th. 8 . 3 . 1). L e t us denote by Pn the o r t h o g o n a l p r o j e c t i o n f r o m F onto the l i n e a r s p a n of

{el,.

such that

.., e n 1.

(1)

i f n z N 1 , Pn(K) c R

(2)

if

x E K (here d

E

(3)

I?

N 2 and N 6 IN 3

There a r e three numbers N

1

> N 2 ’ IIx - Pn ( x ) l & b q d E ( K ,

i s the u s u a l d i s t a n c e i n E ) . if n > N

3 ’

(h(x) - h(Pn(x,)I

<

& y

cE

0) f o r e v e r y

for every x F K

In the s e q u e l N d e n o t e s a n u m b e r l a r g e r t h a n N 1 , N Z , N g ,

3 (z dE(K,CE n))-’

and

sup

x E K

Ilx

.

if n tco n = l denotes

an orthonormal

.

313

An approximation theorem

b a s i s of E

such that

e

and

-1

e

-P IlepllE

]to" P=l

c If

Itrn

n n=l

t h e o r t h o g o n a l projectior, ( i n E ) f r o m E t o the n T h e r e f o r e if x E F and n g N , l i n e a r s p a n of the s e t I f , , , , f n (x) for some n'(n) (2 ( x ) = P,(x) If x E F and n > N , Qn(x) = P n n YE) We der.ote by Q

..

.

with

N 5 n ' ( n ) 5 n.

If

n 2 N

1.

and x E K i t follows f r o m (Z),

T h e r e f o r e , f r o m t h e above c h o i c e of N ,

if

We defir.e K = {xEfl s u c h that l i x l l E ~ n and n and t h e choice of N we obtain :

11

that :

g N and x € K , we h a v e :

dE(x,CE 0)2

1

;I

.From(5)

Pseudo-convex domains

314

is a I l i l b e r t s p a c e and f r o m ( 2 ) . T h e r e f o r e , f r o m (4),

since E

f r o b which it follows t h a t i f n > N

We s e t n 2 N

Rn

0 n Q n ( E ) (Qn ( E ) i s the l i n e a r s p a n of

{f

. ., f n 1).

If

i t follows f r o m (6) and ( 7 ) t h a t :

Now we apply the c o n s t r u c t i o n of l e m m a ( 1 4 . 2 . 2 ) in the s e p a r a b l e H i l b e r t space E,

f o r the b a s i s

the function h

’ON

.

{f \tm and s t a r t i n g a t t h e index N , n n=l

and w i t h

T h u s one obtains a n infinite s e q u e n c e h n ( q n n ) , i f

n > N , with f o r e v e r y n > N

,

’ hntl

hN = h/

nN

= h

I?

4-2 n

a nd

(91

A s i n the proof of th. 14.2.3 we s e t , if x N

h(x) =

lim n e t

E 62

h,(Q,(x)) M

:

375

An approximation theorem N

and h E

Since Q

x(n) .

If x 6 K i t follows f r o m (8) and (9) that

= P

on K ar.d f r o m ( 3 ) w e obtain, if x E K

N

N

1 h (x) - h(x) I c N

e

.

:

CHAPTER 15 THE 3 EQUATION

ABSTRACT.

E b e a compkxleDFN s p a c e a n d 0 be a

THEOREM 1 . -

p s e u d o - c o n v e x open s u b s e t of E of type ( 0 , l ) s bf = F . In o t h e r --__0

-

When

0

,thereexists

n u c l e a r 1. c . s .

THEOREM 2. E

type,

,

(ha )

= E this

c l o s e d differential form

function

f

=nR

such t h a t

in t h e following way

&J

-

b

___j

'

0,1, closed

0 .

:

E be a c o m p l e x n u c l e a r 1. c . s .

bf = F

(n)-

r e s u l t s e x t e n d s t o the c a s e E i s a n y

T h e n t h e r e is a - C

such that

00

00

-

c l o s e d d i f f e r e n t i a l f o r m of type on -

C

a C

w o r d s the foll_owing s e q u e n c e is e x a c t :

- -m ~ (0) 80

. IfFis

03

a n d F 2 Cco

0 , l a n d of u n i f o r m bounded tEe_

function f o n E , of u n i f o r m bounded

.

In o t h e r w o r d s t h e following s e q u e n c e is e x a c t :

316

377

Differential forms

4

15. 1

Differential f o r m and

-

a

operator

If E is a c o m p l e x b. v s the l i n e a r s p a c e of the on E , i. e

x

i

skew-symmetric

p

€IN we d e n o t e by I\

.,P

1,

0,

EE:

. ( a ) i s the s i g n a t u r e of the p e r m u t a t i o n

where

{I,

if

(E) P p - a n t i l i n e a r bounded f o r m s A

and i f

of t h e

0

set

and

i f x . ~ E f o r l s i r ; ; p , if y c E and 1

€~a .

( E ) i s equipped with Ao,p t h e topology of 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 of E , o r with i t s n a t u r a l bornology

15. 1. 1 a

C

a3

,

of equibounded s e t s

D i f f e r e n t i a l f o r m s . - If

d i f f e r e n t i a l f o r m w on 0

,

n

is a

of t y p e

7

.

E - o p e n s e t we

( 0 , p)

,as a

C

a3

define mapping

( E ) . We d e n o t e by 6 ( n ) the l i n e a r s p a c e of P 0, P Cm d i f f e r e n t i a l f o r m s of type ( 0 , p) on Q . 6 (0) i s equipped with 0, P i t s n a t - u r a l t o p o l o g i e s , s e e 4.4. 1. from

into

15. 1. 2

where x 6

A

The

n,

0,

-

a

operator

.-

W e define

a linear operator

yi E E i f 1 s i s p t l , and w h e r e the

h a t on y

k

-

aP

from

means

The i equation

378

that y

k

is o m i t t e d . I n case p = 0 we set g

c o m p l e x valued C valid

00

0,0

( 0 ) a s the

s p a c e of

functions on 0 a n d the above f o r m u l a r e m a i n s

,

By definition, closed i f

-

a n e l e m e n t F of

a F = 0 . We denote by

'

8

0,P

(n)is

s a i d t o be

0,p , c l o s e d P t h e l i n e a r s p a c e of c l o s e d f o r m s . Since t h i s b r i n g s no confusion,

note

a

i n s t e a d of

-

a

P

. We denote

m o r e simply A

(E) by A ( E ) .

0,1

we

309

A review of Hormander's L2 estimates

Q

1 5 . 2.

A r e v i e w of H o r m a n d e r ' s L F i r s t we

basic w = (w

e s t i m a t e s and e x i s t e n c e t h e o r e m s .

r e c a l l th. 4. 4. 1 of H o r m a n d e r

[ I ] , which p l a y s a

.

.

We denote by z = (z , z ) and 1' * . n n , wn ) e l e m e n t s of (T, W e d e n o t e by dh. t h e L e b e s g u e mea-

role in this chapter

...

2

1' n s u r e on (T, 5

.

R

2n

15. 2. 1

THEOREM ( H o r m a n d e r ) 61 be a p s e u d o - c o n v e x o p e n n be a C 2 function on fl a n d l e t c > o be a s u b s e t of (T, , K T n constant such that for e v e r y z , w C ,

T h e n for e v e r y c l o s e d (0, 1) f o r m g on fl t h a t

t h e r e e x i s t s a function u on

-

fl s u c h t h a t a u

satisfies

= g in the

s e n s e of

d i s t r i b u t i o n s and t h a t

1 5 . 2. 2.

R e m a r k : If

n is a C2 function on 0 c(T, which is p l u r i -

$

s u b h a r m o n i c a n d i f tp is t h e f u n c t i o n d e f i n e d by

then

y JI

s a t i s f i e s (1) with c =

1 2

(because j, k = l

i s p l u r i s u b h a r m o n i c , s e e H o r m a n d e r [ 11

4

a 2..a r J

2

0 since

k

2. 6 , a n d s i n c e

The a equation

380

n -

j, k = l

15. 2. 3 . R e m a r k . s(z)= Q

-

a z .J .

If Q

1 z 1 2 - L o g (d(z,Cn))

1

J

azk

k

2

j =1

is a pseudo convex open s u b s e t of

n

C , if

i s a continuous p l u r i s u b h a r m o n i c function on

such that, f o r e v e r y a > 0 , the s e t

'a

= [ z E 0 s u c h that

s(z)

}


i s r e l a t i v e l y c o m p a c t i n fi

Now we r e c a l l a c l a s s i c a l p r o p e r t y of p l u r i s u b h a r m o n i c functions.

15. 2. 4. in

Theorem (Hormander

~ ( n s)u c h

11 th. 2. 6. 3)

that h 2 0 , h(z) = 0 if

d e p e n d s only on

\ z 1I , . . . , 1 znI

i s a p l u r i s u b h a r m o n i c function i n

i s plurisubharmonic, to u ( z ) i f

r

u

E

is

i\ > 1 .

and that

,

and u

Cw where d ( z ,

d e c r e a s e s t o 0 and z E

.-

L e t h be a function

We

assume that h 1 . If u

h ( z ) d), ( z )

#

CQ

-cot then,

)>

E ,

u

(z)

decreases

, and u - u z 0 . E

.

THEOREM ( H o r m a n d e r , a c o n s e q u e n c e of 15. 2. 1) Let 0 n be a pseudo c o n v e x oTen s u b s e t of C , l e t $ be a continuous p l u r i 15. 2. 5.

s u b h a r m o n i c function on

If

n

and l e t

g i s a c l o s e d Coo ( 0 , l ) form of type ( 0 , l ) on

such that

A review of Hormander's L2 estimates

then t h e r e e x i s t s a u E 8

(n)

s u c h that

-

au

= g

38 1

i n t h e s e n s e of d i s -

t r i b u t i o n s and that

proof, have

-

We

apply (15. 2. 4) with $

a function

F r o m (15. 2. 2 . ) , depends upon

E

4E '

We

in p l a c e

of

set

I

VE

and

, a n d thus w e

u

.

s a t i s f i e s (1) with s o m e c z 2 In n a ( E ) ( a ( E1 > 0

na

i s defined in (15. 2. 3 )). If

E

-+

n

0 we m a y

c h o o s e n u m b e r s a(E ) > 0 l a r g e enough s o t h a t n

We may a s s u m e a(E n ) - t

a3

if n

F r o m (15.2. l ) , for e v e r y nEIN

4

R =

un

t h e r e e x i s t s a function u

such that

a u n = g!na(E

tco , s o that

) 11

i n the s e n s e of d i s t r i b u t i o n s a n d that

Oa(E

n on

n)

n a (n)~

R e a equation

382

.

(the second m e m b e r m a k e s s e n s e s i n c e 2 cp) Now t h e r e i s a ta, to3 'En , that we s t i l l denote by { u n 3 , which subsequence of [ u n 3 n=1 n=1 c o n v e r g e s locally i n the weak s e n s e in Q , t o a function which i s 2 locally L i n n. We have

i n the s e n s e of d i s t r i b u t i o n s and

Now u E & ( n ) b e c a u s e e a c h solution u locally L

-

au= g

belongs n e c e s s a r i l y to 8

( s e e Lelong [ 11c h a p t e r 5

(n),

2

in

fl of the equation

since g is a C

0

form

: t h i s a l s o follows f r o m H o r m a n d e r f 11

th. 4.2. 5 and the proof of c o r o l l a r y

4. 2. 6 )

.

383

Integration in Hilberr spaces

6. 15. 3.

A r e v i e w on I n t e g r a t i o n i n H i l b e r t s p a c e s .

In the following s e c t i o n we s h a l l u s e I n t e g r a t i o n T h e o r y a c c o r d i n g to the G a u s s m e a s u r e i n a r e a l s e p a r a b l e H i l b e r t s p a c e H

,

s o we s t a t e

h e r e the r e s u l t s we n e e d ,

15. 3. 1.

The Gauss pro-measure.

on the family 2

-

A system [ p

1

2 /-

L

where

=

of m e a s u r e s

of a l l f i n i t e d i m e n s i o n a l s u b s p a c e s of H is said to

be a p r o - m e a s u r e i f i t c o h e r e n t , i.e.

b L

3

L L€L

YL1

.

If L E 1:

if

L c L i m p l i e s that 1 2

the G a u s s m e a s u r e p

L

on L i s defined by

1

/I 11

d e n o t e s the n o r m i n H I n the d i m e n s i o n of

L and

Lebesgue m e a s u r e o n L =. IRn ( with the s c a l a r p r o d u c t of H )

0

the

.

C l e a r l y , i n the above f o r m u l a , f is a c o m p l e x valued function on L s u c h n From t h a t the second m e m b e r e x i s t s as a L e b e s g u e i n t e g r a l o v e r R

.

the formula

i t follows i m m e d i a t e l y t h a t this f a m i l y

1u L 3 LCS

is coherent

It i s c a l l e d t h e G a u s s - p r o m e a s u r e on H a n d we denote i t by \J 15. 3. 2.

I m a g e of a p r o - m e a s u r e by a l i n e a r o p e r a t o r

,

-

.

L e t T be a

self adjoint continuous l i n e a r m a p p i n g f r o m H i n t o H and let

{u

3 LE

be a s y s t e m of m e a s u r e s on t h e f a m i l y

L e 1 , we define the image

T, L

of

uL

c

.

For every

through T i n the following

way.

F i r s t notice t h a t i f L is finite d i m e n s i o n a l t h e n the orthogonal

The a equation

384

is finite c o d i m e n s i o n a l and t h e r e f o r e T

s p a c e L'

codimensional, s o (T-l(L' )) -1 L' = ( T ( L ' ) ) '

.

-1

(L') i s a l s o finite

is finite d i m e n s i o n a l

.

So if L E d:

t h e orthogonal p r o j e c L tion f r o m H onto L and i f f i s a function defined on L , f o p o T L i s a function defined on L' and we s e t :

E S.

Now i f we denote by p

( T h i s definition i s given i n a m u c h m o r e g e n e r a l s e t t i n g i n B o u r b a k i [ 2 ]

6 . 6. 2 H

/L

; for

u s , s i n c e we

i s i s o m o r p h i c to L')

a r e i n a H i l b e r t s p a c e , t h e quotient s p a c e

.

W e a d m i t ( s e e B o u r b a k i [23 9 6 . 11, T ' Skorohod [ 13 $! 4 ) that i f T i s a H i l b e r t - S c h m i d t i n j e c t i v e self

15. 3. 3.

The m e a s u r e

adjoint o p e r a t o r on H ( T will be e v e n n u c l e a r i n t h e the i m a g e , denoted

by p T , of the G a u s s p r o m e a s u r e

sequel ) , then

, through T ,

is a Radon m e a s u r e on the s p a c e H equipped with t h e weak

gy u

(H, H')

topolo-

.

T h i s , i m p l i e d t h a t t h e c o n v e r g e n c e t h e o r e m s of the c l a s s i c a l L e b e s g u e T h e o r y r e m a i n valid f o r the i n t e g r a t i o n on H a c c o r d i n g t o the m e a s u r e

FT

*

P r a c t i c a l l y , i n o r d e r t o i n t e g r a t e a f u n c t i o n f on H a c c o r d i n g

to the m e a s u r e

( and i n o r d e r t o p r o v e that f i s i n t e g r a b l e ) we T do the f o l l o w i n g . Since T is i n j e c t i v e self adjoint, i t follows f r o m P i e t s c h [I

3

th. 8. 3. 1.

that there i s an orthonormal basis

[e

3

tor)

of n=1 0 since

n

H m a d e of e i g e n v e c t o r s f o r T , i. e. T ( e n ) = 1, e with 1 # n too for) T i s injective S o , i f x = c x e € H I then T(x) = c 1 x e n n n n n n= 1 n: 1 too where [ 1 E e s i n c e T is H i l b e r t - S c h m i d t F r o m (15. 3. 2 . ) , t h e n=l a d m i t t e d f a c t t h a t u T is a measure (we m a y a p p l y F a t o u ' s lemma ) ,

.

.

385

integration in Hilbert spaces

and i f f(x) =

f (

lim

N+t

03

N

c

n=l

xn e n )

- a l m o s t e v e r y w h e r e in H ( f o r i n s t a n c e t h i s i s t r u e i f f i s continuous UT in H ) we a r e l e d t o c o n s i d e r the i n t e g r a l s

Then, f r o m F a t o u ’ s l e m m a , i f t h e s e q u e n c e (I )

1f 1

i s I-I

T

N=l

i s bounded,

- integrable and

We u s u a l l y obtain

f f(x) dv T ( x ) by application of t h e monotone c o n v e r -

H T h e o r e m or of the d o m i n a t e d c o n v e r g e n c e t h e o r e m , a s u s u a l

gence

.A below .

in Lebesgue Integration Theory tation i s given i n (15. 3. 5)

15. 3. 4.

-

The m e a s u r e LJ i s not t r a n s l a T n tion i n v a r i a n t ( a s i s L e b e s g u e m e a s u r e on IR ) , but we have (Skorohod [ i 3 ch. 3

The d e n s i t y of t r a n s l a t i o n .

f i r s t e x a m p l e of a p r a t i c a l c o m p u -

5 . 16

th. 2 ) that i f

z E T ( H ) the

translated measure

(B) = u T ( B - z ) i f B i s a B o r e l i a n s u b s e t of by , J T , z t o uT .vith a d e n s i t y

that we

(defined T, z H) i s e q u i v a l e n t ,J

give explicitely b e l o w . T h e r e is a n o r t h o n o r m a l b a s i s ( e

vectors for T (Pietsch [

11

to3

n

3

n=1

of H m a d e of e i g e n -

th. 8. 3. 1) : T (en) = 1, en a n d 1,

s i n c e T i s i n j e c t i v e and s e l f - a d j o i n t .

#

0

,

The a equation ~

386

If z

,

x € T ( H ) , then < T

a n d is well defined

.

-1

z, T

-1 x > i s the s c a l a r p r o d u c t i n H

Now i f z E T (H) a n d f o r any s u i t a b l e x E H we s e t :

16 t h a t t h i s function of x g H

and i t i s p r o v e d i n Skorohod [ 1 1 chap 3. $

u T - a l m o s t e v e r y w h e r e , f o r e v e r y fixed z cT(H)and t h a t :

is defined

When H i s a complex s e p a r a b l e H i l b e r t s p a c e and s i n c e i n t e g r a t i o n is done in the real underlying s p a c e , w e obtain : 2

f T 15. 3. 5. LEMMA. -

2

11

T -1z \ \ - 2 R e c T

-

2

-

z, T

2

We set

I =

exp (2 R e < T

-1

z, T

-1 x 7 ) dllT(x)

and, f r o m F a t o u ’ s l e m m a , i t suffices t o p r o v e that :

too

If

-1

F o r any z cT(H) the function p T ( z , . ) U

and -

proof.

( z , x ) = e x p t - - 1(

{ en)

n=1

i s a n o r t h o n o r m a l b a s i s i n H I we

set :

-1 x>

n L

2

3.

( uT)

387

Integration in Hilbert spaces

T

T

-1

-1

x =

too

c

n=1 too

c

Z =

n= 1

X Z

n

e

n e

where X

n

n

where Z

n

n

= x f iyn a n d xn, yn fTR

n

= u t i v andu v GIR. n n n' n

F o r e v e r y N E I N we s e t :

We have :

N

n

IN =

i=1

Ai

where

Setting

u.1

=

u.

l

=

-

I

\127i

[ 'R

exp (

- 'j m

j'lR "i=P we obtain

and, if N

- -1

exp (-

2

x2 t 2 i

1

2 2 ( ( xi-2ui) -4 u i ) ) dxi

2

U. X. 1

1

) dxi

j

1 2 2 e x p ( - - x2 t 2 u ) dxi = e x p 2 u 2 i i i '

2 2 A.1 = e x p 2 ( u i t v i ) ,

tcu , f r o m F a t o u ' s

I

lemma ,

lim inf I

n

= exp 2

11 T -1z 11 2

I

The a equation

388

Q

. 15.4.

-

A basic existence result.

Let H be a s e p a r a b l e

c o m p l e x H i l b e r t s p a c e and let T be a n u c l e a r i n j e c t i v e self adjoint o p e r a t o r on H ( t h e r e f o r e

T h a s d e n s e r a n g e ) . We denote b y

the r a n g e of T , equipped with t h e s c a l a r p r o d u c t

H c H T

Let G be a s e p a r a b l e H i l b e r t s p a c e s u c h t h a t H i s contained i n G with c o m p a c t i n c l u s i o n m a p . We a s s u m e t h a t t h e r e e x i s t s a n o r t h o n o r -

mat b a s i s { e 3 f w n n=1 tw

is a t o t a l orthogonal

n=l

n

of H m a d e of e i g e n v e c t o r s f o r T , system i n G

Basic existence l e m m a .

-

Let Q -

s u b s e t of G a n d let F be a c l o s e d C

on Q . f

* on

We

set

T ( e n ) = i nen,

.

> 0 f r o m t h e p r o p e r t i e s of T

15. 4. 1.

.

s u c h that

m

be a p s e u d o - c o n v e x open

d i f f e r e n t i a l f o r m of type (0, 1)

T h e n t h e r e e x i s t s a locally bounded a n d ficitely n H T ( for the topology of H

T

) s u c h that

Coo function

a f*=

F on

Q R HT' Remark.

n

-

It w i l l be p r o v e d i n

5.

15. 5 t h a t t h e function f* is Cm

RHT. The e n d of t h i s s e c t i o n i s devoted to t h e p r o o f of lemma

We denote by d t h e d i s t a n c e i n G, ( d ( x , y) =

11 x - y

I\

)

15. 4. 1.

.

15. 4. 2.

LEMMA. -Th_ere e x i s t s a continuous p l u r i s u b h a r m o n i c function

epon nn

H s u c h that f o r all x E

Q

nH:

A basic existence result

proof.

-

We c o n s i d e r the

S

n

=

389

sets

{ x E R n H such that

(x) s n

3

where

Then

nnH=

to3

u

'n n= 1 c o m p a c t i n G . We set

the s e t s S a r e i n c r e a s i n g a n d r e l a t i v e l y n

M n = SUP XE

Let

x

I( F ( x ) 11

*

A

(H)

be a positive, i n c r e a s i n g , convex function of c l a s s C

such that

1

2 f o r all n = 1,2,.

f o r all

sn

X E

61 n H

.. .

.

x (n-1)

2

log M

2

onR,

n

Then

We s e t cp =

properties ( f r o m Hormander

o

a n d cp

h a s all t h e r e q u e s t e d

[11 th. 2. 6. 7 a n d def 2. 6. 8 , v

p l u r i s u b h a r m o n i c , and continuous, on

n).

is

The a equation

390

15.4. 3.

P r o o f of

set { el,.

. . , en 3

-

15.4. 1.

W e denote by H

n

t h e l i n e a r s p a n of t h e

a n d b y Pn t h e o r t h o g o n a l p r o j e c t i o n f r o m H

H n . We define a mapping

T

n

onto

by :

T

an A

H

~

n

c

i=1

We denote by

N

~e C G

x i zi ei

*

t h e p s e u d o convex open s u b s e t of

'n

n

defined by :

N

W e define a c l o s e d Cm d i f f e r e n t i a l f o r m F of t y p e (0, 1) on n Fn(z). y = F (T if

..,

z E

where

nn

and y c C

\I (Ian

C l e a r l y , if

n

.

We

n

Z)

.

Tn y

set

d e n o t e s t h e E u c l i d i a n n o r m on

n ,

u

ZE

n

N

n

.

We

we have :

T h e r e f o r e , i f z c o n , it follows f r o m

15.4. 2. t h a t :

set

an by

N

:

391

A basic existence result

If

a 2 n denotes the Lebesgue m e a s u r e on IR

2n

-

C n it follows f r o m (1)

that :

N

F r o m (15. 2. 5) t h e e x i s t s a GO3 function f

n

on

nn

N

such that :

We set

and we define a c y l i n d e r function f n on 0,

by :

nn and,

if z c

i f z E 61

n'

Therefore f

We d e n o t e by

,,the

n

is C

00

on

n

and Y E H I

G a u s s p r o - m e a s u r e on the H i l b e r t s p a c e H

( c o n s i d e r i n g i t s r e a l underlying s t r u c t u r e ) , s e e (15. 3. 1 ) and we d e n o t e by

r-

i t s i m a g e by the o p e r a t o r T (15. 3. 2 and

We have :

15. 3 . 3. )

.

-

The a equation

392

T h e r e f o r e , f o r e v e r y nclN

Now l e t u s o b s e r v e t h e following : i f x

~n fl H ,

if

6 (x,

c(n n H))

d e n o t e s the d i s t a n c e in H between x a n d t h e c o m p l e m e n t of 0 n H i n H

,

then

f o r n l a r g e e n o u g h , i f B(x, 1 ) d e n o t e s t h e c l o s e d b a l l i n H of c e n t e r too x and r a d i u s )< We u s e t h i s r e m a r k f o r a d e n s e s e q u e n c e { x 1 n n= 1 in and we d e n o t e by B the above b a l l of c e n t e r x X n n Any B i s c o m p a c t f o r t h e weak topology 0 (H, HI) , t h e r e f o r e u X T n measurable R e c a l l that t h e c l o s e d unit b a l l of t h e H i l b e r t s p a c e

.

.

n

.

2 L (B, 1~ T)i s weakly c o m p a c t , s o a diagonal p r o c e s s g i v e s t h e e x i s t e n c e

of a subsequence

of the

sequence

1

2 ( - v o pn ) ]

too

which i s defined on B X n= 1 n f o r k l a r g e enough (depending on n) a n d w h i c h , f o r e v e r y n G IN , 2 is weakly c o n v e r g e n t i n L (Bx , uT) to s o m e function n 2 E L ( Bx U T ) . We s e t gB X n n {f,.

exp

7

fB Now, i f z c R pL{

T'

let

ned i n s o m e b a l l Bx

n

.

= gB X

n

X

exp

(+I.

n

> o be small enough s o that B ( z , 6

.

)

i s contai-

393

A basic existence result

Let

denotes

the d e n s i t y of t r a n s l a t i o n (15. 3. 4)

by B the above b a l l Bx

a n d by B n

the ball

E

.

B(o,

Then, i f we d e n o t e C )

:

with

.

Since p (z, ) exp of f

5 E L2 (

z tB

€

, u,)

,

a

-0

if k

+

t m by d e f i n i t i o n

B '

2

2

From (3) :

F o r k l a r g e enough the functions ep o z +B

E

,

i f k -.,

a r e u n i f o r m l y bounded on k t h e r e f o r e f r o m t h e t h e o r e m of d o m i n a t e d c o n v e r g e n c e , a -+ o

+ OD.

Therefore :

The a equation

394

when k

-

m .

Now we c o n s i d e r t h e C of C

00

f u n c t i o n on t h e o p e n u n i t b a l l A(0, 1)

defined by

if x E B E .

T h e Cauchy i n t e g r a l f o r m u l a f o r Cc0 f u n c t i o n s ( H o r m a n d e r

[ 1 ] th. 1.2. 1) g i v e s :

(5)

fn(z)

J

2h

-.

i e db 2n ie i0 dr d6 fn(ztxe t2.r ,f Eifn(ztrxe ).xe 2h

0

0

0

Now w e i n t e g r a t e in x on t h e b a l l B&

and f o r t h e m e a s u r e p T : from

t h e r o t a t i o n i n v a r i a n c e in ( 1 5.4.4) below, w e o b t a i n :

t h a t w e m a y w r i t e , i f z E 0 n HT

:

1

395

A basic existence result

F r o m (4)and

( 6 ) it follows t h a t the s e q u e n c e (f a ) kEIN ‘Onverges

pointwise on fl

n HT

to a function denoted by f

\

in t h e s e q u e l .

Now l e t u s c o n s i d e r a closed ball B1 of HT, contained in S? , 1 and such t h a t , f o r s o m e & 3 0 , B t B is contained i n s o m e ball B & X n 1 of t h e s e q u e n c e u s e d above. F r o m ( 6 ) it follows t h a t if z E B ,

1

F r o m (3) :

F u r t h e r m o r e it follows f r o m l e m m a (15. 3. 5) t h a t

is bounded u n i f o r m l y in n

EN

and

E

.

*

.

a r e bounded u n i f o r m l y i n B

T h e r e f o r e t h e functions f

A s a consequence, f

B

1

The last integral in the 1 second m e m b e r of ( 7 ) is bounded u n i f o r m l y in n E IN and z f B z

‘k is bounded on B1 and t h e r e f o r e f

*

.

1

is l o c a l l y

-

The a equation

396

bounded on

0

n HT

( f o r t h e topology of H )

T

Now w e p r o v e that f

*

d i m e n s i o n a l s u b s p a c e of H

i s finitely

If T' from ( 5 ) that the r e s t r i c t i o r . of f

of

Ji(n

?)( f * F

'n

nL),(f ) "k/n nL

'n nL

)= F

/hi r7L

-

*

f+ '0nL

C"

. Let

L be a g i v e n finite

n n L , i t follows i m m e d i a t e l y

z

to 0

n

L is continuous. In t h e s e n s e

and t h e r e f o r e f r o m ( 2 ) we have

in D n L and i n t h e s e n s e of

& ( n n L ) . Since

is Co3 i t follows f r o m t l z e hypoellipticity of the

"L

finite d i m e n s i o n ( s e e t h e end of ( 1 5 . 2 . 5 ) )

15.4.4

that f

Rotation i n v a r i a n c e of s o m e i n t e g r a l s .

5 and if f is a n i n t e g r a b--__le cylinder

f o r any z E H

Proof.

.

-

/nr L

is

o p e r a t o r in

cO'

.

In the notations of t h i s --

function defined on H

I?

,

we have :

-__

.

TC(X

and we s e t i f

-

*

8

1+iy1 ) e 1+ . . . + ( xn +iyn )e n 3 = X 1(x 1t i y 1) e 1t . . . + X n( x n + i yn ) en

1 = ( 1) *

...,An),

By definition t h e second m e m b e r of t h e equality ( 1 5 . 4 . 4 ) i s :

A basic existence result

397

dx

We d o t h e change of v a r i a b l e X and w e have x

RtY.E

= X

the t r a n s f o r m a t i o n ( x

Integration in

I - s R2n

2

a

t i Y

a

a dy k.‘

= (xA t iya ) e

i4

( f o r a n y fixed

t Y 2 and t h e a b s o l u t e v a l u e of the Jacobian of

e

1

y

a

1515 n

+

(

x

~

p

y ) A 14t
is 1

. Therefore

8 gives :

foT(XtiY 1 1’“”

and t h e r e f o r e I

i.1 ( f ) T

.

n

X n t i Y n ) n e x p { - -2 ( X R t Ya ) a =1 1

2

2

1

dXR d Y Q

e)

398

The bequation

0

15.5

An hypoellipticity r e s u l t

15.5. 1 Basic hypoellipticity l e m m a . s p a c e and l e t F

_

_

I

_

B(o, a ) -of E ,

~

k z

C

00

-

&At E

be a complex Banach

differential f o r m of type ( 0 , l ) on a n open ball

which is bounded on B(o, a) together with i t s s u c c e s s i v e

be a finitely d e r i v a t i v e s . L e t f --

Co3 function on B(o, a ) which is

bounded on B(o, u) and such that z f = F on the i n t e r s e c t i o n with B(o, a )

--

of any finite dimensional subspaceof E . Then f is C

m

on

B(o,B)

b f = F.

15.5.2

If EIR denotes the r e a l space a s s o c i a t e d

Proof of 14.5. 1 . -

to E we a r e going to prove that f ' ( x ) 6 L(E with

p< a

, the s e t

1

i f ' ( x ) l,xll..g
IR

) and that f o r any @ > o

i s a bounded s u b s e t of L ( E

IR)

We have :

f ' ( x ) . y = 6 f ( x ) . y -t b f ( x ) . y

where

The set

j

5 f(x) 2

is bounded in L ( E IlXIl-=~

to prove a majorizatior. of

bf(x). We s e t :

IR

) s i n c e s f = F.

It r e m a i n s

399

An hypoellipticity result

and w e c o n s i d e r t h e

C

m

function

+ ( z ) = f(x

if

p

IIxIIs

denote by

ad

l l y I 1 S 1.

If

r'<

on h ( o , g - 8 ) defined by

f

zy)

F L - ~,

w e s e t A = A ( o , r ' ) and w e

b h its bouzdary o r i e n t e d in t h e positive s e n s e . Cauchy's

integral formula for

Coo functions ( H o r m a n d e r [ 1

1th

1.2.1) g i v e s :

thus

F r o m Stokes formula :

Therefore

2 i n

From t h e boundedness of f , an M Irp(t)

I

>

o d e p e c d e n t on r ' ,

, [ _bcD ( t ) 1 and bt

1- b 2cp bt b t

dt t

dt CLli

dtAdt

bt

F and its d e r i v a t i v e s

OR

B(o,

but independent o n x a n d y

8) ,

t h e r e is

, such that

( t ) ' a r e bounded above by M i f t

E E.0

1.

The a equation

400 or t €

4. T h e r e f o r e

T h e r e f o r e the s e t

is bounded i n L ( E

R

) , T h e end of t h e

proof is a n i m m e d i a t e induction on the o r d e r of t h e d e r i v a t i v e : w e s e t ( n - 1) g = f and we p r o v e t h a t {g'(x).y 1 i s bounded i n L((n'l)E ) when / / x j ( s p < e t and

IIy 1 1 s1

15.5. 3 COROLLARY. function f

3(

. The computations a r e the same.

JR

It follows i m m e d i a t e l v f r o m 15.5. 1 t h a t the

obtained in 1 5 . 4 . 1 i s C

03

on 0 n H

T '

40 1

Scale of Hilbert spaces

$

The b

15.6

equation in a s c a l e of H i l b e r t s p a c e s

T h e r e s u l t of t h i s s e c t i o n is a c o n s e q u e n c e of t h e e x i s t e n c e r e s u l t 15.4. 1 and of t h e hypoellipticity r e s u l t 15.5. 1

. In t u r n it w i l l be t h e 0

basic f a c t in the p r o o f s of the r e s o l u t i o n of the

o p e r a t o r given in

the following s e c t i o n s . Let

be a n i n c r e a s i n g s e q u e n c e of t h r e e s e p a r a b l e H i l b e r t s p a c e s with n u c l e a r i n c l u s i o n s . L e t R be a pseudo-convex open s u b s e t of H a closed C

and l e t F be 2 d i f f e r e n t i a l f o r m of type (0, 1 ) on 0. A s u s u a l we e q u i p

m

0 n Ho with the topology induced by H 0 -

15.6.1 LEMMA. that -

b f * = F on

Proof.

-

There exists a

n

I

-

pH

C

03

function f

7y

R nH

0

such --

0 '

Hi Go d e n o t e s t h e c l o s u r e of H i n H . , equipped w i t h 0

If

the s c a l a r p r o d u c t induced by t h a t of H. , w e have t h e i n c l u s i o n

Ho c

HI

fi0

-

H2

c Ho

and t h e s e i n c l u s i o n s a r e n u c l e a r with d e n s e r a n g e ( a s t h e c o m p o s i t i o n - H2 p r o d u c t of a n u c l e a r mapping and a p r o j e c t i o n ) . We s e t G = H

A n u b l e a r mapping is of type Q 1 ( P i e t s c h Pietsch [ 1 ] i s of type

11

th. 8 . 2 . 7 ,

1'"

t

1

3

0

the composed i n j e c t i o n U ,

form H

0

to

C ,

. F r o m the S p e c t r a l D e c o m p o s i t i o n T h e o r e m ( P i e t s c h

] th. 8 . 3 . 1) and f r o m P i e t s c h [ 1 ] th. 8 . 3 . 2 ,

mapping U ,

.

th. 8 . 3 . 3 ) , s o from

applied t o t h e too +a3 there exist 2 orthonormal bases (el ) n n = l and ( f n ) n = l

The a equation ~

402

pf H o and G r e s p e c t i v e l y s u c h t h a t , f o r e v e r y x E H

0 '

too

ha

n

<x,el

z

n Ho

f

n

n = l

Since

u

is injective ,

Atn

#o

f o r e v e r y n. We s e t

We have

and

We s e t :

2 fa3

H

=

{

x n er, E G s u c h t h a t (xn )too n=l E Q21

n = l

equipped with the s c a l a r p r o d u c t

< Z xn e n , C y n e n >H

Cx

n

*

,

403

Scale of Hilbert spaces

Therefore H

iB

tw a H i l b e r t s p a c e with ( e ) a s an orthonormal basis, n n=l

t h e n a t u r a l inclusion f r o m H into G is n u c l e a r . Now l e t u s define t h e operator

T on H by :

T ( e ) = h e n n n tm

1

T is n u c l e a r s i n c e ( A ) 6 l. n n=l r a n g e . Since e '

r,

=

n

e

n

and

(e'

T is i n j e c t i v e , self adjoint, with d e n s e tm

n)n = 1

we c h e c k i m m e d i a t e l y t h a t E T = H o

15.4. 1 and 15.5. 1.

.

.

i s a n o r t h o n o r m a l b a s i s of H

0'

N o w l e m m a 15.6. 1 follows f r o m

-

The a vquation

404

$ 15.7

E x i s t e n c e of

C

00

open s u b s e t s

-

15.7.1 T H E O R E M .

of D F N s p a c e s

L e t E be a c o m p l e x n u c l e a r Silva s p a c e and R

be a ps~!u!o convex open s u b s e t of E. d i f f e r e n t i a l f o r m on solution of

Ef = F

.

s o l u t i o n s in pseudo-convex

R

.

Let

F be a c l o s e d C

Then there exists a

C

Q3

cn

(-0, 1)

function f c n 0

In o t h e r w o r d s the following is a n exact sequence_ of F r C c h e t Schwartz s p a c e s :

P roof.

-

Since E

is a n u c l e a r Silva s p a c e i t is t h e inductive l i m i t of

a n i n c r e a s i n g s e q u e n c e of H i l b e r t s p a c e s E

with a n u c l e a r i n j e c t i o n n We m a y a s s u m e t h a t , f o r e v e r y n ,

-

f o r e v e r y n ( s e e 0.6.10). n Entl t h e inclusion of E into E m a y be f a c t o r i z e d a s follows : n nt1

E n = H0, n

-

H l,n

HZ,n = E n t l

-+

w h e r e a l l t h e s e mappings a r e n u c l e a r i n c l u s i o n s and w h e r e t h e s p a c e s

H.

a r e s e p a r a b l e H i l b e r t s p a c e s . It is e a s y t o p r o v e t h a t ( i f E i s any n Silva s p a c e ) t h e r e e x i s t s a n i n c r e a s i n g e x h a u s t i v e s e q u e n c e of c o m p a c t 1,

subsets K

of

R ,

w h e r e we m a y a s s u m e t h a t K

In the s e q u e l of t h e proof we s e t R(n) =

R nE

n

.

n

is compact i n

Now we c o n s i d e r t h e r e s t r i c t i o n of F t o 0 n E l e m m a ( 1 5 . 6 . 1) t h e r e e x i s t s a on

n(n) ,

Co3 function f

-k

n

nt1

En

.

. From

on O(n) c E

n

s u c h that

405

Existence of C a, solutions

Ff

ji

n

=

F

.

In o r d e r t o s t a r t a n induction w e s e t f

0

-

9

f2

2+

. f 3 -f 2 %

is defined and

C

00

(f -f ) = o on n(2), t h u s f 3 - f 2 is h o l o m o r p h i c or! R(2). 3 2 F r o m t h , 1 4 . 4 . 1 this h o l o m o r p h i c fur,ction m a y be a p p r o x i m a t e d u n i f o r m l y on n ( 2 ) .

OR

jc

K2 by h o l o m o r p h i c f u n c t i o n s on 'i-2

holomorphic function P

2

Ex (nn E 3 )

n E3'

Therefore , there is a

such that:

$(

sup (f3(x)-f ( X ) - P , ( X ) 2 x E K2

I

.

1 522

We s e t

f

3

= f

Y - P 3 2

and w e h a v e :

i s Ccn on P ( 3 ) = R

f3 if

3

= F

on ~ ( 3 )( s i n c e

1 If3(x)-f2fx) 5 2 SUP x E K2 2

I

rE3

SP 2

Df

n

= F and

on n ( n )

0)

*

By a n obvious ifiduction w e obtair, a s e q u e n c e

n ( ~ )s u c h t h a t

=

(f ) of n

C

00

f u n c t i o n s on

-

The a equation

406

F o r every

Thus

for

x f 61

fn(x)

t h e r e is s o m e

i s defined f o r

n

l a r g e enough s u c h t h a t xE K nc n ( n ) . l a r g e enough a n d n

l a r g e enough. We s e t

n

f ( x ) = lim fn(x) n + too We notice that .t, -+ to3

-

b(fnta-

on

fn) = 0

- fn)

, the functions

(fnt.t, s i n c e e v e r y c o m p a c t s u b s e t of

-

n(n) , t h u s

fn+a a r e convergent t o

n(n)

is c o n t a i n e d i n

l a r g e enough. T h e r e f o r e

f

n f

K

P

E x(Q(n)). When

-

f

in K(n(n)) n , f o r some p

f = (f-f ) t f n n

is a

on af

Co3

n .

function o n

Furthermore

n(n) f

-

.

Since t h i s h o l d s f o r a n y

fnET(61(n))

and

-

bf =

n

F

n , f on

is

Cm

O(n) , h e n c e

= F on n(n), i . e .

-

b f = F

on

0 .

m

Existence of C a, solutions

$ 15.8

15.8.1 - THEOREM. be a c l o s e d

Co3

Existence_&

Let

E

Co3

407

solutions in nuclear

be a c o m p l e x n u c l e a r A . c . s .

e .c . s .

and l e t

F

( 0 , l ) d i f f e r e n t i a l f o r m of-uniform bounded type o n

( s e e D e f . 1.6.1). T h e n t h e r e is a bounded t y p e , s u c h that b f = F

function

Ca

on

f

.

E

, of u n i f o r m

E

In o t h e r w o r d s a n d with obvious n o t a t i o n s , t h e following s e q u e n c e isexact -

15.8.2

-

:

R e m a r k : If

E

i s a Silva s p a c e ,

t h e n , from T h e o r e m 1 . 6 . 3 , a n y

Cm

A(E)

E

mapping f r o m

of u n i f o r m bounded t y p e . T h u s we r e c o v e r t h e c a s e

into

n= E

-

R e m a r k : E v e n if

E

(Theorem

F

i s of u n i f o r m bounded type i s

i n d i s p e n s a b l e in o r d e r to e n s u r e t h e e x i s t e n c e of a solution

of

= F . M e i s e - V o g t 1 3 1 proved t h a t t h e

5

equation is not

ble i n n u c l e a r F r e ' c h e t s p a c e s even with continuous n o r m s

15.8.4

is

is a n u c l e a r F r g c h e t s p a c e we s h a l l s e e

in 1 6 . 2 . 3 t h a t t h e a s s u m p t i o n t h a t

-

A(E)

as a p a r t i c u l a r c a s e of T h e o r e m 15.8.1.

15.7.1)

15.8.3

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

-

Proof of T h e o r e m 15.8.1 :

Since

F

of

always resolu-

.

is of u n i f o r m bounded type

( 1 . 6 . 2 ) t h e r e is a convex b a l a n c e d o-neighbourhood the following d i a g r a m is c o m m u t a t i v e :

fEB(E)

V

in

E

such that

The a equation

408

where *

E 'v

V

I

, and w h e r e

F'8b;0,1;

E

denotes the n a t u r a l injection f r o m M

*

%; 0 , l ;c l o s e d ( E v )

Fc c l o s e d (Ev )

into i t s completion

V

i s the continuation of

.

T h e r e a r e convex balanced 0-neighbourhoods in

E

v0 c v 1 c v 2 = v such that the n a t u r a l inclusions E

i L

V

E

i

V

1

1 -E

a r e n u c l e a r mappings and t h a t the s p a c e s

V

E

s p a c e s . P a s s i n g to the completions we have

a r e separable pre-Hilbert

Iz

h

c l ? +i 0 e V d i 1k v .

E V

If t h e mappings

V.

0

+ 1 (k=O,1) k

V

1

a r e not injective ( i . e . i n c a s e

E

d o e s not

a d m i t any continuous n o r m ) we c o n s i d e r quotient s p a c e s . T h e n it suffices t o apply l e m m a 15.6.1.

I

CHAPTER 16

SOME APPLICATIONS OF THE

ABSTRACT

-

2 EQUATION

F i r s t we state and p r o v e t h e e x i s t e n c e of a solution of

t h e first Cousin p r o b l e m , in a pseudo convex open s u b s e t of a D F N space,as a s t r a i g h t f o r w a r d c o n s e q u e n c e of t h e e x i s t e n c e of C of t h e

-

a

equation i n t h e s e s p a c e s

03

solutions

.

T h e n we e x p o s e a c o u n t e r e x a m p l e proving t h a t t h e f i r s t Cousin p r o b l e m may have no solution i n s o m e F r e ' c h e t s p a c e s ( s u c h a s am) a n d 00 solution i n t h e s e that t h e r e f o r e t h e a equation h a s i n g e n e r a l no C spaces. T h e n u s i n g t h e r e s o l u t i o n of t h e

-

a

equation a n d t h e d i v i s i o n of

d i s t r i b u t i o n s by holomorphic functions, we p r o v e t h e following r e s u l t which g i v e s s o m e knowledge of t h e s o l u t i o n s of t h e h o m o g e n e o u s convolution e q u a t i o n s i n the s p a c e E x p E when E is a c o m p l e x n u c l e a r 1. C. s. THEOREM.

-

If

E is a complex nuclear

lution o p e r a t o r on t h e s p a c e E x p E

tor -

@

.

p

@

is a convo-

, t h e n a n y solution of t h e homogeneous

equation @ f = 0 is t h e F o u r i e r t r a n s f o r m that pU = 0 , where

1.c. s. a n d i f

.

of a n e l e m e n t U €4' ' ( E ) s u c h

d e n o t e s t h e c h a r a c t e r i s t i c function of t h e o p e r a -

Some applications

410

5 16. 1. 1.

14. 1

THEOREM.

Solution of the first Cousin p r o b l e m .

-

-

n

E be a c o m p l e x DFN space,

convex open s u b s e t of E

& { n .) '*

trn

n = u

n j . If

j = l

S.

J*k

open s u b s e t s of 61 s u c h t h a t

j=1

J

( Q j n 0,)

EX

a pseudo-

wherej,k=1,2,..

and if, f o r a l l i, j, k,

'j, k =

gi, j

+ gj,k

f

gk, j

J

nk9

blin n . 0

gk,i = 0 in

x ( a .)

E

then one can find functions g . J

-

J

s u c h that, f o r all j and k

gj, k = gk

-

gj

in

n

Rk

fl

,

.

For s h o r t we c l a s s i c a l l y s t a t e t h i s r e s u l t by s a y i n g t h a t t h e first Cousin s e t (g. ., 1s J

p r o b l e m i n 0 h a s a s o l u t i o n , and the Cousin data Proof.

-

.

ni)

i s c:rlled a

I t i s a d i r e c t consequence of t h 5. 3. 1. on e x i s t e n c e of C

p a r t i t i o n s of t h e unity i n solutions of t h e

-

a

a

equation

15. 7 . 1. on e x i s t e n c e of

a n d of the

.

F r o m t h 5. 3. 1. a n d from 5. 3. 4.

t h a t supp cp

I Ic

ai .

..

F o r e v e r y k = 1.2,..

hk =

t v =1

Fp,

.

f

v=l we c h o o s e i G I N

.

we s e t

fa3

V

there

J

3

l i s t e d i n th. 5. 3. 1. F o r e v e r y v = 1, 2 , .

a3

, that j=1 with t h e p r o p e r t i e s

e x i s t s a p a r t i t i o n of unity s u b o r d i n a t e t o t h e c o v e r i n g is t h e r e exists a s e q u e n c e of functions f

C

00

gi , k V

U

such

41 I

Solution of the first Cousin problem

( t h i s s u m i s locally finite) and h E (B (61 )

k

since g k,i

in 0 . n J

v

’

nk .

gi

v

= 0

, j “j,k

.

k

.

F u r t h e m o r e w e have :

This implies that

T h e r e f o r e t h e r e is a

c

03

c l o s e d ( 0 , l ) f o r m ~r on 61

k

s u c h that

in

nk

f o r e v e r y k = 1, 2 , .

function

in 0

.

..

F r o m t h 1 5 . 7. 1.

u € 8 (61) s u c h that

Now t h e functions g k = h k t u

have all t h e r e q u e s t e d p r o p e r t i e s

.

there exists a

Some applications

412

8 16. 2. 1.

A counterexample.

16. 2 .

Very strongly convergent

sequences.

-

-

If E i s 1.

C.S.

f e n 1+0° of non z e r o e l e m e n t s of E is s a i d t o be v e r y s t r o n g l y n=1 I,] '0° of ( r e a l o r c o m p l e x ) n u m b e r s , c o n v e r g e n t if f o r e v e r y s e q u e n c e n=1 tm t h e sequence { X e 3 i s a null s e q u e n c e i n E n n n=1

a sequence

.

E

Example. -

am

and e

11

=

(0,

, 0,1,

.. ) .

0,.

M o r e g e n e r a l l y any

order n Fre'chet s p a c e without continuous n o r m h a s v e r y s t r o n g l y convergent sequences. 16. 2. 2. PROPOSITION.

-

E be a c o m p l e x 1. c. s. which a d m i t s a

U be a c o n n e c t e d open

v e r y s t r o n g l y c o n v e r g e n t sequence, a n d l e t

s u b s e t of E

.

Let

(gi, j, U i ) be a Cousin d a t a on U w h e r e t h e

, a r e h o l o m o r p h i c and continuous i n U . ( s e e 16. 1. 1. ) J T h e n the Cousin p r o b l e m is not a l w a y s s o l v a b l e (with functions g.

functions

g.

1,

holomorphic a n d continuous o n R .) 16. 2. 3.

.

If f u r t h e r m o r e U is a Lindelof s p a c e a n d E a

Remark.-

n u c l e a r 1. c. s.

then we p r o v e d in 5. 3. 1.

t i o n s of unity. T h e r e f o r e , f r o m t h e proof is not solvable i n U

16.2.4.

P r o o f of

.

t h e e x i s t e n c e of C

of

16. 1. l . , t h e

(in t h e s e n s e n e e d e d i n t h e proof of

16. 2 . 2 .

-

-

a

m

parti-

equation

16. 1. 1. )

.

We denote by U a c o n n e c t e d open s u b -

t h e l i n e a r s p a n of { e l , . . . , e n } ( w e a s s u r e t h a t n t h e points e a r e l i n e a r l y independent) a n d by F a d e c r e a s i n g s e n n quence of c l o s e d l i n e a r s u b s p a c e s of E s u c h t h a t F i s a topologin c a l c o m p l e m e n t of E . We set n s e t of E , by E

U 2 = ({ a e l + w )

aE

and (Im a ) c

w EFl

11 4

fl U ) .

413

A counterexample

If n > 2 w e set

f un 3

too

i s a n open c o v e r

n=1

of U

.

If

1 n-ml #

1 , t h e n U n fl U m =

$,

and

n z 2 , any

For

Z E E m a y be w r i t t e n i n

a unique way

n

z= c

p=1

U

n

P

t w

E Fn .

with w If n , m

ap(z) e

2

2 w e define a continuous h o l o m o r p h i c function g

fl Urn

Clearly

by :

gn, m = - gm, n

f o r e v e r y n, m and g

Cousin p r o b l e m h a s a

solution i n U m a d e of continuous h o l o m o r p h i c functions f o r e v e r y n and m

, gn, m = g m

in Un

n

Um

.

Let u s notice that a ‘

n

-

(2).

on

.t gj, kt gk, i= 0 f o r e v e r y i, J

i , j a n d k . L e t u s a d m i t by a b s u r d t h a t t h i s i. e.

n, m

g

n

gn em= 0 i f m

#

n and

in U

n

,

Some applications

414

a In(z) e = 1 for e v e r y n and e v e r y z c E n

we

If z c U n and I m a l ( z )

#

n

)

.

set

We h a v e , i f n

#

1,

Furthermorethereis

(if

.

not

1

a

(2)-

1

a w

n

would be a holomorphic

i n

Now, l e t us notice t h a t

if z E U

and that t h e r e f o r e f o r m u l a function f

E U n c U s u c h that

n

n

n um

(1) defines a continuous holomorphic

on the set

( z G U such that which i s a connected open f r o m ( 2 ) and l e m m a

16. 2. 5.

function of z E U

-

subset in U

16. 2 . 5 below

a,(.)

.

#

n for every

n >1 )

Now the c o n t r a d i c t i o n follows

.

L e t E be a c o m p l e x 1. c. s. with a v e r y sJronglyconvcz too gant_-sequen-c3-- e n ] a n d l e t U be a connected open s u b s e t of E n=1 Then i f - f i s a continuous holomorphic function on U , t h e r e

LEMMA.

.

415

A counterexample

is a n a t u r a l i n t e g e r N s u c h t h a t

f'(z)

.

en = 0

for every Z C U and every proof.

-

n > N

F r o m C a u c h y ' s i n t e g r a l f o r m u l a 2. 2. 7, f o r e v e r y XoE U V in E

t h e r e i s a o-neighborhood family

( f'(x)]

is a n equicontinuous

XFX - t V 0

F o r every sequence {

),

t h e r e f o r e the s e t [ f ' ( x )

such that X

n ),

1t o o e

n

}

t V c U and the

s u b s e t of I ( E ) .

, the sequence { 1

n=1

n

0

i s bounded i n XEX t V

n

a.

too

n=1

i s null,

This implies

0

n

€IN

that f ' ( x ) e n = 0 f o r e v e r y x E x t V

U i s c o n n e c t e d , t h e uniqueness the p r o o f .

16.2.6

i f n is large

enough

.

Since

of h o l o m o r p h i c continuation c o m p l e t e s

I

Remark.

-

T h e a b o v e c o u n t e r e x a m p l e is valid i n a l l n u c l e a r

Fre'chet s p a c e s without continuous n o r m . A d i f f e r e n t c o u n t e r e x a m p l e , valid in s o m e F r e ' c h e t s p a c e s w i t h continuous n o r m s w a s r e c e n t l y obtained i n Meise-Vogt [ 3 ]

.

Some applications

416

4

16. '3.

On t h e solutions of s o m e homogeneous convolution e q u a t i o n s

We c o m p l e t e t h e study of t h e convolution e q u a t i o n s i n E x p E , when E i s a complex n u c l e a r

1. c. s . , by proving

t h e homogsneous convolution e q u a t i o n s a complex n u c l e a r b.

V.S.

.

A s in

5

of t h e s o l u t i o n s of

10. 1 , ue denote by F

( s e p a r a t e d by its d u a l ) and we denote by

convex b a l a n c e d open set i n F

sxk(n),

a property

.

If

@

na

i s a convolution o p e r a t o r on

10. 1. 4, s e e a l s o 10. 3. 6 , that t h e r e i s s o m e t s u c h t h a t (9 = T * , t h a t p = ;F ( T ) ExS(n) i s c a l -

we r e c a l l f r o m

T € ( zK$(n))'

l e d the c h a r a c t e r i s t i c function of Q , a n d t h a t t h e following d i a g r a m is c o m m u t a t i v e

where, i f E

ExFC'S(n) , U ( e ) cK'S(n) i s U(e)

(i. e. ( p .

e )(cp)

=

e

defined by t h e f o r m u l a :

p.e

(p c p ) f o r any ~p E ~ ~ ( Q .) )

In t h i s s e c t i o n w e a r e g o i n g to p r o v e :

Homogeneous convolution equations

26. 3 . 1. Q

THEOREM.

-

Let -

r a t o r on 3

~

S

('n )

.If

Is(n)

-

R e m a r k and Comments

p a r t i t i o n s of unity i n m a n y bornology of of

6

I (

If

and let

i s a convolution ope-

function,

of the h o m o g e n e o u s e q u a t i o n 8 f = 0 -i-s

t h e F o u r i e r t r a n s f o r m of a n e l e m e n t U E & '(

16. 3 . 2.

(g

p c K s ( n ) denotes i t s c h a r a c t e r i s t i c

and i f

then any solution f E 3 X

F be a c o m p l e x n u c l e a r b. v. s.

open s u b s e t of F

b e a convex b a l a n c e d

417

.-

usual

n)

such that

pU = 0

.

F r o m th. 5. 3 . 1. t h e r e exist C 1. c. s.

.

If

00

F is t h e Von N e u m a n n

s u c h a 1. c. s. one may define t h e s u p p o r t of a n e l e m e n t

0 ) a s i t i s c l a s s i c a l l y done i n t h e finite d i m e n s i o n a l c a s e

Then t h e p r o p e r t y t h a t

pU = 0 i m p l i e s t h a t t h e s u p p o r t

.

of U is c o n t a i -

n e d i n t h e c l o s u r e ( f o r t h e topology of t h e 1. c. s. i n which w e h a v e p a r -1 p ( 0 ) of z e r o e s of p , t i t i o n s of unity ) of t h e s e t

the s e t

In t h e one d i m e n s i o n a l c a s e and i f p f 0 , i. e. Q # 0 , t h e n -1 p ( 0 ) i s m a d e of i s o l a t e d p o i n t s and t h e n th. 16. 3 . 1 h a s a

v e r y p r e c i s e and f a m i l i a r f o r m : i f f C x ( C ) and i f convolution o p e r a t o r on Exp

of p

and f o r e v e r y k = 1, 2 , .

tiplicity

o r d e r of z

mensional

cgf = 0

on

k'

i s a non z e r o

,

(E

a n d t h e c h a r a c t e r i s t i c function

a n d i s a n e n t i r e function

(9

p

.

L e t { zk}+03 d e n o t e t h e s e t of z e r o e s k =1 m 2 1 d e n o t e t h e mul. , let

(E

.

i s d e f i n e d by

T h e n i t follows

k

f r o m th. 16. 3. 1 i n t h e o n e d i -

c a s e t h a t e v e r y solution f of t h e h o m o n e n e o u s e q u a t i o n

h a s the

form

Some applications

418

where P

j

i s a polynomial of d e g r e s

m

k-1

'

Now, the s e q u e l of t h i s s e c t i o n i s devoted t o the proof

of T h

16. 3. I

a n d , f o r t h i s , we begin by proving :

16. 3. 3.

Division lemma.

l e t Q -be a -

'

-

L A F be a c o m p l e x S c h w a r t z b. v. s. and

connected M-open

s u b s e t of

F

.

Then for every

--

'0, 1, c l o s e d ( Q ) a n d e v e r y p E ~ , ( Q ) , with p f 0

"'0, proof.

-

1,closed

(Q)

with p S = T

.

w e first p r o v e t h a t t h e m a p p i n g

A s i n 11. 3. 2

'

0,1, c l o s e d ( * )

'

0,1, c l o s e d

P*V i s continuous

'

0, 1, c l o s e d

.

'0,1,

c l o s e d (n1 cp

A f u n d a m e n t a l s y s t e m of o - n e i g h b o r h o o d s i n

(0) i s m a d e of the s e t s

w h e r e K i s a strictly c o m p a c t s u b s e t subset

, t h e r e is a n

of E,

where

E>

0 and n

€IN .

of Q , w h e r e L is a bounded A s usual we

set

Homogeneous convolution equations

For every

y e E a n d e v e r y epEd

i s equivalent to : rg YE

Now i f W (K, L,

E

0, 1, c l o s e d

V( K, L,

E

(n), we

11. 3 . 2

c o n s i d e r t h e mapping:

, n) f o r e v e r y y E L .

, n) i s a given o-neigborhood

follow5 f r o m l e m m a

419

in ' 0 , I, c l o s e d

that t h e r e is a V ( K ' , L',

E

I,

(n)tit

n') such

that, i f h € 6 (n), then

T h e r e f o r e , f r o m w h a t p r e c e e d s , and i f L

c L' ,

which p r o v e s the continuity of the above m a p p i n g e n d of the proof i s quite s i m i l a r to

11. 3 . 3.

p q

I

-+

p . Now t h e

Some applications

420

16. 3 . 4 .

F i r s t s t e p in the proof

s p a c e . F r o m 4. 2.4 s p a c e and s i n c e

closed

FrCchet Schwartz space

.

(n)

We a s s u m e F i s a DFN a Fre'chet Schwartz

i s a c l o s e d s u b s p a c e , i t is a l s o a

F r o m t h . 15.7. 1. we h a v e the e x a c t s e q u e n c e

Since t h e s e s p a c e s a r e F r P c h e t - S c h w a r t z

ils

-

, @ , , , ( a ) is

and 4. 3 . 4

'0,1,

16. 3. 1.

of

spaces, the transposed

se-

a l s o e x a c t ( s i n c e any F r e ' c h e t S c h w a r t z s p a c e is reflexive,o. 6. 7 - 8 and

f r o m the s u r j e c t i o n t h e o r e m i n F r g c h e t s p a c e s u s e d i n the proof Let f E 3 Since

t

~ (n) ' be a solution of @ f = 0 S

function of t h e o p e r a t o r

@

hE5CS("))

W e set

S €6

i is s u r j e c t i v e t h e r e e x i s t s a n

S(h) = $ (h) f o r e v e r y

(i. e.

.

.

If p

, and f o r a n y h

(p S)(ih) = S(p (ih)) = S ( i ( p h ) ) - (

t

'(n)

%

=

of

12. 6. 3 ) .

z-l(f)cXts(n).

s u c h that

t

i (S) =

denotes the c h a r a c t e r i s t i c

exs(n),

i ( S ) ) (p h) = * ( p

we h a v e :

h) = ( p$)(h) = 0

.

T h e r e f o r e p S i s n u l l on I m i c ' ( n ) t t tand a s a consequence p S E K e r i But K e r i = Im a , therefore t t h e r e exists a Y € 8 (n) s u c h that p S = ( a ) ( Y ) . 0, 1, c l o s e d since

p$= 0

( b e c a u s e @ f = 0)

.

F r o m (16. 3. 3 . ) and i f p is t r i v i a l ), t h e r e e x i s t s set

p

0 (if p

t

0, then

'0, 1, c l o s e d

u= U E g ' ( n ) and

=

=

0 and

1 6 . 3. 1

s u c h that p X = Y

.

We

Lr(X) .

s -

t t i ( U ) = i (S) ( s i n c e i

(n)

@

O

%=

0 ) . T h e r e f o r e U ( h ) = S(h)

42 1

Homogeneous convolution equations

I n o r d e r to p r o v e t h 16. 3. 1 i n c a s e F i s a D F N s p a c e i t r e m a i n s p U = 0, i. e.

to p r o v e that

only

( p U ) (h) = 0 f o r a11 h € 8 ( 0 ). We have

and

(recall

-

a

16. 3. 5 Q

#

0

.

p = 0 since p

txc,(n)).T h e r e f o r e

Second s t e p in the proof of and

,

a s in

f r o m t h e proof 10. 3. 6

16. 3. 1. -

We a s s u m e now

15. 3. 1, t h a t F i s a n u c l e a r b. v. s. that t h e r e e x i s t s a f a m i l y (8 .) 1

s p a c e s s u c h that (1)

of D F N

i '

zxIs(0 ) i s t h e b o r n o l o g i c a l inductive limit of t h e ( 3 x c ' R( fl q)i E I (i. e. e v e r y bounded s u b s e t of zxc'S ( 0 )

i s contained and bounded i n s o m e n a t u r a l inclusion m a p p i n g 3~ all i € 1 ) and

(3) f o r a l l i E 1 range in space.

It follows

F i s t h e b o r n o l o g i c a l inductive limit of t h e s p a c e s 8

(2)

spaces

i €1

.

that

a x '(0n 8 i)

,

I (

-

3x'(0 0 8 .)

R 0 8 .)

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

3

and c o n v e r s e l y t h e

x

Is(

@

to

n) i s

bounded for

axc'( 0 n&?1.)

has

and i s a non z e r o convolution o p e r a t o r on t h i s

422

Some applications

F o r every i

E I , we

denote by ( i ) the injective n a t u r a l mapping

z x lS ( a ) . L e t n o w f be a n e l e m e n t of Is(n) s u c h that 8 f = 0 . T h e r e is a n index i0E I a n d t h e r e is e l e m e n t fo€3K '(0 n 8 , ) such t h a t f = (i0 )(f 0 ) . F r o m the first

f r o m 3 K c l ( Q n 8 . ) into an

step 1 6 . 3 . 4

0

there exists a U

where p .

pi, Uo = 0 r e s t r i c t i o n of

8 to

10

0

E % ' ( nn 8 , )

s u c h that f = 3 U a n d 0

0

3K

'(n n

8. ) 0

t h e beginning of t h i s s e c t i o n , p i

0

. Clearly,

f r o m the d i a g r a m at

is the r e s t r i c t i o n of

) (.(P) =

~ / na8 .

0

if rp

E

p to R

n 8. . 15

We denote by r the r e s t r i c t i o n mapping f r o m 8

& ( a n 8,

0

denotes the c h a r a c t e r i s t i c function of the

8

(n))

( n ) into

a n d b y t r its t r a n s p o s e d

10

U

We have the following c o m m u t a t i v e d i a g r a m (with s o m e s i m p l i f i e d obvious n o t a t i o n s )

We

set

t

u = r ( u0 )

BIBLIOGRAPHIC NOTES

F i r s t we quote t h e s o u r c e of t h e t h e o r e m s p r e s e n t e d i n t h e book, a n d t h e n we give s o m e m o r e r e f e r e n c e s of p r e v i o u s o r r e l a t e d r e s u l t s . T h e r e f o r e n u m e r o u s i m p o r t a n t p a p e r s c o n n e c t e d with t h e t o p i c s d i s c u s s e d a r e not m e n t i o n e d . CHAPTER 1 T h e o r e m s 1 . 4 .7-8 w e r e obtained i n C o l o m b e a u [3,17] a n d t h e o r e m s 1 . 6 . 3 w a s obtained in C o l - m b e a u - M u j i c a [l] . T h e concept of differentiable m a p p i n g s in t h e s e n s e of 8 1.1 a n d 6 1 . 2 i s i n Sebastigo e Silva [l, 21 in t h e c a s e of t h e Von N e u m a n n b o r nologies of l o c a l l y convex s p a c e s . T h e n h e p l a c e d h i s t h e o r y i n i t s n a t u r a l s e t t i n g of bornological v e c t o r s p a c e s in S e b a s t i c o e Silva [3]. T h e c o n Cco m a p p i n g s in the e n l a r g e d s e n s e c o n s i d e r e d in $ 1 . 4 w a s i n c e p t of t r o d u c e d in Sebastia40 e Silva [ Z ] . T h e concept of d i f f e r e n t i a b l e m a p p i n g s in t h e s e n s e of $ 1.1 a n d 8 1 . 2 w a s a l s o i n t r o d u c e d i n M a r i n e s c u [l] in the s e t t i n g of bornological v e c t o r s p a c e s ( p o l y n o r m e d s p a c e s i n M a r i n e s c u [ l ] ' s t e r m i n o l o g y ) . T h e r e i s a slight d i f f e r e n c e b e t w e e n M a r i n e s c u ' s definition a n d t h e definition in $ 1.1, 0 1 . 2 , s e e C o l o m b e a u [ 2 ] , p . 20 a n d p.31-32. F o r f u r t h e r c o m p a r i s o n r e s u l t s between t h e t h r e e c o n c e p t s of differentiability of 3 1 . 1 , 0 1 . 2 , 8 1 . 4 a n d $ 1. 5 we r e f e r to C o l o m b e a u [ 3 , 4,171 . F o r the g e n e r a l l i t t e r a t u r e on d i f f e r e n t i a b l e m a p p i n g s between l o c a l l y convex s p a c e s we r e f e r t o t h e e x c e l l e n t s u r v e y s of A v e r b u c k Smolyanov [ Z ] a n d N a s h e d [l] t h a t contain e x c e l l e n t b i b l i o g r a p h i e s f o r p a p e r s a n t e r i o r to 1969. Among t h e v e r y n u m e r o u s books a n d a r t i c l e s on t h i s s u j e c t l e t u s f u r t h e r m o r e quote Averbuck-Smolyanov [l] , B u c h e r F r b l i c h e r [l] , K e l l e r [l] , Y a m a m u r o [l, 21, M e i s e [l], C o l o m b e a u Meise [l]. In t h i s book we did n o t c o n s i d e r t h e v e r y i m p o r t a n t t o p i c s of I m p l i c i t F u n c t i o n s a n d O r d i n a r y D i f f e r e n t i a l E q u a t i o n s . In t h e c a s e of Banach s p a c e s v e r y g e n e r a l r e s u l t s w e r e obtained a l r e a d y i n t h e t h i r t i e s ( s e e C a r t a n [l] o r DieudonnC [l] f o r i n s t a n c e ) a n d a r e now v e r y c l a s s i c a l . In the c a s e of non n o r m e d s p a c e s t h e s i t u a t i o n i s c o n s i d e r a b l y m o r e c o m p l i c a t e d a n d does not s e e m t o be c o m p l e t e l y c l a r i f i e d at p r e s e n t . So we j u s t s k e t c h h e r e t h i s topic a n d give s o m e r e f e r e n c e s . L e t u s f i r s t a s s u m e the e x i s t e n c e of a continuous i m p l i c i t f u n c tion o r of a continuous i n v e r s e , when t h e g i v e n f u n c t i o n s a r e d i f f e r e n t i a b l e In t h e Banach s p a c e c a s e t h i s i m p l i c i t function o r t h i s i n v e r s e m a p i s diff e r e n t i a b l e (Nachbin [43, P r o p o s i t i o n 16.16 a n d 22.10 f o r i n s t a n c e ) . A c o u n t e r e x a m p l e i n Averbuck-Smolyanov [ 2 ) shows that t h i s is no l o n g e r t r u e in g e n e r a l l o c a l l y convex s p a c e s . On t h i s topic s e e C o l o m b e a u [ 4 ] , Smolyanov [l]. If one a s s u m e s s o m e p r o p e r t y s t r o n g e r t h a n continuity on t h e i m p l i c i t o r i n v e r s e m a p , one o b t a i n s the d e s i r e d differentiability 423

Bibliographic Notes

424

r e s u l t , s e e C o l o m b e a u [4], t h e o r e m 2 . 2 which i s q u i t e g e n e r a l . Now l e t i s a n - t i m e s d i f f e r e n t i a b l e b i j e c t i o n a n d t h a t its i n v e r s e m a p us assume f f - l i s one t i m e d i f f e r e n t i a b l e ( s a m e s i t u a t i o n a n d r e s u l t s i n t h e c a s e of t h e i m p l i c i t f u n c t i o n ) . In t h e c a s e of B a n a c h s p a c e s f - l i s n - t i m e s d i f f e rentiable. A counterexample in Colombeau [4], $ 6 , example 2 , shows t h a t t h i s i s no l o n g e r t r u e i n g e n e r a l i n non n o r m e d s p a c e s , but t h e o r e m 3 . 2 i n C o l o m b e a u [4] g i v e s a v e r y g e n e r a l c a s e i n which f - l i s n - t i m e s d i f f e r e n t i a b l e . In s h o r t t h e r e s u l t s a r e q u i t e g e n e r a l , but one n e e d s t o a s s u m e m o r e r e s t r i c t i v e a s s u m p t i o n s t h a n in t h e B a n a c h s p a c e s c a s e (the continuity of t h e i m p l i c i t f u n c t i o n i m p l i e s t h e s e p r o p e r t i e s i n t h e B a n a c h s p a c e s c a s e ) . The s i t u a t i o n of t h e e x i s t e n c e r e s u l t s i s m u c h w o r s e : L e t u s now c o n s i d e r t h e p r o b l e m of e x i s t e n c e of a n i m p l i c i t f u n c t i o n o r of a n i n v e r s e of a g i v e n d i f f e r e n t i a b l e m a p , in t h e c o n d i t i o n s t h a t , i n B a n a c h s p a c e s , e n s u r e t h e i r e x i s t e n c e ( C a r t a n [ l ] , Dieudonng [l], . ) . C o u n t e r e x a m p l e s i n E e l l s [l] , P i s a n e l l i [l] , C o l o m b e a u [l] , L o j a s i e w i c z J r . [l] show t h a t t h e v e r y n e a t r e s u l t s of t h e B a n a c h s p a c e s f r a m e w o r k do not r e m a i n v a l i d i n l o c a l l y c o n v e x s p a c e s . T h e r e is a v e r y l a r g e a m o u n t of w o r k s on t h i s t o p i c : people f i r s t t r i e d v a r i o u s g e n e r a l i z a t i o n s of the c l a s s i c a l p r o o f , s e e f o r i n s t a n c e F a l b - J a c o b [l], M a c D e r m o t t [l, 2 , 31, C o l o m b e a u [l, 5, 6 , 7 1 , Y a m a m u r o [ 2 ] , T h e i m p l i c i t function a n d l o c a l i n v e r s i o n t h e o r e m s i n C o l o m b e a u [l, 5, 61 a r e u s e d i n J . A . L e s l i e [ 3 ] f o r a proof of a K u p k a - S m a l e t h e o r e m in t h e r e a l a n a l y t i c c a s e . A d i f f e r e n t m e t h o d w a s i n s p i r e d by t h e " N a s h I m p l i c i t F u n c t i o n T h e o r e m " , s e e J . T . S c h w a r z [l], M o s e r [l] , H a m i l t o n [l] , S e r g e r a e r t [l] , J a c o b o w i t z [ l ] , Z e h n d e r [l], L o j a s i e w i c z - Z e h n d e r [l] T h i s o t h e r kind of i m p l i c i t function t h e o r e m s have c l a s s i c a l p o w e r f u l a p p l i c a t i o n s , f o r i n s t a n c e in t h e proof of t h e e m b e d d i n g of R i e m a n n i a n m a n i f o l d s i n IR" , s e e S c h w a r z [l]

..

... .

.

.

Now l e t us c o n s i d e r " o r d i n a r y " d i f f e r e n t i a l e q u a t i o n s X ' ( t ) = F ( X ( t ) ,t ) , X(to) = Xo ( C a u c h y p r o b l e m s ) . T h e v e r y b e a u t i f u l e x i s t e n c e , u n i q u e n e s s a n d d e p e n d e n c e on p a r a ' m e t e r s a n d d a t a r e s u l t s of t h e B a n a c h s p a c e c a s e t h e o r y ( C a r t a n [l] , Dieudonne [l] , . ) do not r e m a i n v a l i d i n l o c a l l y convex s p a c e s , s e e C o l o m b e a u [2] , P l i s [l], De G i o r g i [l]). V a r i o u s m e t h o d s w e r e u s e d : a c o m p a c t n e s s m e t h o d in Dubinsky [l], a n , i t e r a t i o n m e t h o d in C o l o m b e a u 61, 2 , 61, T r e v e s [2] , L e s l i e [2], V e r y good r e s u l t s a n d a p p l i c a t i o n s c a m e f r o m a v e r y s t r o n g r e i n f o r c e m e n t of t h i s i t e r a t i o n m e t h o d by a n a s t u t e m a j o r i z a t i o n t e c h n i q u e i n t h e s o c a l l e d "Ovcyannikov method" : Ovcyannikov [l, 2 3 , T r e v e s [ 3 , 4 , 51, S t e i m b e r g T r e v e s [l], S t e i m b e r g [l], P i s a n e l l i [2], N i r e m b e r g [l], Du C h a t e a u [l], L a s c a r [2], . T h e s e r e f e r e n c e s c o n c e r n t h e s t u d y at o r d i n a r y points (for a c l a s s i f i c a t i o n of s i n g u l a r i t i e s of o r d i n a r y d i f f e r e n t i a l e q u a t i o n s s e e Wasow [ l ] ) . T h e Ovcyannikov m e t h o d w a s u s e d i n t h e c a s e of " r e g u l a r s i n g u l a r points" i n Baouendi-Goulaouic [l, 21 a n d i n t h e c a s e of " i r r e g u l a r s i n g u l a r points" i n C o l o m b e a u - M C r i l [l]

..

.. .

...

.

B e s i d e s the a b o v e t o p i c s , D i f f e r e n t i a l C a l c u l u s i n a . c . s . h a s a l o t of v a r i o u s a p p l i c a t i o n s , m o s t of t h e m being at p r e s e n t i n f u l l d e v e l o p -

425

Bibliographic Notes

m e n t : f o r i n s t a n c e d i f f e r e n t i a l s t r u c t u r e s i n s e t s of COD o r real a n a l y tic d i f f e o m o r p h i s m s of c o m p a c t R i e m a n n i a n m a n i f o l d s ( L e s l i e [l, 31 ) o r new c o n c e p t s of g e n e r a l i z e d f u n c t i o n s on IE?? , giving a m e a n i n g t o a n y p r o d u c t of d i s t r i b u t i o n s ( C o l o m b e a u [l8] ), j u s t t o quote a few of t h e m . CHAPTER 2 T h e o r e m 2.2.3 w a s obtained in L a z e t [l, 2 3 . T h e o r e m s 2 . 3 . 3 - 4 a r e i n C o l o m b e a u [ 8 ] . T h e o r e m 2 . 4 . 1 and c o r o l l a r i e s w e r e o b t a i n e d i n L a z e t [l, 21, C o l o m b e a u [9,10] T h e c o u n t e r e x a m p l e in 2 . 5 w a s obtained i n C o l o m b e a u [9,10] . T h e o r e m 2 . 6 . 4 w a s o b t a i n e d i n B o c h n a k - S i c i a k [ l , 23 a n d T h e o r e m 2 . 7 . 4 i n C o l o m b e a u - M u j i c a [l]

.

.

T h e concept of Silva h o l o m o r p h i c m a p p i n g s w a s i n t r o d u c e d in S e b a s t i s e Silva [l, 2 , 33, a n d t h e n s t u d i e d i n m o r e d e t a i l s in C o l o m b e a u [ 9 , 1 0 , 8 , 1 , 1 7 ] , C o l o m b e a u - L a z e t [l], L a z e t [l, 23, P i s a n e l l i [3, 43, M a t o s Nachbin [l] , B i a n c h i n i [l] , B i a n c h i n i - P a q u e s - Z a i n e [l], e t c . L e t u s m e n tion a n i c e o r i g i n a l i n t r o d u c t i o n i n M e i s e - V o g t [l] a n d t h a t s e v e r a l a u t h o r s i n t r o d u c e d t h e concept of "hypo-analytic m a p p i n g s " which c o i n c i d e s with t h e one of Silva h o l o m o r p h i c m a p p i n g s i n all "usual" A . c s . .

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T h e c o n c e p t of h o l o m o r p h i c ( = G - a n a l y t i c a n d c o n t i n u o u s ) m a p p i n g s i n l o c a l l y convex s p a c e s h a s b e e n s t u d i e d by a c o n s i d e r a b l e n u m b e r of a u t h o r s , a n d we r e f e r t o D i n e e n ' s r e c e n t book [I] f o r r e f e r e n c e s , a s w e l l a s t o Nachbin [l] f o r a n i n t r o d u c t i o n a n d r e f e r e n c e s . T h e r e a r e a l s o m a n y books a n d v o l u m e s of P r o c e e d i n g s c o n c e r n i n g t h i s s u b j e c t : B a r r o s o [l, 2 , 3, 41, Boland [3], C o e u r 6 [l], Dineen [l] , Hayden-Suffridge [l] , HervC [l], Lelong [ Z , 33, Lelong-Skoda [l], Machado [l] , M a t o s [7], Mazet [l] , Mujica [l] , Nachbin [ Z ] , N o v e r r a z [l, 43, Z a p a t a [l] , e t c . L e t u s a l s o m e n t i o n the s u r v e y a r t i c l e s N a c h b i n [l, 3 , 51, B i e r s t e d t - M e i s e [Z] , C o l o m b e a u - M a t o s [ Z ] , Dineen [lo] Concerning holomorphic functions on n u c l e a r F r k c h e t s p a c e s a n d v e r y i m p o r t a n t c o u n t e r e x a m p l e s s e e M e i s e Vogt [ 2 , 31

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T h e h o l o m o r p h i c r e p r e s e n t a t i o n of F o c k s p a c e s of B o s o n f i e l d s i s e x p o s e d o r u s e d i n B e r e z i n [l] , Novozhilov-Tulub [l], S a r a v a s t i - V a l a t i n [l], R z e w u s k i [l, 2 3 , Dwyer [l, 2 , 3 , 4 J , C o l o m b e a u - P e r r o t [l, 2 , 31, K r 6 e [l, 21, K r g e - R a c z k a [l]

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CHAPTER 3 T h e o r e m s 3 . 1 . 2 , 3 . 2 . 1 , 3 . 2 . 3 a n d 3 . 3 . 1 a r e r e f o r m u l a t i o n s of c l a s s i c a l r e s u l t s . E x a m p l e 3 ' 2 . 4 is t a k e n f r o m C o l o m b e a u [9,10] ; a s i milar c o u n t e r e x a m p l e w a s o b t a i n e d independently in H i r s c h o w i t z [l] E x a m p l e 3 . 3 . 3 i s t a k e n f r o m Colombeau [17] A m o r e g e n e r a l f o r m of t h e o r e m 3 . 4 . 3 - 4 i s in C o l o m b e a u - L a z e t [l] , C o l o m b e a u [lo]

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S o m e l i t t e r a t u r e on Z o r n , H a r t o g s a n d M o n t e l ' s t h e o r e m s i s i n Dineen [ l ] , S e b a s t i c o e Silva [l, 21, Hille [l] , H i l l e - P h i l l i p s [l],

426

Bibliographic Notes

L a z e t [l, 2 3 , Colombeau [l, 9,10,17], Col.ombeau-Lazet [l] , N o v e r r a z [l, 2 1 , B o c h n a k - S i c i a k [l, 21, P i s a n e l l i [3, 43, Matos [I, 23, .

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CHAPTER 4 T h e o r e m s 4 . 2 . 1 - 2 , 4 . 3 . 1 , 4 . 4 . 1 w e r e obtained i n C o l o m b e a u P e r r o t [2,4,5]. T h e o r e m s 4 . 2 . 4 , 4 . 3 . 4 , 4 . 4 . 2 w e r e obtained i n a n unpublished m a n u s c r i p t Colombeau [ll] ; s e e a l s o M e i s e [l] a n d C o l o m b e a u M e i s e [l]. T h e s t r i c t l y c o m p a c t p o r t e d topology w a s i n t r o d u c e d i n BianchiniP a q u e s - Z a i n e [l] a n d p r o p o s i t i o n 4 . 1 . 4 w a s obtained i n C o l o m b e a u - M e i s e P e r r o t [l] N a c h b i n ' s p o r t e d t o p o l o g y w a s defined i n Nachbin [2] f o r hol o m o r p h i c functions on Banach s p a c e s . T h i s last book o r i g i n a t e d a v e r y i m p o r t a n t t r e n d of w o r k s , in p a r t i c u l a r on topologies i n s p a c e s of holom o r p h i c m a p p i n g s , a n d t h i s last topic i s t r e a t e d i n d e t a i l i n t h e r e c e n t book Dineen [l], to which we r e f e r . Let u s j u s t m e n t i o n r e c e n t r e s u l t s i n Dineen [9], Boland-Dineen [4], M e i s e [ 21, Mujica €21

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CHAPTER 5 T h e o r e m 5.1.2 w a s obtained i n the m o r e g e n e r a l c a s e of a f i n i t e l y Runge open s e t 0 in C o l o m b e a u - M e i s e - P e r r o t [l]. T h e o r e m 5 . 2 . 1 w a s obtained in C o l o m b e a u - M e i s e [l], s e e a l s o M e i s e [l] T h e proof of t h e o r e m 5 . 3 . 1 is a n i m m e d i a t e a d a p t a t i o n of a proof i n B o n i c - F r a m p t o n [I].

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G e n e r a l i z a t i o n s of N a c h b i n ' s A p p r o x i m a t i o n T h e o r e m a r e i n P r o l l a [ 2 , 31, G u e r e i r o - P r o l l a [l], s e e a l s o Nachbin [7]

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T h e r e s u l t s in t h i s c h a p t e r w e r e c h o s e n s i n c e we u s e t h e m i n t h i s book. S e v e r a l o t h e r r e s u l t s of e x i s t e n c e a n d a p p r o x i m a t i o n a r e i n c h a p t e r s 6 , 9 t o 16. S o m e r e s u l t s which a r e not i n t h i s book a r e s u r v e y e d i n : Dineen [l], J o s e f s o n Schottenloher [l] T h e y c o n c e r n (1) . , ( 2 ) e x i s t e n c e of ho[ 1 , 2 3 , R u s e k [I], Bayoumi [l], l o m o r p h i c functions with p r e s c r i b e d r a d i u s of c o n v e r e n c e : A r o n T i f , C o e u r 6 [2], K i s e l m a n 11, 2 , 31, Schotten-&rvey in Schottenloher [l] contains a l s o o t h e r t o p i c s r e l a t e d with following c h a p t e r s of t h i s book. E x i s t e n c e r e s u l t s on h o l o m o r p h i c f u n c t i o n s with p r e s c r i b e d v a l u e s a t a n infinite given s e t of points a r e in H e r v i e r [I], Valdivia [l] R e s u l t s on e x t e n s i o n s of h o l o m o r p h i c f u n c t i o n s defined in a c l o s e d s u b s p a c e of a l o c a l l y convex s p a c e a r e i n B o l a n d r 3 ] , A r o n - B e r n e r [l], Colombeau-Mujica Meise-Vogt [3] R e s u l t s of e x i s t e n c e of Co3 holomorphic ma in s with p r e s c r i b e d a s y m p t o t i c e x a n s i o n s a r e i n G b e a T d k & - 5 , 1 6 ] , Colombeau-Mujici&zr.

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m:

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CHAPTER 6 T h e o r e m 6.1.1 w a s obtained in a m o r e r e s t r i c t i v e c a s e i n A r o n Schottenloher [l], Schottenloher [3], a n d in m o r e g e n e r a l i t y i n P a q u e s [l]

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421

Bibliographic Notes

C o r o l l a r y 6.1.4 w a s d i r e c t l y obtained f o r Cn f u n c t i o n s in M e i s e [l] w h e r e a d e t a i l e d proof is g i v e n . A m o r e g e n e r a l f o r m of t h e o r e m 6 . 2 . 1 i s in C o l o m b e a u - M e i s e [l] t h e o r e m 3 . 5 . C o r o l l a r y 6 . 2 . 2 w a s d i r e c t l y obtained in Colombeau [ll] a n d i s a l s o in C o l o m b e a u - M e i s e [l] t h e o r e m 3 . 6 T h e o r e m 6 . 3 . 2 i s i n C o l o m b e a u - M e i s e - P e r r o t [l] A m o r e g e n e r a l f o r m of t h e o r e m 6 . 3 . 3 i s in C o l o m b e a u - M e i s e [l] r e m a r k 4 . 5 .

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F o r o t h e r p a p e r s on t h e A p p r o x i m a t i o n P r o p e r t y of 'Q (Q), X(Q) a n d Cn(Q) s e e P a q u e s [2], A r o n - S c h o t t e n l o h e r [l] , M e i s e [l], B o m b a l Gordon-Gonzalez Llavona [l] . U s e of the " k e r n e l t h e o r e m " 6.1.4 t o a study of l i n e a r o p e r a t o r s o n F o c k s p a c e s of Boson f i e l d s i s done in C o l o m b e a u - P e r r o t [l, 2 , 3 ] , Kre'e [l, 21, K r i e - R a c z k a [l]

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CHAPTER 7 T h e o r e m 7 . 2 . 1 w a s obtained in C o l o m b e a u - P e r r o t [6] but is a l s o a c o n s e q u e n c e of a t h e o r e m in Boland (2, 31, s e e C o l o m b e a u - P e r r o t [6]. T h e o r e m 7 . 4 . 1 is i n A n s e m i l - C o l o m b e a u [l] a n d t h e o r e m " 7 . 4 . 7 i n C o l o m b e a u - P o n t e [l]

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T h e f i r s t r e s u l t o n t h e F o u r i e r - B o r e 1 i s o m o r p h i s m i n infinite d i m e n s i o n w a s obtained by Gupta [l, 2 , 3 ] f o r e n t i r e f u n c t i o n s of n u c l e a r type on Banach s p a c e s . T h i s r e s u l t w a s e x t e n d e d to l o c a l l y convex s p a c e s in M a t o s [ 3 , 4 ] a n d Boland [ 2 , 3 ] . F o r connections b e t w e e n t h e above r e s u l t s of Boland a n d Matos s e e C o l o m b e a u - M a t o s [l] . S p a c e s of h o l o m o r p h i c g e r m s w e r e i n t e n s i v e l y s t u d i e d , s e e t h e book Dineen [l] ; let u s only quote Mujica [l. 2, 31, Dineen [5], B i e r s t e d t M e i s e (1, 21, A r a g o n a [l] , S o r a g g i [l] , Biagioni [l]

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T h e difficulty to extend the c l a s s i c a l P a l e y - W i e n e r - S c h w a r t z t h e o r e m t o infinite d i m e n s i o n w a s pointed out i n Dineen-Nachbin [l] T h e n a P. W .S. t h e o r e m w a s obtained i n A b u a b a r a [l, 21 a n d h i s i d e a w a s a d a p t e d t o the c a s e of n u c l e a r s p a c e s i n A n s e m i l - C o l o m b e a u [l]. O t h e r r e s u l t s w e r e obtained by defining a s u i t a b l e s p a c e of Coo f u n c t i o n s i n C o l o m b e a u Ponte [l] f o r t h e c a s e of n u c l e a r s p a c e s a n d in C o l o m b e a u - P a q u e s [l] f o r the c a s e of Banach s p a c e s . See a l s o o t h e r kinds of P . W . S . t h e o r e m s in t h i s book c h a p t e r 13 ( C o l o m b e a u - P a q u e s [3] ), i n C o l o m b e a u - P a q u e s [a] a n d Gal6 [l]

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CHAPTER 8 T h e o r e m s 8.1.1-2 a r e r e f o r m u l a t i o n s of a r e s u l t obtained i n d e pendently i n Boland [4] a n d W a e l b r o e c k [I] T h e proof g i v e n h e r e is t h a t of C o l o m b e a u - P e r r o t [ 7 ] obtained l a t e r . T h e o r e m s 8 . 2 . 5 - 6 w e r e g i v e n in C o l o m b e a u - M e i s e [2] a n d a r e e x t e n s i o n s of a r e s u l t i n B i e r s t e d t - G r a m s c h Meise [l] T h e o r e m 8 . 3 . 2 i s due to M e i s e [ 3 ] , s e e a l s o C o l o m b e a u - M e i s e

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bl

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428

Bibliographic Notes

M o r e r e c e n t l y in B o r g e n s - M e i s e - V o g t [l, 2, 31 a p p e a r e d a unified proof of all n u c l e a r i t y r e s u l t s p r e s e n t l y known in Infinite D i m e n s i o n a l Holomorphy, a n d i t is e a s y to s e e t h a t t h e m e t h o d s t h e r e a l s o apply t o the s p a c e s K s ( n ) s i n c e the p r o o f s r e l y on l e m m a s dealing with t h e B a n a c h space situation. CHAPTER 9 T h e o r e m 9 . 4 . 1 i s in C o l o m b e a u - P e r r o t [8], a s well a s a p h y s i c a l i n t e r p r e t a t i o n of t h e s e e q u a t i o n s . A d i f f e r e n t proof a n d f u r t h e r r e s u l t s a r e in Colombeau-Matos [2]. CHAPTER 10 T h e o r e m s 1 0 . 2 . 6 and 1 0 . 3 . 4 a r e i n C o l o m b e a u - P e r r o t [9] a n d we expose h e r e the proof of t h i s p a p e r s i n c e we s h a l l n e e d it a l s o i n c h a p t e r 16. Another proof will be given in c h a p t e r 11, a n d will be t a k e n f r o m C o l o m b e a u G a y - P e r r o t [l] A t h i r d proof is i n C o l o m b e a u - M a t o s [ 2 ] .

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O t h e r r e s u l t s on convolution e q u a t i o n s i n s p a c e s of e n t i r e f u n c t i o n s of exponential type a r e i n Boland-Dineen [3] ; they a r e obtained f r o m t h e f i n i t e d i m e n s i o n a l r e s u l t s by a method of t r a n s f i n i t e induction. A g r e a t a m o u n t of r e s u l t s on r e l a t e d e q u a t i o n s i s i n Dwyer [l, 2, 3, 4,5, 6, 7,8] , i m p r o v e d in p a r t i n C o l o m b e a u - D w y e r - P e r r o t [l]

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CHAPTER 11 T h e W e i ' e r s t r a s s p r e p a r a t i o n t h e o r e m s 11.1.3-4 a r e in M a z e t [l], 8 11.2 is taken f r o m S c h w a r t z [4]. T h e o r e m 11.3.1 w a s obtained R a m i s [l]. i n C o l o m b e a u - G a y - P e r r o t [l] P r o p o s i t i o n 11.5.2 a n d t h e o r e m 1 1 . 6 . 2 a r e in Chansolme €13.

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CHAPTER 12

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T h e o r e m s 1 2 . 6 . 1 and 1 2 . 6 . 3 w e r e obtained i n Gupta [l, 2 , 3 ] T h e o r e m s 1 2 . 7 . 1 and 1 2 . 7 . 4 w e r e obtained in C o l o m b e a u - M a t o s [3]. T h e o r e m 1 2 . 7 . 5 w a s obtained i n C o l o m b e a u - P e r r o t [6], t h e o r e m 1 2 . 8 . 2 i n BolandDineen [3] a n d t h e o r e m 1 2 . 8 . 3 i n C o l o m b e a u - P a q u e s [3)

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P a r t i c u l a r c a s e s of t h e o r e m s 12.7.1, 1 2 . 7 . 4 a n d 1 2 . 7 . 5 a r e i n Matos [5], B e r n e r [l) , Boland [2, 39 A l r e a d y i n 1970 M a t o s e x t e n d e d G u p t a ' s r e s u l t s [l, 2 , 31 to l o c a l l y convex s p a c e s i n M a t o s [3, 41 T h e n Boland [2, 31 obtained r e s u l t s i n n u c l e a r s p a c e s . F o r c o n n e c t i o n s between t h e s e l a s t r e s u l t s s e e C o l o m b e a u - M a t o s [l] . L e t u s a l s o m e n t i o n Matos [ 6 , 8 ] , a s u r v e y a n d f u r t h e r r e f e r e n c e s in C o l o m b e a u - M a t o s [ 2 ) . Convolution e q u a t i o n s i n s p a c e s of h o l o m o r p h i c g e r m s a r e s t u d i e d i n Biagioni[l]

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CHAPTER 13 T h e o r e m 1 3 . 4 . 1 i s i n C o l o m b e a u - P a q u e s [3]. T h e s e r e s u l t s , a s well

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429

Bibliographic Notes

a s o t h e r r e s u l t s , w e r e obtained independently with a d i f f e r e n t proof by S c h w e r d t f e g e r [l] . O t h e r r e s u l t s a r e i n A n s e m i l - P e r r o t [l]

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CHAPTER 14 $1 r e v i e w s c l a s s i c a l definitions on pseudo-convexity i n l o c a l l y convex s p a c e s . T h i s m a t e r i a l i s given i n m u c h m o r e d e t a i l i n N o v e r r a z [l, 43 T h e o r e m 1 4 . 2 . 3 i s in G r u m a n - K i s e l m a n [l] , t h e o r e m 1 4 . 2 . 5 i s in Colombeau-Mujica [3] but follows a l s o e a s i l y f r o m S c h o t t e n l o h e r €43 . T h e o r e m 1 4 . 3 . 8 i s t a k e n f r o m N o v e r r a z [3,4] a n d t h e o r e m 14.4.1 f r o m C o l o m b e a u - P e r r o t [ll] w h e r e it is a l e m m a f o r the s o l u t i o n of t h e 6 e q u a t i o n

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T h e L k v i p r o b l e m w a s s o l v e d in s e p a r a b l e H i l b e r t s p a c e s a n d i n l i n e a r s p a c e s equipped with t h e finite d i m e n s i o n a l bornology by G r u m a n [l], t h e n i n B a n a c h s p a c e s with b a s i s i n G r u m a n - K i s e l m a n [l]. A c o u n t e r e x a m p l e i n non s e p a r a b l e B a n a c h s p a c e s is in Josefson[3].The L k v i p r o b l e m w a s t h e n s o l v e d in Silva s p a c e s with b a s i s i n Pome's [l]. V e r y g e n e r a l r e s u l t s w e r e obtained in Schottenloher [4] f o r d o m a i n s s p r e a d o v e r locally convex s p a c e s with a S c h a u d e r d e c o m p o s i t i o n . A f t e r t h e s e l a s t r e s u l t s w e r e obtained, o t h e r p r o o f s i n F r C c h e t s p a c e s with b a s i s a n d Silva s p a c e s with b a s i s w e r e published i n D i n e e n - N o v e r r a z - S c h o t t e n l o h e r [l], a s w e l l a s t h e c a s e of all n u c l e a r Silva s p a c e s i n C o l o m b e a u - M u j i c a [ 3 ] , S e e a l s o Mujica [4] f o r a s u r v e y a n d new r e s u l t s . See t h e H i s t o r i c a l N o t e s i n Dineen [l] f o r t h e evolution of t h i s p r o b l e m i n infinite d i m e n s i o n . CHAPTER 15 T h e o r e m 1 5 . 7 . 1 i s i n C o l o m b e a u - P e r r o t [ll], ( s e e a l s o N o s s k e equa[l]). A w e a k e r r e s u l t w a s i n Raboin [7] a n d t h e r e s o l u t i o n of t h e t i o n i n t h e whole s p a c e w a s in C o l o m b e a u - P e r r o t [lo] T h e o r e m 1 5 . 8 . l is in C o l o m b e a u - M u j i c a [l].

.

T h e 5 equation i n s e p a r a b l e H i l b e r t s p a c e s w a s s t u d i e d by H e n r i c h [l] when t h e s e c o n d m e m b e r h a s a polynomial g r o w t h . He u s e s i n t e g r a t i o n t h e o r y a c c o r d i n g to G a u s s m e a s u r e i n H i l b e r t s p a c e s a n d h e o b t a i n s solutions defined on a d e n s e s u b s p a c e . L a t e r Raboin el, 2, 3 , 4 , 5, 6,7] s t u d i e d t h e b equation i n a r b i t r a r y pseudo-convex open s u b s e t s of s e p a 2 r a b l e H i l b e r t s p a c e s , without g r o w t h condition. He u s e s H ( 3 r m a n d e r ' s L e s t i m a t e s a n d i n t e g r a t i o n t h e o r y a c c o r d i n g to G a u s s m e a s u r e . When t h e s e c o n d m e m b e r i s Coo a n d of bounded t y p e , h e o b t a i n e d e x i s t e n c e of C 1 solutions defined o n a d e n s e s u b s p a c e . H i s t h e o r e m is published in Raboin [l, 2, 3, 4, 5, 6, 71 i n which i t s proof i s s k e t c h e d at v a r i o u s l e v e l s . It is v e r y c l o s e t o a n i m p r o v e m e n t of the l e m m a 1 5 . 4 . 1 of t h i s book. I n p a r t i c u l a r he obtained a s c o n s e q u e n c e s s o m e e x i s t e n c e r e s u l t in n u c l e a r Silva s p a c e s roblem in these with b a s i s (Raboin [7] ) a n d t h e solution of t h e f i r s t C o u s i n ps p a c e s (Raboin [6, 71). T h e n i n C o l o m b e a u - P e r r o t [lo] the 0 e q u a t i o n w a s s o l v e d , i n the c a s e of all n u c l e a r Silva s p a c e s , but only i n the whole s p a c e . F o r t h i s a c o m p l i c a t e d i m p r o v e m e n t of t h e a b o v e R a b o i n ' s proof w a s u s e d t o obtain i n t h e H i l b e r t i a n f r a m e w o r k a Cw solution on a d e n s e s u b s p a c e , which w a s u s e d a s a l e m m a . T h e n t h i s r e s u l t w a s extended t o a r b i t r a r y pseudo-convex open s u b s e t s of n u c l e a r Silva s p a c e s i n C o l o m b e a u -

430

Bibliographic Notes

P e r r o t [ll], and independently in N o s s k e el]. L a t e r a l a r g e p a r t of R a b o i n ' s proof a n d all i t s long i m p r o v e m e n t i n C o l o m b e a u - P e r r o t [lo] w e r e r e p l a c e d by a c o n s i d e r a b l y s h o r t e r proof of a n hypoellipticity r e s u l t due t o Mazet [2] ( l e m m a 15.5.1 of t h i s book). Still l a t e r a t e c h n i c a l a s s u m p t i o n on the s e c o n d m e m b e r , c o n s i d e r e d i n Raboin [7], w a s p r o v e d t o be a l w a y s t r u e i n Colombeau-Mujica [l] ( t h e o r e m 1 . 6 . 3 of t h i s book), S O t h a t t h e conjunction of Raboin [ 7 ] , Mazet [2] a n d Colombeau-Mujica [l] g i v e s in t h e p a r t i c u l a r c a s e of n u c l e a r Silva s p a c e s with b a s i s a proof d i f f e r e n t f r o m the f o r m e r proofs i n C o l o m b e a u - P e r r o t [ll] a n d N o s s k e [l] (that do not u s e the b a s i s a s s u m p t i o n ) . T h i s c o n c e r n e d the c a s e of 0 , l f o r m s ; the c a s e of p , q f o r m s with q > 1 r e m a i n s unsolved. V e r y i n t e r e s ting c o u n t e r e x a m p l e s of a different n a t u r e , i n Dineen [ 8 ] and M e i s e Vogt [3], show that even t h e c a s e of 0 , l f o r m s is not in g e n e r a l s o l vable in n u c l e a r F r k c h e t s p a c e s . F i n a l l y we m u s t mention that a " p e r sonal" a p p r e c i a t i o n on the r e s p e c t i v e contributions of s o m e of t h e s e a u t h o r s was published in K r a m m [l]. CHAPTER 16 T h e o r e m 16.1.1 is a s t a n d a r d consequence of t h e o r e m 15.7.1 a n d was f o r m e r l y obtained, a s a l r e a d y quoted a b o v e , i n t h e p a r t i c u l a r c a s e of s p a c e s with b a s i s i n Raboin [6, 74. T h e c o u n t e r e x a m p l e i n $ 16.2 i s taken f r o m Dineen [ 8 ] . T h e o r e m 16.3.1 w a s published i n C o l o m b e a u G a y - P e r r o t [l] . O t h e r applications a r e in K r a m m

111.

BIBLIOGRAPHY

A ' k o m p l e t e " b i b l i o g r a p h y on t h e s u b j e c t of t h e book would include m o r e t h a n 1000 i t e m s , s o t h e p r e s e n t b i b l i o g r a p h y is q u i t e u n c o m plete. F o r p a p e r s o n H o l o m o r p h y w e r e f e r t o t h e v e r y good b i b l i o g r a p h y of the r e c e n t D i n e e n ' s book [ 13. F o r p a p e r s o n D i f f e r e n t i a l C a l c u l u s wre r e f e r t o t h e v e r y g o o d - but now m o r e t h a n 1 0 y e a r s o l d - b i b l i o g r a p h i e s in A v e r b u c k - S m o l y a n o v [ 2 ] and N a s h e d [ 1 3 .

T. A b u a b a r a 1. 2.

On t h e P a l e y - W i e n e r - S c h w a r t z t h e o r e m i n infinite d i m e n s i o n A t t i A c a d . Naz. L i n c e i 6 8 , 1977, p 192-194. A v e r s i o n of t h e P a l e y - W i e n e r - S c h w a r t z t h e o r e m i n infinite d i m e n s i o n . A d v a n c e s i n H o l o m o r p h y , J . A . B a r r o s o e d . , N o r t h Holland Math. S t u d i e s 34, 1 9 7 9 , p 1 - 2 9 .

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J. M. A n s e m i l , J . F. C o l o m b e a u The Paley-Wiener-Schwartz theorem in nuclear spaces. 1 R e v u e R o u m . d e Math. P u r e s Appl. 2 6 , 2 , 1981, p.169-181. J. M. A n s e m i l , B . P e r r o t 1 C a f u n c t i o n s i n infinite d i m e n s i o n and l i n e a r p a r t i a l differential difference equations with constant coefficients, preprint, J.Aragona

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H o l o m o r p h i c a l l y s i g n i f i c a n t p r o p e r t i e s of s p a c e s of holomorphic g e r m s . Advances in Holomorphy, J . A . B a r r o s o e d . , N o r t h Holland Math S t u d i e s 34, 1979, p 3 1 - 4 6

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E n t i r e f u n c t i o n s of unbounded t y p e on a B a n a c h s p a c e . Boll U n . Mat. I t a l . 9 , 1974, p 2 8 - 3 1 .

R . A r m , P. B e r n e r A Hahn-Banach extension t h e o r e m f o r analytic mappings. 1 Bull.. S O C . Math. F r a n c e 106, 1978, p 3 - 2 4 . R . A r o n , M. S c h o t t e n l o h e r C o m p a c t h o l o m o r p h i c m a p p i n g s on Bar.ach s p a c e s and 1 t h e A p p r o x i m a t i o n P r o p e r t y . J . F u n c t . A n a . 21, 1976 p. 7 - 3 0 . 43 1

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INDEX

c o - S c h w a r t z s p a c e s , 22 C o u s i n p r o b l e m I, 410 C r e a t i o n o p e r a t o r s , 110

A n a l y t i c (function o r m a p ) : a n a l y t i c (G -analy t i c t c ontinuous), 8 8 G - a n a l y t i c , 79 h y p o a n a l y t i c , 425 r e a l a n a l y t i c , 104 Silva a n a l y t i c , 8 3 Silva a n a l y t i c in t h e e n l a r g e d s e n s e , 88 of n u c l e a r bounded t y p e , 283 of unifiorm bounded t y p e , 105 A n i h i l a t i o n o p e r a t o r s , 110 A p p r o x i m a t i o n t h e o r e m s 143. 1 4 5 , 1 4 9 , 1 6 6 , 2 2 9 ,3 1 7 , 3 2 1 , 3 4 2 , 3 4 7 , 362, 372. A s y m p t o t i c e x p a n s i o n s 426 B a s e of b o u n d e d s e t s , 10 B a s e of o - n e i g h b o r h o o d s , 4 B a s i s (Schauder), 354 B i p o l a r t h e o r e m , 17 Bounded s e t s , 7, 1 0 B o r e l ( F o u r i e r ) t r a n s f o r m 167, 1 6 9 , 196. B o r n i v o r o u s s e t s , 12 B o r n o l o g i c a l d u a l , 11 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 s , 2 B o r n o l o g i c a l s u b s p a c e , 12 B o r n o l o g i c a l topology, 12 Bornological v e c t o r s p a c e , 10 B o s o n f i e l d s , 109 Cauchy s y s t e m , 8 Cauchy's integral f o r m u l a , 86 C n and C a m a p s : s e e d i f f e r e n . tia bility . c o m p a c t m a p p i n g s , 19 c o m p a c t s u b s e t 6 of a F r k c h e t s p a c e , 29 c o m p l e t e n e s s , 9, 134 c o - n u c l e a r s p a c e s , 22 convolution e q u a t i o n s 206, 208, 223, 266,277,326

D. F. N. s p a c e s , 2 8 D i f f e r e n t i a l f o r m s , 377 D i f f e r e n t i a b l e m a p p i n g s , 4 5 , 4 7 , 48, 52, 54, 6 1 , 6 9 , 72, 92 D i v i s i o n of d i s t r i b u t i o n s , 2 4 5 , 252, 262, 268, 272 D i v i s i o n ( o t h e r r e s u l t s ) , 213, 231, 308, 327 D i v i s i o n ( W e i ' e r s t r a s s t h . ) , 246 D o m a i n of e x i s t e n c e , 348 D o m a i n of H o l o m o r p h y , 348 e q u a t i o n , 3 7 6 , 3 7 9 , 388, 398, 401, 404,407 b o p e r a t o r , 377 Exact sequence, 376,404,407,416 E x i s t e n c e d o m a i n , 348 E x i s t e n c e r e s u l t , 154, 222, 235, 266, 3 1 7 , 321, 324, 342, 346, 354, 3 7 9 , 388,401,404,407,410 F i n i t e d iffe r enc e pa r t ia 1 d iffe r e ntia 1 e q u a t i o n s , o p e r a t o r s , 326 F i n i t e l y d i f f e r e n t i a ble m a p p i n g , 5 4 , 288,398 F o c k s p a c e , 109 Fourier-Bore1 t r a n s f o r m , s e e Bore1 F o u r i e r t r a n s f o r m , 167, 186 F r C c h e t s p a c e , 25 G a t e a u x a n a l y t i c , 79 Gateaux differentiable, 54 Gauss m e a s u r e , 383 G e r m s ( H o l o m o r p h i c ) , 182

453

454

Index

H a r t o g s ' t h e o r e m , 123 N u c l e a r i t y of h s ( O , F ) , K ( a , F); H o l o m o r p h i c g e r m s , 182 1 9 3 , 2 0 1 , 428 Holomorphic mapping, s e e analytic N u c l e a r i t y of d(0,F ) , 205 mappings H o l o m o r p h i c r e p r e s e n t a t i o n of FOCI s p a c e s , 109 Paley-W iener -Schwartz t h e o r e m s H o l o m o r p h y ( d o m a i n of), 348 1 8 6 , 3 3 7 , 427 HLirmander's L2 t h e o r y , 3 7 9 P a r t i t i o n of u n i t y , 154 Hypoellipticity r e s u l t , 398 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 , 348 P o l y n o m i a l s , 35, 101 P o l a r b.v. s . , 13 Imaginary exponential polynomials, Polarity, 8 327 P o l a r i z a t i o n f o r m u l a , 36 I m p l i c i t functions t h e o r e m s , 423, 42. P r o j e c t i v e limit, 7 Inductive l i m i t , 1 1 , 26 P s e u d o c o n v e x i t y , 347, 348 I n t e g r a t i o n ( i n H i l b e r t s p a c e s ) , 383 K e r n e l t h e o r e m s , 158, 161, 164

2 3 s p a c e s , 26 L6vi p r o b l e m , 347, 3 4 8 , 3 5 4 Locally convex s p a c e s , 4

M a c k e y - A r e n s t h e o r e m , 18 M a c k e y c l o s u r e topology, 12 M a c k e y c o n v e r g e n t s e q u e n c e s , 14 Ma c key ' s t h e o r e in,17 M a c k e y topology, 17 M e a n v a l u e t h e o r e m 55 M e t r i z a b l e s p a c e s , 25 M o n t e l ' s t h e o r e m , 126 M u l t i l i n e a r m a p s , 35 Na c h bin ' s a p p r o x i m a t i o n t h e o r e m 151 N a c h b i n ' s p o r t e d topology, 132, 426 Naturally reflexive spaces, i 5 N o r m a l c o n v e r g e n c e , 101 N u c l e a r bornology (of a F r 6 c h e t space), 34 N u c l e a r m a p p i n g s , 19 N u c l e a r Silva s p a c e s ; 28 N u c l e a r s p a c e s , 22 N u c l e a r s u b s e t s , 29

Quasi nuclear mappings, 20 R d d i u s of c o n v e r g e n c e , 426 Rapidly d e c r e a s i n g sequences, 34 R e f l e x i v e s p a c e s , 1 5 , 16 R e f l e x i v i t y of 3c ( 0 , F) and &(0, F), S 142 R e p r e s e n t a t i o n of F o c k s p a c e s ( H o l o m o r p h i c ) , 109 R u n g e a p p r o x i m a t i o n t h e o r e m , 362 S e m i - M o n t e l 1. c . s . , 126 S c h w a r t z E - p r o d u c t , 157, 159, 164 S c h w a r t z p r o p e r t y , 138 Schrvartz s p a c e s , 22 Silva spaces, 27-28 S t r i c t l y c o m p a c t b o r m l o g y , 33 S t r i c t l y c o m p a c t s e t s , 33 S u p p o r t (of a n e l e m e n t of 6'(!2)),417 T a y l o r ' s f o r m u l a s , 55 Tdylor series expansion, 8 7 T o p o l o g i c a l b . v . s . , 27 T o p o l o g i e s on X s ( n , F) and d ( 0 , F), 129, 131,426 U n i q u e n e s s of a n a l y t i c c o n t i n u a t i o n 82

index

V i t a l i ' s t h e o r e m , 128

MI e i'e r s t r a s s d iv i s ion t h e o r e m , 2 46 W ei'erstrass preparation theorem, 246 Z o r n ' s t h e o r e m , 118

.

455

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