Foundations of Complex Analysis in Non Locally Convex Spaces: Function Theory Without Convexity Condition

FOUNDATIONS OF COMPLEX ANALYSIS IN NON LOCALLY CONVEX SPACES FUNCTION THEORY WITHOUT CONVEXITY CONDITION

NORTH-HOLLAND MATHEMATICS STUDIES 193 (Continuation of the Notas de Matem&tica)

S a u l LUBKIN University of Rochester New York, U.S.A. Editor:

ELSEVIER 2003 Amsterdam

- Boston - Heidelberg - London - New York - Oxford

Paris - San Diego - San Francisco - Singapore - Sydney-

Tokyo

FOUNDATIONS OF COMPLEX ANALYSIS IN NON LOCALLY CONVEX SPACES FUNCTION THEORY WITHOUT CONVEXITY CONDITION

Aboubakr BAYOUMI

King Saud University Riyadh, Saudi Arabia

ELSEVIER 2003 Amsterdam

- Boston - Heidelberg

Paris - San Diego - San Francisco

- London - New York - Oxford - Singapore

- Sydney-

Tokyo

ELSEVIER B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands 9 2003 Elsevier B.V. All rights reserved. This work is protected under copyright by Elsevier, and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws. Permission of the Publisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery. Special rates are available for educational institutions that wish to make photocopies for non-profit educational classroom use. Permissions may be sought directly from Elsevier's Science & Technology Rights Department in Oxford, UK: phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-mail: [email protected]. You may also complete your request on-line via the Elsevier homepage (http://www.elsevier.com), by selecting 'Customer Support' and then 'Obtaining Permissions'. In the USA, users may clear permissions and make payments through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA; phone: (+1) (978) 7508400, fax: (+1) (978) 7504744, and in the UK through the Copyright Licensing Agency Rapid Clearance Service (CLARCS), 90 Tottenham Court Road, London W1P 0LP, UK; phone: (+44) 207 631 5555; fax: (+44) 207 631 5500. Other countries may have a local reprographic rights agency for payments. Derivative Works Tables of contents may be reproduced for intemal circulation, but permission of Elsevier is required for external resale or distribution of such material. Permission of the Publisher is required for all other derivative works, including compilations and translations. Electronic Storage or Usage Permission of the Publisher is required to store or use electronically any material contained in this work, including any chapter or part of a chapter. Except as outlined above, no part of this work may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the Publisher. Address permissions requests to: Elsevier's Science & Technology Rights Department, at the phone, fax and e-mail addresses noted above. Notice No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made. First edition 2003 Library of Congress Cataloging in Publication Data A catalog record from the Library of Congress has been applied for.

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To EGYPT To Cairo University To Iman, Mohammad and Youmna

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Preface Infinite Dimensional Holomorphic has a long history as a field of mathematics and yet its strong foundation for non locally convex spaces in the realm of contemporary mathematics is relatively new. In fact, the linear and topological properties of non locally convex topological spaces have received much attention in recent years. However the holomorphic properties of such spaces have been neglected. In 1978 we started our study and research towards the foundations of some of the holomorphic properties of certain non locally convex spaces at Uppsala University. During the last 25 years we have devoted our efforts to different fundamental holomorphic problems in several classes of topological spaces in the absence of convexity condition, and succeeded to solve most of them, like Levi problem and bounding sets problems and many other problems which may represent the backbone of Complex and Functional Analysis. I am now glad to present a collection of most of the published results in this field in this book " F o u n d a t i o n s o f C o m p l e x A n a l y s i s in N o n L o c a l l y C o n v e x S p a c e s " which may be the first book written in this new field. The book may also be entitled: F u n c t i o n s t h e o r y w i t h o u t c o n v e x i t y c o n d i t i o n . In fact, several of the results obtained show the s h a r p c o n t r a s t between it and the field of Holomorphy of locally convex spaces. In addition, new concepts and terminologies have been established for the first time by which we have been able to overcome the absence of non convexity and nonlinearity in our cases. Furthermore, these foundations verify, unlike what was previously believed by some mathematicians, t h a t there is no theory of holomorphy we can deal with it without convexity condition, see[175]. As it was expected we have encountered some obstacles and difficulties in developing this "field at the beginning. This normally arises from the absence of convexity of the fundamental base of neighbourhoods of non locally convex spaces. Therefore, it was essential first for us to generalize several of the important theorems in functional analysis to be suitable and helpful to tackle and solve the problems which had already appeared. Those theorems have been generalized for non linearity and non convexity. As for Functional Analysis we have been able to obtain some generalized

viii forms, as for example, of: H a h n B a n a c h t h e o r e m s , B r o u w e r ' s F i x e d Point T h e o r e m , L a g r a n g e M e a n - V a l u e T h e o r e m s , K a k u t a n i ' F i x e d Point T h e o r e m , B a n a c h S t e i n h a u s t h e o r e m and K r e i n - M i l m a n t h e orem. Regarding Infinite Dimensional C o m p l e x A n a l y s i s , w i t h o u t conv e x i t y c o n d i t i o n , we have introduced and studied the new concept of Quasi-Differentiability and its relations to the classical Fr~chet and G~teaux Differentiability. As a consequence we have obtained new and general versions, as for example, of the following Theorems: F u n d a m e n t a l T h e o r e m of C a l c u l u s , M e a n - V a l u e T h e o r e m for I n t e g r a l s , and L a g r a n g e

Mean-Value Inequalities. Also, the Levi p r o b l e m s o l u t i o n s in different classes of non locally convex spaces have been obtained which means the proof of c e r t a i n r i c h n e s s of t h e class of h o l o m o r p h i c f u n c t i o n s d e f i n e d on t h e s e s classes. During the study of Bounding and w e a k l y - b o u n d i n g (or L i m i t e d ) sets, it is showed that they are different, in several classes of non locally convex spaces. The role of the new terminology of what we have c a l l e d " p - e x t r e m e p o i n t s " or quasi extreme points of non convex sets, which arises from the geometry of the unit balls of our spaces, is rather important. It has applications in several branches of pure and applied mathematics such as: Complex Analysis, Functional Analysis, Numerical Analysis, Non Convex Analysis, Nonlinear Programming, and Differential Equations. Also the role of Quasi -Differentiability is very important and fundamental to build and establish this field of Mathematical science which I consider it sometimes a new field. We have given some applications to illustrate the significance of some of the above mentioned results in this book.Therefore, the book a i m s also to present some of the recent contributions to the theory of Functional analysis and Complex analysis without convexity condition.We hope that this purpose has been successfully achieved according to the given arrangement of our text. This text is mainly addressed to Complex analysis for topological spaces which are not necessarily locally convex with emphasis to locally bounded and locally p-convex spaces (0 < p < 1). We believe, however, that it will also attract some functional analysis and topology oriented readers because a great part of the material presented here has a functional analysis and topological flavour. Each chapter contains o p e n u n s o l v e d p r o b l e m s , so

ix that we hope that the present text will prove interesting for researchers, M.Sc. and Ph.D. students, and be successful enough to e m p h a s i z e t h e i m p o r t a n c e of this field to Pure~ A p p l i e d a n d I n d u s t r i a l m a t h e m a t i c s in particular, and to the other branches of science in general. Aboubakr Bayoumi K i n g S a u d University, R i y a d h

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xi

Acknowledgments The author would like to thank prof.C.O.Kisleman at Uppsala University for his encouraging support and his never failing interest in this field, to whom I am gratefully indebted for his friendship since I got my Ph.D. at Uppsala University in 1979. I am thankfull to him for valuable discussions. It is with pleasure that I express my gratitude to all my teachers at Mathematics Department of Uppsala University for their kindness and cooperation; Special thanks to my friends Mats Essen, Benget Josefson, Urban Cegrel, Gunner Berg, and Life Abrahmson. Also to my teachers and colleagues at Mathematics Department of Cairo University I want to express my deeply and sincerely felt thanks for their kindness and friendship, from where I got my B.Sc.degree. I want to express my appreciation to my colleagues at King Saud University for fruitful discussions and kind interest during the preparation of this book.

Ch.1 Fundamental Theorems (Functional Analysis) i Ch.2 Theory of Polynomials in F-Spaces

L I

......

/I

inm.v.s, ii

Levi Problem

........

L_J I

....................................................

I t

I ...............

Ch.9 Bounding and Weakly-Bounding Sets in T.V.S. i............................c,:i0

Ch.3 Fixed points and p-Extreme points -

chi4

...........................................

Quasi (Bayoumi) Differential ............................Calculus

' ............

Ch.6 Higher Quasi-Differential in F-spaces Ch.7

QuasiHolomorphic Maps

---I l t 1

.

Ch.5

Generalized MeanGener Valu, lue Theorems RealI &Complex Spaces

.

.

.

.

.

.

.

.

_

.....

i .................................................

Ch.8 New Versions of main Theorems

Contents 1

2

FUNDAMENTAL THEOREMS IN F-SPACES 1.1 L I N E A R MAPPINGS . . . . . . . . . . . . . . . . . . . . 1.1.1 Linearity and Boundedness ............. 1.1.2 The Space L(E,F) . . . . . . . . . . . . . . . . . . 1.2 HAHN-BANACH THEOREMS ............. 1.2.1 The Main Results . . . . . . . . . . . . . . . . . . 1.2.2 Hahn-Banach Theorem in Locally Bounded Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Failure of Polynomial and Holomorphic Extensions in F-Spaces ............

1 1 2 4

7 8 11 13

1.2.4

Hahn-Banach Theorem in Locally Pseudoconvex Spaces . . . . . . . . . . . . . . . . . . . . . . .

14

1.2.5 1.2.6

Hahn-Banach Theorem in t.v.s ........... Examples of F-spaces and Non F-spaces

17 18

1.3

OPEN

1.4

UNIFORM

THEORY

MAPPING

THEOREM

BOUNDEDNESS

OF POLYNOMIALS

.....

.............. PRINCIPLE

20 ......

25

IN F-SPACES

29

2.1

MULTILINEAR MAPS . . . . . . . . . . . . . . . . . . . 2.1.1 Continuous Multilinear Maps ........... 2.1.2 Locally Bounded Spaces L(E1,...,Em;F) ...... 2.1.3 Examples of Bilinear Maps ............. 2.1.4 Natural Isometry . . . . . . . . . . . . . . . . . . .

29 29 32 33 34

2.2

POLYNOMIALS 2.2.1 Symmetric

35 35

2.2.2

Multilinear

2.2.3 2.2.4

Polynomials Continuous

2.2.5

The

OF P-NORMED SPACES ...... Multilinear Maps ........... Formula

. . . . . . . . . . . . . . . . .

36

. . . . . . . . . . . . . . . . . . . . . . . Homogeneuous Polynomials ....

37 40

Generalized

Universal xiii

Constant

Trl, m q / p

m!

. . .

42

xiv

CONTENTS 2.2.6 2.2.7

The Space P(E,F) Banach-Steinhaus

. . . . . . . . . . . . . . . . . . .

Theorem

for P o l y n o m i a l s . .

45 46

3

FIXED-POINT AND P-EXTREME POINT 49 3.1 p - E X T R E M E P O I N T I N N O N L O C A L L Y C O N V E X SPACES ............................. 50 3.1.1 P r o p e r t i e s o f p - E x t r e m e P o i n t s ......... 54 3.1.2 G e n e r a l i z e d M i l m a n ' s T h e o r e m .......... 56 3.1.3 A p p l i c a t i o n s ...................... 58 3.2 GENERALIZED FIXED POINT THEOREM .... 62 3.2.1 G e n e r a l i z e d B r o u w e r s ' s F i x e d P o i n t T h e o r e m 63 3.2.2 Generalized Kakutani~s Fixed Point Theorem 66 3.3 GENERALIZED KREIN-MILMAN THEOREM . . 68 3.3.1 G e n e r a l i z e d K r e i n - M i l m a n T h e o r e m . . . . . . . 69 3.3.2 S e p a r a t i o n T h e o r e m s in S o m e S e q u e n c e FSpaces .......................... 72

4

QUASI-DIFFERENTIAL CALCULUS 4.1 Q U A S I - D I F F E R E N T I A B L E MAPS ........... 4.1.1 Q u a s i - D i f f e r e n t i a b l e M a p s in F - S p a c e s . . . . . 4.1.2 P r o p e r t i e s o f Q u a s i - D i f f e r e n t i a l s 4.1.3 Quasi-Differentials of Multilinear Maps ..... 4.1.4 Q u a s i - D i f f e r e n t i a l s o f P o l y n o m i a l s . . . . . . . . . 4.1.5 I n v e r s e M a p p i n g T h e o r e m ............. 4.1.6 R e a l a n d C o m p l e x C a s e s . . . . . . . . . . . . . .

77 77 78 83 84 85 87

5

GENERALIZED MEAN-VALUE THEOREM 89 5.1 M E A N - V A L U E T H E O R E M IN REAL SPACES . . 89 5.2 MEAN-VALUE THEOREM I N C O M P L E X S P A C E S 94 5.2.1 M e a n - V a l u e I n e q u a l i t y . . . . . . . . . . . . . . . . 94 5.2.2 Applications ...................... 96 5.2.3 Examples of Sequence Spaces ........... 99

6

HIGHER QUASI-DIFFERENTIAL IN F-SPACES 101 6.1 S C H W A R T Z S Y M M E T R I C THEOREM ....... 101 6.2 H I G H E R Q U A S I - D I F F E R E N T I A L S .......... 106 6.2.1 T h e Q u a s i - D i f f e r e n t i a l s Dmf a n d dmf . . . . . . . 107 6.3 G E N E R A L S C H W A R T Z S Y M M E T R I C THEOREM 109 6.4 D I R E C T I O N A L DERIVATIVES ............. 111 6.5 Q U A S I A N D F R I ~ C H E T D I F F E R E N T I A L S ..... 115

CONTENTS

7

6.5.1

Finite Dimensional

7.3

6.5.2

Infinite Dimensional

POWER

Case

116

..............

118

SERIES

IN F-SPACES

Series

.............

136

7.2.1

Power

7.2.2 7.2.3

Uniform and Normal Convergence ........ Generalized Cauchy-Hadamard Formula .....

138 139

7.2.4 7.2.5 7.2.6 7.2.7

R a d i u s o f N o r m a l C o n v e r g e n c e Pn . . . . . . . . R a d i u s o f A b s o l u t e C o n v e r g e n c e Pa . . . . . Uniqueness of Power Series ............. Quasi-Differentials of Power Series ........

144 144 146 147

. . . . . . . . . . . . . . . . . . . . . .

QUASI-ANALYTIC MAPS ................. 7.3.1 Quasi-Analytic and Quasi-Holomorphic 7.3.2

Principle

7.3.3

Integral

of Quasi-Analytic Domain

QA(U)

. . .

Maps ....

153

Continuation

...............

154

BOLZANO'S INTERMEDIATE THEOREM 8.2.1 Finite Dimensional Spaces ............. 8.2.2 Degree Theory .....................

8.3

INTEGRAL

8.2.3

BOUNDING 9.1

157 . . . 157

R i e m a n n I n t e g r a t i o n o n [~,/~] . . . . . . . . . . . Curvilinear Integrals ................. Fundamental Theorem of Calculus ........

8.2

Infinite Dimensional MEAN-VALUE

AND

WEAKLY-BOUNDING

BOUNDING SETS 9.1.1 Bounding Sets 9.1.2 Bounding Sets 9.1.3 Bounding Sets

. . in in in

.....

Spaces ............. THEOREM

136

151 . 151

NEW VERSIONS OF MAIN THEOREMS 8.1 F U N D A M E N T A L THEOREM OF CALCULUS 8.1.1 8.1.2 8.1.3

9

Case ...............

QUASI-HOLOMORPHIC MAPS 123 7.1 F I N I T E E X P A N S I O N S AND TAYLOR'S FORMULA124 7.1.1 Finite Expansion .................... 125 7.1.2 Taylor's Formula .................... 128 7.1.3 Quasi-Differential of Taylor Polynomials .... 129 7.1.4 General Mean-Value Theorem ........... 130 7.1.5 Taylor's Formula with Lagrange Remainder . . 133 7.2

8

xv

158 160 161 163 163 164 171

....... SETS

176 179

. . . . . . . . . . . . . . . . . . . . 180 L o c a l l y B o u n d e d F - S p a c e s . . 181 S e p a r a b l e M e t r i c S p a c e s . . . 186 Locally Pseudoconvex S p a c e s 189

xvi

CONTENTS 9.1.4

9.2

B o u n d i n g S e t s in N o n L o c a l l y P s e u d o c o n v e x Spaces .......................... WEAKLY-BOUNDING (LIMITED) SETS ......

191 ' 192

9.2.1 9.2.2

9.3

9.4

W e a k l y - B o u n d i n g S e t in L o c a l l y B o u n d e d Spaces192 T h r e e D i f f e r e n t C l a s s e s of H o l o m o r p h i c F u n c tions ........................... 196 9.2.3 E x a m p l e s o f H o l o m o r p h i c F u n c t i o n s . . . . . . . 199 P R O P E R T I E S OF B O U N D I N G A N D L I M I T E D SETS206 9.3.1 B o u n d i n g S e t s a n d C o m p a c t L i n e a r M a p s . . . 206 9.3.2 P r o p e r t i e s of T h e D i f f e r e n t W e a k l y - B o u n d i n g Sets ........................... 209 9.3.3 W e a k * C o n v e r g e n c e in t h e D u a l o f a p - B a n a c h S p a c e A s D i f f e r e n t f r o m N o r m C o n v e r g e n c e . . 211 HOLOMORPHIC COMPLETION ............ 217 9.4.1 H o l o m o r p h i c C o m p l e t i o n in F - S p a c e s . . . . . . 217 9.4.2 H o l o m o r p h i c E x t e n s i o n P r o b l e m ......... 221

10 L E V I P R O B L E M I N T O P L O G I C A L S P A C E S 227 10.1 L E V I P R O B L E M A N D R A D I U S O F C O N V E R G E N C E 2 2 9 10.1.1 P B - S p a c e s ....................... 231 10.1.2 P r o p e r t i e s o f t h e R a d i u s of C o n v e r g e n c e . . . . 234 10.1.3 T h e L e v i P r o b l e m in P B - S p a c e s . . . . . . . . . . 241 10.2 L E V I P R O B L E M ( G R U M A N - K I S E L M A N APPROACH) 243 10.3 L E V I P R O B L E M ( S U R J E C T I V E LIMIT APPROACH)253 10.4 L E V I P R O B L E M ( Q U O T I E N T MAP APPROACH) 258 Bibliography Notations Index

Chapter I

FUNDAMENTAL THEOREMS IN F-SPACES This chapter is devoted to the introduction of some of the fundamental theorems in functional analysis in their new versions. Those versions may be suitable to tackle several types of problems in Infinite Dimensional Holomorphy, in the absence of convexity condition. In fact this chapter consists of ~ parts deal with: I - L i n e a r m a p pings, I I - H a h n - B a n a c h t h e o r e m s , I I I - O p e n m a p p i n g theorem, and I V - U n i f o r m boundedness principles. The well known results of the open mapping theorems have been introduced with different proofs, which may be similar to the classical ones. The results on Hahn-Banach theorems and of the uniform boundedness principles are totally new and the reader may not find them in any other books on functional or complex analysis. They have several applications as we we shall see here in our book.

1,1

LINEAR MAPPINGS

Linear maps between p-normed spaces and linear functional on a p-normed space play a central role in functional analysis and the study of calculus and holomorphy. In this chapter we study the interrelationship between continuity and boundedness. As a consequence we prove the three basic principles of p-Banach spaces, namely the Hahn-Banach theorem, the Open Mapping Theorem and the Uniform Boundedness Theorem, see parts II,III &IV. All the spaces considered in this book unless otherwise stated are corn-

2

CHAPTER

1 FUNDAMENTAL

THEOREMS

IN F-SPACES

plex locally bounded spaces (Lbs), so they are p-normed spaces for some 0 < p _ 1, and consequently they are quasi-normed spaces, Rolewicz [186]. In particular E and F will denote the p-normed and the q-normed spaces respectively, where (0 < p, q _ 1). 1.1.1

Linearity and Boundedness

D e f i n i t i o n 1 A p - n o r m on a vector space E over K is a mapping from E to ~+ satisfying (i) Ilxll = 0 iS and only if x=O (ii) II~xll- I~lPllxll, So~ every A E K, x E E

I1.11

(iii) IIx + Yll -< Ilxll + IlYlI, for every x, y ~ E. D e f i n i t i o n 2 A q u a s i - n o r m on a vector space E function II" II on E satisfying: (a) Ilxl[ - 0 if and only if x -- 0

is a real non-negative

(b) Iltxll- Itl Ilxll

(~) IIx + Yll < o- (llxll + IlYll) for all x , y E E, t E K , and some constant ~ > 1, which is independents of x, y. The smallest a for which (c) holds is called the q u a s i - n o r m c o n s t a n t

oS (E, I1 II). The sets {x" Ilxll < E}, E > 0, form a base of neighbourhoods of 0 for a metrizable vector topology on E. If (E, I1" [[) is quasi-normed, then there exists 0 < p _ 1 and a p- norm [1" lip which determine the topology of E. The function ]]. [[1/p is then a quasi-norm on E which determines the topology of E. D e f i n i t i o n 3 Let E and F be p-normed, and q-normed spaces respectively over the same field K . A map A " E --~ F is said to be l i n e a r if A ( t x + y) = t A ( x ) + A(y) for all x, y in E

and for all t i n K .

A linear map may not be continuous in general as the following example shows.

1.1 L I N E A R

MAPPINGS

3

E x a m p l e 1 Let E = C ( I , K ) be the p- normed space of all continuous functions on the unit interval I - [0,1] under the sup p-norm: Ilfllp = supxe/If(x)lp. Let F - C~(I,tr) be the p-norrned subspace of E consisting of those functions f which have continuous derivative dr. Then the linear map f ---, d f of F into E is not continuous. In fact, if we set f ~ ( x ) = (sin n x ) / n , then the sequence (f~) converges to O, whereas the sequence (df~) does not converge to O.A D e f i n i t i o n 4 Let U C E be a nonempty set of a p-normed space E and let F be a q-normed space, (0 < p , q <_ 1). A linear mapping A : U --~ F is said to be b o u n d e d if there is a real number C such that for all x C U,

(1.1)

IIAxll 1/q <_ ClJxlJ1/p i.e.

[IAxII <_ Cllixll q/p,

C~-C

q

F o r m u l a (1.1) shows t h a t a b o u n d e d linear m a p p i n g m a p s b o u n d e d sets in U to b o u n d e d sets in F. T h e smallest possible C such t h a t (1.1) still holds for all x E U will be called the n o r m of A, d e n o t e d by IIAII-Thus

IIAII- sup(llAxl[1/q/llxll ~/p)

(1.2)

xEU

xr

If U = {0}, we define IIAII = 0. In this case, A - 0 since A(0) = 0. It is useful to note t h a t

IIAxll 1/q ~ IIAIIIIxll lip

(~.3)

This formula will be applied quite frequently. Let A be a b o u n d e d linear m a p p i n g as defined by (1.1). alternative formula for the n o r m of A is

IIAII- sup IIAxll~/q

Then

an

(1.4)

xEU

11~11=1

x/lIxlll/p

In fact, if we set y where x # 0, t h e n Ilyll = 1. Since A is linear, IIAII - sup~ev IIAxll~/q/llxll ~/p - sup IIA(x/IIxlI1/p)II1/q = sup ~ o IIA(y)II 1/q ~r Ilyll=l So we m a y use either of the equivalent formulas (1.2) or (1.4) for the n o r m of A. T h e following t h e o r e m gives some equivalent conditions for the continuity of a linear map.

4

CHAPTER

1 FUNDAMENTAL

THEOREMS

IN F-SPACES

T h e o r e m 1 [26]

Given E p,q < 1) and

and F

p-normed and q-normed spaces respectively, (0 < A:E~F

a linear map, the following conditions are equivalent : (a) A is continuous. (b) A is continuous at 0. (c) There exists M such that IIA(x)I I < MIIxll q/p, x e E. (d) A maps bounded sets onto bounded sets. P r o o f . The implications (a) =~ (b) and (c) r are obvious. (b) =~ (c): If A is continuous at 0, then there exists a closed ball BE(O, r) in E with the centre at the origin with radius r such that the image A(BE) is contained in the unit ball BE(O, 1) of F, i.e. A(BE(O,r)) C_ BE(O, 1). Thus if x ~ - 0 , then we havellA(r 1/p x )11 < 1 and hence ii~ll~/p - ,

IIA(~)II ~ MI Ixl Iq/p where M -

(1.5)

(~)P/q. If x - 0 , this inequality holds trivially.

(c) =v (a): If (c) holds, then

IIA(x)- A(y)II _< M I I x - Yll q/p for every x and y in E; hence A is uniformly continuous on E. II E x a m p l e 2 If p = q, then in (c) we have the following particular case similar to that be given for the normed spaces,

IIA(~)II _< M Ilxll.

1.1.2

T h e S p a c e L ( E , F)

We denote the vector space of all continuous linear maps between E and F by L(E, F). Members of this space are called b o u n d e d l i n e a r m a p s in view of Theorem 1, (c) and (d) and Definition 4. Given A e L(E, F), we can also prove

IIAII- inf{M; IIA(x)ll _< MIIxll p/~ for ~U x e E}

(1.6)

1.1 L I N E A R

MAPPINGS

5

By Theorem 1, IIAII is well-defined and

tlA(x)ll <_ [[A[ll[x[I q/p. It is easy to show that

IIAII = sup{liA(x)ll; Ilxll _< 1} = sup{[lA(x)[; Ilxll = 1} see (1.2), (1.4). As defined above, [[-I[ makes L ( E , F ) a q - n o r m e d s p a c e . This q-norm will be called the u s u a l F - n o r m on L(E, F). If the F - n o r m s on E, F are replaced by equivalent F - n o r m s respectively, it is easy to show that the usual F - n o r m on L ( E , F ) is replaced by an equivalent F-norm. Thus the topology on L ( E , F ) depends only on the topologies on E, F. 2 [26] If F is a q-Banach space, then L ( E , F )

Theorem

Proof.

Let

is also a q-Banach space.

(An) be a Cauchy sequence in L(E, F), and let x E E. Then

]JAm(x) - A~(x)[[ = [[(Am - An)(x)[[ < ][Am

-

An][[[x[[q/p.

So (An(x)) is a Cauchy sequence in F . Since F is complete, (An(x)) converges to an element y in F. If we define A: E --* F by A(x) = y, then A is clearly linear. To see t h a t A is continuous, we note that [[A(x)[] = [] limAn(x)l ] < (sup I]dn[[).]]x]] q/p n

since (A~) is a Cauchy sequence, the sup is finite, hence A is continuous by Theorem 1 (c). Now we show that An ~ A with respect to the usual q-norm on L(E, F). Let e > 0. Choose N such that [ [ A m - A,~[[ < e whenever m, n > N. Then II(A~ - A)(x)I I = 1 ~ II(A~ - A~)(x)ll < r

q/p.

So IIA - A l l - , 0.m

Remark

1 If p = q in Theorem 2, then L(E, F)

But if F = K, space.

is a q-Banach space.

then L ( E , F ) = L(E) and it turns out to be a Banach

6

CHAPTER

1 FUNDAMENTAL

THEOREMS

IN F-SPACES

Linear E x t e n s i o n T h e o r e m Let E be a p-normed space and M M is also a subspace. Thus if

a subspace. T h e n the closure M - o f

x = lim xn, y = lim yn n

n

then x + y = lim(x~ + yn), n

rx -

lim rXn n

where r E K .

Hence M

isasubspaceof

E.

Theorem 3 Let E be a p-normed space, M a subspace of E, and F be a q-Banach space. I f A'M--,F is a continuous linear map, then there exists a unique extension of A continuous linear map X'M-~F

to a

such that

II A I1<11 A' I1_11 A II q P r o o f . The uniqueness of A' is clear from the continuity. We show its existence. Let x E M - and let x - limn Xn with xn in M. Then

II A(xn)

- A(xm) II1/q_
II-II

x ~ - xm II ~Ip .

Hence ( A ( x n ) ) is a Cauchy sequence in F , and since F is complete, ( A ( x n ) ) has a limit in F which we denote by A' (x). The value A ' ( x ) is well-defined, for if x - limn Yn with y~ E F then limn A ( x n ) - l i m , ~ A(yn). Thus we have a well defined map A ~ : M - ~ F. It remains to show t h a t A ~ is a continuous linear map. If y E M - and y - lim Yn with Yn E M , then for r E K,

x + y -- linm(xn + Yn)

and

r x - lim(rx),n

1.2

HAHN-BANACH

THEOREMS

7

Therefore, A'(x + y) = lim A(x~ + yn) = lim A(xn) + lim A(yn) n

n

= A'(x) + A' (y) Similarly,

A' (rx) = rA' (x). Hence A' is linear. For x e M, we have A'(x) = A(x) show that A ~ is a linear extension of A. Finally, we have

trivially. This

II A'(x~) I1= lim II A(xn) II, since the q-norm is a continuous function. From

II ~.~. II

A(x~)I1~/~<11AII. II ~ II~/~ A(x,~)I1~/~<__11AII~ . II x~ II

(see subsection 1.1.1), it follows that

II A'(x)IIp/q
IIp . II

limx~

I1-[I

A

IIp . II x II

Hence A ~ is continuous and II A' I1~11 A IIq .

Since M C M - and A ' = which shows

A on M, we have easily

II A

I1~11 A'

II,

II A I1~11 A' I1~11 A IIq .m Remark2 If E - K , i.e., q = 1, we have II A I1=11 A' II. Also when p - q = 1 we get this classical result for the extension to the closure of subspaces in normed spaces, see[69].

1.2

HAHN-BANACH

THEOREMS

The extension and separation problems for linear functionals have been studied in locally convex spaces by several authors. In fact, the importance of the further study of the Hahn-Banach Theorems arises from its wide applications to several branches of sciences such as

8

CHAPTER

1 FUNDAMENTAL

THEOREMS

IN F-SPACES

Mathematics (Complex and Functional analysis) and Physics ( Thermodynamics ), see Feinber &Lavine [110]. Our goal in this part is the establishment of extension and separation theorems for linear functionals in different classes of topological vector spaces which are not necessarily locally convex, such as locally pseudoconvex spaces and F-spaces (complete metric spaces with F-norms defined by translation invarient metrics). More precisely, we devote this part to give a new and general version of the Hahn-Banach Theorem in vector spaces suitable for application to non locally convex spaces. 1.2.1

The Main Results

D e f i n i t i o n 5 Let E be a vector space over a scalar field K . A non-negative real-valued function q on E is called a s u b l i n e a r f u n c t i o n a l o f type cr if it satisfies, (i) + y) < + q(y)], x, y e E, (ii) q(tx) = tq(x), x e E, t > 0 where a >_ 1 here and throughout this book. q is called a b s o l u t e l y s u b l i n e a r o f t y p e a if (iO is replaced by q(tx) = Itlq(x) for scalars t. It is obvious that a quasi-seminorm is an example of such q, see [207] and [186]. The following theorem provides us with a new version of the HahnBanach Theorem in vector spaces. We extend a linear functional f which is majorized on a subspace M C E by a sublinear functional q of type a defined on E. The majorization will not remain fixed under the extension if a > 1. T h e o r e m 4 [18](1990) Let E be a real vector space, M a subspace of E with c o d i m M - n, and q a sublinear functional of type ~r. Suppose f is a linear functional on M such that, f ( x ) <_ q(x)

On

for all x e M.

Then, there exists a linear extension ] M and f ( x ) < crnq(z)

(1.7)

defined on E for which f - ]

for all z E E.

(1.8)

1.2

HAHN-BANACH

THEOREMS

9

If G = 1 we obtain the classical Hahn-Banach theorem.

P r o o f . Assume first that codim M - 1. Let b E E - M and M1 be the subspace spanned by M U { b } . We claim that there exists a linear extension fl on M1 satisfying, f l ( x ) _< oq(x). (1.9) For any Yl, Y2 E M we have: f(Yl) - f(Y2) -- f ( Y l - Y2) _< q(Yl + b - Y2 - b) < G[q(yl + b) + q ( - Y 2 - b)] By gathering the terms of Yl on one side and that of y2 on the other side we have, -Gq(-Y2 -b)-

f(Y2) _< oq(yl + b) - f(Yl)

(1.10)

Assume yl is held fixed and Y2 is allowed to vary over M. Then, s u p { - c r q ( - y 2 - b ) - f(Y2); Y2 E M}, denoted by a, exists. Similarly, if we keep y2 fixed and allow Yl to vary over M, inf{aq(yl + b ) - f ( Y l ) ; y l C M } , denoted by/3, exists. From (1.10) it is clear that a _
a_<7_
(1.11)

we have - b) - f ( y ) <_ fl/<_ oq(y + b) - f ( y )

Every element x C M1 can be written as x - Ab+y, where b E E - M , a n d y C M. Now define a real-valued function: fl(x) - fl(Ab + y) - A7 + f ( y )

(1.12) A E 1~

(1.13)

where 7 is given by (1.11). Obviously, fl is linear on M1 and fl (x) - f ( x ) on M; and fl (x) <_ crq(x) for x ~ MI" Assume that A > 0. By (1.12) we have 9/+ f (Y) <_ oq(y + b)

Replacing y by ~ in (1.14) we get, Y Y -t, + f ( ~ ) _ Gq(~ + b)

(1.14)

10

CHAPTER

1 FUNDAMENTAL

THEOREMS

IN F-SPACES

Thus A-y + f (y) <_ aq(y + )~b) which shows that f l (x) < aq(x) < 0 we similarly get by (1.12) that fl(x) < crq(x) on M1.

on M1. If

Now if codim M - n, we repeat the process of extension of ] n steps to get the extension ] with the required property; that is, ](x) < anq(x) on E. Finally, if ~r = 1, that is, if q is sublinear of type 1, then we have the classical case where the proof can be completed without the restriction that codim M is finite. In fact, we shall apply the Zorn's lemma on the set of all linear extensions f with f - f on M and f _ f on D], the domain of f, see [163]. This completes the proof of theorem. II R e m a r k 3 It is to be noted that, if codim M = c~ and a > 1, then the process of extensions will naturally be repeated infinitely many times and we get a n T c~. Therefore, the Zorn's lemma cannot be applied for that case. In fact, this will give account to the results concerning the failure of the Hahn-Banach theorems in non-locally convex F-spaces, given by Kalton /131]. The following theorem has principal applications as we shall see in the next sections. In fact it represents the generalized form of the above theorem where E could be a complex vector space. T h e o r e m 5 [18] (1990)

Let E be a real or complex vector space and q an absolutely sublinear functional of type a on E. Furthermore, let f be a linear functional defined on a subspace M of E with codimension n, and satisfies, I f ( x ) I~ q(x)

for all x e M.

Then, there exists a linear extension f on M and I f ( x ) I< a~q(x)

(1.15)

defined on E for which f = f

for all x e E.

(1.16)

If a = 1 we have the classical Hahn-Banach theorem. P r o o f . The proof can be established in an analogous manner with the classical case using Theorem 4, see for example Kreysig [[163],p.214]. II

1.2 H A H N - B A N A C H T H E O R E M S 1.2.2

11

H a h n - B a n a c h T h e o r e m in L o c a l l y B o u n d e d

Spaces

Definition 6 A topological vector space E is said to have the H a h n - B a nach e x t e n s i o n p r o p e r t y ( H B E P ) if every continuous linear functional f on a closed subspace has a continuous linear extension to the whole space. E is said to have the H a h n - B a n a c h s e p a r a t i o n property, ( H B S P ) if E ~ separates points of E from closed subspaces not containing them. Definition 7 (k-spaces)An F-space E is said to be a k-space i f " whenever G is an F-space and H C_ G is a subspace of G of dimension one such that G / H ~_ E then H

is complemented in G, that is G~_H@E.

In fact the concept of the k-spaces is used in solving the three-space p r o b l e m which arises in [[132], ch.5]. It stats that: Assume (p) is some property of an F-space and E has a closed subspace M such that M and E / M have the property (p). Does E necessarily have the property (p) ? In this section, we study, as applications of T h e o r e m 5, the H B E P and H B S P in locally bounded F-spaces E.

Theorem 6 ( E x t e n s i o n T h e o r e m in Lbs)[18] (1990) Let f be a continuous linear functional on a closed subspace M of a locally bounded F-spaces E, and let c o d i m M - n. Then f has a linear extension f to the whole of E satisfying,

ll:llM-< ll]lIE_< :II/II-

(1.17)

Consequently,/or c r - 1, [[filM = [[f[[M for any closed subspace M . P r o o f . The case of M = {0} is obvious, for f - 0 on E is the extension of f - 0 on M. So assume M ~ {0}, and let q(x) = [[f[[M[IX[[1/p for all x e E , where E is metrized by I[-[[, a p-norm on E. Hence q is an absolutely sublinear functional of type a. Since I f ( x ) [ < [[f [[M [IX[[1/p -- q(x), we can apply T h e o r e m 5 to get a linear extension f on E with

[](x)[~

o'nq(x) =

,x

E,

12

CHAPTER

1 FUNDAMENTAL

i.e.~

THEOREMS

IN F-SPACES

Ilfl] _< ~IIIIIM

(1.18)

Since under extension the norm of f cannot be decreased, we shall also have

II/IIE _> IlflIM. This completes the proof. I Kondia [165] proved that : "If M is a p r o p e r s u b s p a c e of a locally b o u n d e d F - s p a c e E, t h e n for e v e r y e > 0, t h e r e is y E E w i t h IlyII = 1 s u c h t h a t t h e e l e m e n t s x of M will s a t i s f y I I x - Yll >- 1 - e". We use this result to prove the following separation theorem. Theorem

7 ( S e p a r a t i o n T h e o r e m in Lbs)[18](1990)

Let M be a closed subspace of a locally bounded F-space E with finite codimension n, and b C E - M. Then, there is a continuous linear functional f on E, satisfying, 1 <_ IlfllE

<--G", f

IM--0

and

f (b) - d 1/p

(1.19)

where d-

Consequently p = 1, and M

inf IIY - bilyEM

E / M has the point separation property. is any closed subspace then IIfllE- 1.

P r o o f . Let M1 be the subspace spanned by M bounded linear functional f by,

f (x) - f (y + ab) - ad 1/p

x E M1,

In particular, if

a n d b . On

M1 definea

y C M, a C K

(1.20)

We show that f satisfies (1.19) after will be extended to E. The linearity of f is obvious. Since M is closed, d > 0 so that f =/: 0. Now a = 0 gives f(y) = 0 on M. For a = 1 and b ~r 0 we have

I(b)Let a -r 0, using (1.20) and noting that ~ If(x)l-

lal dup - [ a I inf IlY - bll lip ~yCM

i.e., ]]fllM~ <-Ilxll 1/p. Hence f

E M

we obtain

loll II- y -bll oL

l/p-

IlY + ozbl] lip

is bounded on M1 with IlfllM1 <- 1.

1.2

HAHN-BANACH

THEOREMS

13

We also show that IIflIM~ -> 1. By the definition of the infimum, M contains a sequence (Yn) such that IlYn - b l l --* d, let xn = Y n - b. Then we have f ( x ~ ) = - d 1/p when a = - 1. Also

IlfllMx-

sup

If(x)I

>

If(xn)I

d I/p

d I/p

-

-

Hence IlfllM~ -- 1. By theorem 6, we can extend f continuously to f on E such that

llfllM1<

llfIIE< o~llfllM~

i.e.,

1 _< II]llE _< cr'~ This completes the proof. II 1.2.3

Failure of Polynomial and Holomorphic E x t e n s i o n s in F - S p a c e s

Stiles [201] proved that the space lp, (1 > p > 0), does not have the HBEP. He showed that a subspace M of lp, with an/1-precompact unit ball, cannot be extended to the whole of lp. Shapiro [190] generalized this result to every non locally convex F-spaces E with basis. Kalton [131] dropped the assumption that E has basis in Shapiro's result. We use Kalton's result to study the extension and separation properties for polynomials and holomorphic functions. D e f i n i t i o n 8 A topological vector space E is said to have the p o l y n o m i a l H a h n - B a n a c h e x t e n s i o n p r o p e r t y ( P - H B E P ) if every continuous polynomial on a closed subspace can be extended continuously to the whole space. Similarly, E is said to have the H o l o m o r p h i c H a h n - B a n a c h ext e n s i o n p r o p e r t y ( H - H B E P ) if every holomorphic function (continuous and Gdteaux analytic) on a closed subspace can be extended holomorphically to the whole space. The following interesting theorem is a consequence of Kalton'result [131]. T h e o r e m 8 [18](1990)

Assume E is a non locally convex F-space. Then E polynomial Hahn-Banach extension property.

does not have the

14

CHAPTER

1 FUNDAMENTAL

THEOREMS

IN F-SPACES

P r o o f . By Kalton's result [131] mentioned above, there exists a continuous linear functional f on a closed subspace M of E which does not have the continuous extension to E. Since continuous linear functionals are continuous polynomials, f represents the required continuous polynomial on M which does not have a continuous extension to E. I Let us now consider the holomorphic extension in F-spaces. That is " W h e t h e r every holomorphic function on a subspace c a n b e e x t e n d e d t o a h o l o m o r p h i c f u n c t i o n o n E ".

M

of E

In fact, several authors studied holomorphic extension in locally convex spaces. For example Dineen [82] showed that, not every holomorphic function on the closed subspace in lcr of co can be extended holomorphically to l~. That is l ~ does not have H-HBEP. Aron and Berner [[8] ,propl.1] characterized those holomorphic functions f on co which can be extended holomorphically from co to l~. He proved that f should be bounded on bounded sets. In non locally convex spaces we studied the holomorphic extension when M is either dense or a closed subspace of certain metric spaces, (see section 9.4, Ch.9, on holomorphic extension problems). In what follows we extend some of their results to F-spaces. T h e o r e m 9 [18](1990) A s s u m e E is a non locally convex F-space. Then E holomorphic Hahn-Banach extension property.

does not have the

P r o o f . Since a continuous linear functional on a closed subspace M of E is holomorphic, the conclusion will be reached if we apply the above Kalton's result [131]. I 1.2.4

Hahn-Banach Pseudoconvex

Theorem Spaces

in L o c a l l y

Let E be a locally pseudoconvex topological vector space (Lps), that is, the topology of E is defined by a fundamental family of absolutely pseudoconvex neighbourhoods of the origin. D e f i n i t i o n 9 A subset A of a vector space E is called a b s o l u t e l y p s e u d o c o n v e x if it is absolutely p-convex for some i >_ p > O. Here A is absolutely p-convex if ax + by E A whenever x, y E A and lal p + Ib]p <_ 1, a, b E (T.

1.2

HAHN-BANACH

THEOREMS

15

In fact, the topology of a Lps E can be determined by a family of Piseminorms IIxll~, i E A, which turns out to be metrizable if the family is countable. In this section we study the H B E P and H B S P in Lps The following theorem gives a general extension version in a Hausdorff Lps. T h e o r e m 10 ( E x t e n s i o n T h e o r e m in Lps)[19](1995)

Let E be a Hausdorff locally pseudoconvex space. If f is a continuous linear functional defined on a closed subspace M of E with codimension n, then f has a continuous linear extension ] to E, satisfying

If(x)l ~ cr'~qu(x), x e E where qu is the p-homogeneous Minkowsky functional of an absolutely pconvex O-neighborhood U in E. P r o o f . Since f is continuous on M, V-

{x e M; If(x)I < 1}

is a 0-neighborhood in M. There exists an absolutely p-convex 0- neighborhood U in E for some 1 > p > 0, such that U A M C V. T h e p homogenous gauge qv of U is a continuous p-seminorm on E, (Waelbroeck [207] ), such that If(x)l < qv(x) on M. This is because U M M C_ V, and E is defined by a family of pi-seminorm (qi), i E A. By Theorem 5, there exists an extension f of f to E such that I/(x)l < anqu(x), and f is then continuous. Notice that

If(x) - f(y)l ~ onqu( x

-

Y)"

This completes the proof. I R e m a r k 4 (i) If E is a locally pseudoconvex F-space and codim M < c~, then E / M is a k-space( by [131], Th. 5.2). Let us recall that: An Fspace E is k-space if whenever G is an F- space and H is a subspace of G with dimension one such that G / H ~_ E then H is complemented in G, i.e. G~_H@E. (ii) The above theorem can also proved as follows: The topology of E is defined by a family of continuous p-semi-norms (qa). The relative topology of M is generated by the family of restrictions q-~ of qa to M (i. e. Cta(x) = q~ (x) for x e M).

16

CHAPTER

1 FUNDAMENTAL

THEOREMS

IN F-SPACES

A linear functional f on E is continuous if and only if there exists 7 > 0 and finitely many ql, ..., q~ from (qa) such that

If(x)l < ~ max qi(x) l
for all x C M.

Hence applying Theorem 5, there exists an extension f of f to E such that If(z)l < -)'a n max qi(x) l
We notice that, I f ( x ) - / ( Y ) I - <

for all z E E.

~/crnq(x - Y ) , where q(x) - maxl
The next theorem extends one of the classical Hahn-Banach separation theorems to Lps. Theorem

11 ( S e p a r a t i o n T h e o r e m in Lps)[18](1995)

Let M be a closed subspace of a Hausdorff locally pseudoconvex space E with finite codimension and b E E - M. Then there exists a continuous linear functional f on E such that II f

IIM-O

and

f (b)-l.

(1.21)

Consequently, M is weakly closed. If M is locally convex then f will be continuous for any closed subspace M. Further, if E is an F-space then E / M has the point separation property. P r o o f . On the space M1 spanned by M and b, we define a linear function

f by /(x +

e C, x e M).

=

(1.22)

It is obvious that f ( z ) = O o h M a n d f ( b ) = l . The hyperplane b + M is closed and does not contain 0. Hence there exists an absolutely p-convex closed neighborhood U of 0 which does not intersect b. Since U = {x; q(x) < 1} for some p-seminorm q, (1 _> p > 0) (e.g. q could be the p-homogenous gauge qv of U), we have q(x + c~b) > 1 for x E M. In case a :/: 0, we obtain for x E M,

If(x + ab)l -

< [ctlql/p(b + x / a ) - ql/p(ab + x)

(1.23)

and obviously this inequality holds also for a = 0. Consequently, f is continuous on M1. Therefore we can extend f to E according to Theorem 7 to obtain the linear extension with the required property. T h a t is,

IlflIM

--

0

and

f(b) = 1.

1.2

HAHN-BANACH

THEOREMS

17

If codim M = n, the steps of extensions will be finite to have a continuous functional on E. As M is weakly closed the quotient E / M has the separation property, see [[132], lemma 4.1]. This completes the proof of theorem. m R e m a r k 5 Allan Shawkat gave an example of a non locally convex space E with the HBEP, where E has a non countable algebraic basis, and E is not a metrizable space, see [107]. Also Gergory and Shapiro [111] showed that if the Makey topology and the weak topology on a vector space are not the same, then there are many non locally convex topologies, lying between them with the HBEP.

1.2.5

Hahn-Banach

Theorem

in t.v.s.

Let E be a topological vector space (t.v.s.). Recall that a function q" E --, R+ is called a s u b l i n e a r f u n c t i o n a l of t y p e a, if it satisfies q(x + y) <_ a(q(x) + q(y)) for some a > 1 and q(tx) - [ t l q ( x ) , x, y E E, t EfT. If [[x[[ is a p-norm then [[x[[1/p is an example of such a functional. In what follows we give a generalization of the classical Hahn-Banach theorem using Theorem 5. m T h e o r e m 12 [19]( L i n e a r E x t e n s i o n i n t . v . s . ) ( 1 9 9 5 ) Let E be a topological vector space and q a continuous sublinear functional of type a. If f is a continuous linear functional on a closed subspace M with codimension n such that

If (x)l <_q(x),

for all

xe M

then, there exists a continuous linear extension f

I] (x)l <

~q(x),

on E

such that

fo~ ~zl x e E

,v

and f -

f

on M .

P r o o f . Since E is a vector space, there exists a linear extension f such that ] ( x ) - f ( x ) for a l l x e M and If(x)[
I] (x) - ] (y)j < ~ q ( x which completes the proof, m

y) -~ o, as x -~ y

18 C H A P T E R

1 FUNDAMENTAL

THEOREMS IN F-SPACES

The results obtained here together with the other results [107],[111],[131] and [190] show that it may be difficult to preserve the norm or the continuity of the extended functional during the extension process. From Kalton's result [131]we can easily deduce that non locally convex spaces should not be metrizable to have the HBEP. In fact, according to Remark 5, Allan Shawkat gave an example of a non locally convex space E with the H B E P where E has a non-countable algebraic basis. Regarding non-locally convex metrizable spaces, the extension and separation problems should then be discussed for subspaces instead of the spaces themselves. Consequently we may ask about t h o s e s u b s p a c e s M~ of a g i v e n n o n - locally spaces~ t h a t m a y h a v e t h e e x t e n s i o n p r o p e r t y . In that case, M is said to have the subspace extension property (SH B E P ) . For example, complemented subspaces and subspaces with finite codimensions have the S-HBEP (see Theorems 5 and 6). On the other hand, a subspace of lp(0 < p <_ 1) whose unit ball is 11- precompact will not have the S-HBEP, see Stiles [201]. P r o b l e m 1. In view of the present results we could ask about those properties of a subspace M of a locally bounded space E so that M has the S-HBEP i.e. all continuous functionals on M have continuous extension to E. For the Hardy space Hp, (0 < p < 1), Kalton and Trautman obtained a specified condition on M C_ H p to have the S-HBEP.

1.2.6

Examples of F-spaces and Non F-spaces

In this section we give some examples of non locally convex spaces which have either Shauder bases or bounded approximation property (b.a.p.), where our theorems here can be applied.

E x a m p l e 3 The Inductive Space Ul_>p_>0/p We define the q-topology on the space Ul>_p>olp to be the strongest vector topology such that each injection ip " lp ---, U lp is continuous. The space Ul>_p>_olpwith this q-topology is a complete separable non locally convex Lps and has the unit vectors (en) as its symmetric Schauder basis.

1.2

HAHN-BANACH

THEOREMS

19

A sequence converges in Olp if and only if it is contained and converges in some lp. A set is compact in Ulp if and only if it is contained and compact in some lp; no closed infinite dimensional subspace of Dlp is contained in lp; and no infinite dimensional subspace of Ulp is metrizable. Hence Ulp itself is not metrizable, Stiles [202]. Since E is Lps, Theorem 10 implies that every continuous linear functional on a closed subspace M of E has a linear extension to E which turns out to be continuous if codlin M < oo. A

E x a m p l e 4 Space o f m a p p i n g s with r a p i d l y decreasing a p p r o x i m a tion numbers Let E be a Banach space and L(E) be the space of all continuous linear mappings of E equipped with the linear mappings norm. For T 6 L(E) we define the rth-approximate number by a~(T) = inf { l I T - SI]; S 6 L(E), with dim S(E) A r } On the set O0

lp(E) = {T 6 L(E);

~-~[ar(T)] p < oo},

(0 < p _< 1),

r=l

we define a quasi-norm II" II given by IITI] = (lITIIp) l/p, where

O0

JJTlip r--1

This quasi-norm satisfies

lIT + sll <_ 22/p-l(I]TII § IIsll) ( see Pietsch, [[18~], p.126] ). Therefore, a process of n steps extension of a given continuous linear function f on a subspace M of codimension n of lp(E), yields, by Theorem 6, a continuous linear extension f satisfying IlfliM ~ Ilfililp(E) ~ 2(2/P-1)~IIfIIM, i.e. with a-

2 (2/p-l)

20

CHAPTER

1.3

OPEN

1 FUNDAMENTAL

MAPPING

THEOREMS

IN F-SPACES

THEOREM

The open mapping theorem is one of the fundamental theorems in Functional Analysis which has applications in Complex Analysis and other branches of science. We shall study this theorem for linear maps from an F- space to another. E is an F-space if its topology induced by a complete inveriant metric. The proof of this theorem, which is given on the line of that for normed spaces, is an application of the B a i r e c a t e g o r y t h e o r e m which we now recall, for such spaces. Theorem

13

(Baire)

An F-space E cannot be represented as the countable union of closed subsets all having empty interiors. Theorem

14 [32] ( Open M a p p i n g T h e o r e m o f locally bounded spaces)

Let E and F be p-Banach and q-Banach spaces respectively. If A E L(E, F) is a surjection, that is A.E~F is a continuous linear mapping from E onto F, then A ping.

is an open map-

P r o o f . Let

and 1

(1.24)

Sn = A(Bn) - A(BE(O,-~-~)). Since E = Un>l nB1 and A

F

:

is a surjection,

Un> 1nS1 -- Un> 1nS1.

By the Baire category theorem, there is an integer k such that kS1 has non empty interior. Thus for some point ~ E F and some a > 0, BF(~, a) C kS1, where BF(~, a) is the open ball in F centered at ~ with radius a. Since the mapping y ~ ky is a homeomorphism, $1 contains an open ball BE(q, r) for some r > 0 and r/E $1. Then

BF(0, ~) c S] - V c S-~ - S-~ c 2 ~ / ~

- &

1.3 O P E N

MAPPING

where So -'- 21/qs1.

THEOREM

21

Hence BE(O, r) C r

and

2-1/qBF(O, r/2) C ~,~1 because

BE(O, r) = r l / q B F ( O , 1).

(1.25)

So inductively, we get the following inclusion 9

BF(0, Now let y E F Xl E B1 with

r

all n.

c

such t h a t Ilyll < r / 2 . Since y E

~1

(1.26)

by

(1.26), there exists

r

Ily - A(z )II <

22+q"

T h e n y - A(Xl)E $2, so there exists x2 c B2 such t h a t

IlY - A(xl) - A(x2)]l <

23+q -

Inductively, we can find Xn E Bn such t h a t ]]y - A(xl) . . . . .

A(x~) I] <

2n+l+q "

Since E Ilxnl] < E ~1 = 1, E xn converges to a point in B0 = BE(O, 1),

A(E

= E A(x~) = y.

Thus y E So. This shows t h a t BE(O, ~) C So. Similarly, we have

r

BF(0, 2n+q+l ) C Sn.

(1.27)

We now show t h a t A is an open m a p p i n g 9 Let U be an open subset of E , x an arbitrary m e m b e r of U, and y = A(x). T h e n for some n, x + B n C U, h e n c e y + S n C A(U) since S,~ contains a neighborhood of 0 by (1.27). A(U) is a neighborhood of y; hence A(U) is open. I

22

CHAPTER

1 FUNDAMENTAL

THEOREMS

IN F-SPACES

C o r o l l a r y 1 Let E

and F be p-Banach and q-Banach spaces respectively, and let A E L ( E , F ) be a bijection. Then A is a homeomorphism and hence A is a topological isomorphism

P r o o f . Since the inverse A -1 of A is continuous by the above Theorem 14, A is a homeomorphism. However, for a continuous linear mapping between Fr6chet spaces we have the following similar result, the proof is mainly similar to that as given above for Theorem 14. I Theorem

15 (Open M a p p i n g between F - s p a c e s )

Let E and F an open mapping.

be F-spaces.

P r o o f . Let U C E

If A E L(E; F)

is surjective, then A

is

be a neighborhood of zero.

We first show that the closure A(U) contains a neighborhood of zero in F. Since the addition x - y is a continuous function, there exists a neighborhood of zero M such that M - M C U. Since M

is a neighborhood of zero (x)

E -- Un= lnM. So, and as A

is surjective,

F-

A(E) - Un~__lnA(M).

By the Bair category theorem, we have at least a set h A ( M ) non-empty open set V. Hence

contains

a

I ( V - V) C A ( M ) - A ( M ) C A ( M ) - A ( M ) C A(U) n and the set ~1( V - V)

is of course a neighborhood of zero in F

For any e > 0, we write

BE( ) = {x E; Ilxll <

and

B g ( e ) = {y E F; ]IYI] < e}

where I1" II is the F - n o r m defined by the metric of E

or of F .

Let e0 > 0 and let (ei) be a sequence of positive numbers such t h a t E i = I ei < e0. As we have already shown in above theorem, there exists a s e q u e n c e (ri) of positive numbers such t h a t OO

1.3 OPEN

MAPPING

THEOREM

23

(i-

A(BE(ei)) C BF(ri)

1,2,...).

(1.28)

Let y be a r b i t r a r y in BF(ro). We shall claim t h a t there is an x E BE(2eo) such t h a t

A(x) - y.

(1.29)

From (1.28), there is xo C BE(CO) such t h a t Ily - A(xo)ll

Taking i -- 1

i.e. y - A ( x o ) e BF(rl). Xl E BE(el) such t h a t

Ily

-

rl

in (1.28) we find t h a t there is

A(xo) - A ( x l ) ll < r2.

So by repeating this process we m a y define a sequence (xn) in BE(e~) the property

with

n

IlY - A(~-'~ xi)II < r~+l. i--0 Since E

is complete, the series ~ Xn

is convergent to x

(1.30) and moreover

Ilxll < 2 0. As (1.30)implies (1.29), the image of a neighborhood of zero in E contains a neighborhood of zero in F. Now let G be open subset of E. Let x C G be arbitrary and let N be a neighborhood of zero in E such t h a t x + N C G. Let M be s u c h a neighborhood in F with M C A ( N ) . Then A(x) + M C A(x) + A ( N ) - A ( x § N ) C A(G). Therefore A(G) contains a neighborhood of each of its points, t h a t is the set A(G) is open. m As a corollary of the above T h e o r e m 15 we get. C o r o l l a r y 2 ( B a n a c h ) L e t a continuous linear mapping A maps an F-space E onto an F-space F in one-to-one way. Then the inverse mapping A -1 is continuous.

24

CHAPTER

1 FUNDAMENTAL

THEOREMS

IN F-SPACES

E x a m p l e 5 The above Theorem 15 cannot be applied if E = L p, 0 < p < 1, a n d F =~T, since E' = {0}, and hence there is no A C (/.2;r which is a surjection. However, every separable locally bounded space E with a p- homogeneous norm is an image of the space 1p by a continuous linear map. In fact, this continuous linear map A : lp --~ E is defined by A ( { t n } ) - ~-~.nC~=ltnXn, where {Xn} is dense in the unit ball of E, {tn}E1 p, see Rolewicz [ [186], Th. III 2.9]. Therefore d is open by the open mapping Theorem 15.A

1.4 UNIFORM

BOUNDEDNESS

UNIFORM

1.4

PRINCIPLE

BOUNDEDNESS

25

PRINCIPLE

As usual E and F denote F-spaces. If a subset B of L ( E , F) is bounded, i.e. sup{llAII ; A E B } _ _ M for some M > 0 , then V x e E, 3Mx > 0 such that

sup{llA(x)ll;

A e B} < Mx.

That is, B is bounded pointwise on E. (We may also say that each o r b i t F(x) is bounded). For a p-Banach space E we shall show that the converse is also true, i.e. if sup{llA(x)lI,A E B } < M~ for each x E E, then sup{llAII;A C B} < M. That is, B is bounded with respect to the F-norm. Thus an extension of the B a n a c h - S t e i n h a u s T h e o r e m ( w h i c h is known as the p r i n c i p l e of u n i f o r m b o u n d e d n e s s ) is given here to locally bounded F-spaces and the proof is analogus to that for Banach spaces. T h e o r e m 16 [26](Generalized B a n a c h - S t e i n h a u s T h e o r e m ) Let E be a p-Banach space and F a q-normed space. Given a family {Ak}kEI in L ( E , F ) the following are equivalent"

(a) {Ak}ke, is an equicontinuous family, i.e. for any e > O, there exists 5 > 0 such that Ilxll < 5 =:> IIA,,:(x)ll < ,~ Vk e I.

D) {Ak}k~,

is bounded pointwise, i.e. for each x E E,

Mx > O such that

IIA~(x)ll <_ M~ (c)

{Ak}kel

vk,~x. there exists M > 0

is uniformly bounded, i.e.

that

IIAkll <_ M

there exists

such

V keI.

9 (a) => (b). Assuming the statement (a), we can find a 5 > 0 such that Ilxll < ~ ~ IIAk(x)ll < 1 for all k E I. If x =/= 0, then Proof.

51/Px

lIAr( ilxil l/p) II < 1 , i.e. for all k E I.

IIAk(*)ll <

Ilxll~/p

~a/v

= M~

26

CHAPTER

1 FUNDAMENTAL

THEOREMS

IN F-SPACES

(b) =~ (c). For each natural number n, let

D n = {x e E;llAk(x)l I < n

Vk e I } .

Since each Ak is continuous, it follows that D,~ is closed. By (b) we have E = Un>l Dn. Then the Baire category theorem ensures that s o m e Dn contains a closed ball BE(~, r). Consequently we have

ilAk(x)l I <_ n for all x E B ( ~ , r )

and k E I .

This implies that if Ilxll <_r, then

IIAk(x)ll <_ IIAk(x + ~)11 + IIAk(~)ll-< 2n for all k C I; note that I I x - ~11-< r, Therefore iiAkllq < 2n _ M

IIAkll q Ilxllq/p

B(~, r) C_ Dn. for all k e I,

since

D]Ak(x)ll <

see part I, to get IIAkll < (2n)~/q -- M lZq ~

(c) =~ (a)" Let e > 0 have

r l / p

~

be given and let 5 - ~ .

.

Then for Ilxll
IIAk(x)ll <_ Mllxll q/p < e for all k E I; so the family {Ak}kd

is equicontinuous, m

For a subset B of L(E, F), where E and F are Fr@chet spaces (complete metric spaces), we have the following.result, see Rolewicz [186]. Theorem

Let E {Ak}k~I

17

[186]

Then every bounded family B = and F be Fr~chet spaces. in L ( E , F ) is equicontinuous.

P r o o f . First we claim that a closed absorbing set V open set. Since V is absorbing,

of E

contains an

OO

E - Un= inV. By the Baire theorem, the space E is of second category. Therefore, there is a positive integer no such that noV is of the second category. Since noV ~UcV. is a closed set, it contains an open set U. Hence ~00 Secondly we prove that the bounded family B is equicontinuous. Lete>0 and let

u, - nAeB{X e E; IIA(x)ll _< c}

1.4 U N I F O R M

BOUNDEDNESS

PRINCIPLE

27

Since the m a p s A are continuous, the set [7] is closed. Let us show t h a t [71 is an absorbing set. In fact, if x E E is arbitrary, then the hypothesis implies t h a t the set { A ( x ) ; A e B} is bounded. Hence there exists a positive n u m b e r a such t h a t for b with 0 < b < a, we get

]]bA(x)ll<e

VA.

Therefore bx E U1. From the first part, the set [7] contains an open set U2. Let xo E [72. T h e set {A(xo); A E B} is bounded. Hence there is a positive n u m b e r b with Ibl < 1 such t h a t libA(xo)]] < e. T h u s bxo e U1. Let U - b ( U 2 - x o ) . Hence U is a neighborhood of zero. Let x E U. T h e n x = b y - bxo, where y e V2. Therefore IIA(x)]] < libA(xo)]l + IIbA(y)l] <

+ sup

li bz ii.

I1~11<~ Hence the continuity of multiplication by scalars implies the Theorem. i

This Page Intentionally Left Blank

Chapter 2

T H E O R Y OF POLYNOMIALS F-SPACES

IN

Our approach here in the study of polynomials is the multilinear mappings between spaces which are not necessarily locally convex. Continuous polynomial mappings with respect to the given topologies on non locally convex spaces play a fundamental role in our study. Therefore we think it is more convenient to study the continuity of the multilinear maps in details. We have paid also attention to study and check the validity of the B a n a c h S t e i n h a u s t h e o r e m f o r p o l y n o m i a l s between locally bounded F-spaces. A generalized u n i v e r s a l c o n s t a n t appears relating the continuous mhomogeneous polynomials and the corresponding symmetric m-linear maps.

2.1

MULTILINEAR

MAPS

Multilinear maps play a fundamental role in our study of Polynomials, Differentiable maps and Holomorphy. In this part we present the basic facts about multilinear maps of p-Banach spaces (complete locally bounded Fspaces) and give conditions for a multilinear map to be continuous. We also introduce the p-normed space of all continuous multilinear maps. 2.1.1

Continuous

Multilinear

Maps

Let us first deal with the concept of the product of different p-normed spaces and its topology. 29

30

CHAPTER

2 THEORY

OF POLYNOMIALS

IN F-SPACES

D e f i n i t i o n 10 ( P r o d u c t o f p - n o r m e d s p a c e s ) L e t Ei be a finite family of pi-normed spaces, 1 < i < m, over the same field K , whose pi-norms are denoted by I]-]]p,, and let E be their product, i.e.

E - H~E~ E is called the C a r t e s i a n p r o d u c t of Ei. Then E is a vector space over K with coordinatewise addition and multiplication. Thus the F-norms:

]](Xl,''',Xm)[]or

sup ]]xi]]l(p', l
]] (Xl," " " ,Xm)ll

_~. __1 ~-~. ]]xi][l(p,

m i=1

are equivalent. A n y of these F - n o r m s is naturally equivalent to a p-norm on E if Pi are all equal to p. That is, if each Ei is p-normed so is the product E. D e f i n i t i o n 11 ( M u l t i l i n e a r M a p ) I f Ei are different p-normed spaces, 1 < i < m, over the the same field K, then the map,

A'E1

•215

is said to be m u l t i l i n e a r if it is linear in each variable separately. This, of course, does not imply that A is linear on the product vector space E. So we say that A is multilinear if each partial map, Xi --* A ( x l , ' " , x i , ' " X m ) is a linear map of Ei into F. If m = 2, a multilinear map is called bilinear. In general, a multilinear map on the product of m pi- normed spaces is called an m - l i n e a r m a p .

It is fruitful to note that several results which are true for linear maps can be extended easily to multilinear maps. T h e o r e m 18 [36] Let Ei be pi-normed spaces 1 <_ i <_ m, and F is a q-normed space. Let A " E1 • "" • Em --* F be a multilinear map. Then the following are equivalent:

2.1 MULTILINEAR

MAPS

31

( a ) A is c o n t i n u o u s . (b) A is c o n t i n u o u s at the origin ( 0 , - . - , 0 ) . ( c ) T h e r e exists M > 0 s u c h t h a t f o r e v e r y (x l, E1 •

,x~) ~ HT~E~ =

xEm,

IIA(xl,...

,xm)l111q <_ M l l x l l l

I/p~ ' ' ' llxmll I/pm.

~1

Proof. I n this p r o o f we use the F - n o r m I1~11p r o d u c t E1 • • Era, where x - - (Xl,." " , X m ) .

m

Y~i:l IIx~llY p'

o,~ th~

I t is obvious t h a t ( a ) ::~

(b) .

(b) =~ (c): Assume A is continuous at the origin. Then there exists a number r such t h a t Ilx[[ _< r implies t h a t [[A(x)l I _< 1. Then the inequality in (c) trivially holds if for some i, x~ = 0. For any vector x = ( x ~ , . . . , X m ) , rx i if each xi # 0, set zi = iix~ll~m. T h e n we have

IIA(z:,-..

, z ~ ) l l 1/q <_ m m.

But

IIA(zl,

rxl

...

,zm)ll ~lq = IIA(llx~lll/p,, _

r m

_

-

and hence

llA(x~,...,x~)ll

rXm

, iix.~lll/. ~)11 l/q

llx11111p~..,

llx~ll~/,',,, IIA( xl,''',xm)ll

I/q

~/" < MIIxlII lips''' llxmll ~/'m

where M -- ( ~ - ) . (c) ::~ (a)" To show the continuity of A at any point a - (al, consider

A(x) - A(a)

9 9 9 ,am),

= A ( x l - h i , x2, . . . , Xm) + A ( a l , x2 - a2, x3, . . . Xm) + 9" + A ( a l , . . . ,

a m - l , Xm -- a m ) .

If I1~- ~11 < 1, llA(x) - A(a)ll ~ Mq[llx~ -a~llq/P~.llx211q/p= ''" llx.~llq/p'~

+llal II q/'l I1~1 - ~211q/':'~ 11~311~/p3 "" IIx,,,ll q/p'" +-"

+ Ila, II q/'l

< MqKm-lllx

_

9 9 Ila~-I 9 II~/P'"-II~,~ - a-,ll q/''']

~llq;

we

32

CHAPTER

2 THEORY

OF POLYNOMIALS

i.e. IlA(x) - A(a)ll 1/q <_ M K ( m - 1 ) / q l l x - a l l that A is continuous. I

IN F-SPACES

with K - Ilallq + 1. This proves

C o r o l l a r y 3 Every multilinear map of a product of finite dimensional pinormed spaces into a q-normed space is continuous. P r o o f . Since every linear map on a finite dimensional space is continuous, the corollary follows from the preceding Theorem 18. I T h e o r e m 19 [36] Let Ei be a pi-normed space (1 <_ i <_ rn) and F A rnultilinear map A" E1 x ... x E,~--, F

is continuous if and only if f o A

is continuous

be a q-normed space.

for every f E F'..

P r o o f . Suppose that A is not continuous. Then there exists a sequence (ak) in E1 • 2 1 5 Em satisfying Ilakll ~ 0 and IIA(ak)ll > k for all k" If ( f o A ) is continuous for all f E F'~ then the sequence (f[A(ak)]) is bounded for every f in F ' , so the sequence (A(ak)) must be bounded by Theorem 16 of uniform boundedness. This contradicts the fact that IIA(ak)ll > k; thus A is continuous. II

2.1.2

Locally Bounded

S p a c e s L ( E 1 . . . , E m ; F)

Let us denote by L ( E 1 , . . . , Era;F) the set of all continuous multilinear maps of E1 x ... x Em into F. Then L ( E I , . . . , Era;F) becomes a vector space under the pointwise addition and scalar multiplication For any A E L ( E 1 , . . . , E,~; F), we define

IIAII -

sup{ IlA(x)II; Ilxll~ _< 1}

Then by Theorem 18 (c) IIA(x) II -< IIAll Ilxlll q/pl 9 9 9IIx~ll q/pro

and IIAII is the smallest M q > 0 satisfying the inequality in Theorem 18 (c) and IIAII is a q-norm on L(E1, " " , E m ; F ) . In case that we are concerned with continuity of multilinear mappings we write the vector space of all (algebraic) multilinear mapping, from

2.1 M U L T I L I N E A R M A P S E1 x . . . x Em we write

to F

33

by L a ( E 1 , ' " , E m ,

F). When E - E1 = "" = Era,

L ( m E , F) - L(E1, " " ,Era; F), La( E , F ) - La(E1, " ' , E m ; F ) m

o

T h e o r e m 20 [36] If F is p-Banach space, then L ( E 1 , . . . , Em; F) is a p-Banach space for any pi-normed spaces Ei (1 < i _< m). P r o o f . This is modeled on after the proof of Theorem 1 (subsection 1.1.1). m

2.1.3

Examples

Example 6 . (a) Let E , F

of Bilinear Maps

and G

r

be p-no~vned spaces. Then the map

L(E; F) • L(F; G) --~ L(E; G)

defined by

r

g) - g o f

is bilinear, and furthermore, it is continuous since IIg o fll <- Ilgllllfll.If Ilfll <- I and Ilgll <- I then llgo fll <-1, hence ll~ll <- I.A (b) Consider the coordinate map P~ : K n --~ K defined by Pi(x) = xi. Define Pi • Pj : K n • K n --* K by (I~ • Pj)(x, y) = xiyj where i , j = 1,...,n. Then a bilinear map f : K n • K n ---. K can be written uniquely in the

form f(x,y) = Eaij(Pi

• Pj)(x,y)

with s o m e aij in K . Therefore, the bilinear maps I~ • Pj i , j - 1 , . . . , n f o r m a basis for the space L ( E ; K ) where E - Kn.A (c) The inner product of Rn, the multiplication of complex numbers in (T, and the cross product in ~3 are all continuous bilinear maps.A

34

CHAPTER

2.1.4

Natural

2 THEORY

OF POLYNOMIALS

IN F=SPACES

Isometry

When we deal with the differential calculus and other applications we often need to identify

L(E1,'",Em;F)

with

L(E,L(E2,...,Em;F))

under an isometric isomorphism. For simplicity we consider the bilinear case and discuss here such an isometry between L(E1,E2; F) and L(E1; L(E2; F)) since the general case can be studied similarly. Here, as usual, E~ are p~normed spaces and F is a q-normed space. T h e o r e m 21 [36] Let E~ be Pi- normed spaces (1 < i ~ 2), and F be a q-normed space. Let r L ( E 1 , E 2 ; F ) ~ L ( E 1 ; L ( E 2 ; F ) ) be defined by

r

- A(x, y)

where A E L ( E 1 , E 2 ; F ) , x E E1 and y E E2. isometry i.e. r preserves the F-norms.

Then r

is an isomorphic

P r o o f . Obviously r is linear, and IIr < IIAIIqlIxlIq/P~IlYlIq/P~. Hence for each x e El, r e L ( E 2 ; F ) and IIr < IIAIIqlIxlIq/P~; consequently ~(A) e L ( E 1 ; L ( E 2 , F ) ) and ]I~(A)II < IIAII. Therefore ~p is a continuous linear map with II~il <- 1. On the other hand, let a " L ( E 1 ; L ( E 2 ; F ) ) --~ L ( E 1 , E 2 ; F ) be such that ~ ( B ) ( ~ , y) - B(x)(y). Then a is linear and a ( B ) is bilinear for each B in L(E1; L(E2; F)). Since IIo~(B)(x,y) II<_ IIBllq[Ixilq/pl [[yllq/p2 , o~(B) c:_ L(E1,E2; F) and ][(~(B)[I _< liB[I; so Ilall _< 1 and c~ is continuous. It is clear that c~ and r are inverse to each other; thus, i - I1~ o r < I1~1111r

and we conclude that I1~11- I1r 1 since I1~11 < 1, ~nd Irr < 1. This implies that r is an isomorphic isometry. II We can now state the general form of the preceding theorem T h e o r e m 22 [36] There exists an isomorphic isometry

L(E1, . . . ,Era; F) ~- L(E1; L(E2; L(E3, " " ,n(Em; F) . . .))) from the space of continuous multilinear maps to the space of rrL- times repeatedly continuous linear maps. m

2.2 P O L Y N O M I A L S

2.2

OF P-NORMED

POLYNOMIALS

SPACES

OF P-NORMED

35

SPACES

Polynomials are used to approximate all holomorphic mappings. In fact they represent the simplest holomorphic functions. So their properties should be investigated first and carefully. We define a homogenous polynomial on a p- normed space as the restriction of a symmetric multilinear map to be the diagonomial one. In this Part, we study several necessary and sufficient conditions for a polynomial to be continuous, and we extend the BanachSteinhaus theorem to the space of m-homogeneous polynomials on locally bounded F-spaces E. 2.2.1

Symmetric Multilinear Maps E~

If the pi-normed spaces E 1 , ' " , E m are equal to a p- normed space we recall that, i ( m E ; F) = i ( E 1 , " "E,~; F). D e f i n i t i o n 12 An m-linear map A : E m --, F

A(xl,"" ,Xm) = A(xa(1),

is called s y m m e t r i c if

,xo(m)),

for any permutation a of ( 1 , . . . , m ) . Let Las(mE; F) denote the vector space of all a l g e b r a i c s y m m e t r i c m - l i n e a r m a p s of E m into F, and let

Ls(mE; F) = L(mE, F) N Las(mE, F). Then it is easy to check that Ls(mE; F) is a closed subspace of L(mE; F) and hence Ls(mE; F) is a p-Banach space if F is a p-Banach space. For m = 0 we write L ~ = Ls(~ - F, and for simplicity, when F = A" we write

L(mE) = L ( m E ; K ) , L s ( m E ) = n s ( m E , K ) .

D e f i n i t i o n 13 ( S y m m e t r i z a t i o n ) F o r each m- linear map A : E m - , F we define an m - l i n e a r m a p As : E m --* F by

1 A ~ ( x l , . . . ,Xm) = roT.~

A ( x o ( 1 ) , ' ' ' , xa(m))

(2.1)

a

where the summation is over the m! permutation cr of ( 1 , . . . , m). As is a symmetric m-linear map, and it is called the s y m m e t r i z a t i o n of A.

36

CHAPTER

2 THEORY

OF POLYNOMIALS

IN F-SPACES

If A E L s ( m E , F), then it is clear t h a t A = As. We also note t h a t the m a p A ~ A8 is a continuous m a p of L('~E, F) onto Ls(mE; F) since

IIA [[ 2.2.2

IIAII-

Multilinear Formula

The multilinear formula will be generalized from elementary algebra. Let A E L a ( E 1 , ' " , E m ; F ) , O < _ n _ < m , and ( x i , " - , x n ) E E l X ' " x We define A ( x l , . . . , x n ) by the following formulas 9 (1) If n - m , A(xi,'",xn)-A(xl,"',Xm). (2) If n < m , A ( x l , ' " , x n ) " En+l X "" x Em'-* F is a m a p p i n g defined by (X~+l,""", xm) ~ A ( x l , " " x~, X~+l,""", xm). T h a t is, A ( x l , . . . , x ~ ) is an ( m - n ) - l i n e a r m a p of En+l x . . . - xn, we write into F . W h e n A E LaB(mE, F) and x - Xl - ' "

A x ~ - A ( x l , . . . , x,~).

X Em

(2.2)

We define, for convenience A x ~ - a. This shows t h a t

Ax nELl(

t o w n

E,F) o

alln, 0_n<m.

D e f i n i t i o n 14 Let A E La(mE, F) 0 < k <_ m, X l , ' " , x k h i , . . . , nk E N with nl + . . . + nk - n < m. Then we define

Ax~ where Xl

Lemma

9' ' X ~k k ~

appears nl

A( X l , ' ' ' ~ X l ,

X2~'''~X2,''',

times, Xk appears nk

Xk~'''~Xk

E E

)

times, and so on.

1 ( M u l t i n o m i a l F o r m u l a ) If A e La(~E, F),

then

and

2.2 P O L Y N O M I A L S

OF P-NORMED

SPACES

37

nl ! n[ nk[ Ax~l . . . x k

A ( x l + . . . + xk) n = ~

-

where the s u m m a t i o n is over all k-tuples ( n l , . . . , nk)

(2.3)

satisfying nl + ' . " +

nk --n.

P r o o f . The procedure is obvious for n - 0 and n - 1. We shall prove the l e m m a by induction. Assuming the formula is valid for a certain n > l , one can readily establish it for n + 1. Note t h a t if A E La(n+lE; F ) t h e n one can write

A ( x l + . . . + xk) ~+1 = A ( x l + . . . + x k ) ( x l + . . . + Xk) ~ where in this case nl 4-... + nk = n + 1 in (2.3). m As a special case of the Lemma, familiar binomial formula.

when k - 2,

we have the following

Binomial Formula A ( x + y)~ = ~

A x k y ~-k.

(2.4)

k--O

2.2.3

Polynomials

D e f i n i t i o n 15 ( H o m o g e n e o u s p o l y n o m i a l s ) L e t E , F be vector spaces over the same field K . A mapping P : E ~ F is said to be an m h o m o g e n e o u s p o l y n o m i a l or a h o m o g e n e o u s p o l y n o m i a l o f degree m if there exists an m-linear map A : E m ~ F such that

P(x) - Ax m

(2.5)

for all x E E . For m - 1, an m-homogeneous polynomial is simply a linear m a p of E into F . It is agreed t h a t a homogeneous polynomial of degree 0 is a constant map. If P " E --~ F is an m-homogeneous polynomial, then P ( r x ) = r m p ( x ) for any r E K. We shall denote by Pa(mE; F ) the vector space of all m- h o m o g e n e o u s p o l y n o m i a l s of E into F . T h e index "a" is to indicate t h a t the m- homogeneous polynomials in this space are not necessarily continuous. For convenience, we agree to write F = Pa(~ F).

38

CHAPTER

2 THEORY

OF POLYNOMIALS

IN F-SPACES

Polarization Formula

The following theorem provides us with an interesting formula. It relates m- homogeneous polynomials and symmetric m-linear maps. T h e o r e m 23

[26] ( P o l a r i z a t i o n

Formula)

Let E and F be p - n o r m e d and q-normed spaces, and A " E m --~ F be a s y m m e t r i c m - linear map. Then

1 A ( X l , " " ,Xm) = m!2m ~

el, " " , emA(elXl + " " + emXm) m

(2.6)

where the s u m m a t i o n is taken for all ei E { - 1 , +1}, i = 1 , . . . , m.

P r o o f . By the multinomial formula, we have rrt! A(ElXl)n hi!... urn!

A ( e l X l + " " + emXm) m =

(nl,...,nm)

m!

= ~n1!:::nm! Multiply both sides by ~

s

En,~ A x ~ I

1 . . . (EnXm)n.~

. . . xnm .

we get

s163

~i=-I-1

e l " ' " emA(elXl + ... + em X m) m e~=d:l

= m! ~ / . r

1 n l I .urn! 1 "--

~

E~1+1 ..E~nnm+lAx~l..Xn'~ "

ei=+ 1 l
Clearly ~--~e~ 1+1"" "enn m + l --" 0 if ni = 0 for some 1 _< i < m. Since e2...em2 = 2 m we get the desired result; ei==t=l

i.e.

1 m!2 m

el.." emA(elXl + ' "

+ e~Xm) "~ = A ( x l , ...,xm). m

ei =:1::1

l
The following theorem shows the unique correspondence between the classes P a ( m E , F ) and LaB(mE, F).

2.2 P O L Y N O M I A L S

OF P-NORMED

SPACES

39

T h e o r e m 24 [26] I f P E p a ( m E , F), then there exists a unique symmetric m-linear map A E L a s ( m E , F) such that P(x) = Ax m for all x E E.

P r o o f . Since P E P ( m E ; F), by definition, P ( x ) = A x m for some A E L a ( m E , F). Then the symmetrization As of A satisfies P ( x ) = A s x m.

The uniqueness of such a symmetric m-linear map is a consequence of the polarization formula. I We have just seen that there is a unique correspondence between the m- homogeneous polynomial P and the symmetric m-linear map A, so to emphasize this we write

The following theorem is self-evident. T h e o r e m 25 The mapping A E Las(mE, F) ~ ft E P~(mE, F) is a vector space isomorphism.

E x a m p l e 7 Let E = K

and A : E m --. F

be any m-linear map of the

form A ( ) q , " ' , A m ) = aAl"")~m, where alkl " . " Am denotes (ikl,.. . Am)a. polynomial P : K ~ F is of the f o r m P ( x ) = a)~m

a E F,

Therefore, an m-homogeneous

a E F

In this way, if F = K , we get the classical m-homogeneous polynomial of K into K . This example motivates and justifies our definition of m homogeneous polynomial. A

40

CHAPTER

2 THEORY

OF POLYNOMIALS

IN F-SPACES

D e f i n i t i o n 16 ( P o l y n o m i a l s ) A mapping P : E --. F is called a p o l y such n o m i a l if there exists m and Pk E Pa(kE, F), k -- O, 1 , . . . , m that P - Po + PI + " " + Pm. The addition of this is pointwise, and if Pm ~ 0 we say that

P

is a

polynomial of d e g r e e m. The vector space of all polynomials from E to F with respect to pointwise vector operation will be denoted by P a ( E , F ) .

Theorem 26 Let P : E --, F be a non-zero polynomial of degree m between locally bounded F-spaces E, F. Then the representation

(e.7)

P - - Po + P~ + . . . + Pm is unique.

P r o o f . It suffices to show that implies

P = Po+P~ +... +Pro = 0

Note that if P ( x ) = O

for x E E ,

Po = . . .

= P.~ = O.

then we have for any r E K

P ( r x ) = Po(x) + rP~ (x) . . . + r m P m ( x ) = 0

Dividing through by r m if r ~ 0 and making r --. co, we obtain Pm = 0 . By induction we then get Po - P1 = "'" Pro-1 - O. I 2.2.4

Continuous Homogeneuous Polynomials

Let us denote by p ( m E , F), the vector space of all continuous m-homogeneous polynomials from a p-normed space E to a q-normed space F with respect to all pointwise vector operations. We write F = P ( ~ as before. As a criterion for the continuity of an m-homogeneous polynomial we present the following theorem. T h e o r e m 27 [25] For P E P~(mE; F ) , let A E La~('~E; F) be such that P - A . following statements are equivalent.

(a) A E L,(mE; F). (b) P E P ( m E ; F ) . (c) P is continuous at the origin.

The

2.2 P O L Y N O M I A L S

OF P-NORMED

SPACES

41

(d) There exists a constant M > 0 such that

IIP(x)ll ~ MIIxll mq/p.

(2.8)

P r o o f . The implications (a) =~ (b) =~ (c) =~(d) are evident. We claim that (d) =~ (a)" If IIP(x)ll < M for Irxll < 1, it follows from the polarization formula that

IIA(x~,-.. ,xm)ll ~ M/m! for IIx~ll ~ ~1, i - 1,...,m, and hence A E L~(mE; F). i It is clear that if P E p(mE; F), then (

sup

IIP(~) ll/ ll~ll~/~ ;

0, x e

"1

= sup{llP(~)ll; Ilzll < 1} = inf{M > 0; [[P(x)[[ < MI[xH[mq/p}. Let IIPII be the common value of these equalites. Then IIPII is a q-norm on P(mE, F) if F is a q-normed space. This q-norm will be considered throughout the book. It is easy to see that (2.9)

IIP(x) ll ~ IIPIIIIxllmp/~ . i Martin's theorem

The following theorem extends M a r t i n ' s t h e o r e m to a non locally convex situation, see C h a t [[69], Th.4.13]. T h e o r e m 28 [26]( Generalized M a r t i n ' s t h e o r e m ) For a p-normed space E and a q- normed space F, the mapping

A e Ls(mE; F) --* A e p(mE; F) is a vector space isomorphism and a homeomorphism of the first onto the second space. Moreover,

mmq/p

I1-~11 _< IIAII _<

m!

IIAIr

(2.10)

42

CHAPTER

2 THEORY

OF POLYNOMIALS

IN F-SPACES

P r o o f . The assertion is trivially true for m = 0, 1. So we assume t h a t m > 1. It is clear t h a t the m a p p i n g is a vector space isomorphism, and it remains to show t h a t the inequality (2.10) holds. We have II~ill <_ ]IA[I since II~ill <_ IIAllllxll mq/p. We use the polarization formula to get the other inequality

IIA(xl,...,x~)ll

1

~_ rn!2~ ~ IIA(~lXl + ' " 1 < m!2 ~ ~

1

-< m!2~ ~

+ ~=x~)ll + ~mXmll)ma/p

IIAIl(llelXl + "

IIAIl(llxlll + " " + Ilxmll) ~ / "

1

-< ~i IIA.II(llxlll + - . . + IIx~ll) ~q/p since we have 2 m terms in the above summation. 1,. 9 9 m, we obtain

Hence if

mmq/p

IIAII _< m----7-11AII.

IIx{ll <_ 1,

i

m

(2.11)

II Corollary 4

If F is a q-Banach space, then P(mE; F) is a q-Banach

space.

2.2.5

The Generalized

The m a p

Universal

Constant

A C Ls(mE; F) ---* A ~_ p ( m E , F)

m-w/, m!

considered above is not in

general an isometry. We also notice t h a t the coefficient mmq/p ,~! in Theorem 28 depends only on m, p and q and not on the p-normed space E and the q-normed space F. Therefore we can consider mmq/P/m! as a universal c o n s t a n t relating the continuous m-homogeneous polynomialg and the corresponding symmetric m- linear maps. Of course this constant turns out to be equal to ram~m! if E and F are p-normed spaces. The following example shows t h a t we cannot replace the constant mmq/P/m! by a smaller one. 8 [26](2003). Consider the F-space E - 1p (1 > p > 0) of all sequences x~- (xn) of ~ , ~ b ~ x , e I ; ~ c h that Ilxll = E F Ix, I" < ~ . Fo~ ,~ > O, l~t

Example

2.2 P O L Y N O M I A L S A " E m --~K

OF P-NORMED

SPACES

43

be the m - l i n e a r map defined by ~'~

99 .

~

x ~)

_

~

X l Z 2

2

9 . .Xm;

i.e. the product of the diagonal of x 1 -

(xi~xl

...

X m1

...)~

x 2 --

(X 2,x2,

...,xm, 2

. ..),

X~

Then

A

is continuous.

=

m

(X~,X~,...,Xlyt,.

The s y m m e t r i z a t i o n 1

As(xl"

" " x m ) -- -~. E

..).

As

of A

is now given by

Xl(1) " " x ~ m ) (T

where the s u m m a t i o n is over all p e r m u t a t i o n s We claim that IIA~II __ ~1. . In fact 1

o-(1)

IIA~(x ~, ..,x'~)ll 9 < ~., ~-~'(Ix~

~

I... Ix,~m)l)<

of {1, 2 , . . . , m}.

1

~--~.llxlll1/p''- Ilxmll 1/"

o-

Hence

1 I[A~[[ _< ~.,.

But

1

A s ( e l , . . . , e m) --

m!"

where e i -- ( 0 , . . . 0, 1 , 0 , . . ").

Therefore

IIA~]ILet A s ( x ) - A s ( x , . . . , x).

l/m!

.

(2.12)

Then As e P ( ' ~ E ; K ) A~(x)

-

~l

and

. . . x~

where x - ( x l , " ' , X m , ' " "). Since the geometric m e a n of positive numbers is always less than or equal to the arithmetic mean, we have

I1.s p -IXlF"

1

IXml p ~ m'~

(Ix1

ip

-t-'''-I-IxmlP) ~.

Thus ]I-A~II _< 1 / m "~/p. If we take

= (1/,~1/~,..-, 1/m ~/~, 0---),

44

CHAPTER

2 THEORY

OF POLYNOMIALS

IN F-SPACES

where 1 / m 1/p appears in the first m terms of x, we obtain

IIA (x)llThis shows that IIAs II --

1

1

mm/p.

(2.13)

and hence, mm/p

where q - 1

here A.

R e m a r k 6 I f E is a real Hilbert space and F is a Banach space then the mapping A E L s ( m E , F ) ~ A E P ( ' ~ E , F) becomes an isometry. I f p = 1 mm then the universal constant will equal -~. and the result is due to Nachbin

We generalize the polarization formula for the sake of using it in the next section. G e n e r a l i z e d P o l a r i z a t i o n Formula. For locally bounded spaces E, F, and f " E - - . F,

1 ~ m ( f ) = m!2 m

el... E

I(ElXl + " " +

, Xl,

let

,xmeE

where the s u m m a t i o n is over all ek - 1 o r - 1 , k - 1 , . - . , m . I f Ak " E k --~ F is a symmetric k-linear map and Pk -- Ak,

(2.14)

then

[

_ ~ Am(xl,...,xm) ~m (Pk)

[0

ifk-m

(2.15)

i l k < m.

We shall call ~ m ( f ) the p o l a r i z a t i o n of f with respect to X l , ' " , xm. We have shown the formula for the case k - m in Section 2.2.3 If k < m, a straightforward but lengthy computation shows that ~m(Pk) = 0 (see the proof of Theorem 23).

2.2 P O L Y N O M I A L S 2.2.6

The Space

OF P-NORMED

SPACES

45

P(E, F)

We are now concerned with continuous polynomials from a p-normed space E into a q-normed space F. Let P ( E , F ) denotes the vector space of all continuous polynomials from E into F. The following theorem provides criterion for the continuity of a polynomial P.

T h e o r e m 29 [32](2000) Let P " E ~ F be a polynomial of degree E into a q-normed space F such that P-

Po + P~ + ' "

~t

from a p-normed space

+ Pm.

Then the following are equivalent : (a) P 0 , / : ' I , ' " , P , ~ are continuous. (b) P is continuous. (c) P is continuous at the origin. (d) P is bounded on the unit ball.

P r o o f . The implications (a) =~(b) =~ (c) =~ (d) are trivial. It remains to show (d) ~ (a). This will be shown by induction. We use the general polarization formula above. We note that ~Pm(P) - ~m (Po) + " " + ~m (Pro) - ~Pm(Pro) or ~m(P)

--

Am(Xl,

. " . , Xm).

where -Am - Pro. Since P is bounded on the unit ball, we have JJP(x)J I < M on the unit ball Ilxll < 1 , for s o m e M > 0 . Now

1 IIA,~(xl,... , x m ) l l - II~,~(f)ll < ~.v ][P(elXl + . . . + CmXm)]] Hence if Ilxlll 1/p + . . . + Ilx.~ll 1/p < 1, then I l A m ( X l , " " , xm)ll

M m!

which shows that Am is continuous and hence Pm - - 2~m is continuous. As P - P m is also bounded on the unit ball, repeating the same argument as above, we can show that Pro-1 is continuous. Inductively, therefore we conclude that P0, P 1 , " " Pm are continuous, m

46

CHAPTER

2 T H E O R Y OF P O L Y N O M I A L S I N F - S P A C E S

Corollary 5 Every polynomial on a finite dimensional space is continuous. Proof.

This is obvious, since every m-linear m a p on a finite dimensional space is continuous. I

2.2.7

Banach-Steinhaus

T h e o r e m for P o l y n o m i a l s

The following theorem extends B a n a c h - S t e i n h a u s t y p e t h e o r e m to homogeneous polynomials between locally bounded F-spaces which are not necessarily locally convex.

T h e o r e m 30 [26](2002) Let E be a p-Banach space, F be a q- normed space and (Pn) be a sequence in p(mE; F). If P is the pointwise limit of the sequence (Pn), then P E p(mE; F). P r o o f . Let A , E Ls(mE, F) be such that A , - P , where Pn E P('~E; F). The polarization formula, (subsection 2.2.3) implies t h a t lim A n ( x 1 , ' " , x,~) exists at each point ( x l , . . . , Xm) of E m, let A = lira A , . T h e n A E L~(mE, F). We claim t h a t A E Ls(mE, F).

Since

L(mE, F) "" L(E,L(m-IE; F ) ) , if we consider (An) as a sequence in L(E; L(m-IE;F)), by the BanachSteinhaus theorem for linear mappings, we obtain A E L ( ( E ; L('~-IE, F)). Hence A E L,(mE, F) a s An E Ls(mE, F) for each n. Now we have

P(x) - lira Pn(x) - lim An(x) - lira A n ( x , ' " , x) = A ( x , . - . , x ) - ~i.(x). T h a t is, P -

A.

This completes the proof. I

2.2 P O L Y N O M I A L S

OF P-NORMED

SPACES

47

In the next chapters we will have new concepts to complete our building of the theory of holomorphy without convexity condition.The following organogram (Schedule) contains these new concepts.

New concepts in the Book ,.

i

.,

p.ExtremePoints and p-ExtremeSets (1995)

I

Arcsegment A (1995)

!

I

I

Boundedand Weaklybounded HolomorphicFunctions (1990)

Quasi-Differential Calculus (2000)

I

Boundingand Weakly-bounding (Limitedsets) (1990) !

!

Quasi.Differential or BayoumiDifferential (1995)

Ouasi-Holomorphy and Quasi-HolomorphicMaps (2000) ,

I

(~-QuasiDifferentialbility (1999)

I

Quasi-Analyticity and Quasi-AnalyticMaps

1 Higher Quasi.Differentiability

(2000)

(2000)

I

PB-Spaces (or BayoumiSpaces) and PB.property (1978)

Chapter 3

FIXED-POINT AND P-EXTREME POINT Fixed point theorems represent an important area in the growing field of nonlinear mappings. Indeed, they play an important role in many parts of analysis and topology. They have many applications, for example, in the field of linear algebraic equations, ordinary and partial differential equations, and integral equations. This chapter is devoted to extend the fundamental fixed point theorems to some non locally convex spaces. W e e x t e n d the f i x e d p o i n t t h e o r e m o f B r o u w e r to p - c o n v e x c o m p a c t sets o f locally pc o n v e x F-spaces. In fact this represents also a generalization of Schauder's theorem of convex sets in normed spaces and of Kakutani in locally convex spaces. Further we generalize K a k u t a n i ' s fixed p o i n t s f o r e q u i c o n t i n u o u s group o f aJ~fine m a p s o f a p - c o n v e x c o m p a c t set. In addition, we introduce and study the new concept, which we have called, " p - e x t r e m e p o i n t s ". This concept may have applications to different branches of mathematics. We use it here to generalize M i l m a n ' s T h e o r e m to p-convex sets of locally p-convex spaces. We also make use of it to extend K r i e n - M i l m a n ' s T h e o r e m to p-extreme points of a locally p-convex space. Applications to p-extreme points in complex and functional analysis and mathematical programming are given. 49

50

CHAPTER

3 FIXED-POINT

AND

P-EXTREME

POINT

3.1

p-EXTREME POINT IN NON LOCALLY CONVEX SPACES

We introduce and study in this part the new concept which we have called " p - e x t r e m e p o i n t s " and use it to extend M i l m a n , s t h e o r e m to locally p-convex spaces E. T h a t is, to study the p-extreme points of the pconvex hull of a compact set K of E. This concept may have applications in different branches of pure and applied Mathematics. In general, a p o i n t o f a s e t A in a vector space X is said to b e a pe x t r e m e p o i n t , (1 _> p > 0), if it is not an internal point of an arc segment in A. The idea of introducing the concept of p-extreme points of a non convex set in a non locally convex space X arises from the geometry of the closed unit ball of X. Fortunately this has also led us to obtain other fundamental results in functional and complex analysis. Therefore, our goal in this part is to study the properties of the p-extreme points of some non-convex sets in locally bounded and locally p-convex spaces and to give some applications to non convex analysis and Operations research. Additionally, the set of p-extreme points of the closed unit ball Blp of lp will be obtained. As a counterexample, we prove t h a t in LP[0, 1] (1 > p > 0), the closed unit ball BLp has no p-extreme points. D e f i n i t i o n 17 (Arc Segment)Let U C_X x, y C U. The set A~ -

{81/px

a set in a vector space X

Jr- tl/py; s + t -- 1}

= {sx+ty;

sp + t p - l }

and

(3.1) (3.2)

is said to be the a r c s e g m e n t j o i n i n g x , y , (1 > p > 0). I f p - 1, and we restrict our attention to s + t - 1, then A y turns out to be the line segment j o i n i n g x and y.

We proved in ([20], Proposition) that the unit ball BE(O, 1) of any locally bounded F-space E contains all its arc segments A g for every x, y E BE(0, 1).

3.1 P - E X T R E M E

POINT IN NON LOCALLY CONVEX

SPACES51

D e f i n i t i o n 18 ( p - c o n v e x s e t ) A set A

in a vector space is said to be if for every x, y E A, s, t >_ O, we have

p-convex

(1 - t)l/Px + tl/py E A,

whenever 0 <_ t <_ 1..

(3.3)

This is equivalent to saying that, for every x, y E A, sx + ty E A, whenever 8 p -t- t p -- 1

(3.4)

A is said to be a b s o l u t e l y p - c o n v e x if for every x, y E A, sx + ty E A for I s Ip + i t IP_< 1, (Rolewicz [186] ). If p definitions of convex and balanced convex sets.

1, we have of course the

For example, the unit ball ]~lp(O, 1) of the space 1p (1 _> p > 0) is pconvex. It is in fact absolutely p-convex and also polynomially convex, see the author [[15], Proposition]. D e f i n i t i o n 19 ( p - c o n v e x hull) If X is a topological vector space and A C_ X , the closed p - c o n v e x h u l l o f A denoted by Cp(A) is the smallest closed p-convex set containing A. D e f i n i t i o n 2 0 ( p - e x t r e m e s e t s a n d p - e x t r e m e p o i n t s ) Let K be a set in a vector space X . A nonempty set E of K is called a p - e x t r e m e s e t of K , if no point of E is an internal point of any arc-segment whose end points are in K , except when both end the points are in E. We say that z is an internal point of A~ if z r x nor z 7~ y. Analytically, the condition can be expressed as follows: If x, y E K, O < s , t and

sx + ty E E,

whenever s p + t p <_ I,

(3.5)

then x E E , y E E . In particular, ( 1 - t)l/px + tl/py E E, 0 < t < 1, implies x E E, y E E. The sets that z E are such

x

-

p - e x t r e m e p o i n t s of K , denoted by E p ( K ) , are the p-extreme consist of just one point. That is, K is a p-extreme point of K if the following holds: If x, y E K that z - sx + ty, s p + t p - l , O<s,t (3.6) y

[=

+

52

CHAPTER

3 FIXED-POINT

AND

P-EXTREME

POINT

T h i s is equivalent to: 1

1

If z -- s-~x + t~y, then

x-y

1

-(s 5 +t

1)__1 ] z

s + t --1,

. In p a r t i c u l a r if 1

1

z-(1-t)~x+t~y, we o b t a i n

x -- y

O < s,t

O<_t<_ 1

[-- ((1 - t) !p + t !p )-lz ]

In other words, a p-extreme point z of a p-convex set A is a point in A that is not an internal point of any arc-segment A~ in A; that is, z ~ x or z # y. Note that x - y - z if p - 1 . 21 ( p - c o n v e x c o m b i n a t i o n ) L e t A be p-convex and X l , . . . , X n E n n A, and ti ~_ O, Y~I tp - 1. Then ~-~1 tixi is said to be a p - c o n v e x c o m b i n a t i o n of {x~}. If E It~lp <_ 1, instead of, th~n E t~x~ i~ ~aid to b~ th~ a b s o l u t e l y p - c o n v e x It is easy to see that ~-~=1 tixi E A for a p- convex set A. Definition

s p a c e ) A topological vector space is said to be locally p - c o n v e x if the origin has a f u n d a m e n t a l set of absolutely p-convex O-neighborhoods. This topology can be determined by ps e m i n o r m s which are defined in the obvious way. Definition

22 ( L o c a l l y p - c o n v e x

In w h a t follows we show t h a t the p-convex hull of a c o m p a c t set of a c e r t a i n non locally convex space is also c o m p a c t . 1 I f X is locally p-convex F-space and K C_ X is compact then the closed p-convex hull C p ( K ) is also compact.

Proposition

P r o o f . Let us first claim t h a t C p ( K ) is p r e c o m p a c t t h a t is t o t a l l y b o u n d e d whenever K is p r e c o m p a c t . Let U be a n e i g h b o r h o o d of 0 in X . T h e n there exists a p-convex n e i g h b o r h o o d of 0, V in X such V + V C U and there exists a finite set F such t h a t K C F + V. Let

K1 - CpK, If

F1 - CpF.

{Yl, Y2,-.-, Y~} are points of F , t h e n F1 is the image of the c o m p a c t

set

{(tl, . . . , tn); ti >_ O, ~--~ tP~ - 1} C_R n

3.1 P - E X T R E M E

POINT IN NON LOCALLY CONVEX

SPACES53

under the m a p p i n g n

( t l , . . . , t n ) --~ ~

tiYi,

i=1

and is, therefore, compact. k 8 i x i , where si >_ 0, ~ i =k 1 siP - l a n d x i E K . If x E K 1 , then x - ~-~i=1 For each i, there is zi E F such t h a t (xi - z i ) E V. Decompose x into a sum (1)

x--x+x where , (1) X-

k 8iZi , x(2) -- ~-~-i=1 k ~--~-i=1

(2)

8i(Xi- Zi).

Now

x (1) e F 1 and x (2) E V

since V is p-convex. Thus K1 C F1 + V. But F1 is compact, so there exist a finite set F2 such that F1 C F2 + V. Hence

K1 C_ F1 + V C_ F2 + V + V c_ F2 + U and K1 is precompact. Since a subset of an F - s p a c e is compact if and only if it closed and p r e c o m p a c t we obtain the desired result. I T h e concept of p-convex sets can be reformulated as follows 9 A subset A of a vector space X is p - c o n v e x if it contains its arc segment A y whenever its end points x, y lie in A. T h a t is, if 0 < s, t, and x, y E A, then

A~ - {sx + ty;

sp + tp - 1 }

c A.

A is said to be a b s o l u t e l y p - c o n v e x if x, y E A and s, t E K, then

A'J-

{sx + ty;

Is Ip + I t IP< 1} c A.

(3.7)

First, using the above concept, we can prove t h a t it is equivalent to the first classical one, of Definition 18. In fact, the following two lemmas characterize the absolutely p-convex sets. T h e y will be useful in different places of the book.

2 A subset A of a vector space X is absolutely p-convex i.e. for every x, y E A, and s, t E K with It]p + lsl p <_ I we have

Lemma

tx+syE if and only if A~cA for every x, y E A.

A.

54

CHAPTER

3 FIXED-POINT

AND

P-EXTREME

POINT

P r o o f . r )" It is obvious t h a t if tx + sy C A y for every x, y E A and every t, s E C with It]p + Is]p <_ 1, t h e n A is balanced and p-convex. In fact, if s - 0, t h e n t x E AVItIP <__ 1, i.e. tA C_ AVItl _< 1. Also for 0_< t_< 1, (1 - t)l/px + tl/Py E A, since (1 - t) + t - 1. =~)" A s s u m e A is absolutely p-convex. If either s - 0 or t - 0, t h e n

tx+syEA;ifs~O,t~O,

then

~

xEA,

~[

yEA

and

Itl t I~l ~ ] (Itl + 181) Itl + 18I lt-I x + Itl + I ~ l ~ y e A,

tx + 8y -

i.e. AYC_A. This is because ]tiP + ]sip <_ 1 implies

I~ Itl I" + Itl I~l + Isl Itl + Isl

-

-

Itl"+1~1" (ItI + Isl) p -< 1.

I

Lemma

3 Let A C_ X (A ~ r

be an absolutely p-convex set. Then

(i) O e A . (ii) tA C_ sA when Itl _< Isl. (iii) Ei~=l tiA C_ (Ein__l It~l)A. P r o o f . (i) (ii) If s (iii) for If s - t

is n -

clear. t x c A. So tx c sA. 0, trivial; if s :/: 0 and x C A, t h e n (s) - 2, we have sA + tA C_ (Is[ + It[)A. 0, this is trivial, otherwise s t is I + it-------~ A + is I + itl A c_ sA + tA c_ (Isl + Iti)A.

I n d u c t i o n gives the general result. I

3.1.1

Properties

of p-Extreme

Points

T h e concept of p - e x t r e m e p o i n t s seems to be a n a t u r a l extension of the e x t r e m e points. However it solves several problems which m a y arise in non locally convex spaces, which m a y not be solved by the classical concept of e x t r e m e points.

3.1 P - E X T R E M E

POINT IN NON LOCALLYCONVEX

SPACES55

In fact, and as we have mentioned the geometry of, for example, the closed unit balls of lp, (1 > p > 0), helped us to introduce the concept of the arc segment A b, and the new terminology of p-extreme points. The following lemma characterizes the new concept of the p-extreme points of some non-convex sets in a vector space X . Lemma4 Let X be a vector space and l > p > O. Let K be a n o n e m p t y p-convex subset of X and z E K . T h e n the following s t a t e m e n t s are equivalent : (i) z is a p - e x t r e m e point of K . (ii) Let x, y E K be such that x # y, and t, s C [0,1]. I f z=sx+ty,sP+tP=l,

then

s=0

or t = 0 .

Equivalently, if z - (1 - t) 1/px + tl/Py, then we have either t - 0 o r t = (iii) I f x, y E K are such that 1 z -

Ov)

t -

+ y),

then

x -

y -

(2)I/p-Iz;

1.

(3.8)

( 89 1/p

{z }

P r o o f . This follows from the definition of the p-extreme points, noticing t h a t z could not be an internal point of any arc segment A y joining x, y inK. I P r o p o s i t i o n 2 Let X be a locally p-convex space and K be a p - c o n v e x subset of X . Then K ~ Ep(K) = r That is, any interior point of K , when it exists, is not a p- extreme point of K . P r o o f . For K ~ = r the result is trivial. Assume now t h a t K ~ ~ r and t h a t x E K ~ So we can assume t h a t K contains 0. Let x C K ~ T h e n there exists an absolutely p-convex 0-neighborhood V in X such t h a t VCK

with x + V C K .

The m a p ~ ~ /~x f r o m t ~ i n t o X is continuous, say, at ~0 T h e n for this x-neighborhood x + V, there is an r > 0 such t h a t :

(3.9) 1_ 1 2~ .

56

CHAPTER

3 FIXED-POINT

1 --1

]A-2~

[_
AND P-EXTREME

1

impliesAxC2~

_i x

+Y

POINT

,(AC~)

(3.10)

In particular, we have 1

(21/p-l- r ) X -

(21/p-1 -t- r ) x -- 2~ -1 x c V C K ,

21-1 x c V c K .

That is, rx C V C K,

-rx

(3.11)

E V C K. 1

Notice that (3.10) can also imply Ax + 2~-1x E V. 1

In fact

Ax C

1

2~-1x + V, implies

-Ax E 2~-1x + V, since V is balanced and because of 1

1

(3.11). Hence we have Ax C - 2 ~ - 1 x + V, to get Ax + 2~-1x E V. Now we have 1

(2 l/p-1 + r ) x -- 2~ -1 x E V C K ,

1

(2~ - 1 - r ) x + 2 ~ - l x C V C K .

Consequently, X -:

1

21/P

[

(2 l/p-1 + r ) x - - 2 - P

1[ (2 l/p-1 +

91/p

~-1

x-t-(2

r ) x -Jr- (2 l/p-1

1/p-1

]

1_1]

r ) x + 2~

x ,

-- 7")X ,

which implies that x is not a p-extreme point of K. I

3.1.2

Generalized

Milman's

Theorem

Let K be a p-convex compact set in a locally p-space X. It may happen that its closed p-convex hull Cp(K) has p-extreme points which are not in K. However the next theorem shows that this cannot occur if Cp(K) is compact. As for example if X is an F-space then Cp(K) is compact whenever K C X is compact. Consequently this situation occurs in no F-spaces. Let us first state the following lemma whose proof is similar to that of the classical one. L e m m a 5 (1) If A 1 , . . . , An are p-convex compact sets in a topological vector space X , then Cp(A1 U . . . U An) is compact.

3.1 P - E X T R E M E

POINT IN NON LOCALLYCONVEX

SPACES57

(2) If X is an F-space and K is compact, then Cp(K) is compact. P r o o f . (1) Let S - { ( t l , . . . , t n ) ; t j A-

>_O,t~+t~+...+tPn-1}

and put

A1 x A2.. x An.

Define f " S x A ---, X by

f(t, a) - tla I + . . . +tnan Put

K-

(3.12)

f(s • A).

It is clear that K is compact and that K C_ Cp(A1 U A2.. U An). We claim that this inclusion is actually an equality. If (t,a),(s,b) are inS• and ifc~_~0,~__0, c~p+~p-lthen,

a f ( t , a) + 3 f ( s , b) - f(u, c)

(3.13)

where u - (aPtP + 3psp)llp E S, because

atiai + ~sibi

c A~.

(3.14)

This shows that K is p-convex. Since Aj C K for each j, the p-convexity of K implies that Cp(A1 U A2...[-JAn) C_ K ( t a k e t j 1 ill ( 3 . 1 2 ) , t i - - O for i # j ) . (2) Let S be the set in R n+l consisting of all t - (tl,t2,...,tn+l) w i t h t i > _ 0 a n d ~ t ~ P - 1 . Let K b e c o m p a c t , K C R n+l But x E C p ( K ) if and only if X -- t l X l + ... + tn+lXn+l for some t C S and xj c K , ( 1 < j < n + l ) . In other words, Cp(K) the image of S x K n+l under the continuous mapping : ( t l , . . . , t n + l , X l , .., X n + l ) --+ t l X l + ... + tn+lXn+l. Thus Cp(K) is compact, m T h e o r e m 31

is

(Generalized Milman Theorem)[2~(2001)

If K is compact in a locally p-convex space X and if Cp(K) compact, then every p-extreme point of Cp(g) lies in K, that is C_ K.

is also

(3.15)

58

CHAPTER

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POINT

P r o o f . Suppose t h a t some p-extreme point a r Cp(K) is not in K. Then there is an absolutely p-convex 0-neighborhood V in X such that

(a+V) AK-r Choose x l , . . . , Xn in K

(3.16)

so that K C Ul(Xj + v ) . Each set

Aj -- Cp(K A (xj + V))

(1 _< i < n)

(3.17)

is absolutely p-convex and also compact since Aj c Cp(K). Also note t h a t K C A1 U . . . U An. But L e m m a 5 shows t h a t m

Cp(K) C Cp(A1 U ... U An) -- Cp(A1 U ... U An)

(3.18)

N

The opposite inclusion holds also, because Aj C Cp(K) fore,

for each j. There-

Cp(K) - Cp(A1 U ... U An).

(3.19)

In particular, a = tlYl + . . . + tnyn for each yj in some Aj, where each tj > 0 and E1n tjp _ 1. Now the grouping

t2y2 + . . . + tnYn a -- flY1 + (1 - tPl)1/p ~2 ~-~ ~--t-tP) llp

(3.20) !

exhibits a as a p-convex combination of two points of Cp(K) by (3.18). Note t h a t the denominator equals (1 - t~) 1/v. Since a is a p-extreme point of Cv(K), we conclude yl - (t2y2 + ... + t~y~)/(t~ + ... + t{) 1/p and hence a -- (tl + (1 - t~)l/p)y I

e

A1.

Thus for some j, m

a E Aj C xj + V C K + V

(3.21)

which contradicts (3.16). Note that Aj C xj + V by (3.17). I 3.1.3

Applications

Although the notion of the p-extreme point seems to be a purely algebraic one, we have the several interesting applications in non convex analysis and m a t h e m a t i c a l programming. Below we introduce some of them.

3.1 P - E X T R E M E

POINT IN NON LOCALLYCONVEX

SPACES59

A p p l i c a t i o n s to N o n C o n v e x A n a l y s i s

It may be useful to give examples of p-extreme points of the closed unit ball in some concrete p-Banach spaces. The following theorem studies the set of p-extreme points in the F-space l P ( 1 > p > 0). The result for 11 is similar and well known. T h e o r e m 32 [27](2000) In

lp, the set of p-extreme points of the closed unit ball Blp

is given

by Ep(Blp) = {Ae("); I)~1- 1, n -

(3.22)

1,2,...}.

Consequently, the p-convex hall Cp(A) of the following set A represents a p-extreme set of Bl~,

1)p !

A-

x(n);x(n)-((n

_l ) nth place ~ 0 , " " . ) , n E N }

"

P r o o f . Suppose IAI- 1 and )~e(n) = Then

1

21/p (x + y), 1

)~ -

2--~/p ( Xn nu Yn )

where x - (Xn), y - (Yn) e Blp.

and 0 -

1

2-~(xj+YJ) jCn.

As A, with IAI - 1, is a p-extreme point of Bz~ - Bzp NK, K is the scalar field, it follows that x~ - Yn and hence 1 - I A I< }--~ I xk IP< 1 and 1 - I A I< ~ l Yk IP~ 1: that is, xj - - y j - - 0 (for j ~ n). Consequently, 21/p/~e (n) -- x -- y.

Thus /~e (n) e E p ( B l p ) , if I A I-- 1. Conversely, if r / - (r/j) e Ep(Blp), then Ilrlll- 1; To see this we claim that there is some n such that r/j -- 0, for all j 7~ n. Hence, r/ - (~j) -- ~n e(n) with I r/n I-- 1, for we have II~ll- 1. Assum, on the contrary that there are j and k with j < k such that

rjj r 0 and r~k r O. Then j

0<~lrlkl i=1

p-#
and

c~

~lnjl p-l-#. i>j

60

CHAPTER

3 FIXED-POINT

AND

P-EXTREME

POINT

Let X m

1 ~(~1,"

"", ~j, 0, 0 , . . . ) ,

Y-

#l/p

1 (1 - #)l/p (0, 0 , . . . ?]j-t-l, f]j+2, "")"

T h e n x, y E Blp are such t h a t x # y

and (r/j) - #l/Px + (1 - ~ ) l / p y

where p e (0, 1). This contradicts the a s s u m p t i o n t h a t (~j) e Ep(Blp), see L e m m a 4(iii). Finally, we note first t h a t the elements of A satisfy, 1

d(x (~), O) - n . - - 1. n

T h e y lie on the surface of the ball Bzp in different places. T h e set Cp(A) is a part of OBzp, for Blp is an absolutely p-convex closed set. m 7 The set A mentioned in the above theorem has played a basic role in bringing out the difference between the theory of holomorphic functions in locally and non locally convex spaces (see Ch.9, section 9.2). m

Remark

In w h a t follows we introduce a closed p-convex set BLp(O, 1) without p - e x t r e m e points. For the unit ball of L 1 the result is well known. T h e o r e m 33 [27](2000)

The closed unit ball BLp(1 > p > O) of Lp [0, 1] points.

has no p-extreme

Proof. It suffices to consider any f E BL, with llfllp - 1 since the interior points will n a t u r a l l y be non p-extreme points. In fact one can choose c E [ 0 , 1 ] such t h a t f o f f l p d t - 1 / 2 , f~lffPdt-1/2. Now let

g-

21/p f.x[O,c], h -

21/Pf.x[c,11

where x[0,c], X[c,l] are respectively the characteristic functions of [0, c] and [c, 1]. T h e n Ilgllp = Ilhl[p- 1 are such t h a t 1

1

f - 2-i~/pg + 2-g~h, g ~r h Hence f

is not a p - e x t r e m e point of BLp. m

3.1 P - E X T R E M E

POINT IN NON LOCALLY CONVEX

SPACES61

Applications to Operations Research In the following we give another application in m a t h e m a t i c a l p r o g r a m m i n g of the p-extreme points. We solve a system of nonlinear inequalities and find the maximum or the minimum of a given nonlinear function restricted to this system.

Example 9 Consider the following example of a nonlinear system of inequalities y < ( x - 1) 2 y < ( x + 1) 2 (3.23) y > - ( x + 1) 2 Y >_ -(~

m

1) 2

Figure 1: The d o m a i n of i n t e r s e c t i o n of t h e four inequalities (3.23) In fact this can be solved if we first consider the four following arcs drawn in the four different quadrant anticlockwise.

y-x2+2x-1

=0

y- x2 - 2x- 1 = 0

(3.24)

y+x2+2x+l=O y+x2-2x+l

=0

These four arcs in R 2 represent the graphs of the four intersecting parabolas whose vertices are (• 0), (0, i l ) , see the figure. The shaded area represents the solution of the system of inequalities (3.23). Since the p-extreme points of the shaded area are (0, i l ) , (=t=l,0), the maximum of the given function will be taken at some of these points. For example, the maximum of the nonlinear function :

62

CHAPTER

3 FIXED-POINT

AND

P-EXTREME

POINT

f ( x , y) -- x 2 - 3y 3

(3.25)

restricted to the system (3.23) will be equal to 3 and is taken at the p-extreme point (0, - 1). The above example is simple but an interesting one to explain the benefit of our new concept, the p-extreme points. However the concept of p-extreme points can be used widely to solve a general problem in mathematical programing. For example the nonlinear programming problem (NLP) can be expressed as follows :

E x a m p l e 10 Find the values of the decision variables that

X l ~X2 , ...Xn

such

max (or rain) z -- f ( X l , X 2 , . . . x n ) s.t. gl ( Xl, X2, ...Xn) ( <_ ,--,or _>)bl

g2( Xl, X2, ...Xn)

( ~ ,--, or >_)b2

gin( Xl, X2, ...Xn)

( ~ , - - , or ~ ) b m .

(3.26)

As in the linear programming, f(Xl,X2,...x,~) is the NLP's object function, and gl( Xl, x2,...Xn) (_<,-,or >)bl., ..., gm( Xl, x2, ...xn) (_<, = ,or _>)bin are the constraints. NLP with no constraints is an u n c o n s t r a i n e d NLP. The solution will then be obtained whenever the p-extreme points of D, the intersection domain of the constraints, are obtained.

3.2

GENERALIZED

FIXED POINT THEOREM

Fixed point theorems play an important role in many parts of analysis and topology. They have many applications, for example, in the fields of linear algebraic equations, ordinary and partial differential equations, and integral equations. In fact, one of these basic theorems is due to Brouwer which is considered as one of the most celebrated theorems in mathematical analysis. Fixed points of continuous and holomorphic functions on compact convex sets in locally convex spaces have been studied by several authors, see for example ,[108] and [188].[206]

3.2

GENERALIZED

FIXED POINT THEOREM

63

We devote this section to extend Schauder's and Tychnoff's Fixed Point versions, of Brouwer's Fixed point theorem, to p-convex sets K in locally p-convex spaces X, (1 > p > 0), by proving that K has the fixed point property. We also generalize Kakutani's Fixed Point Theorem to locally p-convex F-spaces whose dual separates points. We prove that an equicontinuous group of affine maps of K has a fixed point.These depend on making use of the new terminology, the p-extreme points of non-convex sets, see the preceding part. In [24], we studied the set of fixed points of holomorphic mappings of some non convex bounded domains U in r n and proved that it represented an analytic submanifold of U. Moreover, in [37], we studied the fixed points theorem of holomorphic mappings f of a bounded Riemannian manifold M and proved that the set of all fixed points of f is a complex submanifold of M. D e f i n i t i o n 23 Let X and Y be vector spaces and T " K --. Y satisfies T ( s x + ty) = s T ( x ) + t T ( y ) whenever x, y e K , 0 < s, t. Then T is said to be a j ~ n e . Let us recall that a point z E A is said to be a p-extreme point of A if z is not an internal point of the arc segment A y for x, y E A. That is, if z - s x + ty, x , y E A, 0 < s , t , s p + t p - l t h e n x - y [ = ( s + t ) - l z ] . Ep(A) denotes the set of p-extreme points of A, see the preceding part. D e f i n i t i o n 24 Let X be a vector space and K C X . A set G is said to be a n e q u i c o n t i n u o u s g r o u p of maps of a set K if it takes K into K , and G is a group. That is, every T C G is a 1-1 map of K into K whose inverse T -1 also belongs to G (so T maps K onto K ) and that T1T2 E G whenever T1, T2 E G. Here (T1T2)x = T1 (T2x). We note that the composition of two affine maps is affine. Also, by equicontinuity we mean that, to every neighborhood W of 0 in X corresponds a neighborhood V of O in X such that T x - T y E W whenever x E K , y C K , x-yCV andTCG.

3.2.1

Generalized

Brouwers's

Fixed Point Theorem

The infinite-dimensional version of Brouwer's theorem, concerning the fixed point property of closed balls in ~ , of convex sets of normed spaces has been given by Schauder [188] and of locally convex spaces by Tychnoff [205].

64

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POINT

In what follows, we generalize their fixed point theorems to p - c o n v e x s e t s in l o c a l l y p - c o n v e x F - s p a c e s , need not be locally convex, (1 > p >

0). The problem is nonlinear and involves Minkowsky p-functionals. T h e o r e m 34 ( G e n e r a l i z e d B r o u w e r ' s theorem)[28](2000). If K is a nonempty compact p-convex set in a locally p-convex F-space X , and f " K ~ K is continuous, then f ( a ) - a for some a E K. P r o o f . C o n t r a r y to the conclusion, we assume f fixes no point of K. T h e n its graph G - {(x, f ( x ) ) E X • X; x E K } (3.27) is disjoint from the diagonal A of X x X and is compact. Hence there is an absolutely p-convex neighborhood V in X such t h a t G + (V x V) does not intersect A. In particular we get

f(x) ~ x + V

(x E K ) .

(3.28)

Let qv be p-Minkowsky functional of V. Now qv is continuous on X and q v ( x ) < 1 if and only if x E V, see [207]. Define a ( x ) -- max{0, 1 - q v ( x ) } Choose X l , . . . ,

Xn E

K

(x E x).

(3.29)

so t h a t the sets xi + V (1 _ i _< n) cover K, i.e. K C U~xi + V.

P u t a i ( z ) - c ~ ( x - xi)

and define

I OZi(X) I lip ~i(X) -- O~1(X) ~- : : :--nc tin(X)

(X E K, 1 < i <

n)

(3.30)

For every x E K , we note t h a t the pth power of the d e n o m i n a t o r is positive. Let L - C p { X l , . . . , X n } , the p-convex hall of xi (1 < i < n). T h e n the function g defined by g'K--~L n

g(x) - ~-~,fli(x)xi 1 is a continuous m a p from K This is also true for g o f.

(x E K )

(3.31)

into the compact finite dimensional L C K.

3.2

GENERALIZED

FIXED POINT THEOREM

65

Note t h a t ~-]-1/3~p - 1 and ~-]~1/3i -< 1. Assume t h a t ~-].~ ( f l ~ - / 3 i ) - 7. Hence 0 < 7 < 1. Now, since the different topologies on a finite dimensional space are equivalent, Brouwer's fixed point asserts t h a t there is an x* E L such t h a t

g ( f ( x * ) ) - x* As ~i(x) - 0

outside xi + V, we find t h a t

n

x-gtx)

(3.32)

n

- x.

Z tx)x 1

n

n

1

1

-

(x e K).

1

Hence, n

(1- 7)x-

g(x) - ~--],fli(x)(x- xi)

(x E K )

(3.33)

1

is a p-convex combination of vectors x (1-9/)x-g(x)

EV,

In particular, this is true for x (1- ?)f(x*)-

xi E V.

Thus

for e v e r y x E K .

f(x*). We therefore have g ( f ( x * ) ) E V.

Therefore,

( 1 - 7)f(x*) E x* + If, i.e.

)~f(x*) E x* + V

(3.34)

for some 0 < )~ < 1, where A - 1 - ~ / contrary to (3.28). This is because, the absolutely p-convex 0-neighborhood V of our metrizable space X can be chosen so t h a t A f ( x ) ~ x + V whenever it satisfies f ( x ) ~ x + V. In fact and as E is an F-space we m a y choose V - B(0, e) with e < min {d(x, f ( x ) ) , d ( x , ) ~ f ( x ) ) } . I The following well known l e m m a will be useful in proving the generalized K a k u t a n i ' s fixed point theorem, see Rudin [187]. L e m m a 6 Suppose that A and B are topological spaces, B is compact, 7~ is the natural projection of A • B onto A and E C A • B. I f a E A lies lies in the closure of E for some b E B.

I

66

3.2.2

CHAPTER

3 FIXED-POINT

AND P-EXTREME

POINT

Generalized Kakutani's Fixed Point Theorem

Let X be a locally p-convex space which is not necessarily locally convex whose dual separates points. In this section we study the extension of the Kakutani's fixed point theorem to compact p-convex sets of X.

T h e o r e m 35 (Generalized Kakutani's Fixed Point Theorem)J28] A s s u m e that: (1) X is a locally p - c o n v e x space whose dual separates points, and K is a n o n e m p t y compact p-convex set in X such that K equals the closed p - c o n v e x hull of its p - e x t r e m e points, that is C p ( E p ( K ) ) -- K . (2) G is an equicontinuous group of ajfine maps taking K into K . Then G has a c o m m o n fixed point in K . That is, there exists a E K such that

Ta=a for every T E G.

Proof. HC_K

Let ~ be the collection of all nonempty compact p-convex sets such t h a t T ( H ) C_H for e v e r y T E G . That is, -{HCK;

H~r

CH, everyTEG}

Partially order ~ by the set inclusion. Notice that ~ ~ r since K E ~. By Hausdorff's maximality theorem, ~ contains a maximal totally ordered subcollection ~0. The intersection Q of all members of ~0, i.e.

Q - NTE Fro T is a minimal member of ~. The desired result will be obtained by showing that Q contains only one point. Assume, on the contrary, that there exist x E Q, y E Q, x ~- y. Then there is a neighborhood V0 of 0 in X such that x - y /~V0. Let W0 be associated to V0 by the equicontinuity of G. If T x - T y were in W0, for some T E G, then x - y -- T -1 ( T x ) - T -1 ( T y )

would be in V0, a contradiction. Therefore, we conclude that for no T E G isTx-Ty inW0. Put 1 z - 2- (x + v)-

3.2

GENERALIZED

FIXED

POINT

THEOREM

67

Then z C Q as Q is p-convex. We note that the intersection of p-convex sets is p-convex. Define G(z) - {Tz; T e G}.

This "G-orbit of z" is G-invariant; that is, every T C G maps G(z) into itself, and consequently for its closure Kp - G(z); and therefore Cp(Kp) is a nonempty G-invariant, compact p- convex subset of Q. By minimally of

Q, Cp(Kp) - Q.

Let a be a p-extreme point of Q. Since Q is compact and Q - Cp(Kp), Theorem 31 shows that a lies in the closure Kp - G(z) . Define the set E-

{(Tz, T x , Ty) . T C G} c Q x Q • Q.

Since a E G(z) and Q x Q is compact, the above Lemma 6 shows that there exists a point (x*, y*) E Q x Q so that (a, y*, x*) lies in the closure o r E , (just t a k e A = Q , B = Q x Q andb=(x*,y*). Since 2-1/PT(z) - T ( x ) + T(y) for every T C G, it follows that 2-1/Pa x* + y* and this implies that x* = y* because a is a p-extreme point of Q. But T x - T y /~W, for every T E G; hence x * - y * f W , that is, x* ~ y*, and we have a contradiction. I R e m a r k 8 The set K - A b, the arc segment between a, b E X , satisfies the hypothesis of the preceding Theorem 35. Note that K - Cp(Ep(K)). E x a m p l e 11 T h e spaces l(pn). O0

[Xn[p"
l ( p n ) -- { X -- { X n } ; Xn e (~,

(1 > _ p n > O )

1

with the F - n o r m IixII = E Ix, Ip are F-spaces with Schauder basis (e,). They are locally bounded, and hence p-normed if Pn /-*0, see Rolewicz [186]. As the closed unit ball Bzp. (0, 1), Pn /--*0 is absolutely p-convex for some (0 < p < 1), then the obtained generalized Brouwer's theorem 3~ asserts any continuous f " Blp" (0, 1) --~ Btp,. (0, 1) has a fixed point x e Bzp" (0, 1), that is with f ( x ) - x, where Bl~p,(0, 1) is the n-dimensional unit ball; (see [186]). A

68

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3 FIXED-POINT

AND

P-EXTREME

POINT

E x a m p l e 12 T h e O r l i c z s p a c e s l(r Let r be a continuous unbounded subadditive increasing function on [0, oc] with r = 0 if and only if t - O. We define Ir as O0

r

lr - - { X - - ( X n ) , x n E (~;

< ~}.

n=l

Shapiro [191] proved that if the function f ( t ) : t-PC(T) is a monotonic decreasing in (0, ~ ) for some (0 < p < 1), then every closed norm-bounded subset of 1r is compact. In connection with the topological properties of lr we can apply, for example, the generalized Kakutani's theorem to every equicontinuous group of affine map taking B~r 1) --~ B~r 1). We may take G to be the set of all linear isometrics on K - B~r 1); that is

a-

{T. K --, K i~ li~a~, Ilxll~- IITxll,}.

We note that K - C,(E,(K),

for K - Bl~ since it is proved in Theorem 32 that E p ( K ) - {)~ej; I)~1 -

1, ( ~j ) ~ ba~ of l~ }. In [15] we proved that Bt(r polynomially convex. Now T~(x) - c~x, Ic~[- 1, x e Bh(0, 1), are elements of G. So G has a fixed point. Note that for a certain r as r - I t l p, lr is a locally bounded p-norrned space. A

3.3

GENERALIZED

KREIN-MILMAN

THEOREM We extend the Krein-Milman Theorem to locally p-convex spaces whose dual separates points. Also some separation theorems to separate certain types of disjoint non convex sets are given. More precisely this part is devoted to establish a G e n e r a l i z e d K r e i n M i l m a n T h e o r e m , suitable to apply to non convex sets as the p-convex sets. We also present new separation theorems for disjoint p-convex sets and for disjoint logarithmically convex sets in certain p-normed spaces by making use of continuous polynomials (1 > p > 0). In fact, the importance of the concept of the extreme points and KreinMilman Theorem arises from its wide applications to different branches of

3.3

GENERALIZED

KREIN-MILMAN

THEOREM

69

m a t h e m a t i c s such as functional analysis, convex analysis, linear programming and complex analysis. We found ourself in a need of a new version of the Krein-Milman Theorem after introducing the new concept of p-extreme points in the preceding part in order to solve some problems in complex analysis for non locally convex spaces.

3.3.1

Generalized Krein-Milman

Theorem

The following theorem generalizes K r e i n - M i l m a n T h e o r e m to sequence locally bounded spaces which are not necessarily locally convex. In fact, it shows t h a t under certain conditions the set of p-extreme points Ep(K) of a compact set K is quite a large set. Let us first state the following lemma. Let X be a topological vector space whose dual X ' separates points of X. Then we have the following separation theorem which will be useful to obtain the Krein Milman theorem.

L e m m a 7 Assume X is a topological vector space whose dual X ' separates points, and B is compact in X . Then for every xo ~ B, there exists an aj~ne functional T on X such that Re T(xo) < sup Re T(x). xEB

P r o o f . It is enough to prove this for real scalars. From the assumption, for any y E B, there exists T A E X ~ such t h a t

T ~ (U) # T ~ (x0) Since B is compact, there exists a finite cover, say Bi = B(xi,e), i = 1, 2...,n to every infinite cover of B. T h a t is,

B c u?B(x,, ~). Choose Ti E X ~ such that Ti(x0) = 0, Ti(xi) = 1 Any x E B should be in some B(xi, e). By the continuity of Ti we have

I T~(x)- T~(x~)I=1 T~(~)Hence

1

I<

70

CHAPTER

3 FIXED-POINT

Ti(xi) > l - e

AND

P-EXTREME

POINT

and T/ (x0) - 0.

Now set T - 7'1 + . . . T ~

where T/(x~) - 1, T~ (x0) - 0.

It follows, from Ti(xi) - 1, t h a t

T(x) > l - e ,

T(xo) - O.

Since x is a r b i t r a r y we have

T(x).

T(x0) < xEB

m Before we state the Krein Milman theorem, let us also prove the following lemma.

L e m m a 8 Let X be a locally p-convex space. Further let K be a compact p-convex set in X , and F the collection of all compact p-extreme sets of K. Then, the intersection S of any nonempty subcollection o f F is a member of F, unless S = r i.e. F has the finite intersection property.

P r o o f . Let S - N c ~ c I A~ -r r be an intersection of an a r b i t r a r y subcollection of F , for some index set I. Let x, y C K. Now if 1

1

z -- (1 - t)~x + t~y C S,

(3.35)

thenzEA~ for a l l a E I . Hencex, yCA~ f o r A ~ is a p - e x t r e m e s e t o f K . T h a t is, x, y E S to show t h a t S i s a p - e x t r e m e s e t o f K . 1

T h e o r e m 36 (Generalized Krein-Milman Theorem)J23] (1999) Assume X is a locally p-convex space whose dual separates points. If K is a nonempty compact p-convex set in X , then K is the closed p-convex hull of the set of its p-extreme points, i.e.

K - Cp(Ep(K)).

(3.36)

3.3

GENERALIZED

KREIN-MILMAN

THEOREM

71

P r o o f . Let F be the collection of all compact p-extreme sets of K. Since K C F , F % r We shall use the following property of F : (*) If S c F , # = maxRe T

S

is the m a x i m u m of R e T ( x ) on S where T E X ~ and

ST = {x e S; R e T ( x ) = #},

(3.37)

then ST C F. To prove (*), assume 1 (_~)l/p(y + x) _ z e ST ,

x e K , y e K, ( t - l ~ 2 ) .

(3.38)

Since z E S, S c F , we have x E S , y E S. Hence R e T ( x ) < #, R e T ( y ) < p. As R e T ( z ) = p and T is linear, we conclude that: R e T ( x ) = # = R e T ( y ) . In fact, 1

T(z) - (-~)I/p[T(y) + T(x)]. Therefore R e T ( z ) - ( 89 + R e T ( x ) ) - #. Since S is a p-extreme set, R e T ( x ) - R e T ( y ) 21/P-1# ~_ #.

(3.39) 21/P-lReT(z) -

Hence x C ST and y C ST. This proves t h a t ST E F. Choose some S C F. Let F ~ be the collection of all members of F t h a t are subsets of S. Since S c F ~, F ~ ~ r Partially order F ~ by set inclusion. Let ~t be a maximal totally ordered subcollection of F ~ and let M be the intersection of all members of f~. Since ~t is a collection of compact sets with the finite intersection property, it follows that M :fi r by L e m m a 8, M E F . The maximality of ~t implies t h a t no proper subset of M belongs to F . It follows now from (*) t h a t every T E X ~, is constant on M. Since X ~ separates points on X , M has only one point. Therefore M is a p-extreme point of K. We have therefore proved t h a t

E p ( K ) N S 7/= r

(3.40)

for every S E F. In other words, every p-extreme set of K , like S, contains one p-extreme point of K.

72

CHAPTER

3 FIXED-POINT

AND P-EXTREME

POINT

Since K is compact and p-convex, by the assumption of p-convexity of K, we have obtained the inclusion

(3.41)

Cp(Ep(K)) c K.

To claim by contradiction the other inclusion, assume x0 C K were not in Cp(Ep(K)). Then T E X ' exists such that

(3.42)

Re IT(x) + ~] < Re IT(x0) +/3],

where ~ is a scalar, for every x e Cp(Ep(K)), by lemma 7. If KT defined as in (*), then K T E F. Our choice of T shows that

is

KT = { x C K ; R e T = # } -

{xCK;Re(T+~)-supRe(T+~)}K

is disjoint from Cp(Ep(K)), where # - suPK Re T. This contradicts (3.40). This completes the proof. I

3.3.2

Separation F-Spaces

Theorems

in

Some Sequence

We present, in this section, new separation theorems which deal with disjoint p-convex sets, and with disjoint logarithmically convex sets, in sequence locally bounded spaces by using continuous polynomials, (1 > p > 0). Let us first introduce the following definitions. D e f i n i t i o n 25 ( R e i n h a r d t a n d l o g a r i t h m i c a l l y c o n v e x d o m a i n s ) . An open set U C (Fn is called a R e i n h a r d t d o m a i n if ( z l , . . . ,Zn) E U implies (eie~zl,...,eie~zn) E U for arbitrary real 0 1 , . . . , 0 n . It is said to be l o g a r i t h m i c a l l y c o n v e x if it has the properties of a convergence domain, that is, (i) u* - {t e

~; (~,..., ~)

e u}

i~ o p ~

a~d c o ~ ~

~ ~.

(ii) If ~ E U*, then ~T E U* if ~]j < ~j for every j 5ii) z C U if and only if there exists t E U* such that [zy I < etj, (j - 1 , . . . , n), see Hormander [118]. A set U of a topological vector space X is said to be l o g a r i t h m i c a l l y c o n v e x if U N Xn is logarithmically convex as a subset of every finite dimensional Xn of X .

3.3 G E N E R A L I Z E D K R E I N - M I L M A N

THEOREM

73

Definition 26 (Space o f C o n v e x type). A p-normed space X is said to be of c o n v e x type if for every n E ~ and every p-convex set A in Xn, the set ~-I(A)

-

{x-

(Xl,...,Xn)

(exl,...,e xn) E A}

E Xn;

(3.43)

is convex in the finite-dimension space Xn --"~(Tn, where

~(Xl,-.-,

Xn)

--

(eXl,

. . . ,

e xn).

For examples of such spaces of convex type, see the author [23],. Let us recall t h a t a function P 9E --, (T is said to be a homogeneous polynomial if P(x) = A ( x , . . . , x) for a symmetric m-linear m a p A on E.

Sequence Spaces with p-Convex Balls Let us remind you t h a t the p-convex hull of a compact set of a certain non locally space is also compact. More precisely 9 If X is a locally p-convex F - s p a c e and K C_ X is compact then the closed p-convex hull Cp(K)is also compact. T h e following theorem extends the classical separation t h e o r e m of convex sets in a locally convex space to p-convex sets of sequence locally b o u n d e d spaces which m a y not be locally convex.

T h e o r e m 37 [22] (1999) Suppose A and B are disjoint, nonempty p-convex sets (0 < p <_ 1) in a p-normed space of convex type and with Schauder basis, 0 l B . If A is open, then there exists a continuous monomial P E P ( m X ) and c~ E ~ such that

Vx E A, Vy E B

ReP(x)
(3.44)

P r o o f . Let b E B \ A be arbitrary. We can assume bj >_ 0, by multiplying bj by complex numbers of modulus one and ignoring coordinate for bj = O. The construction of P is a finite dimensional problem. This is because for every b ~ A there exists n E/lg such t h a t 7rn(b) f_Trn(A) = A N X n where Xn C_X of dimension n. Consider the m a p p i n g ~ : ~ n __, ~ n defined by 9 9 ( t l , . . . , tn)

-

-

(etl,

... , e tn)

(3.45)

74

CHAPTER

3 FIXED-POINT

where t - ( t l , . . . , tn) e B,~ i.e.,

Let A*

~n.

--

AND

yg-l(Trn(A))

P-EXTREME

An, B*

-

A* -

{t elan; ( e t ~ , . . . , e t~) e A~}

B* -

{t e ~ n ; ( e t l , . . . , e t~) e Bn}

--

POINT

yg-l(Trn(B))

-

Since X is of convex type, A* and B* are convex sets in ~ n and A* is open as ~p is convex continuous. Apply now the classical s e p a r a t i o n t h e o r e m on the disjoint convex sets A* and B*, see the a u t h o r [15] and[23], we get a continuous functional o n ~ n s u c h t h a t for some a E/~, sup ~(a*) < c~ _< Reu(b*)

(3.46)

a'cA*

for every a* C A*, where b* - ( b ~ , . . . , bn) - (log bl, log b 2 , . . . , log b,~). Here takes the form n

p(t) -- ~ m j t j

-- m l t l + . . . + mntn,

t E ~n

(3.47)

j=l where necessarily mj >_ 0 , j - 1 , . . . , n , point s with sj <_ tj, see [118].

since A*,B* contain with t every

B u t the h y p e r p l a n e {t E ~n; }--~j~--1m j t j - v(b*)} can be moved so t h a t all rnj become rational numbers, j - , 1 , . . . , n. T h u s there exists a positive integer m such t h a t rnrnj = kj are all non-negative integers. T h e monomial P(x)

- - Xkll . . . X nkn,

x C g,n

(3.48)

satisfies the desired inequality. In fact,

P(b) - b ~ . . . b ~ " - e k~b~ . . . ~-k~*, - e "~'(b*)

(3.49)

and ReacA

P(a) <_ sup

IP(a)l-

ernu(a*) < P(b),

a C A, a* E A*

aEA

Pick c to be the right h a n d end point of u(A*) and p u t now c~ - e c we get,

n P(x) <

<_ n P(y),

xcA,

yEB.

I

3.3 G E N E R A L I Z E D K R E I N - M I L M A N

THEOREM

75

R e m a r k 9 L e m m a 7 at the beginning of the the previous part which deals also with some type of separation of compacts sets in topological vector spaces.

E x a m p l e 13 We note that the spaces oo

IXn [pn < (:X:)}, (1 :> Pn > O)

l(pn) -- { x -- (Xn); Xn ~ (~, 1

with the F - n o r m Ilxll- E Ix l F- spaces with Schauder basis. For example, if p , /--~0, it is locally bounded otherwise it is not even locally pseudoconvex, (see Rolewicz [[186], p. 151]). Now the balls oo

-

e

Ix l

<

> 0,

1

are absolutely p-convex, when l(p,~) is a locally bounded space and metrized by a p-norm.

R e m a r k 10 In fact the unit ball B x of any locally bounded space X is absolutely p-convex and of convex type, see the author [15] and[20]. So we can apply the preceding Theorem 37 to all locally bounded spaces with Schauder bases. A

Sequence Spaces with Logarithmically Convex Balls We give in this section another form of separation theorem to separate logarithmically convex sets which are not necessarily convex, by using continuous monomials. Note that not every logarithmically convex domain is p-convex, see [[15], Remark]. T h e o r e m 38 [22](1999) Assume that A and B are disjoint, nonempty, logarithmically convex Reinhardt domains of a locally bounded F-space X with Schauder basis (ej). Then there exists a continuous monomial P ( x ) and a C ~ such that

Re P ( x ) < a <_ R e P ( y )

Vx E A, Vy E B.

P r o o f . We follow the proof of Theorem 37, noticing that

(3.50)

76

CHAPTER

3 FIXED-POINT

A* - ~-1 (An), B*

AND

-

~-1

P-EXTREME

POINT

(Bn)

are convex subsets of ~n, since they are logarithmically convex domains by assumption, m E x a m p l e 14 Let X - 1r the Orlicz space, and B be the unit ball of X . Then every disjoint logarithmically convex Reinhardt domain A can be separated from B by a continuous monomial P ( x ) , that is, ReP(x) < c < ReP(y)

Vx C A, Vy C B.

We note that the Orlicz space l~ is an F-space with basis and it is metrized subadditive, increasing function on [0, (x~], with r = 0 if and only if t=0. If the function f ( t ) = t - ' r is monotone decreasing in (0, c~) for some (1 > p :> 0), Shapiro [189] proved that, every closed, norm- bounded convex subset A of 1r is compact. Now the unit balls Bn of 1r for some r restricted to every finite dimensional space X n is logarithmically convex, see [23]. So it is logarithmically convex in such 1r Hence we can separate its points from other sets in 1r by a continuous polynomial. A

Chapter 4

QUASI-DIFFERENTIAL CALCULUS We introduce in this book, starting from this chapter, a new concept for differentiability of mappings between locally bounded F-spaces which we have called"Quasi-differential, or pq-dij~ferential ''. We shall use it in all this chapter and in other chapters. I t is d i s c o v e r e d in 1 9 9 5 while we were working to have a new form of L a g r a n g e M e a n - Value t h e o r e m in some non locally convex spaces and for more f o u n d a t i o n s o f the t h e o r y o f H o l o m o r p h y w i t h o u t c o n v e x i t y c o n d i t i o n ; it is also called Bayoumi differential by some Mathematicians, see [20], [21], [26] ~r So all the results dealing with this quasi-differentiability are due to the author, see bibliography.

4.1

QUASI-DIFFERENTIABLE MAPS

One of the fundamental concepts of the theory of functions between normed spaces is differentiability where the differentials of Fr~chet and G~teaux appear to be the most appropriate concepts. In this part we study a new and different concept of differentiability. This is what we have called, Q u a s i - D i f f e r e n t i a b i l i t y or p q - D i f f e r e n t i a b i l i t y , for maps f : E --, F between locally bounded F-spaces E, F. In fact, our Quasi differentiability is totally different from the Fr~chet one although it has the same advantage of the Fr~chet one. T h a t is, it is linear and continuous mapping. This new concept of differentiability was discovered by the author in 77

CHAPTER

78

4 QUASI-DIFFERENTIAL

CALCULUS

1995 while he was working on finding theorems for the mean-value in real and complex locally bounded F-spaces, see the author [20], [21] and [35]. So all the results of this chapter are due to the author. 4.1.1

Q u a s i - D i f f e r e n t i a b l e M a p s in F - S p a c e s

Definition respectively f , g : U --. pq-tangent

27 (0 F, to

( 1 9 9 5 ) Let < p,q <_ 1), and a E U, each other at lim

IIf(x)

x---4a

E and F be p-normed and q-normed spaces and U a nonempty open subset of E. For we say that f and g are q u a s i t a n g e n t or a if -

g(x)lll/q/llx

-

all 1/p

-

0

.

That is, for any e > 0 there exists 6 > 0 such that

0 < Ilx- all < 6 =>

Ill(x)- g(x)ll 1/q ~ EIIx- all

9

(4.2)

This is equivalent to 0 < I I x - all < 5 =~ I [ f ( x ) - g(x)ll p <_ el l l x - all q,

s

-

-

(4.3)

s

from which we choose the word pq-tangency in our definition.

Notice t h a t if f, g are quasi-tangent to each other at a then ( f - g) is continuous at a and f ( a ) - g(a); thus if one of these functions is continuous at a the other must be continuous at a. Moreover f, g are quasi-tangent at a if and only if and only if ( f - g) is quasi-tangent to 0 at a. Now it is easy to verify t h a t quasi-tangency at a is an e q u i v a l e n c e r e l a t i o n on the vector space of all mappings from U to F which are continuous at a. In fact, if f and g are pq-tangent at a, and g and h are pq-tangent at a, then f and h are pq-tangent at a. Notice t h a t

ilf(x) _ h(x)lll/q <_ ~(llf(x) - g(x)[] 1/q + ilg(x) - h(x)lll/q),

~ >__1 (4.4)

as a consequence of which II" IIlzq is a quasi-norm. The notation of pq-tangency depends only on the topologies of E not on the p-norm and, q-norm used to define the topologies.

and

4.1 QUASI-DIFFERENTIABLE

MAPS

79

28 [20] ( 1 9 9 5 ) Let E and F be p-normed and q- normed spaces, respectively (0 < p, q < 1), and U a n o n e m p t y open set in E. A mapping f : U ~ F is said to be q u a s i - d i f f e r e n t i a b l e (or p q - d i f f e r e n t i a b l e ) at a C U, if there exists a linear map Ta E L ( E , F ) , such that f and the continuous affine linear map x E E ~ f (a) + Ta(x - a) are pq-tangent at a, that is, Definition

lira ( l l f ( x ) x----+a

Hence,

II/(x)

f(a)

-

T~(x

-

a)llP/llx

-

0 < IIx- all < 5 =:>

for every ~ > O, there exists 5 > O"

f(a)

-

-

Ta(x - a)ll <_ ellx

-

(4.5)

all q) - o.

all q/p

29 I f f " U --, F is pq-differentiable at each point of U, is said to be p q - d i f f e r e n t i a b l e on U.

Definition

f

then

R e m a r k 11 It is to be noted that if E and F are both p-normed spaces or quasi- normed spaces with the same quasi-norm constants then the condition (~.5) turns out to be as the following classical one: lim ( l l f ( z ) -

x---~a

4.1.2

Properties

f(a)

-

T~(x

-

a)ll/llz

- all) - 0.

of Quasi-Differentials

We show in what follows the linearity of the quasi-differentiable mappings. Theorem

39

[21]

The set of the quasi-differentiable mappings at a C U represents a vector space. P r o o f . In fact, if f and g are pq-differentiable at a C U, then there exist linear m a p p i n g s T~, Tg E L ( E , F) such t h a t [If(x) - f ( a ) - T~(x - a)[[ 1/q ~ ~[[X -- all 1/p Iig(x)

-- g(a)

--

T * ( x - a)ll 1/q ~ ~ I I x - a]] 1/p

for a given c > O. Since [[. I]1/q is a quasi norm, we have

II(f + g ) ( x ) - ( f + g ) ( a ) - (T~ + T ~ ) ( x - a)ll ~/q ~_ a ( l l f ( x )

-

f(a)

- T~(x

- a)ll ~/q + ]lg(x) - g ( a ) - T 2 ( x

- a)lll/q).

CHAPTER

80

4 QUASI-DIFFERENTIAL

CALCULUS

So by dividing b o t h sides by I I x - a[[ lip and taking the limit as x ---, a we obtain

D ( f + g)(a) - D r ( a ) + Dg(a)

(4.6)

D()~ f)(a) - AD f (a).

(4.7)

Also Notice t h a t IIA(f(x) - f ( a ) - T ~ ( x - a))ll x/q - I I ( A f ) ( x ) - ( ) ~ f ) ( a ) - A T ~ ( x - a)ll ~/q

So if we divide b o t h sides by I I x - a[[ 1/p and take the limit as x --~ a we o b t a i n the required equality, i T h e following t h e o r e m extends the classical Lipschitzian property to locally b o u n d e d spaces, see [21]

Theorem

40 ( L i p s c h i t z i a n property)J21]

Let E

and F be p-normed and q-normed spaces, respectively, (0 < and U a non-empty open subset of E. If f : U --~ F is pqdifferentiable at a E U, then there exist c > 0 and 5 > 0 such that

p , q <__ 1)

IIf(x) - f ( a ) l l <_ clJx - all q/p fo~ x c u,

I I x - all < ~.

Let T IIf(z) - f ( a ) ] ] -

Proof.

I n particular, it f o l l o w s that f

(4.8) is c o n t i n u o u s at a.

D f ( a ) and A(x) -- f ( z ) - f ( a ) - T ( x - a ) for x E U. T h e n l i T ( x - a ) + A(x)II <_ IlTllqllz-a[Iq/P + IIA(x)[i because IlT(x - a)l [ _< Ilrl[qllx -- al[q/p,

see (1.3). Since l i m z ~ a IlA(x)iiP/]]x- allq -- O, and A ( a ) -- 0, given e -- 1, there exists S > 0 such t h a t if IIx - all _< 5, x E U, t h e n

ilZX(x)ll _< IIx- all ~/~. Hence

I I f ( x ) - f(a)] I <_ (IITIIq +

1)llx-ailq/p

= ~ l l x - all ~/~ where c-

IITll q + 1. i

T h e following t h e o r e m has useful applications.

4.1 Q U A S I - D I F F E R E N T I A B L E

MAPS

81

41 [21] Let f : U --~ F be pq- differentiable at a E U and let T E L ( F , G) where E, F and U are as in the preceding theorem and G is a q - n o r m e d space. Then ( T o f ) is pq- differentiable at a and Theorem

D ( T o f ) (a) Proof.

=

T o D f (a).

(4.9)

Since

[IT[f(x)- f(a)-Df(a)(x-a)]II

1/q <_ I l T i l l / q l l f ( x ) - f ( a ) - D f ( a ) ( x - a ) l l

1/q,

dividing b o t h sides by IIx - a l l ~/p a n d t a k i n g t h e limit as x --+ a we o b t a i n II(T o f ) ( x ) - ( T o f ) ( a ) - (T o D f ) ( a ) ( x - a)lll/q/ll x - all ~/p <_

IITlll/qllf(x) - f ( a ) - D f ( a ) ( x - a)ill/q/llx - all 1/p -+ 0 i.e. D ( T o f ) ( a ) = ( T o D f ) ( a ) . I 30 A f u n c t i o n f : E --~ F is said to be l o c a l l y c o n s t a n t i f f is a constant f u n c t i o n on every neighborhood of each point of E . B y a constant map o n E we mean a map x E E --~ b E F, for a fixed b in F. Definition

Theorem

Let E p,q<_l).

4 2 [21]

and F

be p - n o r m e d and q- normed spaces respectively, (0 <

(a) If f : E ~ F

is a constant map, then D f = 0; i.e.D f (x) = 0 Vx E E.

(b) I f A E L ( E , F ) ,

then D A DA(x)

Proof. we have

is the constant map satisfying :

A,

(a) Let f be c o n s t a n t on E . IIf(a)

-

- Df(x)(a -

Vx E

(4.10)

E.

H e n c e for its q u a s i - d i f f e r e n t i a b i l i t y

x)lll/q/llx-

-+ 0

i.e. I I D f ( x ) ( a - x)lll/q/llx - all 1/p < IlDf(x)llll x - aJlllP/llx - all lIp --+ 0 to have IIDf(x)ll -+ o from which D f ( x ) = 0 for all x E E .

82

CHAPTER

4 QUASI-DIFFERENTIAL

CALCULUS

(b) Let A E L ( E , F ) . Then ]]A(x) - A ( a ) - D A ( a ) ( x - a ) i i l / q / I I

= ]IA(x - a) - D A ( a ) ( x - a ) i i l / q / I I = IIA - DA(a)l[ii(x - a ) i i l / p / i i x

x x -

ai] I / p aiI 1 / p

- aiI l i p --+ o

i.e. D A ( a ) - A for a l l a C E . I We shall write for simplicity the constant map D A " x C E --, A E L ( E , F ) by D A A. However the rule D A A should not be confused with the formula d(**) = e ~ in calculus. In fact the rule D A - A should be compared dx with d (cx) -- c by noticing that K ' - K .

T h e o r e m 43 ( M a p p i n g i n t o a p r o d u c t s p a c e ) [ 2 1 ] ( 1 9 9 6 ) Let U C E be open in a p - n o r m e d space E and Fn be p n - n o r m e d spaces (n = 1 , . . . , m ) . A mapping f : U --, F1 • • Fn is quasi-differentiable at a if and only if each coordinate map fn = 7rn o f is quasi-differentiable at a. In this case D f ( a ) = ( D f l ( a ) , . . . , D fro(a)) where 7rn is the projection f r o m F1 x . . . x Fm

(4.11)

onto F~.

Proof. Let Fn be pn-normed spaces for n - i , . . . , m , and let F = F1 x ... x Fro. Then F is an F-slJace and the topology induced by an F - n o r m on F is the product topology. Let 7rn " F ~ Fn

be the projection onto the nth factor F~ and let u~ " F ~ - - , F

be the natural embedding map defined by Un(Xn)

--

( O , O . . . , O, X n ,

O, O, . . . , O )

Then both 7rn and un are continuous linear maps and 7~n O U n

-~Un O Trn

-

-

-

-

1F~ 1F

(the identity map on Fn) (the identity map on F).

4.1 Q U A S I - D I F F E R E N T I A B L E MAPS

83

Let U be open subset of E and let f 9U --~ F; and let f n be the nth coordinate map. Then m

7r,~o f 9U - ~ Fn

m

s-

o s-

o

n=l

(Sl,...,sm).

n--1

So if f is quasi-differentiable at a, then (u~ o 7r~) is quasi-differentiable at a by Theorem 41 and we obtain D f ( a ) -- ~

Un o

D f n ( a ) -- ( D f l ( a ) , . . . ,

Df,~(a)).

Conversely, if each f , is differentiable at a, then f is clearly differentiable at a. This completes the proof, m 4.1.3

Quasi-Differentials of Multilinear M a p s

We note that the product rule (fg)' - f' g + fg'

can be derived from the following theorem. Observe that the map x e U ~ f(x)g(x) e K

is a composition of the following two maps: x e U ~ ( f ( x ) , g ( x ) ) e K 2, (Yl, y2) C K 2 ~ ylY2 E K , where the first map taking values in the product space K 2 and the second being a continuous bilinear map(see subsection 2.1.3, p. 3a).

Theorem 44

[29] (2000)

Let E 1 , E 2 , . . . , E r a be p~-normed spaces, (i = 1 , . . . , m ) and F be a qnormed space. I f A C L ( E 1 , . . . , Era; F) is a continuous m-linear mapping, then A is quasi-differentiable at ( a l , . . . , a m ) . Furthermore

m

DA(al,

...,am)(Xl, ...,Xm)

-

A ( a l , ...ai-1, x~, a~+l,..., am),

-

1

i.e.

D A ( a ) " E1 x ... x Em ~ L ( E 1 , . . . E m , F)

(4.12)

CHAPTER 4 QUASI-DIFFERENTIAL CALCULUS

84

P r o o f . For m = 1, the result is given by T h e o r e m 42(b). If m = 2, we want to claim

D A ( a l , a2)(Xl, x2) = A ( x l , a2) + A ( a l , x2) Since

A ( x l , x 2 ) = A(al, a2) + A ( h l , a2) + A(al, h2) + A ( h l , h2) where h i = Xl - a l ,

h2 = x2 - a 2 ,

IlA(xl,x2)

we get

- A ( a l , a 2 ) - A ( x l - al,a2) - A ( a l , x 2 - a2)[I 1/q

< IlAll[[Xl - allll/m[lx2 - a2111/p2 < where we used the F - n o r m El x E 2 . If we write

Ilxll

IIAllllx-

- sup{llXlll lips,

all 2

IIx2ll 1/p~ }

for x - ( x l , x 2 ) e

Aa(Xl,X2) = A ( x l , a 2 ) + A ( a l , x 2 ) . t h e n Aa C

L(E1

IIA(x)

-

x E2; F) and

A(a) - Aa(x - a)ll 1/q <

IlAIIIIx

-

all~/P~llx2

-

a2111/p2

<_ IIAllllx-all 2. Hence

D A ( a ) = Aa. T h e p r o o f for the general case can be c o m p l e t e d by induction. II

4.1.4

Quasi-Differentials of P o l y n o m i a l s

T h e following t h e o r e m shows t h a t the quasi-differential of a p o l y n o m i a l is a linear m a p .

Theorem 45 [29] (2000,) If P c p ( m E , F)

and P -

A,

then

D P ( x ) - m A x "~-1

i . e . , D P " iF_,---, L(E; F) - P ( 1 E " F).

(4.13)

4.1 Q U A S I - D I F F E R E N T I A B L E

MAPS

85

P r o o f . We use the multinomial formula (Lemma 1) which in particular has the binomial formula (2.4)

A ( x + y)'~ = A ( x + y , . . . , x

Axm-iy i

+ y) = i=0

where x, y E E. Now

lira [[A(x +

y ) ' ~ - A x "~

- mAx'~-lylll/q/lly[ll/P - 0

y--~0

and if we compare it with lira IIP(x + y ) - P ( x ) - D P ( x ) ( y ) l l / l l y l l

y---~O

1/p - 0

we get D P ( x ) - m A x m-1. m

4.1.5

Inverse Mapping Theorem

Let U, V be open subsets of E and F p-normed and q-normed spaces respectively. If the mapping f : U ~ V is a homeomorphisms and is quasidifferentiable on U, it does not necessarily follow that f is a diffeomorphism, i.e. the inverse f - 1 of f is quasi-differentiable on V. Consider for example, the function f : ~ --~ ~ defined by f(x)

-

x 3.

Then f is a homeomorphism of class C 1 but the inverse g = f - 1 quasi-differentiable at the origin.

is not

The following theorem gives a sufficient condition to have inverse quasidifferentiable maps between locally bounded F-spaces.

T h e o r e m 46

( I n v e r s e m a p p i n g theorem)[29](2000) Let U and V be open subsets of E and F, p-normed and q-normed spaces respectively, and let f : U ~ V be a homeomorphism. A s s u m e that f is pq-differentiable at a point a C U. Then the inverse g - f - 1 is qp-differentiable at the point b - f ( a ) if and only if D r ( a ) C L ( E , F ) is a homeomorphism of E onto F; i.e. D f ( a ) is a topological isomorphism. In that case,

86

CHAPTER

4 QUASI-DIFFERENTIAL

Dg(b) - ( D f ( a ) ) -1.

CALCULUS

(4.14)

P r o o f . If g is qp-differentiable at b = f ( a ) then by the chain rule and T h e o r e m 42, we have g o f = 1v, f o g = 1v and

Dg(b) o D f (a) = 1E,

D f(a) o Dg(b) = l r

which proves t h a t D r ( a ) is a topological isomorphism of E onto F . Conversely, assume t h a t D f ( a ) is topological isomorphism from E into F . We want to show t h a t 9 is qp-differentiable at b = f(a). We will first prove t h a t g has the Lipschitzian property at b. For simplicity, let A = D r ( a ) and A ( x ) = f (x) - f (a) - A ( x - a) for x E U. we obtain

Since A -1 E L ( F ; E ) ,

x-

a-

by applying A -1

A-l(f(x)

(4.15) to b o t h sides of (4.15)

- f(a) - A ( x ) )

I I x - all ~/p ~< ~c(llf(x) - f(a)ll 1/q + II/X(x)ll l/q)

(4.16) (4.17)

with c - I I A - 1 1 1 . Since f is pq-differentiable at a, for e - 1 / 2 a , there is by definition, r > 0 such t h a t B(a, r) C U and IIAxll 1/q <_ I l x - alll/p/2ccr for x E B ( a , r ) . Hence from (4.17) we get

IIx - all 1/p <_ 2~cllf(x)

- f ( a ) l l 1/q

(4.18)

Since g is also continuous, for some s > 0, g(B(b, s)) C B(a, r). If we set in (4.18), we have

x = g(y)

IIg(Y) - g(b)ll lip <- 2crclly - bll 1/q.

(4.19)

for y C B(b, s), which shows t h a t g is Lipschitzian at b. Now we show t h a t g is qp-differentiable. From the relation (4.16), we have

IIx-

a-

A-l(f(x)

- f(a))ll 1/p < cllAxl] 1/q.

(4.20)

4.1 Q U A S I - D I F F E R E N T I A B L E M A P S

87

Since f is pq-differentiable at a, for any e > 0 there exists r > 0 such t h a t B(a,r) C U, and if x e B(a,r), IIAxlI1/q< e i l x - all 1/p. Hence (4.20) becomes ]Ix - a - A - l ( f ( x ) - f(a))]] lip < c llx - all lip for x E B(a,r). Now choose a > 0 such t h a t B ( b , a ) C U and g(B(b, 5 ) ) C T h e n if y E B(b, 5) and x = g(y), using (4.19) we obtain

IIg(Y) - g(b) - A - l ( y - b)ll 1/p <_ coe]lg(y ) - g(b)]l 1/q ~

2o-c2~lly

-

B(a,r).

b]l 1/q.

This completes the proof. I

4.1.6

R e a l and C o m p l e x Cases

If E and F are two complex F - n o r m e d spaces, they can be considered as real F - n o r m e d spaces since tg is a subfield of (F. If we denote these real F - n o r m e d spaces by E~ and F~, then L(E; F) becomes a subspace of L(E~; F~) since a complex linear m a p is a f o r t i o r i a real linear map. Let U be an open subset of E, f : U --~ F and let a C U. We say t h a t f is complex (respectively real) quasi-differentiable at a if f is quasi-differentiable at a for the vector space s t r u c t u r e over ~T (respectively /~). Since L ( E ; F ) C L(E~;E~), it is clear t h a t if f is complex quasidifferentiable at a then it is real quasi-differentiable at a and its quasidifferential in the real sense is equal to its quasi-differentiable D f ( a ) in the complex sense. However, the converse is not true as shown by the following example. If we let E - F - C , then E~ - F~ - t g 2, and the function f " z C ~ --+ 2 C (/' is real quasi-differentiable, but not complex quasi-differentiable. If f is real quasi-differentiable at a and D f ( a ) C L(E~; E~) then for f to be complex quasi-differentiable at a it is necessary and sufficient t h a t Dr(a) should belong to the subspace L(E, F) of L(E~, F~). A function which is complex quasi-differentiable on an open set is said to be q u a s i - h o l o m o r p h i c . We will s t u d y such functions in Ch.5.

This Page Intentionally Left Blank

Chapter 5

GENERALIZED MEAN-VALUE THEOREM The importance of further study of the Lagrange Mean-Value Theorems and Inequalities arises from its wide applications to different branches of Mathematics and Physics, see for example [69], [173] and [178]. In this chapter we give general forms of the Lagrange M e a n - V a l u e T h e o r e m s a n d M e a n - V a l u e I n e q u a l i t i e s in locally bounded F-spaces which are not necessarily locally convex spaces. More precisely, we establish here a new version of the M e a n - V a l u e T h e o r e m in real locally bounded spaces, which implies the classical one in normed spaces. In addition we prove n e w general f o r m s o f the M e a n V a l u e - I n e q u a l i t i e s suitable to apply to quasi-differentiable maps between non locally convex F-spaces. Several applications of these results will also be given. For example we give a condition for interchanging the order of differentiation and limit of a sequence of quasi-differentiable maps of locally bounded F-spaces. Our tool here is the new concept of differentiability, that is, the quasi- differentiability, see the preceding chapter and some of the papers of the author [20] ~J[21].

5.1

MEAN-VALUE

THEOREM

IN REAL SPACES

In this part we prove two extended forms of the M e a n - V a l u e T h e o r e m of L a g r a n g e for real locally bounded F-spaces. We begin by recalling the classical formulation of the Lagrange Mean89

90

CHAPTER

5 GENERALIZED

MEAN-VALUE

THEOREM

Value theorem in real normed spaces. T h e o r e m 47 [173] Let E and F be real n o r m e d spaces and U be an open subset of E containing the line segment [a, b]. If f : U --+ ~ is a differentiable mapping, then there exists c E (a, b) such that f(b) - f(b) = D f ( c ) ( b - a). m

Let us recall that, by an arc segment A b between a, b C E, a p-normed space, we mean the set 1

1

Ab-{(1-A)~a+A~b;

0<_A__I}.

A b is called the (closed) arc segment joining a and b. The following theorem, obtained by the author[20], gives a general version for the Mean-Value theorem in real locally bounded spaces (Lbs). T h e o r e m 48 ( M e a n - V a l u e t h e o r e m i n real Lbs)[20](1995) Let E be a real p - n o r m e d space and U be open in E which contains the arc segment A b. I f f " U --~ ~ is a quasi-differentiable mapping on U, then there exists c E A b such that A-~1 - I

f (b) - f (a) = D f (c)(

-1~ - I

p

a),

p

1 >p>0,

(5.1)

f o r some A E (0, 1). I f p - 1, we have A b - [ a , b] and the classical m e a n value theorem for n o r m e d spaces is obtained.

P r o o f . Let p : ~ +

--~ E

be defined by 1

=

1

+

(1 -

(5.2)

so that ~(0) - a, ~(1) - b, and ~([0, 1]) - A b. Since U is open and ~p is continuous, there is a number ~ > 0 such that p ( - 5 , 1 + 5) C U. By the chain rule it follows that

5.1 M E A N - V A L U E

THEOREM

1

( f o ~)'()~) - D f ( b ) ~

IN REAL

i

1

)~-

1

+ (1 - A)~a)(

SPACES

91

i

(1 -- A); -1 b-

a).

P

(5.3)

P

If we apply the classical m e a n value t h e o r e m to ( f o ~), we deduce t h a t there exists A0 E (0, 1) such t h a t f(~;(1)) - f ( ~ ( 0 ) ) - ( f o ~;)'(A0)(1 - 0), and by

(5.3) we have,

f(b) - f(a)

-

i

i

b

i--1

D f ( A ~ b + (I - A 0 ) ~ a ) ( ~ / ~

- - ( 1 - )~0) 1-1)

(5.4)

_. . . . Df(c)()~o/P-1 .. b _ (I - ,k0) i - 1 a)

P 1

(5.5)

P

1

where c - ) ~ b + (1 - A0)~a E A b C U. If we p u t p - 1, we obtain A b - [a, b], and the classical mean-value t h e o r e m in n o r m e d spaces is o b t a i n e d . T h i s completes the p r o o f , i We shall now see t h a t the locally b o u n d e d spaces E which are not locally convex have balls BE satisfying the hypotheses of the preceding T h e o r e m . Let us recall t h a t , a subset U of E is said to be a b s o l u t e l y p - c o n v e x if for e v e r y x , y E U and for every #1, #2 E ~ with [#11p + I # 2 1 p <__ 1 we have tt 1x + P2Y E U. L e m m a 9 Let E be a p - n o r m e d space with the unit ball B E . Then, f o r every a, b E B E the arc segment A b joining a and b lies entirely in B E . Proof. Pl,P2

We first show t h a t BE is absolutely p-convex. Let x , y E B E and E ~ w i t h Ipil p + ]#2[ p <_ 1. T h e n

II,lX + ,2yll _< 1,11"llxll + I l'llyli _ I, lp + 1,21 p < 1. Hence #i x + #2Y E B E . Secondly, let a, b E BE and # i , # 2 as above.

As BE is absolutely p1

convex it follows t h a t #1 b + p2a E B E .

So take in p a r t i c u l a r ;t i - / k ~

and

92

CHAPTER

#2-(1-A)~ That is,

1

5 GENERALIZED

with0KA<

MEAN-VALUE

THEOREM

1

1

1, w e g e t t h e p o i n t s x - A ~ b + ( 1 - A ) ~

o f A b.

A~ c BE. I

We use Theorem 7 ( C h . 1 ) to prove a special form of the Mean-Value Inequality in the next theorem. Let us recall it first"Let M be a closed subspace of a locally bounded F-space F, with finite codimension n and b E F / M , the complement of M in F. Then there is a continuous linear functional f on F such that

_< II]llF -<

JIM --0

and

(5.6)

](b) - d l / P ( b , M )

for some a > 1, where d(b, M ) - infvcM IIY -- bl[, the distance from b to M , and d(x, O) -]]xll, the p - n o r m on F".

T h e o r e m 49 [20] (1995) Let E and F be real p-normed spaces (1 > p > 0) and M be a closed subspace of F with codimension n. Let U be open in E which contains the arc segment A b for a, b C U. If f 9U ---, F is quasi-differentiable on U such that f ( A b) C F / M , then there exists c e A b such that 1

1

dl/P(f(b) - f ( a ) , M ) <_ a n D f ( c ) ( A - ~ - I b P

(1 -- A)F -1 a) p

1/p

(5.7)

for some constant cr > 1. In particular if p - 1, we get

I I f ( b ) - f ( a ) l l ~ IIDf (c)(b - a)l I

[a, hi. P r o o f . Let Yo - f ( b ) - f ( a ) . If Y0 is the zero vector in F, then the result is trivial. If Y0 ~ 0, then by the result of the Hahn-Banach separation theorem mentioned in the above Theorem 7, and since Y0 ~ M, there exists E

F! ~

the dual space of F,

such that 1

1 < i19911F< crn,

9~(Y0) ----d~(y0, M).

5.1 M E A N - V A L U E

THEOREM

IN REAL SPACES

93

Let A 9U --~ ~ be given by A - ~ o f. Obviously A is quasi-differentiable on U and 1

A(b) - A(a) -

~(yo) - d-~ (Y0, M ) ,

DA(x) -

r o D f (x).

It follows by applying theorem 48 to A t h a t there is c C A b such t h a t

A(b) - A(a)

-

(r o D f ( c ) ) ( A~-I P I

i.e., d i / p ( f ( b ) -

f(a),M)

< [[~[[F"

DI(c)( I

<

O-n

D f ( c ) ( A,- ~ p

(1 - A)~ -1 a ) P

b1

P

(1 - a) P

b-

(1-/~) ~ - 1

1

1/p

Z

b-

-I

1/p

i

p

a)

a)

Note in this connection t h a t [[T(x)l ] < ][Tl[][x[]1/p for any T e F ' , x e E, see ch.1, part I. If E and F are normed spaces, i.e. p - 1, we get the classical result [ I f ( b ) - f(a)[[ < ] [ D f ( c ) ( b - a)]] by using the classical H a h n - B a n a c h separation theorem where for every Y0 E F there exists ~ E F t such t h a t ~(Y0) - [ [ ~ ] [ and ][qp[[- 1, A b - [ a , b] C U. This completes the proof, m We shall now see t h a t the locally b o u n d e d spaces E which are not locally convex have balls BE satisfying the hypotheses of the theorems of this part. Example

15 T h e O r l i c z s p a c e s l~

We define l~ as given in (subsection 3.2.2, p.68 )" Oo

-

{x -

e 1

In connection with the topological properties of l~, we can apply the Mean Value T h e o r e m 48 to a differential m a p p i n g as f 9 Bz~ --~ ~ and T h e o r e m 49 to a differential m a p p i n g like g 9 Bl~ --* l~. Note t h a t for certain ~ as ~(t) - [ t l P , l~ is a locally b o u n d e d space.

94

CHAPTER

Example

5 GENERALIZED

MEAN-VALUE

THEOREM

16 T h e s p a c e s l(p,). The spaces Oo

l(p

) = {x =

xn E i~, ~

Ixnl p= < oc}

(1 > Pn > O)

1

with the F - n o r m Ilxfl = E a~e F-spaces with Schauder bases (en). They are locally bounded if Pn /--,0, otherwise they are not even locally pseudoconvex, see (Ch 3, p.67 ). Since Bl(p.),Pn /--*0 is absolutely pconvex for some (1 > p > 0 ), it contains the arc segment A b whenever a, b E B. Theorem ~8 and Theorem ~9, can be applied here for example for differentiable mappings as f " Bz(p.) --* 1~ and g " Bt(p.) --+ l(pn) respectively.

5.2 5.2.1

MEAN-VALUE SPACES Mean-Value

THEOREM

IN COMPLEX

Inequality

In this part we study and discuss in detail the Mean-Value theorems in complex locally bounded F-spaces. These have been obtained by the author using the new concept of quasi-differentiability (or what we have also called the pq-differentiability). The following theorem gives a general formula of the Mean-Value inequality for pq-differentiable mappings between locally bounded F-spaces. T h e o r e m 50 [21] ( M e a n - V a l u e I n e q u a l i t y ) ( 1 9 9 6 ) Let E and F be locally bounded F-spaces, whose topologies are metrized by p-norm and q-norm respectively, (0 < p, q < 1). Let U C E be open in E such that U contains the arc segment A~, for a, b E U. If f : U ~ F is pq- differentiable on U, then

Ilf(b) - f(a)l / _< sup IIDf(x)iiq.lI b - all q/p.

(5.8)

xE A b

P r o o f . If f ( a ) - f(b) or the mapping x --~ IIDf(x)ll is unbounded on Aba, then the theorem is evident. So assume that M - sup~cAb IiDf(x)ll q < cr Let e > 0 and let

X - {x E Ab; IIf(t) -- f(a)ll < ( M + e ) l l t - all q/p, t E A~}. Notice that a E X; and if x E X then Aax C X. Denote c = sup X.

5.2

MEAN-VALUE

THEOREM

IN COMPLEX

SPACES

95

We claim t h a t c = b. If c < b, t h e n by pq-differentiability of f at c, for every ~ > 0, there exists 5 > 0 such t h a t B(c, 5) C U and Ilf(t) - f(c) - D f ( c ) ( t Hence i f t , t' c B(c,~) N A b, (5.4) we have

c)] I <_ e l l t - cll q/p,

t e B(c, 5)

a < c, t < c < t' < b, t h e n t

cX

(5.4) and from

II/(t') - f ( a ) l I _< IIf(t') - f(c)l I + IIf(c) - f ( t ) l I + Ill(t) - f(a)l I

+ IIf(t') - f ( c ) - D f ( c ) ( t ' - c ) l l

<_ I I D f ( c ) ( t ' - c ) l l

c)ll + IIf(t) - f ( c ) - D f ( c ) ( t -

+lIDf(c)(t-

+llf(t)-

c)ll

f(a)ll

< IlOf(c)llqllt '

-

c] q/p + ellt'

-

cl] q/p + IlDf(c)llqllt - cll~

+ellt - cll q/p + ( M + e)li t - all q/p <_ (llDf(c)ll q + c)(llt' - cll q/p + IIc - tll q/p) + ( M + ~)llt - all N < (M + E)llt' - all q/p, as t ~ t' e B~(c) n Aba . Note from the properties of t, t' t h a t as t ~ t' we get lit - c l l - 0 - l i t ' - cll. In fact, l i t - t'll ~/p ~ o implies t h a t l i t - cll 1/p - 0; and consequently we have l i t ' - cll 1/p --, 0 from l i t ' - cll 1/p <_ or(lit' - t[I 1/p + lit eli l/p) - o. This shows t h a t t ~ E X and also c E X , c o n t r a d i c t i n g c = sup X . Hence c = b, and -

IIf(b) - f(a)l I <_ Mllb -

allq/P

However, this can also be shown t h a t if we choose t, t' arbitrarily close to c noticing t h a t t < c < t ~, t, t' E B(c, 5). I T h e following result is a direct consequence of T h e o r e m 50. Corollary

6 Assume that the hypotheses of the preceding theorem are sat-

isfied and that there exists M > 0 such that IID f (x) lIq <_ M for all x C d b. Then I I f ( b ) - f (a) lI <_ MIIb - all q/p. (5.5) I

In w h a t follows we give a n o t h e r form of the Mean-Value inequality of locally b o u n d e d F-spaces. 51 [21] Let E and F be as in the above Theorem 50 and let U be open subset of E containing the arc segment A b for every a, b C U. If f " U --, F is a pq-differentiable map, then Theorem

96

CHAPTER

5 GENERALIZED

MEAN-VALUE

Ilf(b) - f ( a ) - D f ( a ) ( b - a)ll _< lib - all q/p sup

xeA~

Proof.

THEOREM

I l D f ( x ) - Df(a)ll q

(5.6)

If the m a p g : U --, F is given by

g(x) = f (x) - D f (a)(x) then

Dg(x) = D f (x) - D f (a). If we apply the Mean-Value T h e o r e m 50 to g we get: IIf(b) - f ( a ) - D f ( a ) ( b - a)I I = Iig(b) - g(a)I I

<_ l i b - a l l q/p sup IlD f (x) - D f (a) llq. xEA b

This completes the proof of t h e o r e m . I As we shall see later, the following l e m m a has some applications, where C 1 denotes the class of continuous quasi-differentiable functions. L e m m a 10 ( A p p r o x i m a t i o n L e m m a )Let E and F be p- normed and q-normed spaces and let f 9U ~ F be of class Clq, of continuous quasidifferentiable maps, on U. If a C U, then for any e > 0 there exists 5 > 0 such that (a, 5) C U, and if A ~ C B(a, 5) for Xl,X2 C B(a, 5), then [if(x1) - f(x2) -- D f ( a ) ( x l - x2)[[ ~ ([[[Xl

-

-

X21[q/p.

(5.7)

Proof. Since D f : U ---, L ( E , F) is continuous on U, given e > 0 there exists 5 > 0 such t h a t if ]ix - a i ] < 5, t h e n x e U and I i D f ( x ) - Of(a)] I < c. Now let Xl,X2 E B(a, 5), so A ~ C B(a, 5) by hypothesis, and then apply T h e o r e m 51 to obtain the desired conclusion. I

5.2.2

Applications

In w h a t follows we give one of the main applications of the quasi-differentiability for m a p s between locally bounded F-spaces.

5.2

MEAN-VALUE

THEOREM

IN COMPLEX

SPACES

97

I n t e r c h a n g i n g t h e O r d e r of Q u a s i - D i f f e r e n t i a t i o n a n d Limit. We now use the Mean Value inequality, T h e o r e m 50, to obtain the following theorem which explicitely states a sufficient condition for interchanging the order of differentiation and the limit of a sequence of quasi-differentiable mappings, (see [69] and [173] for the case of normed spaces ).

T h e o r e m 52 [21] (1996) Let U be an open connected set in a p-normed space E and let (f~) be a sequence of pq-differentiable maps of U into a q-Banach space F, (0 < p,q <_ 1). Assume the following: (a) There exists a point ~ C U such that the sequence (fn({)) converges in F. (b) For each point x C U there is a ball B ( x , r ) C U such that the

(DA)

B(x,

Then, for each x E U, the sequence (f~) converges uniformly on some open ball centered at x. Furthermore, if for each x C U we let f ( x ) = lim f,~(x), then f is quasi-differentiable on U and

(ss)

Df(x) = limDf~(x).

P r o o f . We first show t h a t (f~(x)) converges at each point x in U. Let D = {x E U; fn(X) converges }, then D =fi r by condition (a). We now prove t h a t D is a closed and open set in the connected set U, and hence D = U. In fact, if a C D, by condition (b), there exists r > 0 such t h a t (Dfn) converges uniformly on B ( a , r ) . So let e > 0 and c/r p/q be given. There exists no > 0 such t h a t if n, m > no, I l D f ~ ( y ) - Dfm(Y)ll <

,

all y e B ( a , r ) .

(5.9)

Moreover I ] f ~ ( a ) - f,~(a)ll < e.

(5.10)

Now by applying the Mean Value T h e o r e m 50 to fn - f m , we have, for x c B(a,

IIA(x)-

fm(a))ll < IIx--all qlp

9

sup

yCB(a,r)

IiDf~(y)-Dfm(y)]] q

(5.11)

98

CHAPTER

5 GENERALIZED

MEAN-VALUE

THEOREM

or

IIf~(x)-fm(x)ll <_ IIf~(a)-f~(a)ll+llx-all q/p.

sup

v~B~(~)

IIDf~(u)-Dfm(u)ll q, (5.~2)

hence by (5.10), ] ] f ~ ( x ) - fm(X)l I < 2e if n, m > n0. Thus f n ( x ) i s a Cauchy sequence in the p - S a n a c h space F , which shows t h a t (f~(x)) converges. Therefore, B(a, r) C D, and hence D is open. On the other hand, if x E D, then there exists r > 0 such t h a t ( D f n ) converges uniformly on B(x , r). Let a e B ( x , r ) N D. T h e n again by (5.12) we deduce t h a t ( f ~ ( x ) ) c o n v e r g e s , hence D C D and D is closed. This completes the proof t h a t D - U. Secondly, let f " U ---, F be defined by

f ( x ) = lim f~(x),

for each x E U.

(5.13)

We show t h a t f is quasi-differentiable and t h a t D f ( x ) = l i m D f ~ ( x ) for all x C U. According to (b) let g(x) = l i m D f ~ ( x ) , on U. Consider a E U. So we shall prove t h a t f is quasi-differentiable at a and D f ( a ) = g(a). To show this, we estimate

Ilf(x)

-

f ( a ) - g(a)(x- a)l[ _< Ilf(x)- f(a) - [ f ~ ( x ) - f~(a)]ll + +llf,~(x)- f ~ ( a ) - D f n ( a ) ( x a)ll + +llOA(a)(x - a ) - g ( a ) ( x - a)ll.

Let E > O. Since (Df~) converges uniformly on B(r ,a), there exists no > 0 such that for rn, n > no, we have IIDfn(Y) - Df~(Y)[I < eq for every y E B(a, r), moreover, [ I g ( a ) - D fn(a)l ] < eq (5.14) If we let m ---+ cx) in (5.11), we obtain

I l l ( x ) - f(a) - [ f n ( x ) - A(a)]ll

<

d l l x - all alp.

(5.~5)

Since D f n ( a ) exists, there exists rl < r such t h a t

Ilfr,(X) - f,~(a) - D f,~(a)(x - a)l I <_ eqllx - all q/p

allq/P

(5.16)

whenever I l x < 7'1 and x E U. Therefore, we get the following estimation from the inequalities (5.13) -(5.16)"

[If(x) - f(a) - g(a)(x - a)l I <_ 3 e q l l x - all q/p

5.2

MEAN-VALUE

THEOREM

whenever I l x - allq/p < rl, X C: U, equal to g(a).

IN COMPLEX

SPACES

99

which proves that D f(a) exists and is

Finally, to show that for each a E U, (fn) converges to f uniformly on some open neighborhood of a, we estimate

IIf,~(x)- f ( x ) -

[ f n ( a ) - f(a)][[

Note that, for x E B(a, 5) we have by (5.15) that: lim Ilf~(x) - f ( x ) - [ f ~ ( a )

- f(a)]]l = 0.

x----+a

Hence if (fn(X)) converges for some x E B(a, 5), then the above equality implies that (f~(a)) tends to a limit. This, in turn by the same equality, implies that the sequence (fn(X)) converges for every x E B(a, 5). This completes the proof, m 5.2.3

Examples

of Sequence

Spaces

In section 5.1 Lemma 9 of this chapter, we proved that the open unit ball BE of a real locally bounded space E has the following property : For every a, b E BE, the arc segment A b joining a, b lies entirely in BE. In fact this result can be extended similarly to complex locally bounded spaces E. Therefore, all results of this part can be applied for this general case.

E x a m p l e 17 Let E - lp and F = lq (1 >_ q > p > 0) and f " Bzp --~ lq be

thepq-differentiable mapping defined on the unit ball Bl, by f (x) = E F x~, x = (x~) E lp. Then, for example, Theorem 50 implies for every a, b E Bzp that oo

oo

1

oo

ann[[lq -< sup I[Df(c)[[q(E [bn - anlP) q/p

[I Y~ bn n- ~ 1

c~A~

(5.17)

1

where A b c Blp.A E x a m p l e 18 Let E - l(pn), F - ~ T 1 >_ pn > O, Pn /--*0 where l(p,~) = {x -- (xn); xn E (~, E F Ixnl p~ < c~}. Then E is a locally bounded space and

hence metrized by a p-norm (see Rolewicz [186]). Now if fn " Bl(p~) --~(~ is a sequence of quasi-differentiable mappings which satisfy the hypotheses of Theorem 52, then we conclude that lira fn is quasi-differentiable, and lim(D fn) -- D(lim fn)

100 C H A P T E R

5 GENERALIZED

MEAN-VALUE

space of l(p,~). N o t i c e that (lP) ' ~ lc~ for every 1 >_ p > O. A

THEOREM

Chapter 6

HIGHER Q U A S I - D I F F E R E N T I A L IN F-SPACES In this part we introduce the concept of higher order quasi-differentials for maps between locally bounded F-spaces, and prove an analogy to the theorem, which states that 9 "All h i g h e r quasi-differentials can be c o n s i d e r e d as s y m m e t r i c m u l t i l i n e a r m a p s ". We shall also introduce the Ggteaux differential and study its relationship with the Quasi-differential.

6.1

SCHWARTZ

SYMMETRIC

THEOREM

S e c o n d Differential Let E and F be p-normed and q-normed spaces respectively (0 < p, q _< 1), U an open subset of E and f " U --~ F quasi-differentiable. Then

D f " U ---+L ( E , F). Since L ( E , F) is a q-normed space, let us define the second differential of f.

D e f i n i t i o n 31 Assume that f 9 U ~ F is quasi- differentiable on U. The function f is said to be twice quasi- d i f f e r e n t i a b l e (or quasid i f f e r e n t i a b l e o f order two) at a C U ( or on U ) when the mapping I01

102CHAPTER 6 HIGHER QUASI-DIFFERENTIAL

IN F-SPACES

D f : U --, L(E; F) is quasi-differentiable at a (or on U ), respectively. In this case D(DF)(a) e L(E; L(E; F)) is called the second quasi-differential of f

on U and is denoted by

DZ f 9U ~ L(E; L(E; F)).

This concept of second quasi -differential, obtained by a simple repetition of differentiation of order one, has the apparent disadvantage that the values of the second quasi differentials belongs to the q-normed space L ( E ; L ( E ; F ) ) . In this form this space appears complicated, since it is a q-normed space of continuous linear maps. However, because of its natural isomorphism to L(2E; F) which follows from Theorem 21,(Ch.2, subsection 2.1.4, p.34). So we are able to treat the concept of the second quasidifferentials as follows : Consider D2 f 9U ~ L(E; L(E; F)) and denote by

d2 f (a) e L(2E; F) the element of L(2E; F) that corresponds to D2f by the natural isometric isomorphism given in Theorem 21, we note that the relation between D2f(a) and d2f(a)is characterized by

d2 f(a)(s,t) -

D2 f(a)(s)(t)

for any s, t C E. Therefore, the second quasi-differential

D2 f(a) 9A --, L(E; L(E; F))" corresponds to the second quasi-differential

d2 f 9A --~ L(2E; F).

Thus, quasi-differential of order two now takes values in the q-normed space L(2E; F). We recall that this is done by identifying L(E; L(E; F)) with L(2E; F), the space of all c o n t i n u o u s b i l i n e a r m a p p i n g s of E x E into F, which appears less complicated.

6.1 S C H W A R T Z S Y M M E T R I C

THEOREM

103

R e m a r k 12 In the case of quasi-differentials of order one, we also use the notation df (a) - D f (a), df - D f . We point out here that: without assuming that f is quasi-differentiable on the entire subset, we can say more generally that f is twice quasid i f f e r e n t i a b l e at the point a C V if : 1 - f is quasi-differentiable on a neighborhood of V of a. 2- The mapping D f : V --~ L(E; F) is quasi-differentiable at the point a.

Schwartz S y m m e t r i c T h e o r e m of T w i c e Quasi-Differentiable M a p s In elementary calculus, it is known that if U C ~2, f . U ~ ~, a C U, 02f 6yO~ 02f are continuous on U, then and the partial derivatives O~Oy,

02f

02f

OxOy

OyOx

We will derive in this section a generalization of this symmetric theorem for twice quasi-differentiable functions on locally bounded F-spaces. This can be obtained from the following generalized theorem for normed spaces, (see Chae [69]).

T h e o r e m 53 ( G e n e r a l i z e d S c h w a r t ' s s y m m e t r i c theorem)J29](2003) Let f " U --~ ~ be twice quasi differentiable at a E U. Then d2 f(a) is a symmetric bilinear map. That is, for all k, h E E, d2 f ( a ) ( h , k) -

d2 f ( a ) ( k , h).

(6.2)

P r o o f . Since f is twice quasi-differentiable at a, there is some r > 0 such that B(a, r) C U and f is quasi-differentiable on B(a, r). Consider the function

A (h, k) - f ( a + h + k) - f ( a + h) - f ( a + k) + f ( a ) whereh, kcE are such that a + h + k , a + h , a + k belong t o U . precisely, we may assume [[ h [[_< r / 2 [[ k I[_< r/2. We note that

More

(h, k) - a (k, h) We wish by a suitable process to approximate d2f(a)(h, k) with A (h, k) to show that d2f(a) is symmetrtic and to obtain the desired equality

+'~

I

,

+

L",O

,-~

~ ~

~

0 ~

~

i

---

~.

~

'.-,..

~

~

- q-

IA

~

I

I

+

--

I

I

~

~

~o

r~ I A

II ~

+

~.~

,

'

+

9

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106CHAPTER 6 HIGHER QUASI-DIFFERENTIAL

IN F-SPACES

HIGHER QUASI-DIFFERENTIALS

6.2

The third quasi-differential is the differential of the second, t h e fourth is the differential of the third, and so on. The general definition is best expressed inductively. Let f : U --, F be twice quasi-differentiable. T h e n we have the second quasi-differential

D 2" U ~ L I ( E ; L ( E ; F ) ) d 2" U ~ L2(E;F) By repeating the procedure inductively we can define the m t h quasidifferential of f. Let us first simplify some notations. For m = 1, we write L 1 (E; F) - L(E; F) and for rn = 2, L2(E; F) - L(E; L(E; F)) Inductively we write

Lm(E; F) - L(E; L m - I ( E ; F ) ) For example, if f : U ~ F

is twice quasi-differentiable on U, we have

O f " U ---+La(E; F ) = L(E;F) and D2f 9U ~ L2(E;F) After defining

D i n - i f " U ~ L m-l(E; F ) if D i n - i f

is quasi-differentiable at the point a C U, we let

O r a l ( a ) - D ( D m - l f ) ( a ) e Lm(E; F) which will be called the also have

mthquasi-differential of f at a. Of course, we

D m f : U ~ Lm(E;F) if D m f

exists at each point

a of U. The identity

DP(DqI) = (Din f) is trivially true if p + q = m and D'~f exists. differential D'~f is linear in the sense t h a t

Din(f + g ) =

Din f +

Drag and

Also,

the mthquasi -

Dm(,k f) = ADm f

6.2 H I G H E R Q U A S I - D I F F E R E N T I A L S

107

where ~ E K. For convenience, we i d e n t i f y D ~ - f. For aE U and Xl,...,Xm E E, we have

Dmf(a)(xl)...(xk) e Lm-k(E; F) for 1_< k < m. To avoid the cumbersome parentheses, we often write

Dm f ( a ) ( x l ' ...,xk) - Dm f(a)(xl)...(xk).

D e f i n i t i o n 32 /f D m f

9U ~ Lm(E; F) is continuous, we say that f is of class C q , space of continuously m-quasi-differentiable maps. If f is of class C~n for every m, f is said to be of class C~; in this case f is called i n f i n i t e l y m a n y t i m e s q u a s i - d i f f e r e n t i a b l e .

6.2.1

T h e Quasi-Differentials D'~f and

dmf

As in section 6.1 we shall identify the mth quasi-differential D m f with the corresponding m-linear map dmf. More explicitely, D e f i n i t i o n 33 Let

r

L(mE; F) ---, L'~(E; F)

be the natural isometric isomorphism defined by r Fo? ~

- A(xl, ...,xm).

D'~ f 9U --, L'~(E; F), we define

d'~ f

9U --~ Lm(E; F) to be such

that dmf_ r We call d m f evident that

also the the

m th

o D'~f.

q u a s i - d i f f e r e n t i a l on

d ~ f ( a ) ( x l , ..., Xm) We notice that if m -

D ' ~ f ( a ) ( x l , ..., Xm)

1,

df-Df.

U. Then it is

108CHAPTER

6 HIGHER

QUASI-DIFFERENTIAL

IN F-SPACES

We should always keep in mind that the m th quasi-differentials D'~f and dmf represent two different objects if m:/: 1. We prefer the m th differential d'~f to D m f in this book because the former is much simpler. We should also note that although for rn = p + q we have

Dr~f(a) = DP(Dqf)(a),

we cannot expect to have d'~f(a) = dPdqf (a) are elements of two different spaces. Theorem

54 .

If A c L~(mE; F) and P dkP(x) -\

since the last two objects

I

.]I E p(mE; F), then

m(rn-1)...(m-k+l)Ax

m-k

1

0

Om

L

J

dkP 9E --+ L(kE; F). P r o o f . We have shown this result for k = 1 in Ch.2(subsection 2.2.3, p.38). The rest can be proved by induction. T h e m t h Q u a s i - D i f f e r e n t i a l in R e l a t i o n t o L i n e a r M a p s

For the higher Quasi-Differential, we have statements similar to those obtained with the first differential in relation to the linear maps, see Theorem 41(subsection 4.1.2, p.81). Notice that if A E L(mE; F) and r E L(F; G), then the composite map r c L(mE; G). Let r

L(mE; F) --+E L(mE; G)

be given by r Then it is immediate that r

]1 r

= r is linear. Since

r I1" II A II. II Xl II lip ... II Xm II ~/p

r is also continuous where E is a p-normed space and E,G are q-normed spaces, (0 < p, q _< 1).

6.3 G E N E R A L

SCHWARTZ

SYMMETRIC

THEOREM

109

Theorem 55 For an open subset U of E, let f " U ~ F be m times quasi- differentiable and let r c L ( F ; G ) . Then f o r any a E U we have dm(r f ) ( a ) - r

P r o o f . Use induction on m. Consider the map x C U ~ dm-l(r

By induction we get dm-l(dpof)(x)

-- r

f(x).

Now the composite map of

d rn-lf" U ~ L m - l ( E ; F) and r

L m-1 (E; F) --~ L m-1 (E; G)

is quasi-differentiable by Theorem 44 ( subsection 4.1.3, p.83) and d(r od ~ - 1 f) - r Therefore, for any a, dm(r

6.3

GENERAL THEOREM

- r

SCHWARTZ

I

SYMMETRIC

We have seen in Theorem 53 (section 6.6, p.103 ) that the second quasidifferential d 2 f ( a ) is a symmetric bilinear map. This result can be extended to the mth quasi-differential d m f ( a ) by induction. We need first the following lemma where E and F are p-normed and q-normed spaces respectively. L e m m a 11 Let f " U ~ F be ( m - 1)-quasi-differentiable on U and m - quasi-differentiable at a. For m > 2 and f o r any fixed x2, ...Xm E E, if g" U ~ F is defined by g(x) -- d m - l f ( x ) ( x 2 , . . . , x r a ) ,

110CHAPTER 6 HIGHER QUASI-DIFFERENTIAL

IN F-SPACES

then g is quasi-differentiable at a and dg(a)(x) -

d m f ( a ) ( x , x2, ..., x,~)

for all x E E. P r o o f . The m a p g:U --~ F satisfies the composition

g - )~or 1oD m - 1f. where

r

L ( E m - 1 ; F) --* L m-1 (E; F )

is the natural isometric isomorphism and A" L m - I ( E ; F) --* F is given by A(A) - A(x2, ..., Xm)

for all

A E L m - l ( E ; E).

is clearly continuous and linear. Now

Dg-

n(s

- s162

- ~or

Hence, for all x E E, we have

dg(a)(x)

-

D g ( a ) ( x ) - ()~or

f(a))(x) - A(r

f(a)(x)))

= A ( d m f ( a ) ( x ) ) - d m f ( a ) ( x , x2, ...,xm). which is what set out to prove, m T h e o r e m 56 ( G e n e r a l i z e d S c h w a r z T h e o r e m ) (2002) Let f " U --~ F be m-quasi-differentiable at a E U. Then d m f (a) is a symmetric m-linear map; i.e.

dm f (a) E Ls(mE; F). P r o o f . For m - 2, this was proved in Theorem 53 (section 6.1,p.103). We now proceed by induction on m for m >_ 3. Assume t h a t the theorem has been proved for m - 1. Then d m f ( a ) is the quasi-differential of the m a p p i n g d i n - i f , which by our assumption exists in a neighborhood V of a. By the induction hypothesis, we can assume d i n - i f 9V --~ Ls(mE; F).

F

By the l e m m a 11, p.109, for fixed x3, ..., xm C E, the mapping defined by g(x) -- dm-2 f ( x ) ( x 3 , ..., Xm)

g" V

6.4 D I R E C T I O N A L

DERIVATIVES

111

is quasi-differentiable at a and its quasi-differential at a is given by

dg(x)(y) - dm-l f(x)(y, x3,...,Xm) for all x E V and for all y E E. Similarly, applying the lemma to the map X --+ d m - l f ( x ) ( x 2 ,

..., xm)

where x2,..., Xm are fixed, we obtain

d2g(a)(x2,xl) - dm f ( x ) ( x l , x 2 , ...,Xm) for all X 1 E E .

(6.13)

From (section 6.6, p.103 ) we have

d2g(a) (Xl, X2) -- d2g(a)(x2, Xl) and hence

dmf(a)(x2,xl,x3, ...,Xm) = dmf(a)(xl, ...,Xm). But din-if(a) is symmetric and from (6.13), for Xl E E, the map

(6.14)

xl,...,Xm are fixed but arbitrary, so

dmf(a)(xl) : (x2,...,xm)--+ dmf(a)(xl,...,x,~)

(6.15)

is symmetric. This shows that the multilinear map dmf(a) : E m --+ F is symmetric by(6.14) and (6.15), so the proof is complete. I

6.4

DIRECTIONAL

Definition 34

DERIVATIVES

(Gdteaux Differential)Let U

space E. If for a given a E U and h E E, Of

be open in a p-normed the limit

[ f (a + Ah) - f (a) ]

(616)

exists, where A E K, then f is said to be "Gateaux differentiable at a in the direction of h "; Sometimes we write Of(h,a) instead of ~ (a) " Also ~h(a) is called the G a t e a u x derivative o f f at a in the Oh direction h.

112CHAPTER 6 HIGHER QUASI-DIFFERENTIAL

IN F-SPACES

D e f i n i t i o n 35 If f is Gdteaux differentiable at a in any direction, we say that f is " G d t e a u x d i f f e r e n t i a b l e at a " and the mapping of o f ( ~ ) . h e E -~ ~ ( a )

(6.~7)

e F

is called the " G d t e a u x d i f f e r e n t i a b l e " o r G d t e a u x d e r i v a t i v e .

D e f i n i t i o n 36 I f f is Gdteaux differentiable at each point of direction h, the mapping of

~ u ~

Oh

is called the" d e r i v a t i v e o f f o n

U

of ~(x)

U

in the

(6.18)

~ F

i n the d i r e c t i o n h ".

E x a m p l e 19 We give examples which show that G~teaux derivative is not in general a linear map, and that f may have G~teaux derivative without being continuous. Let us first start with this example of a directional derivative in its simple form. (1)

Let f : K n ~ K

and ej = ( 0 , . . . , 0 , 1 , 0 , . . . 0 ) , j = l , . . . n .

Then

n X--(Xl,...,Xn)--~-'~xje.j 1 and Of

Of

o%-7(x)- ~ (x) which is the partial derivative of f with respect to i th coordinate. the partial derivative is the derivative in the direction ej. (2) Let

Thus

f "K 2 ---, K be defined by

I XlX~ f (x) -

(~+;~) , x # o,

0

,

x-O

Then ~ (0) - f ( h ) , which shows that the G~teaux derivative may not be a linear map. (3) Let f "K 2 --* K

be defined by

d ~r f(x) -

~'

0,

.

x-O

6.4 D I R E C T I O N A L

DERIVATIVES

113

Then f is not continuous. However O0-~h(0) - 0 for all h E K 2. That is, f is Gdteaux differentiable at 0 but f is not continuous at 0. G a t e a u x Differentials. We have shown that G~teaux differentials may not be linear and G~teaux differentiability does not necessarily imply the continuity of the function. We now show that quasi-differentiability is stronger than G~teaux differentiability (or directional differentiability ).

T h e o r e m 57 [29](2002) Let E and F be p-normed and q-normed spaces respectively. If

f :E---~F is quasi-differentiable at a E E, and

then

of

Oh (a) - D f ( a ) h ,

f

is G~teaux differentiable at a

i.e. O f ( a ) - D f ( a ) .

(6.19)

P r o o f . We have

lim II f ( a + x) - f(a)

-

x ---*O

D f ( a ) x II1/q /II x II~/p= o

Replace x by Ah to get the desired result. In fact lim~_~0 II f ( a + A h ) - f ( a ) -

1 II h II1/~

~%II

Df(a)(Ah)111/q /II Ah

f ( a + Ah) - f(a) ~

ADf(a)(h) ill~q_ 0; -

~

i.e., lira [ f ( a + Ah) - f(a)] _ D f ( a ) ( h ) ,~---.0 ,X Hence

m

II~/p

Of Oh (a) - D f ( a ) h .

-

114CHAPTER 6 HIGHER QUASI-DIFFERENTIAL

IN F-SPACES

Corollary 7 Let E and F be p-normed and q-normed spaces respectively, (0 < p , q _ < 1). If f " U ~ F is quasi-differentiable at the point a C U, then df(a + Ah)

d~

Of

I~,--o=~

(a)

(6.20)

=Of(h,a)

That is, the usual derivative of f ( a + A h ) with respect to A at A - O is equal to the G~teaux derivative at a in the direction h. I Proof. Define g" V--, F by g ( A ) - f ( a + A h ) where V is a neighborhood of 0 in K. So by the chain rule we obtain the derivative g' (0) - D f (a)h,

since g ' ( 0 ) - lim~__,0 [g(~)-g(0)]x . Hence, g'(0)- li~ [f(a+ Ah)-f(a)l

_ -~ Of ( a ) - D f ( a ) ( h )

I

T h e o r e m 58 ( C o m m u t a t i v i t y

of directional derivatives)(2002) Let E and F be p-normed and q-normed spaces respectively, and U C E open, (0 < p,q < 1). Let f : U ~ F be twice quasi-differentiable on U. Then for any h, k E E,

of " U ~ F Ok

is quasi-differentiable on U such that for any x C U 02 f d2 OhOk (X) f(x)(h,k). Hence 02 f

02

OhOk (~) - OkOh(x) f Proof. The mapping

of Ok

9x E U ~ D f ( x ) ( k ) ,

(6.21)

S

6.5 Q U A S I A N D F R E C H E T D I F F E R E N T I A L S

115

is the composition of the mappings

x e U ~ D f ( x ) e L(E; F), and

A e L ( E ; F ) ~ A(x) e F Thus

is quasi-differentiable, and its quasi-differential at x is given

by

of

o2f

D(--~-~(x))(h) - OhOk (x). We have then

d2 f ( x ) ( h , k ) _

02 f OhOk (x).

Since d2f(x) is symmetric by Theorem 53 (p.103), this establishes the commutativity of the directional derivatives, m R e m a r k 13 The above Theorem (and hence Schwartz theorem ) generalizes the rule of the classical differential calculus, where for some w - f ( x , y ) , we have

i)2 f OyOx

02 f OxOy

d

6.5

QUASI AND FRECHET DIFFERENTIALS

This part is devoted to studying the relationship between the concepts of Quasi-Differentiability and Fr~chet-Differentiability. These will be called Qdifferentiability and F-Differentiability, and their classes will be denoted by QD and FD, respectively. We have seen, in the preceding part, that quasi-differentiability is a continuous linear mapping on spaces of maps between locally bounded Fspaces. This is similar to the situation for the Fr~chet differentiabilty and normed spaces. That is, both are continuous and satisfy

D ~ ( f + g) - D ~ f + Dag,

Da(,X f) - ,X(D~f)

whenever f, g are simultaenously differentiable in any sense.

116CHAPTER

6 HIGHER QUASI-DIFFERENTIAL

IN F-SPACES

However, our goal here is to prove that: " T h e t w o c o n c e p t s of diff e r e n t i a b i l i t y are t o t a l l y different ". But the new one, t h a t is the Q-differentiability, is more suitable for all F-spaces which are locally convex or not. It may play a role in analysis and applied mathematics, where the classical Fr6chet differentiability can not be used.

6.5.1

Finite Dimensional

Case

The following lemma shows the equivalence between the Q-differentiability and G-differentiability in some finite dimensional spaces.

L e m m a 12 Let E and F be any two finite dimensional p-normed spaces and f be a map between them. Then f is Q-differentiable if and only if f is F-differentiable. P r o o f . Note t h a t lim IIf(x) - f ( a ) - T~(x - a)lll/p _ 0 ~-~ IIx - all lip

(6.22)

if and only if lim ilf(x) - f ( a ) - T~(x - a)l [ = 0

IIx--alf for a continuous linear mapping Ta E L ( E , F), where a , x C E. We know that finite dimensional topologies are equivalent, m The following theorem shows that we still have the linear transformation, defined by the Jacobian matrix, to find out quasi-differentials when we deal with finite dimensionM spaces. T h e o r e m 59 [29](2002) Let E and F be any two finite dimensional p-normed and q-normed spaces with dimensions n and m respectively, (1 > p,q > 0). Let U C E be open and f 9U ---+~ m a function defined by

f (a) - (fl (a), ..., fro(a)), where If of~ ox~ (a) D f(a) matrix

fi " U - - + ~ , ( 1 < i < m ) . f is quasi-differentiable at aC U, each of the partial derivatives exists, (1 < i < m, 1 < j < n). Furthermore, the quasi-differential

9~n ___+~ m is the linear transformation defined by the Jacobian of f at a. II

6.5 QUASI A N D F R ~ C H E T DIFFERENTIALS P r o o f . Let L - D f (a) 9j ~ n ~ s t a n d a r d basis for ~ n and ~m. f at a, so t h a t

Let Assume

117

el,...,en and e l , . . . , e m b e the (aij) is the Jacobian m a t r i x of

~m.

m

Lej - ~

aijei,

l<j
i=1

Since f

is quasi-differentiable at a, it follows that lira

IIf(a + hej) - f ( a ) - L(hej)ll 1/q

1[/"--ej lilt ''lIp

x ----~a

--

O, 1 <_ j <_ n.

Now

1/q lim ]lf(a + hej) - f ( a ) - hL(ej)l] 1/q I n fi(a + hej) - fi(a) - aij [q] x---~a [hi = liI-na ~-~1 h i--1

=0. Hence each term in the sum must tend to zero. Therefore

Ofi(a) = lira f j(a + hej) - f j(a) = a i j , 1 < i < m, 1 < j < n. Oxj ~--,~ h This proves t h a t the theorem, m

Ofi(a) Oxj

exists and is equal to

aij,

completing the proof of

The following is an example of a non quasi-differentiable function. Example

20 Let E = 1p, (1 > p > 0). The function

f : 1p --+ Kdefined by

f(x)-[ Xll

(6.23)

is not quasi-differentiable at O. Notice that lira [] Ilxll - I1011- T o ( x ) I q ] l / q - lira II Xl I -C~lXl I 9--,o [ I x - O[ll/p x-+o ( E l x i I p ) l / p

This shows that the limit does not exist for any constant a since it depends on the way that Xl approaches O. That is, f is not quasi-differentiable at O; i.e. f ~ Q D ( l p , K ) . Remark

14 For E -

11 - lp N K , we can also take 1

f ( x ) - I Xl Ip" lp ~ K tO h a v e

lim X---~O

II Xl

[p -OLXl I __-- lim [ I Xl ]p I Xl [ Xl---+O X1

which does not also exist if p equals, say, 1/2. II

118CHAPTER 6 HIGHER QUASI-DIFFERENTIAL 6.5.2

Infinite Dimensional

IN F-SPACES

Case

T h e f o l l o w i n g e x a m p l e is for a n o n F r 6 c h e t a n d a n o n q u a s i - d i f f e r e n t i a b l e function.

E - lp, F - ~ , a = O. We are going to establish a function f 9E ~ F which is neither Fr~chet differentiable nor quasidifferentiable at a - O, i.e. Example

21 Put

f ~ QD U FD T h a t is

IIf(x)

lim

- f ( a ) - T a ( x - a)[I 1/q

x----,a

# 0

[Ix - a[[1/p

(6.24)

even when p - q - 1. C o n s i d e r the following f u n c t i o n (given as an example, in Ch. 9, subsection 9. 2.1. p. 200)

f(x)

(~1 )(l_l/p).xn,

=

x-

(Xn) E lp,

1 and the the following set which approaches zero in 11

AI-{

1/p

x(n) ; x(n) -- (O' ' " " " O' ( 1

,0,...),noN

I o

Now

IIf(x(~)) - T~ Xn--~O IIx(n) ll lim

=

=

lira

II E T ( n ) 1 (1-1/p)xn -- E C~nXnll

Xn ---*O lira

E Ix~ I, 1 (1-1/p) (n) 1 lip -- E Olnxnll II(n)

(1/~)

n - - - y 0<9

=

lim

I -~1

n----, o o

=

-

1 lip I O~n(n) 1/ n

I 1 - ct(1)l/p-1 177,

1

which does not go to zero. T h a t is, f ~ F D . B y the s a m e way we can show that Let f

= n --. im

Notice that x (~) --. 0 in 11.

f ~ Q D.

be as given above, a f u n c t i o n on lp. N o w

S

6.5 Q U A S I A N D F R E C H E T

lira IIf(x~) - r~

--

Ilxnlll/p

x..-~o

1

lira II E F ( n ) ( ' - I / p ) - x ~ - EF/3~x~II 1/q

(EIx~

x~-~o 1

=

119

DIFFERENTIALS

[p)l/p

1 1/q

lira I n - / ~ (1n ) ~~ (~)~/p

1 1

] - - lim I np ~-~

--~ oc; that is, f ~ QD. Here To(x) - E /3nXn, ( /3n) E l~, and in

-~,~l Xn --->0

lp.A The following theorem shows that the class of

Fr~chet-differentiable

maps may not be a subset of the class of Quasi -differentiable ones. Theorem

60 [33](2002)

Not every Fr~chet-differentiable map f 9E --~ F from a p-normed space E into a q-normed space F, (1 > p, q > 0), is Quasi-differentiable. That is F D ( E ; F) ~= Q D ( E ; F). (6.25) P r o o f . Take E - LP(I),the space of integrable functions on I - [0, 1], and F - K , t h e scalar field, w i t h t h e q-norm, f - 0, the zero function in LP(I). Consider the function G ' L P ( I ) ~ K given by

G ( f ) - llfll -

I f [P dx.

(6.26)

Then lira f~o

I l G ( f ) - G ( o ) - 0)ll 1/q llf-

oll 1/p

Ill)) ~/q = lira y-~o,llflll/p f~o

= lira

1 llfll ' / p - l / "

So the limit does not exist as ~1- ~ 1 > O. Notice that the dual space of /if(I) equals {0}. Therefore, the function G is not quasi-differentiable at f - 0. However, we notice that LI(0, 1) C Lll2(0, 1).

(6.27)

For if f E L1 (0, 1), then

j/O1 [1 f [1/2 . l i d

x < ( ~I [ f l dx)l/2.(

/i

1

ldx)l/2;

120CHAPTER 6 HIGHER QUASI-DIFFERENTIAL

IN F-SPACES

t h a t is, f E LlZ2(0, 1). Therefore w e consider our function f in L I ( 0 , 1). We claim t h a t G(f ) = Ilfll is a Fr6chet differentiable function at 0. Notice t h a t To(x) C (L1) ', the dual space which is isomorphic to Lo~. Now lim

Ila(f)

- G(o) -

f~o

To(f -

0)[ I =

lim I a ( f ) To(f)[ X-~0 IlfllL~ -

[If - OIIL~

lim I Ilfll- T o ( f ) f ~O IlfllL1

=

l _ 0

0 in L 1, where To(f) E (L1) ' -~ L c~ may be taken as equal to I l f l l - f0~ I f dx, since f is considered in L 1 9 This completes the proof of the theorem, m as f ~

[1/2

In what follows we prove t h a t the class of Quasi-differentiable maps may be not contained in the class of Fr~chet-differentiable maps.

T h e o r e m 61 [33](2002) Not every Quasi-differentiable map f from a p-normed space E into a q-normed space F is Fr~chet-differentiable, (1 > p, q > 0), that is

QD(E; F) ~ FD(E; F).

(6.28)

P r o o f . As in the proof of Theorem 60, we take E = LP(I), the space of integrable functions on I = [0, 1], F = K , with the q-norm, and f = 0, the zero function in LP(I). Consider the function G : LP(I) ---,K given by

G(f) - I l f l l -

I f Ip dx.

(6.29)

'O

We have seen t h a t lim

f---,o

I I G ( f ) - G ( 0 ) - 0)ll 1/q

Ilf - oil 1/p

Ilfll ~/q

= lim

f~o Ilflll/p

1 f~o ]lflll/p-1/q

= lim

Now if q1 p1 > 0, the limit will approach zero and G(f) will be a quasi-differentiable function. But if p = q we have non-zero limit since 1

lim f--,0 So G(f)

1

Ilfll~/Itfll~

- 1.

will never be a Fr6chet-differentiable function, m

(6.30)

6.5 Q U A S I A N D F R E C H E T

121

DIFFERENTIALS

R e m a r k 15 The relation between the calsses of Q-differentiability and F-

differentiablity, proved so far, may be summarized by the following diagram Gateaux differentiable ~ .

Qu

-

.

.

.

9

9

ii1

C~F d,

Figure 2: R e l a t i o n between the different calsses of differentiability's.

This Page Intentionally Left Blank

Chapter 7

QUASI-HOLOMORPHIC MAPS The subject of this chapter is the holomorphic mappings of metrizable spaces, in particular locally bounded F-spaces which are not necessarily locally convex. We first review the basic concepts of complex variables with values in Let f 9t2 --~ (~ be a function defined in an open subset ~ of the field (Y of a complex variable with values in (~. We say f is holomorphic if for every point a in ~ there is a convergent series such that oo

f ( a + t) -- ~

ck(a) t k o

holds for all complex numbers t of sufficiently small modulus. This is one of several possible definitions and it is suitable for generalization to vector spaces. D e f i n i t i o n 37 Let open set in E. Then

E

be a topological vector space over (~ and t2

is said to be G d t e a u x a n a l y t i c convergent series such that

if for any a C ~

Oo

f (a + tb) - ~

ck(a, b) t k 0

123

and b E E

an

there is a

124

CHAPTER

7 QUASI-HOLOMORPHIC

MAPS

for It I small. If E is of finite dimension, it is a classical result that such a function is already continuous, in fact it is infinitely differentiable. If E is of infinite dimension it is desirable to impose some kind of regularity: It is convenient to require f to be continuous. See example1, p. 3, of a non continuous linear mapping. Thus we have D e f i n i t i o n 38 We say that f : ~ -.(~, C E open in an infinite dimensiononal space, is h o l o m o r p h i c if f is Gdteaux analytic and continuous. The theory of holomorphic functions on topological spaces has developed rapidly in locally convex spaces in general during the last 50 years. However it is totally neglected for non locally convex spaces till our thesis [11] was written in 1979 containing the solution of the Levi problem in some of such spaces to prove that : T h e r e is s o m e t y p e o f r i c h n e s s to the class o f h o l o m o r p h i c f u n c t i o n s o n s u c h spaces. Therefore we proceed to solve and describe some of the basic problems and the new concepts in the field to which we have consecrated our book.

7.1

FINITE EXPANSIONS FORMULA

AND TAYLOR'S

This chapter is devoted to extend the Taylor~s T h e o r e m to maps between locally bounded F-spaces. It says that : " I f a f u n c t i o n f is m - q u a s i - d i f f e r e n t i a b l e ~ t h e n f m a y b e a p p r o x i m a t e d locally by a p o l y n o m i a l of d e g r e e m". Let us first introduce the following concept. D e f i n i t i o n 39 ( m - q u a s i t a n g e n c y ) L e t E and F be p-normed and qnormed spaces (0 < p, q ~_ 1) respectively and U a nonempty open subset of E. Two functions f,g:U-.F are said to be m - q u a s i - t a n g e n t (or m - ( p q ) - t a n g e n t ) aEU if lira I] f ( x ) - g(x) ]Ip = O. x---~a ]] X - a ]]mq

at a f i x e d p o i n t

7.1 F I N I T E E X P A N S I O N S That is, then

f o r every e > 0

AND TAYLOR~S FORMULA

there e x i s t s 5 > 0

]1 f ( x ) - g ( x )

such that if, II x - a

]]P< e ] i x - a l l

125 II< 5,

mq .

In particular, for m - 1, we regain the definition of quasi-tangency at a point a which was introduced in chapter 4. As in part X we can easily show the following properties 9 If f and g are m - q u a s i - t a n g e n t at a, then, (i) ( f - g) is continuous at a and f (a) = g (a). (ii) f and g are ( m - 1 ) quasi-tangent at a. (iii) ( f - g) is m-quasi-tangent to 0 at a. (iv) m - q u a s i - t a n g e n c y is an e q u i v a l e n c e r e l a t i o n on the vector space of all m a p p i n g s from U to F which are continuous at a. (v) The notion of m-quasi-tangency depends only on the topologies of E and F, not on the F - n o r m s used to induce the topologies.

7.1.1

Finite Expansion

If f 9U --~ F is quasi-differentiable at a point a E U, f is 1-quasi-tangent to an atone linear m a p (i.e. a polynomial of degree <_ 1) at a. T h e n f is "almost" a polynomial of degree <_ 1 on some neighborhood of U. T h a t is,

rE>0

35>0

such t h a t

II x - a II< 5 ~11 f (x) - f (a) - f (a) (x - a) lip<_ e [I x - a IIq We are in general interested in a p p r o x i m a t i n g a function f " U ~ F by polynomials on some neighborhood of a with an infinitely small error. This leads us to the following concept. D e f i n i t i o n 40 ( m - e x p a n s i o n ) A polynomial P E P ( E , F ) of degree <_ m is said to be an m - e x p a n s i o n o f a f u n c t i o n f " U ~ F at p o i n t a E U if P is m-quasi-tangent to f at a, lim [ I f ( x ) - P ( x ) I[p = 0 . II X - a [['~q

x--~a

CHAPTER

126

7 QUASI-HOLOMORPHIC

MAPS

The limit (7.1) expresses an "asymptotic property between f and P as x approaches a ". It is obvious t h a t if P is polynomial of degree _< m, then P is an m-expansion of itself at any point.

U n i q u e n e s s of F i n i t e E x p a n s i o n s For a given function f : U ---, F, it is not clear whether f admits an m-expansion at a. However we shall prove t h a t " e v e r y f u n c t i o n a d m i t s at m o s t o n e m - e x p a n s i o n at a p o i n t " . To show this fact we need the following two lemmas. Let us first recall the polarization formula presented in subsection 2.2.5, p.44 Let U be an open subset of E containing an open ball B (0, 5). If f'U-+F

and X l , --., X m (3_ U are such that II Xl II~/p + . . . + II xm Ill/p< 5, xm is defined by polarization Cm (f) of f with respect to Xl 1 Cm (f) -- m[2 m ~

el...emf (glXl -~-... + s

then the

(7.2)

Ek=•

where the s u m m a t i o n is over all ek - :t:1, k - 1 , . . . , m . We have shown in subsection 2.2.3, p.44 that if P - P o + . . . + P m E P (E; F) with P,~ - fi-m, where Am E Ls (mE; F ) , then Cm (P) - Am (xl, ..., x m ) .

(7.3)

L e m m a 13 Let U be an open subset of E which contains the origin. I f f 9U --~ F is m-tangent to 0 at the origin, then for any e > O, there exists 5 > 0 with B (0, 5) C U such that for any m elements x l , . . . , Xm in U with II xl II1/p + . . . + II Xm Ill~P< 5, the polarization Cm ( f ) of f with respect to Xl , . . . xm satisfies the inequality

II r

~ (11 Xl

II lip §

§

II x~ II~/P) ~/p

7.1 F I N I T E

EXPANSIONS

AND

TAYLOR'S

FORMULA

127

P r o o f . It suffices to show, by the definition of Cm (f) (see p.44), t h a t + emXm)IIlZq< e (il X~ II + ' ' " + II Xr~ li) "~zp

II f (ClXl + . . .

for all ek - +1, k - 1, ..., m. to 0 at the origin. In fact, lim il/(~)-~ ~--,o II~llr~/p

But this is obvious, since f is m - q u a s i - t a n g e n t

= 0

implies t h a t given e > 0

we get for small

v~]u~ of ~, II f (x)lll/~< ~ II x IIm/~. Thus It f(s

+--.

nc s

1/q~_ s

( II Xl II1/p + . . .

)

m

II Xm II lip

~ .

I is a polynomial of degree < m m-quasi-tangent to 0 at a point a C E, then P - O.

Lemma

Proof.

14 If P E P ( E , F )

If P

and if P

is

is m - q u a s i - t a n g e n t to 0 at a point a, i.e. limll P (x) HI]/q /II x -

a lim/P= O,

x---->a

then the polynomial Pa defined by P~ (x) - P (x + is m - q u a s i - t a n g e n t to 0 IIP(~+~)ll alzllm/p~/~. Therefore,

~)

at the origin. Notice t h a t lim

IIP(x)ll~/q - l i m

x---,a iix--all mlp

it is suffices to consider the case where a

--x~O

is the

origin. Let P - Po + P1 + . . . + P,~. T h e n for Am C Ls (mE; F ) , P,~ - -Amwe have from L e m m a 13 and relation (7.3) above t h a t for any e > 0, there exists > 0 such t h a t

II Am (Xl...,Xm) II1/q< _ e

if II

(11 x~

II1/p + . . . +

II xm II~/P)"~/q

(7.4)

]11/p

Zl -~-''" -~- II Xm Ill/p< 6. T h e preceding inequality (7.4) holds for any x, ..,x,~ m-linear map. In fact, we have

II Am (Axl...,Axm)II

~/q- I~Xl"~/q II A ~

since Am is an

(xl...,x~)II 1/q .

Since the inequality (7.4) is valid for e > 0 and for any Xl ~ " ' ~ Xm conclude t h a t A m (Xl,..., X m ) -- 0

we

CHAPTER

128

7 QUASI-HOLOMORPHIC

MAPS

for all Xl..., Xm. Thus Pm = 0. Consequently, the degree of P is less than m. Since P is m-quasi-tangent to 0 at the origin, P is also ( m - 1)quasi-tangent to 0 at the origin. Repeating the above argument, we have P,~-I = 0. Thus by induction we obtain Po = P1... = Pm = O. m

T h e o r e m 62

Let f : U---, F and a E U. If P1 and P2 f at a, then =

are two "m-expansions "of

P2.

P r o o f . Let P = P1 - P 2 . We claim that P = 0. Since both P1 and P2 are m-expansions of f, it follows that P is m-quasi-tangent to 0 at the point a. By Lemma 14, we conclude that P = 0. li 7.1.2

Taylor's Formula

Let E and F be p-normed and q-normed spaces respectively (0 < p, q _< 1) and f : U ~ F, where U is an open subset o r E . If f is m-quasidifferentiable at a C U, then it was shown that

dk f (a) E Ls (kE; F ) for k - 0 , 1 . . . , m . Hence we can associate dkf(a) the u n i q u e k hom o g e n e o u s p o l y n o m i a l which we will denote by dkf (a) (See subsection

2.2.3). Thus

dk f (a) E P (kE; F) and

dkf (a) (x) - dk f (a)x k. As dkf (a) is a k-homogeneous polynomial, the mapping xEE~dkf(a)(x-a) is a p o l y n o m i a l of d e g r e e k.

cF

7.1 F I N I T E E X P A N S I O N S A N D TAYLOR'S F O R M U L A

Definition 41 By a "Taylor p o l y n o m i a l " Tm,l,a a, we mean the polynomial defined by

of order m

129 of f

at

m

Tm,l,a (x) -- ~

~.d f (a) (x - a)

k--o m

= ~

1

~ d k f ( a ) ( x - a) k

k--o

Example 22 If P C P (E, F) is of degree < m, then for every a C E P-

T~,p,~.

To see this we may assume without loss of generality that P is a homogeneous polynomial, since the general case follows from the homogeneous case by adding a finite number of homogeneous polynomials. Let P - ft, where A C Lsm (E; F). Then we have by the binomial formula and Theorem 5~, p. 108. m

P (x) -

ix-ol

A x m - A (a + (x - a) )m k--o m

1

= ~

k

~ d f ( a ) ( x - a) k

-

Tin,p,

a

k=o

which shows that P -

7.1.3

Tm,P,a for any a E E.

Q u a s i - D i f f e r e n t i a l of Taylor P o l y n o m i a l s

If f - U --+ F is m-quasi-differentiable at a point a C U, then df (a) e L (E,F)

and d m - 1 (dr) (a) E L

(m-lE; L (E; F ) ) .

On the other hand, dm f (a) e n (mE; F) .

Therefore we do not have d m f - d m-1 (df) unlike D m f - D m-1 ( O F ) . However, the map r n (mE; F) --+ n (m-lE; L (E; F)) defined by r (A) (Xl,..., Xm-1) (Xm)

-

-

A (xl,..., Xm)

CHAPTER

130

7 QUASI-HOLOMORPHIC

MAPS

is an isometric isomorphism. Under this correspondence, we can identify d m f (a) with d m - 1 (df) (a) with respect to r Then

d m-1 (dr)(a)(xl,..., Xm-1) (Xm)

dmf (a)(Xl,... , Xm)

--

and in particular,

d m-~ ( d r ) ( a ) ( x - a) m-~ (x - a) - d m f ( a ) ( x - a) m .

We have also from Theorem 45 (section 4.1.4, p.84)

d ( d k f (a) ( x - a ) k) = k d k f (a) ( x - a ) k-1 The following theorem gives the q u a s i - d i f f e r e n t i a t i o n of t h e T a y l o r p o l y n o m i a l of o r d e r m of f at a and the f o r m of t h a t o r d e r (m-l) of df a t a.

T h e o r e m 63

Iff

is quasi-differentiable at a, then we have

U--,F

dTm,f,a - Tm-:,df,a. Proof. m 1

m 1

dTm,/,a - d(~-'~ ~.dk f (a) (x - a) k) -- E k=o

m

1 k--1

m-1

= ~

-

a)

k--o

m =

(a))(x

~.d (dk f

d k f ( a ) ( x - a) k-1 -- ~ "

ld k -~. ( d f ) ( a ) ( x -

1

~(k -d 1)w k-1 ( d r ) ( a ) ( x - a) k-1

k = l

a) k

__ T m - l , d f , a

"

(X).

k--o

7.1.4

General Mean-Value

Theorem

For a given function f : U --* F, it is not certain that f admits an mexpansion at a point a in U. However, if f is m - q u a s i - d i f f e r e n t i a b l e , we will see t h a t f a d m i t s a n m - e x p a n s i o n . This result is the so called Taylor~s t h e o r e m . But we first need the following generalization of the Mean-Value theorem (Theorem 50, p.94) for Taylor's theorem.

7.1 F I N I T E

EXPANSIONS

AND

TAYLOR'S

FORMULA

131

T h e o r e m 6 4 (Generalized Mean- Value Theorem)J29](2002) Let U be an open subset of a p - n o r m e d space E such that U contains the arc segment A b and F is q-normed. I f

f "U--,F is quasi-differentiable and p ' [ 0 , 1 ] - - ~

is differentiable such that

II D f [)~l/pb + ( 1 - ) k ) 1/p a] I1<_dp ()~) V ,k C [0,1]. Then

II f(b) - f ( a )

IllZq~ll

b- a

IIlzp

(r

- r

P r o o f . We shall modify the proof given in Theorem 50 (subsection 5.2.1, p.92). If x E A b, the arc segment from a and b, then we denote by A~ E [0, 1] the unique real number satisfying

x - )~/Pb + (1 - )~x) 1/p a. For a given E > 0, let X following condition-

II f (t) - f (a)/ll/q
be the set of all points x E A b satisfying the

b - a II ~/p .

[r (~t) - r (0)1 § e lit - a

II ~/p

(7.5)

for a_< t_< x. T h e n X -7/=C s i n c e a E X. We notice that i f x E X, then A~ C X. Let c - sup X . We prove t h a t c C X and c - b. If c < b, then by quasi-differentiability of f at c, we can find 5 > 0 such that

II b - c I1> ~, B (c, ~) c U and if t C t3 (c, 6), II f (t) - f (c) - D f (c) (t - c) IIlZq< E lit

-

c II1/p/2

or

II f (t) - f

(c) Ill~q<

(11 D f

(c)

II +e/2) lit

- c IIq/p .

(7.6)

132

CHAPTER

7 QUASI-HOLOMORPHIC

MAPS

Since r is also differentiable at Ac E [0, 1], there exists rl > 0 with such t h a t i f l A t - ) ~ c l < r l ,

rl<6/llb-all

I r (~) - r (~)

- r (~c)(~

- ~)1

<_ ~ I ~t - ~c112

or

Ir (,xc)(~xt - ~xc) l _< Ir (,xt) - r (~x~) I +EI,xt - ,x~l/2. Let -y - r1 II b - a t < c < t' . T h e n

f

I1~11 f

II

f (t') -

<

II D f (c)LIII t ' - c

(a)

II < ~,

(r.r)

and let t, t' E B (c, r) A A b such that

(t') - f (c)I1+

II f

(c) - f (t)

II + II f

(t) - f (a)

II

IIq/p + II D f (c)IIII c - t llqlP +

+E 11 t' - t II/2+

II

b - a

II lzp

[r (At) - r (0)] + e

II t

(7.8)

- a IIxzp

by (7.6) and (7.5). Note t h a t IIt'-tll=llt'-cll +llc-t I[, t ' - t = ( ) h \ - A t ) . (b-a), and II D f (c) I1< r (Ac). Also note t h a t ~o is an increasing function since

r

>0.

Keeping this in mind and applying (7.7) to the first two terms in the extreme right h a n d side of (7.8) we obtain after some c o m p u t a t i o n t h a t

II f (t') - f (a) [lllq_
-F- s II t' -

a II lip .

This shows that t ~ E X and also c E X, which contradicts the fact t h a t c = s u p X. Thus c - b, and X - A b. It follows t h a t for any e > 0 we have

II

f (b) - f ( a )

Illzq
b - a

II lzp [r ( 1 )

- r (0)] + e

II

b - a

II lzp

and hence [I f (b) - f (a) Remark

Ilxzq
- r

II

16 If ~o " [0, 1] --+/~ is defined by ~(~) - M~

where M - supxEA ~ II D f ( x ) II, then we obtain the Mean-Value inequality, Theorem 50, p.94, from the preceding theorem.

7.1 F I N I T E E X P A N S I O N S A N D TAYLOR'S F O R M U L A 7.1.5

133

Taylor's F o r m u l a w i t h L a g r a n g e R e m a i n d e r

We now prove the main theory of this part. The following theorem may be called T a y l o r ' s f o r m u l a with Lagrange r e m i n d e r . The forthcoming Theorems 66 & 67 may also be called Taylor's formula.

T h e o r e m 65 ( T a y l o r ' s F o r m u l a ) Let U be an open subset of a p-normed space E such that U contains the arc segment A b. If f" U --~ F is (m + 1)-quasi differentiable on U, then - M II b - a II(m+l)/p II f(b) Tm,df ,~( b) ill~q< (m + 1)! (7.9) where

M-

sup

{11 din+if(t) I!" t

e At}.

(7.10)

Proof. If m - 0, we regain the Mean-Value Theorem 50, (subsection 5.2.1, p.94). We now continue by induction on m. Assume that the theorem is true for r n - 1. If x cA b then by Theorem 64 we have

II df(x) - Tm_l,f,a(X ) II1//q~_~

II x -

a IIm/p sup II din(dr)(t) II(m) I9 Ab tE a

II x - a

M

IImlp

m! Consider

g-f-Tm,f,a. Then g(a) - 0

and

dg - df - Tm-l,df,a. Hence for x C A b, we have

II dg(x) 1

Ill~q<_ 1

M II x-

a IIm/P

m!

Replacing x by A ~ b + ( 1 - A ) ~ a , 0 _< A _< 1, since the arc segment A~a is p-convex set (see Ch.3, section 3.1, p.50), we obtain

134

CHAPTER

7 QUASI-HOLOMORPHIC

MAPS

II dg( A~b + (1 - A)~a) II~'q< M II ,X}b + (1 - A ) l a - a IIm/p -

r

m!

Apply the preceding generalized mean-value theorem to g and the map 1] ~ / ~ defined by i

r

=

M II A~b + (i

i

- / k ) ~ a - a I1~/p

(m +

i)!

to obtain

II g(b) II~/q-II g(b) - g(a) II~/q<

M

II b -

a II(m+~)/p ( m + 1)!

This establishes the proof. I C o r o l l a r y 8 If f : U ~ F is (m + 1)-quasi differentiable on U, then for any a E U,

f

admits the m-expansion Tm,l,a

at a.

The preceding theorem has been obtained by assuming that f is ( m + 1)-quasi differentiable. We weaken this condition and obtain the following theorem. T h e o r e m 66 ( T a y l o r ' s T h e o r e m ) Let U be an open subset of a p-normed space E such that U contains the arc segment A b. If f :U - , F is m-quasi differentiable on U, then

fo~ ~ e A b, II f ( x ) - Tm,f,a(X) II1/q~

Mx II x - a IIm/P m!

where

M~ = sup{ll d~ f ( t ) - d m f ( a )

I1: t C A~} .

P r o o f . Let g-f-T,~,f,~.

Then dk9 -- dk f - T m - k , d k f,a

7.1 FINITE

EXPANSIONS

AND

TAYLOR'S

FORMULA

135

and by Theorem 63, and for k = 0, 1,..., rn dk g(a) -- O.

Therefore,

Tm-l,g,a =

0. We also notice that drag(t) = d~ f (t) - dm f (a).

It follows from the preceding Theorem 65 that

II g(x)

II~/~=llg(~)-T~_~,~,~(x) I1~/~

II x-

a I1~/~ m!

sup II drag(t)

II.

tE A t

This completes the proof. II C o r o l l a r y 9 If f : U ~ F differentiable functions on U, Tm,f,~ at every a C U.

is of class C q of continuously m-quasithen f admits the unique m-expansion

P r o o f . This follows from Lemma 14 and the fact that d m f

is continuous.

m

We present finally T a y l o r ' s f o r m u l a w i t h a s y m p t o t i c p r o p e r t y Theorem

67

.

Let U be an open subset of a p-normed space E. If f :U --~ F is m-quasi differentiable at a point a E U, then f admits the unique m-expansion Tm,f,~ at a. P r o o f . The uniqueness follows from Lemma 14(p.127). It remains to show that Tm,f,a is an m-expansion of f at a. Let g=f-Tm,f,a. Then we know d kg - d k f - T,~_k,dk f,~ and dkg(a) -- 0 for k - 0,1,...m. Theorem 66, for A~ c U,

Thus

Tm-2,.f,a,a - O. By Taylor'

136

CHAPTER

7 QUASI-HOLOMORPHIC

II g(x) II1/q-II g(x)- Tm_2,9,=(x) ill/q~

( m - i)!

MAPS

Mx

where

MX - s ~ p

{ II d m - l g ( t ) I1 t ~ A ~ } .

But dm-lg(t)

-

dm-]f(t)-Tl,,d~_~/,~

=

din-if(t)

-

d~-If(a) - d(dm-]f(a)(t - a).

Since d i n - i f is quasi-differentiable at a, for e > 0 there exists 5 > 0 such that B(a, 5) C U and if x e B(a, 5) then

II : - ~ f ( ~ )

- din-if(a)

Hence, if II x -

-

d(dm-lf(a)(

x -

a)

II1/q~ e II x - a II~/p .

a II< - ~ then Mx _< e II x - a II/P, and

II g(x) II1/q~ II x - a Jim/; ( m - 1)!

i

7.2

POWER

SERIES

IN F-SPACES

This part is devoted mainly to study the basic p r o p e r t i e s of p o w e r series in locally b o u n d e d F - s p a c e s . In particular, we discuss various concepts of the radius of convergence of a power series, namely the r a d i u s of unif o r m c o n v e r g e n c e , the r a d i u s of a b s o l u t e c o n v e r g e n c e and the r a d i u s of n o r m a l c o n v e r g e n c e . We shall then show that a function which is represented by a power series is an i n f i n i t e l y q u a s i o d i f f e r e n t i a b l e f u n c t i o n ( C ~ function). A C ~ function in open set is called q u a s i - h o l o m o r p h i c . Recall that E and F are p-normed and q-normed spaces respectively,

(o < p , q < 7.2.1

1). Power

Series

It is known in classical analysis that a power series is a formal series of polynomials. Maintaining a parallel with classical analysis, we have the following definition of a power series.

7.2 P O W E R

SERIES IN F-SPACES

137

Definition 42 A p o w e r s e i n e s f r o m E to F about a is a formal series of the f o r m Oo

A m ( x - a) m

m--O where Am C L s ( m E , F), rn - O, 1, ... I f we prefer to use polynomials, a power series f r o m E to F about a is a formal series of the f o r m oo

P m ( x - a) m--O

where Pm C p ( m E , F), m - O, 1, ...Both c o e ] f i c i e n t s of the power series.

Am

and Pm will be called

m th

E x a m p l e 23 A polynomial in its general f o r m

Po+P~ +...+P.~ where Pk E P ( k E , F), k series. A

0,1,...m,

is an obvious example of a power

E x a m p l e 24 For r E L ( E , E), we introduce the exponential

~2 expr162

oo r n--i" 0

This is a power series on L ( E , E )

about the origin.A

E x a m p l e 25 I f E=lP, (0 < p < 1) and F - K ,

then

oo

~-~.(Xn) n,

X--(Xl,X2,...)

C

lp

1

is a power series about 0 since P,~(x) - x m is an m-homogenous polynomial f r o m E to F.A

138

CHAPTER

7.2.2

Uniform and Normal

7 QUASI-HOLOMORPHIC

MAPS

Convergence

We now introduce the concept of the radius of uniform convergence for a power series defined between locally bounded F- spaces. D e f i n i t i o n 43 ( R a d i u s o f U n i f o r m C o n v e r g e n c e ) L e t ~-].~=oP m ( x a) be a power series from E to F about a C E . A number p, 0 < p <_ cx~ is said to be the r a d i u s o f u n i f o r m converg e n c e of the series if p is the supremum of all r, 0 < r < c~ such that the power series converges uniformly on B - ( a , r ) , the closure of B - ( a , r ) . Since r can can be O, the radius of uniform convergence always exists. If p is the called the o p e n the power series For example and hence is an

radius of uniform convergence, then the ball B(a, p) is ball of uniform convergence. When p = (x~, we say that c o n v e r g e s u n i f o r m l y on every b o u n d e d s u b s e t of E. the exponential function exp r r E L(E, F), has p = oo entire function if F is a q-Banach space.

It will be useful to introduce the following concept of convergence. Let be a p-normed space and F be a q-Banach space. Consider the q-Banach space B ( X , F) of bounded functions of X into F with the sup q-norm, ] l f i i = s u p { ] ] f ( x ) i i : x e X } .

X

D e f i n i t i o n 44 ( N o r m a l C o n v e r g e n c e ) A series y]~o ~ f~ of functions in B ( X , F) is said to be n o r m a l l y c o n v e r g e n t if Oo

II/nll < o

This implies that, for each x E X the series ~-].o ~ f~(x) is absolutely convergent. Since F is complete, the series ~-]~o ~ f~(x) converges. If f ( x )

is the sum of this series, i.e. if f ( x ) - }-~'~ fn(x), x e X , O<3

Ilfll_< ~ l l f ~ l l , o

and m

lira

[If- ~ T/,-- 0

f n [ [ - O.

then

7.2 P O W E R

S E R I E S IN F - S P A C E S

139

Therefore, ~ o~ fn converges to f uniformly on X. This result is known as the W e i e r s t r a s s M - t e s t . The limit of a uniformly convergent sequence of continuous functions on a topological space is always continuous, and hence the sum of normally convergent series of continuous functions is also continuous. Thus, we have the following diagram showing implications of different concepts of convergence normal~ convergence ~ uni f orm~convergence

(7.11)

absolute convergence ~ convergence

In general, uniform convergence does not imply normal convergence.

7.2.3

Generalized

Cauchy-Hadamard

Formula

The radius of uniform convergence can be computed by the generalized Cauchy -Hadamard formula if the range q-space is complete. We recall that if (a~) is a sequence of real numbers, then the limit superior of (an) is defined by lira sup an -- lim(sup {an, an+l, ...}). The following result extends Cauchy-Hadamard formula to locally bounded F-spaces, see the author [11].

T h e o r e m 68 ( G e n e r a l i z e d Cauchy -Hadamard F o r m u l a . ) [ 1 1 ] ( 1 9 7 9 ) Let F be a q-Banach space. Then the radius of uniform convergence of a power series

P

oo

E

P m ( x - a)

TFL-'-O

is given by

1 = limsup IIPmll~eq. P (we adopt here: p - 0 r

1/p - oo, and p - oo r

(7.12) 1/p - 0).

CHAPTER

140

7 QUASI-HOLOMORPHIC

MAPS

P r o o f . Let 1 E

L - lim sup IIPmll~,.

(7.13)

We consider three cases: (1) A s s u m e L - o e ' W e c l a i m t h a t p-0. In fact, let K > 0 bea given real number. Since L > K, there exists an infinite subset NK of the natural numbers such t h a t 1

IlPmll~

> K

Vm r NK.

Now let r ~1 . We want to show t h a t the series does not converge uniformly on B - (a, r). This implies t h a t p - 0. Using of the q-norm IIP,~II, it is easy to find Ym e E , I l y m l l - 1 such that [IPm(ym)ll > K mq for each m E NK. Let X m - a + r y m . It follows that

IIPm(Xm-

> rmq(r1)mq

a)]] - rmqllPm(ym)l I

m

for m E N K

_ l

This proves t h a t the power series does not converge uniformly on B - ( a , r).

(2)Assumethat Let

0

0
< oo.

and

IIP ll

p for m - O ,

1,.

Then lim s u p ( a m ) ~

-

lim sup(llPmll

-

Lr.

m--~(N~

m--~oo

By the root test of the series of nonnegative numbers, ~ a~ converges if L r < 1. Therefore oo

!

rn--O

for

s

converges normally and < ~. This shows t h a t y ] . ~ = o P , ~ ( x - a ) 1 < hence uniformly on the ball B - - ( a , r) for each r < -~. Thus 7 - P "

7.2 P O W E R

SERIES

IN F-SPACES

On the other hand, if r > ~, then F~ of natural numbers, we have

141

L>

!

IIPmilq >

1 Then for some infinite set r.

1 ~FTM

whenever m E Ft. As in case (1), we can find Ym e E, I l y m l [ - 1, for each m C F~ such t h a t 1 1

IIr.~(ym)ll~ > - -r. m Then for Xm - a + rym, we have 1

IIPm(xm-- a)ll

-- r m l l P ~ ( Y m ) l l ~ >

r~(1) m-

1

which shows t h a t Em~__0P.~(x-a) does not converge uniformly on B - ( a , r ) . This shows t h a t p <_ 1/L. Hence p - 1/L. (3) For L - 0 " Let r > 0 ande-

1 Then there exists N > 0 such t h a t 2rq/v" IIP.~II _< e "~

If x E B - ( a , r ) ,

,for m >_ N.

we have

IIP,~(x- a)ll <__IIPmll.llx- all ~q/p ~ ( erq/P) m--

m

Vm>N.

Notice t h a t IlP,~(x)ll _ IIP~ll.llxll ~q/p for p-homogenous polynomial (see section 2.2.4.). This shows t h a t the power series converges uniformly on B - ( a , r). Since r > 0 is arbitrary, we conclude t h a t p - oe. / C o r o l l a r y 10 I f p is the radius of uniform convergence for a) and O < r < p, then

E~%0Pro(x-

oo

IIP~ll ~ / ~

< o~

(7.14)

rn--0

P r o o f . As in the case (2) of the preceding proof, apply the root test with am -I[Pmll r mq/p t o obtain the result, m

CHAPTER

142

7 QUASI-HOLOMORPHIC

MAPS

F is a q-Banach space, the following statements are equivalent 9 (a) E m ~ = 0 P m ( x - a) converges uniformly on B - ( a , r ) , r > O.

C o r o l l a r y 11 If

1

(b) lira sup IIPmll~ < oo. (c) The sequence ( IlPmll -~ ) is bounded, (d) There exists co > O and c > O such that lIPmll ~ coc mq/p

liP011 ~

(In (d), to avoid

V m - o, 1, 2, ..

(7.15)

1, w~ u ~ the constant co)

P r o o f . We note that the implications (a)=~ (b) =~ (c) =~ (d) are clear. For ( d ) ~ ( a ) ' l e t r > 0 be such that r c o < l . Then OO

oo

I I P m ( x - a)ll _< ~

co(cr) mq/p < c~

(7.16)

m--O

m--O

for x e B(a, r), which shows that the series converges normally on B - ( a , r). We used here the fact, IIPm(x)ll <_ IIPmlI.IlXllmq/p. Hence the radius of convergence is strictly positive, m In the definition of the radius of uniform convergence and the generalized C a t c h y - H a d a m a r d formula, we used the power series ~-~.m~__oPm(x- a) with polynomial coefficients. Let us now use the power series OO

A m ( x - a) m,

with

Am E L~(mE, F).

rn--0

We define the radius of uniform convergence as before. Let p - be the radius of uniform convergence for this series. Then we can prove that m p1- = limsup II Atoll ~

if F

(7.17)

is q-Banach space. In what follows we have the following relations between p and

T h e o r e m 69 [36]

Let F

be a q - B a n a c h space and let

p-.

7.2 P O W E R

SERIES IN F-SPACES

(x)

143

oo

Pm (x - a) = ~ rn--0

Am (x - a) m

m=0

where Pm - f~m, Am E Ls(mE; F), then

_P _
(7.18)

e

in particular, p - p - if F - K. P r o o f . Since by Theorem 28, (subsection 2.2.4 p.41)

mm(q/P)

It Pml] _< II Atoll _<

m!

Pmli,

Vm E~g

we have

p > p- > p lim (m!)l/m

Vm E ~

(7.19)

m

Hence by Stirling's formula" m~ lira = 1 m---,c~ mme-m(27rm)l/2

(7.20)

we have m

lira ~-~ (m!)i/m

~ze

(7.21)

and hence

P<_p-<_p. e

(7.22)

m

The strict inequality in example.

p - <_ p

can be achieved

as in the following

CHAPTER

144

7 QUASI-HOLOMORPHIC

E x a m p l e 26 Let E - l P , ( O < p < 1 ) , F - K . 0, 1, 2, ...there exists A,~ e L , ( m E , F), II A,~ II - 1

and

MAPS

Then for each

m

--

m!

[[ Atoll -

?Ttm. 1 lp

see example 8, p.42. Now by the Cauchy -Hadamard formula, we obtain 1

p- = limsup II A~II 1/~p,

p

--1,

7.2.4

p--e

since lim(m!)l/m = e ,

Radius

m

of Normal

D e f i n i t i o n 45 Let F E~=0Pm(x-a) from 0 <_ r < oo, such that We call Pn the r a d i u s

i.e.

p

Convergence

-

(7.23)

< p. A

p~

be a q - n o r m e d space. Consider the power series E to F, and let p,~ be the supermum of all r, the power series converges normally on B - ( a , r ) . o f the n o r m a l c o n v e r g e n c e of the power series.

In the proof of the generalized Cauchy-Hadamard formula for the radius of the uniform convergence, we in fact proved the following theorem "

T h e o r e m 70 . If F is a q - B a n a c h space, then the radius of the normal convergence is equal to the radius of uniform convergence. 7.2.5

Radius of Absolute

Convergence

D e f i n i t i o n 46 A number p~, 0 <_ Pa <-- oo a b s o l u t e c o n v e r g e n c e for a power series

fla is said to be the r a d i u s o f

oo

P m ( x - b), rn--0

if Pa is the largest number such that the series converges absolutely for II x - b I1< Pa" The radius of absolute convergence is equal to radius of uniform convergence if E - K. It is clear that p <_ fla, but p can be strictly less than #a for some infinite dimensional metric space, as the following example shows.

7.2 P O W E R

Example

SERIES

IN F-SPACES

27 Let E - 1

p, (O < p < 1 ) .

Pro(x)-

(Xm) m,

145 For

(Xl, X2, ...) E l B.

X-

(7.24)

we have

II Frail = 1,

Vm

and hence p = 1. On the other hand, since x m --+ 0 as m --+ oo, the series ~ ( x , ~ ) "~ converges absolutely f o r all x C lp. So this shows that Pa = +oo, and hence P < Pa. A

F o r m u l a for R a d i u s o f A b s o l u t e

Convergence

Pa

D e f i n i t i o n 47 Let F be a q - B a n a c h space.For a p o w e r series ~m~176 P m ( x a) and u E E , Ilull - 1, we define the f u n c t i o n p(u) to satisfy

1

P(~)

(7.2s)

= lim sup IIP~(~)II m

T h e o r e m 71 [36] Let E be a p - n o r m e d space and Then Pa -- inf

F

a q - B a n a c h space, and

{p(u) ; liul[

- 1}.

uCE.

(7.26)

P r o o f . Let R-

inf {p(u);

I1~11- 1}.

We may take a - O. Suppose R > O. T h e n if Ilxll < n, u ~ E, Ilull- 1 and k >_ 0 such t h a t x ku. We now have l i m s u D l , P ~ ( k u ) l l - ~ . q --

kp

kp < - - < 1.

p(~)- R

It follows immediately by the root test t h a t the series converges absolutely. Hence I~ ~_ Pa" On the other hand, if kp > R, we can find 5 > 0 such t h a t Since

we can find

(7.27) E~%0 P-~ (x) (7.28) k p > R +5.

146

CHAPTER

R - inf there exists u e E ,

I1~11- 1,

7 QUASI-HOLOMORPHIC

{p(u);llull

- 1},

such t h a t

kP > R + ~

p(u).

>

Then limsup [[P~(ku)ll&~ and hence the series that

E [IPm(ku)ll

-

kp

p(~)

>1

diverges by the root test.

Pa <- R Hence we have proved Remark

7.2.6

17 If E - K ,

Uniqueness

MAPS

p~ - R ,

This shows

(7.29)

by(7.28) and(7.29), m

the formula for Pa reduces to p. of Power

Series

If two series ~--~=0 P m ( x - a), ~-~m~=0Q m ( x - a) converge absolutely and have the same sum on a n e i g h b o r h o o d of a, then we show t h a t P,~ = Qm for all m - 0, 1, 2, ...This is an i m m e d i a t e consequence of the following theorem.

T h e o r e m 72 [36] If the power series ~-~m~=oP m ( x - a) converges absolutely and its sum is equal to zero on some neighborhood of a, then

Pro-O,

m-O,

1,...

(7.30)

P r o o f . We first prove the t h e o r e m for E - J R . For simplicity assume a -- 0, and let }-~-m=0UrnXm be a power series from K to F , where um E F. Suppose t h a t ~-~,r umxm converges absolutely and its s u m is equal to zero for I x l < ~ for some ~ > 0 . If we put x 0, then u0 - 0. Inductively assume t h a t u 0 - Ul ... = Un--1 -0 for some n > 1. Since }--~-~=0u-r162 - 0, Ilu.~[15 "~q --+ 0 as m --+ oo; hence {llu~ll~q

9~

-

o,

1,2,

...}

(7.31)

7.2 POWER

SERIES

IN F-SPACES

147

is bounded above, say by c. Now for x ~ 0, I x l < 5 , (N:)

I1~11 ~ ~

Ilumll Ix I(r~-~)q<

re=n+1

c Ix Iq

(7.32)

-- (~nq(~_ I X Iq )"

Thus we obtain u,~ = 0 by letting x --~ 0. We are now ready to prove the theorem for the general case. Assume the power series converges absolutely to 0 for Ilxll _< ~, r r 0. Let t e E be fixed but arbitrary. Then for ]A J_< ~/lltll, oo

oo

Pm()~t) - ~ m--O

Pm(t))~ TM - O .

Hence from the case where F - K above, we have Pro(t) - 0 m - O, 1, ...Since t is arbitrary, we can conclude that Pm - 0 m -- O, 1, ..m 7.2.7 Theorem

Quasi-Differentials 73

(7.33)

m--O

of Power

for all for all

Series

[34]

Let F be a q-Banach space and let ~-~m~=oAm(x-a) m be a power series whose radius of uniform convergence p is strictly positive, and let (X)

f(x) - ~

A m ( x - a) m

(7.34)

m--0

be its sum on B ( a , p ) . If b E B ( a , p ) , }-~.k~__oB k ( x - b) k such that

then there exists a power series

0(3

f(x) - ~

B k ( x - b) k

(7.35)

k=0

fo~ IIx- bll < p - lib- all, and its radius of uniform convergence is > p l i b - all; the coefficients Bk are given by

m

k)Am(b-

a)m_ k

.

(7.36)

m--k,

Proof. Let 0 < r < p. T h e n Em~__011A.~llrm~ < oe by Corollary 10, (subsection 7.2.3, p.141) and E m ~ = o A , ~ ( x - a) m converges uniformly on

CHAPTER 7 QUASI-HOLOMORPHIC MAPS

148

B-(a, p). For b e B(a, p), apply the multinomial formula (2.2) ( sbsection 2.2.2, p.36) to Am [(b - a) + (x - b)] m to have oo

m

A m ( x - a) TM -- ~

( k ) A m ( b - a)m-k(x - b ) k

m--O

Then Oo

k ) IIAml "lIb - all

~ llx - bll ~ < IIAmllrm~

rn~-O

for

IIx - bll < r -

lib - all. Thus

oo

oO

][Am(x - b)mll < ~ m=O

Oo

~

IIAmllq lib- all

k

~.llx- bll

m = O k=O oo

~

IIA~llqr~

< +c~.

m--O

Since the iterated series in the preceding inequalities converges, we can interchange the order of summation to get ~

m=

m

k

IlAmllq'llb - all

(~-k)

~ "llx -

bk k

II N < + o o

Therefore, if we set oo

Bk - ~

m

( k ) Am(b-a)(m-k)~,

rn--k

then it follows that

IIBkll <

+oo.

Hence, Bk E Ls(kE; F). Thus we have

Am(x - a) m - ~ m--0

Bk(x - b) k

k=0

and the convergence in the last series being uniform for IIx-bll < r - lib-all. This shows that the radius of uniform convergence for ~k~176B k ( x - b) k is at least p - lib- all since r was arbitrary. This completes the proof, m We are ready now to show, as a consequence of the preceding theorem, that a function represented by a power series is a C q function within the circle of convergence.

7.2 P O W E R

SERIES IN F-SPACES

149

T h e o r e m 74 [36] Let F be a q-Banach space. If ~m~=O A m ( x - a) m converges to f ( x ) on B(a, p), where p is the radius of uniform convergence of the series, then f is C~q function on B(a,p) and

ld~ Am = m!

f(a)

(7.37)

for m - O, 1, 2, ...

d

sup{llAmllr

--

Since f ( x ) - ~-~,~--oA m ( x -

rrttl

Em=0 I I A m l l r

P r o o f . Let 0 < r < p. Then (subsection 7.2.3.p.141). Let

p ;

"

m--O,

1,2,

< oo by Corollary 10

}

....

(7.38)

a) m, we have f(a) - Ao and oo

f(x)

-

f(a)

-

-

Al(x

a)

-- Z

Am(x

-

a) m.

m=2

Hence c~

llf (x) - f (a) - A ~ (x - a) ll <_

d

~

m~

m~ I]x - all

m - - 2 /*

p

2 q

dllx-all

"

r~(r~-IIx-all

P

~ )

for x C B(a, r). This proves that f is quasi-differential at b and D f ( a ) A1. Repeating this argument for b E B(a, p) and oo

f (x) - ~

Bk(x - b) k

k=O

for I I x - bll < p - l i b - all, see the preceding theorem, we conclude that is quasi-differentiable at b, and (X)

Df(b) - B1 - ~

mAm(x-

a) m-1

m--1

This proves that

f is quasi differentiable on B(a, p) and Oo

D f(x) - ~ m--1

Oo

m A m ( x - a) m-1 - ~ m--1

D A m ( x - a) m

f

150

CHAPTER

7 QUASI-HOLOMORPHIC

MAPS

for x C B(a, p). Since lira sup]]mA,~[] 1/m - l i m sup[[Aml[ 1/'~, the radius uniform convergence of the power series ~--~,~0r m A m ( x - a) "~-1 is equal to p. By repeating this argument, we conclude that D f is quasi-differentiable on B ( a , p ) and (x)

(x)

d2 f ( x ) -- ~

m ( m - 1)Am(x - a) m-2 = ~

m=2

d2Am(x - a) m

m=2

where the radius of uniform convergence is equal to p, see the definition of d "~f in subsection 6.2.1, p.107. Notice that

A2 - ~.1 d2 f(a). Inductively, B(a, p) and

we conclude that

f

1 Am = mi. dmf(a),

(7.39)

is infinitely quasi differentiable on for m - O, 1, 2, ...

(7.40)

m T h e o r e m 75 [3~] Let f be as in the above Theorem oo

oo

dk f ( x ) -- ~

dkAm( x - a) m -- ~

m=k

and

oo

dAkPm(x - a) - ~

d^k f (x) m--k

where P ~ - f l

C o r o l l a r y 12 Let f

dkAm+k( x - a)m+ k

m=0

(:x)

for x C B ( a , p )

7~. Then

dAkPm+k(x - a)

m=0

m. II

be as in Theorem 7~ with

ldAk~ k~ (.dAmf)(a)

Pro-fire,

ldAm( 1 -- -~. (k + m)' dAk+mf(a)) .

P r o o f . In the notation of the preceding Theorem 74 with have 1 d/~k(d/~ m 1 1 kU f)(a) - dAmPk+,~ = -m! -dAm((k+m)! m

we have

Pm-

dAk+mf(a))"

Am, we

(7.41)

7.3 Q U A S I - A N A L Y T I C M A P S

151

C o r o l l a r y 13 Let f be as in the Theorem 7~ and a C U. Then for any k, m, we have the following relations between Taylor's expansions Tk,d^mLa -- d Am [Tk+m,L~] 9

(7.42)

m

7.3

QUASI-ANALYTIC MAPS

The theory of holomorphic functions based on complex quasi-differentiability is called here Q u a s i - H o l o m o r p h y . It refers to the extension of Cauchy viewpoint. The theory based on power series is considered here as extension of Wierstrass viewpoint and is called Q u a s i - A n a l y t i c i t y . The view points are equivalent for functions for one complex variable. This part is devoted to the theory of quasi-analytic functions based on the wierstrass approach and to study the p r i n c i p l e of quasi-analytic continuation. As we noticed in our study of one variable, that the Weirstrass view point of analytic functions is more general than the Cauchy view point. It unifies both real and complex cases in development. As usual E and F will denote p-normed and q-normed spaces over the field K respectively (0 _< p, q __ 1), and U will denote a nonempty open subset of E. We will assume that F is always complete, i.e. a q-Banach space.

7.3.1

Q u a s i - A n a l y t i c and Q u a s i - H o l o m o r p h i c M a p s

D e f i n i t i o n 48 ( T a y l o r s e r i e s ) W e say that a function f:U~F has a Taylor Series expansion at a point a C U if there exists a power series O0

cO

( x - a) = 119,--'0

( x - a) rrt-- o

from E to F about a such that the power series converges to f uniformly on the ball B (a, r) C U for some r > O. The series, if it exists, is unique by Theorem 72 (subsection 7.2.~, p.1~6).

152

CHAPTER 7 QUASI-HOLOMORPHIC MAPS

It deserves to point out that if f has a Taylor Series expansion (or representation) at a, then the function f is infinitely quasi-differentiable in a neighborhood of a as shown in Theorem 74(subsection7.2.7, p.149). Moreover we have the following relations:

Pm = --m!l~ f (a)

,

Am - -~dm

(a)

for m = 0, 1, 2 ..... Consequently, if f has a Taylor Series expansion at a, then 1

f (x) - ~

~

1

-~..dmf (a)(x - a) - ~

lrt-- O

m--

--~.dmf (a)(x - a) m O

in some neighborhood of a in U. Since the open ball of uniform convergence of the Taylor Series at a and the domain of this function f could be different, we usually write OO

f (x) _ ~

~1. dAmf (a) (x -- a ) - ~

1 --~..dmf (a)(x - a) m

?Tt"-- O

to indicate formally, the relationship between f and its Taylor Series expansion of a. D e f i n i t i o n 49 A function f : U --~ F

is said to be quasi-analytic(or Bayoumi analytic) on U if f has a Taylor Series expansion at each point a E U. If K = (~, a quasi-analytic mapping will be specially called quasiholomorphic(or Bayoumi holomorphic). It is clear that any linear combination of quasi-analytic mappings is quasi-analytic, and hence we have a v e c t o r s p a c e of all q u a s i - a n a l y t i c m a p p i n g s f r o m U t o F . This space will be denoted by QA (U; F ) . In caseK =(Y, we write QH (U; F) instead of QA (U, F ) . For simplicity, we denote QA (U) = QA (U;K) and QH(U) = QH(U,(T). Since QA ( U ; F ) is a vector subspace of Qc$, ( U , F ) , the space of infinitely q u a s i - d i f f e r e n t i a b l e m a p p i n g s from U to F, we have for f C QA (U; F) the following quasi-differentiable mappings.

din f : x e U ~ d ~ f (x) e L~ (mE; F)

(7.43)

d m f " x E U ~ dmf (x) e P (mE, F)

(7.44)

We also have from Ch.6(sec.6.2 ), the following quasi-differentiable operators

7.3 Q U A S I - A N A L Y T I C M A P S

153

rim: f E QA (U; F) ~ d ~ f E QA (U; L~ (mE; F))

~"

f E QA (U; F ) -~ ~ f

E QA (U; P (mE; F))

R e m a r k 18 (Local Property)Quasi-Analyticity is a local property in the following sense: If f E QA ( U ; F ) and V is a nonempty open subset of U; then f i y E QA (V, F) conversely if U - - U V~, where V is an open nonempty subset of aEI

oc

U and f : U --, F is such that for each a E I, fiv~ E QA ( V ~ , F ) , then f E QA(U; F). The following theorem gives a condition on a power series to be a Taylor series to its sum.

T h e o r e m 76 Let ~ Pm ( x - a) be a power series between locally bounded F-spaces whose radius of uniform convergence r is positive and f (x) be its sum for II x I1< r. Then f is quasi-analytic in B (a,r). Proof. This is a restatement of Theorem 74 (subsection 7.2.7, p.149). I 7.3.2

Principle of Quasi-Analytic Continuation

To solve the problem of q u a s i - a n a l y t i c c o n t i n u a t i o n means: " G i v e n a q u a s i - a n a l y t i c m a p f in a connected open set U a n d a connected open set V c o n t a i n i n g U, find a n a n a l y t i c m a p g o n V s u c h t h a t giu = f ". The following theorem guarantees that such a mapping g is u n i q u e if it exists. This is the principle of analytic continuation in its s t r o n g form.

T h e o r e m 77 ( s t r o n g form)[34] (2002) Let E and F be a p-normed and a q-normed spaces respectively, (0 p, q <_ 1). Let U be a connected open subset of E, f E QA (U; F) and a E U; The following are equivalent : (i) d m f (a) - 0 Vm = 0, 1, 2 ..... (ii) f is identically zero in a neighborhood of a. (iii) f is identically zero in U.

154

CHAPTER 7 QUASI-HOLOMORPHIC MAPS

P r o o f . It is obvious that (iii) =~(i). It remains to show that (i) =v(ii)=~(iii) (i)=~(ii). Assume (i) is satisfied. Since f E QA (V; F ) , the Taylor series at a converges to f uniformly on some neighborhood of a. Thus f is zero identically in this neighborhood. (ii) =v (iii): Let W = {x E U; f (x) = 0} Then by the assumption (ii), W is nonempty and open since the Taylor series of f at each x C W represents f in a neighborhood of x. Furthermore W is closed since f is continuous. Thus W - U, since U is connected and W is a closed and open subset of U. This shows that f is identically zero on U. I The following theorem gives the weak form of the principle of quasianalytic continuation. T h e o r e m 78 (weak form)[34](2002) Let U be a connected open set and f,g ~ QA (U) . If f and g are identical on a nonempty open subset of U, then they are identical on U. P r o o f . Let V be a nonempty subset of U and assume that f and g are identical on V. It follows then from the preceding Theorem 77 that f - g, onU. I C o r o l l a r y 14 Let U and V be two connected open subsets of E, and f C QA (U; F ) . Then there is at most one g E QA (V; F) satisfying f - g on UNV. I R e m a r k 19 If f, g E QA (U), where U is a connected open set i n k and if f - g on an infinite set A with an accumulation point, where A C U, then f-gonU. The analogous statement for dim E >_ 2 does not hold. For example, let g -- ~ 2 , f (Xl, x2) -- Xl and g (Xl, X2) -- 0 Then f , g E QA (E) and f - g on { ( x l , x 2 ) ' X l 0}, but f # g. In general, if M is a vector subspace of E which is not dense in E, then we can find a bounded linear functional f satisfying I] f ]I- 1 and f - 0 on M , (see Ch.1, p.12).

7.3.3

Integral D o m a i n QA (U)

If f, g ~ QA (U) then the multiplication

f.g ~ QA (U) as it can be shown from multiplication of power series. Therefore, QA (U) is a ring.

7.3 Q U A S I - A N A L Y T I C

MAPS

155

T h e o r e m 79

QA (U) is an integral domain if and only if U is connected. P r o o f . It is clear that, if QA (U) is an integral domain then U is connected. Assume that U is connected and f , g C QA (U) are such t h a t f.g = O. If f ~ O. Then there exists a nonempty open subset V of U such t h a t f 5r 0 on V. Thus g must be zero on V. Since g is analytic, g must be zero identically on U by Theorem 77 on the strong form of analytic continuation. In what follows we give an organogram (schedule) which shows the generalized theorems in the book obtained by the author between 1979 (the year he has obtained his Ph.D.) and 2003 (the year of writing the book).

GenralizedTheoremsinourbook (1979.2003) ,,

CompleAnal x ysisi ,I

I

,

Schwatz Levip~oblem 'e~= i Symmetric Steinhaust Theorem Theorem I _L_

!

~vex ~es [/I | Separable T.V.S

,,

I

Boundingi Sets t ! ! I

t

!

I

I

,-

I

,

,

.os i1..msI .anva.ue,,

Lagrange IFund, an oe~,llF,~o,,s a, I ,n,eg~,i' i o-n~ MeanValue nterrTled~I Theorems ofCaculus Theorems[ V~ Thee

ied I We dy'

_

hn' lach

.

.

.

.

1

Realani l Cmplex ! =t~!

bdedSpaces~

....

....

............ 1.

.

.

I .

!

....

Krien . Milman Milman 1 Theorem

J

.

!

l e,s II; , |s, ' B~

,

FunctionalAnalysis

!

,,

]

.

1 ........

F-Spaces i S:~:s~ ! .actor ! Spaces I

Chapter 8

NEW VERSIONS THEOREMS

OF MAIN

This chapter is devoted to v e c t o r - v a l u e d continuous or quasi h o l o m o r phic m a p p i n g s . The reader should be familiar with the two equivalent ways of finding a holomorphic mapping, the one based on complex q u a s i d i f f e r e n t i a b i l i t y , and the other based on convergent p o w e r series. We point out here that the concept of vector-valued holomorphic mappings arises naturally, as in the study of Banach algebras as well as in some other contexts. We have already enlarged the classical definition of holomorphic mappings from complex to vector-valued ones in locally bounded F-spaces. We use some of the new concepts introduced here, as the concept of quasi-differentiability, to present more generalized fundamental theorems in complex and functional analysis such as : F u n d a m e n t a l Theorem of Calculus, Integral M e a n - V a l u e Theorem, and B o l z a n o ' s I n t e r m e d i a t e - V a l u e Theorem.

8.1

FUNDAMENTAL CALCULUS

THEOREM

OF

In this part we extend one of the most celebrated theorems of mathematical analysis, the Fundamental Theorem of Calculus~ to vector-valued functions of p-Banach spaces using the concept of quasi-differentiability. Hence, we have established a new general form of the fundamental theorem 157

158

CHAPTER

8 NEW

VERSIONS

OF MAIN

THEOREMS

of calculus suitable for applying to the integral forms of continuous vector -valued maps of locally bounded F-spaces, not necessarily locally convex. In fact, using the curvilinear integrals, the theory of vector-valued holomorphic mappings, like that of complex-valued holomorphic mappings can be developed most efficiently. We first need to generalize the Riemann Integral to vector- valued functions. 8.1.1

Riemann

Integration

on [a,~]

D e f i n i t i o n 50 ( S t e p M a p s ) . Let I = [a, ~] be a closed interval in ~, and E be a p-Banach space (0 < p < 1) over K . A s t e p m a p

f.

E

is a map for which there exists a partition P. aand elements Y l , . . . , Y n E E

ao <

0~1 < . . .

~

~n

-- ~,

such that

f-

n ~ yiX(OLi_l,OG)

(8.1)

1

where >CA is the characteristic function on the set [a,/3]. Let S(I, E) be the v e c t o r s p a c e of all s t e p m a p s from I to E. Then S ( I , E ) is a v e c t o r s u b s p a c e of the vector space B ( I , E ) of all bounded maps from I to E. Having the p-sup norm, [[fl[ - sup

{[f(x)[[

xEI

on the space B ( I , E ) , B ( I , E) becomes a p - B a n a c h s p a c e . D e f i n i t i o n 51 ( R e g u l a r F u n c t i o n s ) L e t R(I, E) - S(I, E) be the closure of S ( I , E ) in B ( I , E ) . A map in R ( I , E ) is said to be regular. Since a continuous map f " I --~ E is uniformly continuous, we can conclude that

C(I, E) C R(I, E)

(s.2)

8.1 F U N D A M E N T A L

THEOREM

OF CALCULUS

159

Definition 52 (Integra 0 We define the integral of a step map f " [~, ~] --, E,

n

f - ~

yi)~.(C~i__l,(~i ) 1

by

(8.3)

f - ~ ( o ~ i - oLi-1)Yi. 1

It is easy to see that the integral

J Z.S(I,E)-~E is linear and

II

E

fll 1/p <_ IIfll(~-

~).

(8.4)

Note that IIT(x)lll~q < IITll.llxll~p for any T C L ( E , F ) being E , F p-normed and q-normed spaces respectively, see Chapter 1,( section 1.1, p.5) Hence f ~ f is continuous. In fact, 1/p

- II

n

n

1

1

~-~(~- ~-l)y~ll 1/" <__~--~ I~-

_< c m a x

l
lly~llI/p < cllfIl~/p

= (9 - ~). So ,,,~ can ~zt~nd f [

~-11 .lly~ll1/~

to ~ f~,~t~on on th~ clo~,~ R ( I , E ) , of S(• E), by the linear extension,, Theorem 2, and Theorem 3 (see section 1.1, p.5, 6). H ~ ~ if f ~ R(I, E), ~ c ~ d ~ ~

f

~f

and call it the integral o f f. D e f i n i t i o n 53 If a < c < d <_ 19, we define

idf

-- /d~ f

160

CHAPTER

8 NEW VERSIONS

OF M A I N T H E O R E M S

Then for any three points c, d, e in any order, f C R ( I , E), we can prove easily that

8.1.2

lying in I = [c~,/3], and

Curvilinear Integrals

D e f i n i t i o n 54 ( p a t h ) . A p a t h in the complex plane~T is a continuous mapping =

(8.5)

of closed interval I of ~ (not reduced to a point) into (~, which is piecewise continuously differentiable. A path ~ : I ~ (~ is said to be contained in an open set U C (~ if ~/(I) C U, where U may not be convex. The point ~(c~) is called the initial point of the path ~/ and 7(fl) the terminal point. If ~(tl) ~- ~(t2) unless tl = t2 or tl and t2 are the same as c~ and ~, ~ i s said to be s i m p l e . If ~ : I ~ (~ is a path and ~/(~) = ~/(~), then ~/ is said to be closed. If a point a lies in the image of'~, we often say that a is on the path ~.

Let 7 : I ~ ~T be a path and let f :~(I) ~ E be continuous, where E is a complex p-normed space. Then the complex composed function t e I --~ f ( 7 ( t ) ) 7 ' ( t ) e E

is piecewise-continuous and hence its integral exists on I by the definition of integral. E x a m p l e 28 As a particular case is the Arc segment A b - { A1/Pa + (1 joining a and b in(~.

-/~)I/pb);

1 > A > 0}

(8.6)

Let us call it here a p a t h o f t y p e Ab.A

D e f i n i t i o n 55 (Curvilinear Integral )The curvilinear integral of f over the path ~/ is defined by ]~ f (z)dz - / ~ ~ f ( 7 ( t ) ) ~ / ( t ) d t . Note that the curvilinear integral above is an element of E.

(8.7)

8.1 F U N D A M E N T A L

THEOREM

OF CALCULUS

161

The following theorem will show t h a t T(f~ f ( z ) d z ) - f~ T ( f ( z ) ) d z for every T C L(E, F), t h a t is the c o m m u t a t i v i t y between the curvilinear integral and continuous linear maps.

T h e o r e m 80 [22](1999) Let E and F be p-Banach and q-Banach spaces respectively (0 < p, q <_ 1) and T E L ( E , F ) . If ~ : I = [a,/3]--+(T is a path and f : "y(I) -+ E is continuous, then

Proof. P u t

gn = fn O')' for some f~ C S ( I , E ) , where f = l i m f~. Let g = lim gn be the uniform limit of a sequence (gn) of step maps. T h e n each T o gn is a step m a p from I into F. So the sequence (T o gn) converges to T o g and hence Tog C R(I, F). Using the definition of the integral for a step m a p g~ we have

Tgn

T(]a gn).

i.e. f~ T f n - T(f~ fn). Since the integral is continuous, taking the limit we will get the required identity. This completes the proof, m

8.1.3

Fundamental

Theorem

of Calculus

T h e o r e m 81 ( F u n d a m e n t a l T h e o r e m o f Calculus)[g2] (1999) Let 9/" I - [0, 1] --, ~ be a path of type A b, i.e. "y(I)=A b, a, bc4T and let f: 7(Z)--+F br ~onti~o~,~ at a p o ~ t "r(t) of'r(1), wnr162 F i~ a p-Sa~a~h ~p~r the map

H(t) -

f(z)dz J a

Then

f(-y(A))-y'(A)dA --0

is quasi-differentiable at t and H'(t) - f(tl/pb + (1 - t)l/Pa). [ (tl/p-lb - (1 - t)l/p-la)] P i.e.

H'(t)-

f(9/(t)) "y'(t).

(8.9)

V

V

i_~

m

9

~

~"

I~

9

~

~ ~

o

m

~

r~

..

.

I

~

o

c::~

~.

.,"~

~"~"

9~

~

~

.

I

I

I

I

I

~

~'~

-

.

9

1~o

i,--~ o

~-"

~

~ T"

o .

Oo ~

~

~

Oo

I

~

~ ~ ' 1 ~ -~

"J

~

o~

II

-~

~

~ ~

~

t::;I ~

~ ~

~

~

I

I

~

+

"<..

~

IA

I ~.,

~

+

+

-.2

II +

q.

-R

I

--2

II

I

4-

4-

O

rh

IA

IA

o

..,o

I

k~

II

~<

d~

O O

t~

= t~ 9

9

9

t~

L'~

8.2 B O L Z A N O ' S

INTERMEDIATE

THEOREM

163

and V(I)" [a,/3] --~ (T be a path of type A b, a, b e (T, I - [0, 1] C ~, i.e.

V(I) - {x C (T; x - tl/pb + (1 - t)l/Pa,

t E [0, 1]}.

Set H(t) - - / ~ f (z)dz - /: t

f(7(A))v'(A)dA,

?(t) =

e ?(z) e r

--OL

Then using the preceding theorem, we have H' -- f ( v ( t ) ) v ' ( t ) ,

v(t) -- x e r

= f(tl/Pb + (1 - t ) l / p a ) . ( l t l / p - l b 1(1 - t ) l / p - l a ) P P -- (tl/pb + (1 - t)l/pa, 0 , . . . ) . p1( t l / p - l b (1 - t ) l / p - l a ) which is an element of 1p. Now if p - 1, f ( x ) - (x, 0 , . . . ) e 11/2, then for x -- x(t) e 7 ( I ) ,

H'(x) -

(t2b + ( 1 - t) 2a, 0 , . . . ) . 2 ( ( t b - (1 - t)a))

i.e. H ' ( x ) = ( x x ' , 0 , . . . ) .

8.2

BOLZANO'S

INTERMEDIATE

THEOREM

In this part we extend the intermediate-value theorem (or Bolzano's theorem) to locally bounded F-spaces which are not necessarily locally convex. In its simplest form it says t h a t : A r e a l - v a l u e d c o n t i n u o u s m a p f o n a c l o s e d i n t e r v a l [a, b] s u c h t h a t f(a) a n d f(b) h a v e d i f f e r e n t s i g n s , h a s a z e r o in ( a , b ) .

8.2.1

Finite Dimensional

Spaces

Shih [192] extends the intermediate value theorem to the complex plane ~T. He first remarked that Bolzano's theorem can reformulated as follows : Assume a < 0 < b and f(a) < 0 < f(b) and consequently the condition f (a). f (b) < 0 becomes

x. f (x) > O,

for x E OU

(8.12)

164

CHAPTER

8 NEW

VERSIONS

OF MAIN

THEOREMS

where OU denotes the b o u n d a r y of U - (a, b). T h e n f has at least one zero in U. He then proved t h a t " I f U is a b o u n d e d d o m a i n o f (T c o n t a i n i n g t h e o r i g i n , a n d

(s.13) is a n a l y t i c in U a n d c o n t i n u o u s o n Re z - f ( z ) then result. If closed

> 0,

U- and such that for z e / ) U

(8.14)

f h a s o n e z e r o in U. He has used Rouche's theorem to prove that It says that : f and g are each functions which are analytic inside and on a simple contour C and if the strict inequality

If(z)-

g(z)]<1 f ( z ) l

for each z e C

then f and g must have the same total number of zeros (counting multiplicity ) inside C.

8.2.2

Degree

Theory

Shih [193] extends then his above result to analytic functions in d~ n. His approach is an application of degree theory. Therefore it deserves and fruitful to discuss the degree theory first. We point out here t h a t all the results written here in this section have been obtained from the book by Smart [200] of fixed point theorems. Consider a simple closed curve C in ~2 and a m a p p i n g

g . C---~ S 1

Figure 3" A m a p p i n g

of d e g r e e

1 of C o n t o

S1

8.2 B O L Z A N O ' S

INTERMEDIATE

THEOREM

165

In simple cases the following facts are clear from the above diagram. L e m m a 15 Let g : C --~ S 1 have the above figure. Then (1)Most points of S1 are covered a finite number of times nl in the pointwise sense, and a finite number of times n2 in the negative sense. (2) The number n - - n l --n2

is the same at all points where nl

and n2 exist.

g(x)

-

f(x)

II f(x) II

for a non-zero vector field f on C, then the rotation of g about C exactly n.

is

D e f i n i t i o n 56 The number n given in the above lemma is called the degree o f the m a p p i n g f.

In what follows we mention the main properties of the degree( for mapping of sn). L e m m a 16 ( P r o p e r t i e s o f Degree o f M a p s ) (1)The degree is an integer.

(2)The degree is unaltered by continuous deformation(homotopy) of the mapping. deg ( S T ) degI-

1 ,

deg S . deg T

deg(-I)-

(--I)n+l

The following lemma discusses the degree of a mapping f without fixed point. Lemma (1)If

17 ( F u n c t i o n s w i t h o u t f i x e d p o i n t s ) f .S~-,

S~

and has no fixed point, then deg f - ( - 1)n+l. (2)If ( - f )

has no fixed point, then deg f -

1.

166

CHAPTER

8 NEW VERSIONS

OF MAIN THEOREMS

T h e D e g r e e for m a p p i n g s of o p e n sets : Consider the open subset M of a p-Banach space F and a continuous m a p p i n g f of M into F i.e.

f:

M---*F

First ( b o u n d e d o p e n sets) :For the m o m e n t assume that M bounded and t h a t F = ~n. We illustrate the particular case in ~2.

is

i/~ /,.,- .//

f,' ,',,'/

If"tl 11

t

/

~ ~ \

Figure 4" A m a p p i n g of M b y f . region is s h o w n .

T h e v a l u e of d e g r e e in e a c h

We observe in this particular case the following : The images of OM i.e. f ( O M ) divides F into (connected) regions u~; in each region the algebraic number of times that a point is covered is constant (except of relatively few exceptional points where the image is folded ). This is true for all simplicial mappings, or differentiable mappings. In b o t h these cases the exceptional points in a region ui form a set of lower dimension, hence of measure zero.

D e f i n i t i o n 57 Let f of M into ~n i.e.

be a simplicial mapping (or a differentiable mapping) f :M~

~n.

The degree of f (with respect to M ) at a point x E ~ - f ( OM ) is the algebraic number of times (almost all) points are covered, in the region ui containing x. This integer is written

8.2 B O L Z A N O ' S

INTERMEDIATE

THEOREM

167

deg(f , M , x ) or deg(f, M, ui). The degree of an arbitrary, continuous mapping is defined by approximating by simplicial mapping (or differentiable mapping). S e c o n d ( u n b o u n d e d o p e n sets): If M is unbounded open set we must assume that f ( x ) - x is bounded on M; in other words, ( f - I) is compact. If F is infinite dimensional, we must approximate f by finite dimensional mappings; this requires some compactness assumptions. The usual theory defines the degree for mappings f such that ( f - I) is compact. The main properties of the degree are given in what follows 9 L e m m a 18 ( M a i n properties o f the degree) (1)deg ( f , M, x) is an integer, defined if x E f (OM). then deg ( f , M , x ) - i if x E M, and deg ( f , M, x) - O (2)If f - I if x ~ M .

(s) deg (f, UMi, x) -

deg ( f , M~, x) i

if Mi are disjoint regions and both sides of the equation are defined. (4)If deg (f, M , x ) r O, then x e f ( M ) . (5) ( H o m o t o p y ) : If f (x, t) is a continuous function compact mapping of [0, 1] x M into F, that is f ( x , t ) ' [ O , 1] • M --~ F and if x-

f (x,t) ~ y for all x C OM

then deg(I - f(O, .), M, y) - deg(I - f(1, .), M, y) [We say here f(O, .) is fp-homotopic to f ( 1 , . ) ] . (6) (Multiplication)" If f (OM) divides F into regions u~ and if g( f (OM) ) divides F into regions wi then deg(gf , M, wi) -

deg(f, M, ui). deg(9, ui,, wi). i

168

CHAPTER

R e m a r k 20 If f

8 NEW

VERSIONS

has no zeros in M

deg(f, M, 0) - ~

OF MAIN

except

Xl , .., Xn,

THEOREMS then

deg(f, Mi, 0) Also if f is a homeomorphism

where the Mi is a small neighborhood of xi. then it has degree •

In fact, Shih[193] has obtained the following extension of Bolzano's theorem to (T n. T h e o r e m 82 [193] ( 1 9 8 2 ) Let ~

be a bounded domain in (T n containing the origin. Let f : f2- --. 6~ n

be analytic in ~

and continuous

in ~ - , and assume that

Re z - f ( z ) > 0 , Then

f

for z E 0 f t

has exactly one zero in ~.

The proof depends on the following three lemmas. For the first lemma see the book of Bochner & Martin[48] L e m m a 19 Let

f = (fl,,..-, be analytic in a domain ~ of (T n are related by the formula

fn) : ~

~

(T n

Then the real and complex Jacobians

O(uj,_vj) ) _[ det(0fJ 2 det(o(Xk,Yk) ' ~zk) [ , where zk = xk + i yk

and

(8.15)

(8.16)

f j = uj + i vj.

R e m a r k 21 Notice that f :(T n __,(T n can be considered as a map f :~2n

~

j~2n.

Using degree theory, we see that the local degree of a complex analytic function at any preimage of a regular value is always +1. Thus the number of points in the preimage of a regular value of f is always exactly deg f.

8.2 B O L Z A N O ' S

INTERMEDIATE

THEOREM

169

Lemma20 Let A be an ( n x n) complex matrix and d e t A = for sufficiently small e > 0, det(A + eI) r 0

0. Then

(8.17)

where I is the identity matrix.

Proof. By Schur's theorem, there is a non singular matrix P such that p - l A p is triangular and the leading diagonal elements of p - l A p are eigenvalues of A. Since det A = 0, for small enough e > 0 we have

det(A + eI) - d e t ( p - l ( A + e I ) P ) - d e t ( p - 1 A p

+ eI) 7s O.

(8.18)

m

Lemma

21 Let U be an open bounded set in(F ~ and let f, g : U - --, (~ n

(8.19)

be two continuous maps. Let w C ~T n and

(8.20)

w ~ f ( O U ) u g(OU). A s s u m e further that e satisfies

0 < e < min

{llf(z)- ~11;

c ou}.

(8.21)

all z C OU,

(8.22)

z

If I l f ( z ) - g(z)ll < e for then

deg(f, U, w) = deg(g, U, w).

(8.23)

P r o o f . Define the homotopy H : U - x [0, 1] ---, Cn

(8.24)

by H(z,t) :=(1-t)f(z)+tg(z),

forzEU-

andtE

[0,1].

(8.25)

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THEOREMS

By assumption, it is easy to see that H ( z , t ) ~ w for z C OU and t E [0, 1]. By homotopy invariance theorem the result follows, m Proof

of T h e o r e m 82, Define the homotopy H " ~ - • [0, 1] --~ 07 n

by H(z,t)'-(1-t)z+tf(z),

forze~-

andte[0,1].

By assumption, H ( z , 0) # 0 and H ( z , 1) ~= 0, for z e / ) ~ and Re z - . H ( z , t ) > Re z-.(1 - t)z > O,

(s.26)

for z E Ogt and t E (0, 1). By the homotopy invariance theorem, deg(f, ~, 0) - deg(H(z, t), fl, 0) - deg(I, ~, 0) - 1.

(8.27)

Notice that f-~(0) is a compact subvariety of ~t. We claim that f - l ( 0 ) is finite. For, let M be a component subset of f - l ( 0 ) . Then M is a compact subvariety of ~. Since each projection ~ j ( z l , .., .Zn) - zj is analytic on M, ~j is constant by applying the result of Gunning&Rossi[[ll7],pl06]. So M is a singleton; thus f - l ( 0 ) is discrete and hence finite. Now, let ~1,---, ~k denote the zeros of f. Let Aj be a neighborhood of ~j such that the closed sets A~- are pairwise disjoint and Aj C ~. Let

So K is a closed subset of ~t- which does not contain a zero of f. By the excision and additivity properties of the degree, Schwartz [[188],p.86] we have deg(f, ~, 0) - deg(f, ~ - K, 0) - ~

deg(f, Aj,0).

(s.29)

J We claim that deg(f, Aj, 0) _> 1,

for each j.

(8.30)

If deg(J G (f)) =fi 0, where J~j (f) denotes the complex Jacobian matrix of f at ~j, then deg(f, Aj,0) - 1 by Lemma 19. Suppose now that d e t ( J ~ (f)) - 0.

(8.31)

8.2 B O L Z A N O ' S I N T E R M E D I A T E

THEOREM

171

Then by Lemma 20, det(JCj (g)) # 0 when

g(z)-e(z-~j)+f(z),

and e > 0

is small enough.

Consequently it follows from Lemmas 19 and 21 that deg(f, Aj, 0) = deg(g, Aj, 0) _> 1.

(8.32)

By (8.27) and(8.29), the theorem is established, m 8.2.3

Infinite Dimensional

Spaces

In infinite dimensional spaces Wlodraczyk[208], obtained the following extension of Bolzano's intermediate-value theorem for h o l o m o r p h i c m a p s in certain c o m p l e x B a n a c h spaces. B a n a c h Spaces of C o n t i n u o u s Linear M a p s

Let E , F be complex Hilbert spaces; let L(E; F), as before, denotes the Banach space of all continuous linear maps A : E ~ F with the mapping 12orm

IIAII-- sup IIA(x)ll.

(8.33)

L1 (E, F) C L(E; F)

(8.34)

Ilxll_
Let

be a closed linear subspace of L(E; F). T h e o r e m 83 [208](1991)

Assume ft C LI (E, F) is a bounded domain such that 0 C f~ and let f : A e ~ - ~ f (A) e LI (E, F)

(8.35)

Re {A* f ( A ) } > O, for A EOFt

(8.36)

be continuous in ~ - . (1) /f

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OF MAIN

THEOREMS

and ( I - f)(f~) is contained in a compact subset of L(E; F), where A'is the conjugate of A, then f has at least one zero in ft, i.e. f (A) - 0 for at least one A E ft.

(2) If additionally f is holomorphic in f~ and D r ( A ) inverse for A E ft, then f has exactly one zero in f~. Proof.

has a bounded

To prove (1), let

ht(A) - A -

f(t,A),

where f (t, A) - t [A - f(A)] ;

i.e. ht(A) - ( 1 - t)A + t f ( A ) ,

AEft

and0_
(s.37)

Then

ho(A) - A ~ O, hi (A) - f (A) r O,

for A E 0ft.

(8.38)

Moreover,

ht(A)r

for A E 0 f t

and 0 < t < l ,

(8.39)

because

Re {A*ht(A)} -

Re {(1 - t)A*A + tA* f ( A ) }

> (1 - t) Re {A'A} > 0. Since 0 E ft, we apply the homotopy property to ht(A), see L e m m a 18 (5) or Smart[[200], 10.3.7], to obtain d e g ( I - f(0, .), ft, 0) - deg(I - f(1, .), ft, 0) - 1 or equivalently deg(f, ft, 0) - deg(I, ft, 0) - 1. Hence it follows that

f-l(0)

(8.40)

is nonempty.

(2) Since

A ---, D f ( A )

(8.41)

8.2 B O L Z A N O ' S I N T E R M E D I A T E

THEOREM

173

is a holomorphic map of ft into L(L1; L1), see Nachbin[[?l,props.3,p.29], by the inverse map theorem for complex Banach spaces, see Rudin [187], the map f is locally biholomorphic in ft, i.e. for each A C f~, there exists a neighborhood UA of A in ft such that f ( U A ) = VA,

is open in L1,

(8.42)

f - 1 exists and is holomorphic in VA. We now prove that f - l ( 0 ) contains one m a p p i n g : Let us first show that f - 1 (0) is finite. Towards a contradiction, let Ak E f - l ( 0 ) , k = 1,2,... We have h(Ak) = Ak, k = 1,2,...,where h = I - f . Since h is compact and continuous in f~-, there exists a subsequence of (Ak), say (Ak), and A C a - such that

IIA-AkII~0

as k ~ c ~

and

h(A)=

A.

(8.43)

This yields f ( A ) = O and

Re{A*f(A)}=0.

9

(8.44)

Consequently, A C ft. But f is biholomorphic in UA and At: c UA, for k sufficiently large. This yields a contradiction. Thus f - l ( 0 ) is finite. If (8.45)

f - ~ (0) - {A1, ..., An} n

deg(f, ft, 0) - deg(I, f t / U , 0) - ~

deg(f, Uk, 0),

(8.46)

k=l

where Uk are all small neighbourhoods of Ak such that the sets Uk are pairwise disjoint, U~- C ft and U - ft-/U'~= 1Uk. Further we have f - I - h , h has isolated fixed points in Ft; h is compact and L1 (E, F ) is complex. Thus the multiplicity of each eigenvalue of Dh(Ak) = D(I - f)(Ak),

k = 1,2,...,n

is even and by Kransnoseliski [lemma 4.1 and theorem 4.7], this yields deg(f, Uk, 0) = deg(I - h, Uk, 0) = 1, From (8.40) and proof, m

( 8 . 4 7 ) w e deduce that

k = 1, 2, ..., n.

(8.47)

n = 1, which completes the

174

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THEOREMS

C o n t i n u o u s L i n e a r M a p s o n N o n locally c o n v e x S p a c e s Regarding the non locally convex spaces we have obtained the following result for the space lp, ( 0 < p < 1). The proof is similar to that given in the preceding theorem, but making use of our recent results here. Let

Ll(lp;K) C L(lp;K) ~ l ~ be a closed linear subspace of lo~. T h e o r e m 84 [30](2002)

Assume ft c L1 is a bounded domain such that 0 E f~ and let f " Ft- ~ Ll(lp;K)

(8.48)

Re {A* f ( A ) } > 0, for A E 0gt,

(8.49)

be continuous in f t - . (1) If

and ( I - f ) ( ~ ) is contained in a compact subset of L(lp;K), then f has at least one zero in ~, that is f ( A ) - 0 for at least one A E ~. (2) If additionally f is quasi-holomorphic in ~ and D r ( A ) bounded invese for A E ~, then f has exactly one zero in ~. Proof.

(1) Let

ht(A) - A -

f (t, A),

i.e. h t ( A ) - ( 1 - t ) A + t f ( A ) ,

where f (t, A) - t [A - f(A)];

AE~t

and 0 _ < t < l .

Then

no(A ) -- n # 0, h i ( A ) - f ( A ) # 0,

for A E Oft.

M or eover,

ht(A) 7~0, since

has a

for A E 0 f t

and 0 < t <

1,

(8.50)

8.2 B O L Z A N O ' S

INTERMEDIATE

THEOREM

175

Re {A*ht(A)} = Re {(1 - t)A*A + tA*f(A)} > (1 - t) Re {A'A} > 0. Since 0 E gt, we apply the homotopy property to [[200],p.811, to have

ht (A), see Smart

deg(I - f(0, .), ~, 0) = d e g ( I - f(1, .), a , 0) = 1

(8.51)

or equivalently deg(f, a , 0) = deg(I, a , 0) = 1. Hence it follows that

f-l(0)

(8.52)

is nonempty.

(2) Since

A ~ Df(A) is a quasi-holomorphic map of ~ into L(lp;K), by the inverse map theorem for complex p-Banach spaces, the map f is locally quasi-biholomorphic in ~t, that is, f and f - 1 are quasi-holomorphic, see Theorem 46(Ch.4, subsection 4.1.5, p.85). i.e. for each A E ~, there exists a neighborhood UA of A in ~ such that

f(Un) = VA is open in Ll(lp;K), f - 1 exists and is quasi-holomorphic in

(8.53)

VA.

We now prove that f - l ( 0 ) contains one mapping : Let us first show that f - l ( 0 ) is finite. Towards a contradiction, let Ak E f - ~ ( 0 ) , for all k - 1,2,...We have h(Ak) - Ak, k - 1,2, ...,where

h=I-f. Since h is compact and continuous in ~ - , there exists a subsequence of (Ak), say (Ak), and A e ~t- such t h a t

ilA-Akrl 0

as k ~ ~

and

h(A)-A

(8.54)

This yields

f(A) = 0

and

R e { A * f ( A ) } = 0.

(8.55)

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THEOREMS

Consequently, A E ft. But f is biholomorphic in UA and Ak E UA for sufficiently large k. This yields a contradiction. Thus f - l ( 0 ) is finite. If f - 1 (0) -- {A1, ..., A n } ,

(8.56)

then by Smart[J200], propertiesl0.3.h&10.3.6, p.80)], we have n

deg(f, t2, 0) - deg(I, t2/U, 0) - ~

deg(f, Uk, 0)

(8.57)

k=l

where Uk is a small neighborhood of Ak such that the sets Uk are pairwise disjoint, U~- C f~ and U - t2-/(U~=lUk ). Further we have f - I - h , h has isolated fixed points in t2; h is compact and L I ( E , F ) is complex. Thus the multiplicity of each eigenvalue of

D(h)(Ak) - D ( I - f ) ( A k ) ,

k - 1, 2, ..., n

(8.58)

is even by Kransnoseliski[[164]Lemma 4.1 and Theorem 4.7], this yields

deg(f, Uk, 0) - deg(I - h, Uk, 0) - 1 By (8.51), (8.57) and the proof. I

(8.59)

k - 1, 2, ..., n.

we deduce that

(s.59)

n - 1, which completes

R e m a r k 22 Every separable locally bounded space Ep is isomorphic to a quotient space of lp, (0 < p <_ 1), i.e.

Ep ~_ l p / M for some closed subspace M of lp. Therefore the above theorem can be applied to Ep whenever its dual separates the points of Ep. We consider a bounded domain f~ of a closed subspace LI(Ep;K) C L(Ep;K).

8.3

INTEGRAL

MEAN-VALUE

THEOREM

One of the most important theorems of mathematical analysis is the MeanValue Theorem for definite integrals. In the simple case it stats that, if f is a continuous function on an interval [a, b], a, b Eli,, then its integral satisfies

ffa b f dx - f (c).(b - a)

8.3 I N T E G R A L

MEAN-VALUE

THEOREM

177

for at least one c C (a, b). In this part we establish new general forms of the I n t e g r a l M e a n V a l u e T h e o r e m o f v e c t o r - v a l u e d f u n c t i o n s , by using the concept of curvilinear integral over the arc segment Aba in different spaces. R e a l - V a l u e d Functions"

The following theorem extends the classical Mean-Value Theorem for integrals to curvilinear integrals of a real-valued functions on a path 7(I) of type A~ C (F ~_~2 i.e.on (1 -- t)l/pa + tl/pb; 0 < t <_ 1} Theorem

85 (Generalized Mean-Value Theorem for Integrals in

~2) [31] (2000) Let f " 7(I) ---, 1~ be a continuous real-valued function on a path 7 in (F of the type Abe, where I - [0, 1]. Then, there is zo C 7 such that

/•

f ( z ) i dz I -

j['oI

f ( 7 ( ) ~ ) ) l T ' ( ~ ) i d ) ~ - f(zo). I

to/P-lb-

(1 -

p

to)l/p-la

I.

(8.60) In particular, if p - 1, and 7(I) - [a, b], we obtain the classical mean-value theorem, thatis,

i

b f (z)dz

f (zo) (b - a).

P r o o f . Since f 0 7 is continuous, f o 7 C R ( I , ~ )

and the curvilinear integral

~ f (z) l dz [ exists. Let M-

max {f(7(t)). I 7 ' ( t ) l }

tel

rn - min { f(7(t)). I 7'(t) I}-

tCI

Then

rn <_ f(7(t)). I 7 ' ( t ) I _< M

(8.61)

Hence if J(t) - f(7(t)). I 7'(t) I, then by the intermediate value theorem for the function f ( 7 ( t ) ) l T ' ( t ) l o n [0, 1], there exists to C [0, 1] such that

J(t)- f(7(t0)). 17'(t0) I

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THEOREMS

Consequently,

j/0

1 f(~'(t)).

"/(t) l d t -

f('y(to)). I 3/(to) I .(1 - O )

= f[tlo/Pb + (1 - to)l/Pa].

I t~

(1 - to)l/P-la I P

for some zo - to/Pb + (1 - to)a E ~/(I) - A~. Now if p - 1 we get 7(I) - [ a , b], and we have b

fa

f ( z ) d z - f(tob + (1 - to)a). [b - a]

=

f(zo)(b

-

(8.62)

a)

for some zo - tob + ( 1 - to)a E I to obtain on the line segment [a, b] the classical mean-value theorem for definite integrals. II Complex-Valued Theorem

Functions

9

86 ( M e a n - Value T h e o r e m f o r I n t e g r a l s in r

(2000)

Let : " 7(I) --r be a continuous vector-valued function on a path 7(I) of the type Ab~ in a bounded domain U C (T, where I - [0, 1]. Assume II f fl is bounded on -y(I). Then, there is zo E "~(I) such that

f llf(z)dzll-fo

1

1-1

1-1

= IIf(zo)ll.II tg

b-(1

p -to)~

all .

P r o o f . Similar to that given for the preceding theorem. II

Remark

23 The above theorem can be extended to a vector-valued function

f " 7(I) -~ E, from a path of arc type ?(I)

to a p-Banach space E.

Chapter 9

BOUNDING AND WEAKLY- B O UND IN G SETS A subset A of a topological vector space E is called b o u n d i n g if it is bounded for all entire function f on A. That is, bounded for all continuous and Gdteaux-analytic function.

If E --(~n every entire function f on (~n i8 bounded on every bounded subset A

of E. T h a t is e v e r y b o u n d e d set i n (T~ is bounding.

If E is an infinite-dimensional topological vector spaces, Dineen [86] showed that this is not the case by proving that : the closed bounding subsets of separable or reflexive Banach spaces are compact, and of course a bounded set in E need not be compact. T h a t is b o u n d e d s e t s are n o t necessarily bounding. An example of a bounding subset which is not compact is the set A= {ej;j E N} of the unit vectors in the Banach space l~, see Dineen [87]. Josefson [122] extended this result by showing that a subset of co is bounding as a subset of l ~ if it is bounded. He also proved in [122] that: the bounding subsets of any infinite-dimensional locally convex space are nowhere dense. Moreover he proved that the bounding subsets and weakly-bounding subsets in lcr are the same. Here a set A in a topological vector space E, with rich dual space E', is w e a k l y - b o u n d i n g if:

179

180CHAPTER

9 BOUNDING

AND WEAKLY-BOUNDING

SETS

Cn(X) ~ O, x C E, implies suPn I Cn I ~ O, for every sequence (r E'. The weakly bounding sets are sometimes called l i m i t e d sets. By this he gives a negative answer to the question 9 D o e s there exist an i n f i n i t e - d i m e n s i o n a l B a n a c h space s u c h t h a t its u n i t ball is b o u n d i n g or e v e n w e a k l y - b o u n d i n g ? In this chapter we study and discuss in details the above problems in different classes of topological vector spaces E without having any type of convexity condition.

in

However, the i m p o r t a n t and fundamental results in the theory of holomorphy so far achieved by us ( in brief ) is as follows:

I-The concepts of bounding and weakly-bounding subsets are n o t the s a m e , a n d a n e w class o f bounding sets has already appeared. II-There are three d i f f e r e n t classes o f h o l o m o r p h i c f u n c t i o n s , contained in each other properly, which are found and which are same in locally convex spaces. I I I - M a n y properties of bounding and weakly-bounding subsets have been studied. We have given the relations between t h e m , a n d the radius o f c o n v e r g e n c e o f h o l o m o r p h i c f u n c t i o n s a n d the L e v i problem. This study in non locally convex spaces shows a big d i f f e r e n c e and f a r f r o m that for locally convex cases. Consequently one of the points which has been high-lighted is the following" "The t h e o r y o f h o l o m o r p h i c f u n c t i o n s in locally c o n v e x and n o n locally c o n v e x spaces are n o t the s a m e . "

9.1

BOUNDING

SETS

This part is devoted to the study of the bounding sets in different topological vector spaces which are not necessarily locally convex E. They are bounded by all holomophic functions on E. By a holomophic function we mean here c o n t i n u o u s a n d G ~ t e a u x a n a l y t i c . We also study the weaklybounding subsets of E. T h e y are bounded by certain type of holomophic functions. It is shown t h a t they are different from the bounding sets, cont r a r y to the locally convex cases. The study shows that: h o l o m o r p h i c p r o p e r t i e s and non locally convex spaces are different.

of l o c a l l y c o n v e x

In fact, the study of bounding sets is of a great interest. It makes a progress wherever we study it. T h e y arise in several problems in intl-

9.1 B O U N D I N G

SETS

181

nite dimensional complex and functional analysis. For example in analytic continuation and in the construction of the envelope of holomorphy. It is known that the extension of the notion of the holomorplically convex domain to the infinite dimensional case leads to the notion of the bounding sets which was introduced by Alexander [1]. Indeed the bounding and weakly-bounding sets have been studied for locally convex spaces by Dineen [86], [88], Josefson [123], Schottenloher [196].

9.1.1

B o u n d i n g Sets in L o c a l l y B o u n d e d F - S p a c e s

Definition 58 A subset A of a locally bounded space E is said to be bounding if l] f [IA-- sup If (X) I < OO xcA

(9.1)

for all holomorphic functions f on E. By a holomorphic function we mean a continuous and Gdteaux-analytic function.

Definition 59 A set U in E is said to be p o l y n o m i a l l y convex, if there exists a continuous polynomial P on E which separates the points of U and E/U. That is, for every Xo E E / U there exists P E L ( E ) such that

P (Xo) r P (u) T h e o r e m 87 [15] (1979) The closed unit ball Bzv of lp, (0 < p < 1), is polynomially convex. More precisely, for every point b ~ B~p one can find a contiuous monomial P E p ( m lp) such that sup [ P (x) I< 1 <__P (b).

(9.2)

xEBlp

P r o o f . The construction of P is a finite dimension problem. Obviously for every b ~ B~p there exists an n C N such that 7rn(b) ~ 7rn(BLp). We can assume that all bj > 0, j - 1, ..., n by multiplying bj by complex numbers of modulus one and indeed that bj > 0 by ignoring the coordinates for which bi - 0. Consider the mapping r __,/~ defined by ...,

where t -

(tl,...,tn) E R n. Let

-

...,

to)

(9.3)

182CHAPTER

9 BOUNDING

AND WEAKLY-BOUNDING

/

B * -- d/)-l(Trn(Blp)) --

t e ~n; ~

n

eptj < 1

SETS

/

(9.4)

j=l

and b; - 05-l(b)

The set

- (log bl, ..., log

bn).

(9.5)

B* is closed and convex in/~n. This is because n

t --+ ~

e ptj

(9.6)

j=l

is a continuous convex function in JRn. Since b* ~ B*, it follows by the classical Hahn- Banach Theorem that there exists a continuous linear functional T o n ~]~n such that sup T ( t ) < T(b*) tCB*

where T takes the form n

T(t) - ~ mjtj, j=l

t E ~n

where necessarily m j >_ O,j - 1,...n, since point s with 8j < tj. But the hyperplane

/

n

t c ~n; ~ . ~ j t j

B*

- T(b*)

(9.7)

contains with

t

every

/

j--1

can be moved so that all the m j become rational numbers, j - 1, ...n. Thus there exists a positive integer m such that m m j - kj are all non-negative integers. The monomial P(x)

kl xk2 .. kn -- x 1 .x n

(9.8)

satisfies the following" n

P ( b ) -- rIjn=l.bknj -- exp(~--~, kjb;) - e x p ( m T ( b * ) ) ,

j=l

(9.9)

9.1 B O U N D I N G

SETS

183

and sup [ P ( x ) 1 -

sup e x p ( m T ( t ) ) <

xEB

tEB*

P(b).

(9.10)

has the desired property. This completes the proof. I

So P I P ( b )

C o r o l l a r y 15 Let such that the sets D-

E

be a metric vector space with S c h a u d e r basis (ej)

z E ~T'~ ; d( ~ e x p ( z j ) e j

, O) < r

-

j=l

= {z e

d(

...,

O) <

are convex. T h e n E has a polynomially convex unit ball. In particular, this valid if, d ( t x , O) - d(x, O) f o r t E 6~, I t I-- 1.

P r o o f . Notice the formula( 9.3 ) of the convex function r and (8.4) of the sets B*. I E x a m p l e 30 The spaces

l(p~) -

x - - ( x j ) ; x j E (~, d(x,O) -

I xj Ipj< c~

,

l > p,~ > O

(9.12) have the property of the above Corollary. Notice that these spaces are locally bounded spaces if p~ /-+0 otherwise they are not locally p s e u d o c o n v e x spaces, See Rolewicz[186] . If E is a separable locally bounded space, it is known that E is the image of lp for some (0 < p ~< 1) under a continuous linear mapping T. The following theorem asserts that the kernel of the mapping T is a subspace of lp which has some interesting properties. In fact E is isomorphic to a quotient space of lp for some (0 < p ~< 1), see Rolewicz [186] and Stile [202]. A.

184CHAPTER

Theorem

9 BOUNDING

AND

WEAKLY-BOUNDING

SETS

88 [186]

Every separable locally bounded space E with a p-homogenous norm Ilxll,(0 < p ~< 1), is an image of lp by a continuous linear mapping T. More precisely, there exists a continuous linear mapping T.l,

--, E

(9.13)

i.e. T c L(lp; E), from lp onto E, with E ~ _ l p / M ,

where M -

kerT.

P r o o f . Let {Xn} be a sequence dense in the unit ball BE -- {x; Ilxll < 1}. Let T be defined by O0

T(t) - ~

t~x~,

t-(tn)

C lp.

(9.14)

n=l

Since the sequence {Xn} is bounded, the mapping T is continuous. To show that T maps lp onto E, let x C BE. Since {Xn} is dense in 1 BE, we can choose a subsequence {Xnk} and a sequence {tk} with Itk I< ~, and m

1

II ~ tkXnk, --Xll < 2--~. k=l

Let

~

!

tn 0

nk/

elsewhere.

The sequence {t'k} C lp and T({t'k} ) - x. This completes the proof of the theorem, m It is not yet known which of the locally bounded spaces E with phomogeneous norms (0 < p < 1) have polynomially convex balls BE (x.r). However, the fact that E is isomorphic to a quotient space of lp (0 < p < 1) will help to overcome the difficulty arising from the absence of the polynomial convexity of the balls of E. The following theorem characterizes the bounding sets of a big class of separable locally bounded F-spaces.

9.1 B O U N D I N G

Theorem

89

SETS

185

[16](1990)

Let E be a complete separable complex locally bounded space which is isomorphic to a quotient space lp/M (0 < p < 1) with Schauder basis, for some subspace M of lp. Then, the bounding subsets of E are relatively compact. Proof. Since E is isomorphic to a quotient space lp/M with Schauder basis (ej), where M is a subspace of lp (0 < p < 1), it suffices to prove that, the bounding subsets of lp/M are relatively compact whenever lp/M has a basis. Notice t h a t a subspace F which is a Fr6chet space in the topology inherited from a topological vector space on E is closed, see Rudin [[187],p.20]. In fact we need the following l e m m a : Lemma

22 Let M be a closed subspace of lp. Then the balls of the quotient

space lp/M

(B are polynomially convex provided that lp/M has a Schauder basis. P r o o f . Consider the quotient m a p 7r 9lp --, lp/M and assume lp/M has Schauder basis (ej) and 7r is a continuous linear mapping. It m a p s the Fr~chet space lp onto the Fr~chet space lp/M which is metrized by the metric (Trx, Try) -

inf d ( x ' , y ' ) ~'~(~) yleTr(y)

=

Since all the sets

inf ~ l x } - y ~ l x'~Tr(x) j = l

z e (~n; d ((e zl, ..., eZ"), O) < r

p.

are convex in C n r > 0,

n C N, it follows by Corollary 15, p.181, t h a t the balls/~ (0, r) - 7r (B (0, r)) are polynomially convex in lp/M. m P r o o f . of T h e o r e m . We now start proving the theorem. Since the sequence spaces lp/M (0 < p < 1) have polynomially convex balls t h e n the bounding subsets of lp/M are relatively compact. Consequently the bounding subsets of E are also relatively compact. Notice t h a t E is assumed to be isomorphic to lp/M. This completes the proof of Theorem. m C o r o l l a r y 16 [16](1990)The bounding subsets of each complex complete lo-

cally bounded space E with Schauder basis (ej) are relatively compact.

186CHAPTER

9 BOUNDING

AND WEAKLY-BOUNDING

SETS

P r o o f . E is isomorphic to a quotient space lp/F for some subspace F of lp(, 0 < p < 1). The isomorphic T is isometric since E and lp/F are metrizable. Since x in lp/F is uniquely determined by T ( ( e j ) ) , lp/F has a basis. Now apply the preceding Theorem 89 to obtain the desired result. m R e m a r k 24 The assumption that E has a Schauder basis cannot be dropped out of the above corollary 19, Take for example, the space Lp [0, 1] of measurable functions on [0,1],(0 < p < 1). It is a locally bounded separable metrizable space (without Schauder basis). Therefore it is mapped onto a quotient space l p / M with the property that each continuous linear functional which vanishes on M should vanish on the whole space lp, Stile [202]. Hence subsets of L p [0, 1], which may be unbounded, are bounding. 9.1.2

Bounding

Sets in Separable

Metric

Spaces

D e f i n i t i o n 60 A metric vector space is said to have the b o u n d e d approxi m a t i o n property(b.a.p), if there is an equicontinuous family (r of linear mappings Ct : E --~ Ct(E), of finite rank such that for every ~ > 0 and every compact set K in E, there is a t E T with

d(r

x) < e

(9.15)

for all x C K. T h e o r e m 90 [15](1979) Let E be a complete separable complex metric vector with a translation invariant metric d which has the bounded approximation property and is such that the closed balls in E are polynomially convex. Then a subset of E is bounding if and only if it is relatively compact.

P r o o f . A relatively compact set is obviously bounding since any continuous function on a compact set is bounded. Assume that A C E is not relatively compact. Then A is not precompact, for its completion is equal to its closure in E. So there exists a sequence (xj) of elements of A such that

d(xj, xk) > 5/~ when j ~- k

(9.16)

for some suitable number ~. The proof will depend on whether or not (xj) stays close to a finite-dimensional subspace. In fact we have two eases to study.

9.1 B O U N D I N G

SETS

187

Let fit -- I (~t, where (d/gt)te T is the given family of linear mappings, and let B ( x , r) denotes here the closed ball of centre x and radius r. C a s e 1 . F o r s o m e t C T w e h a v e pt(xj) e B(O, 2A),for e v e r y j : In this case the sequence of points (yj), (yj) = r satisfies

d(xj, xk) <_ d(xj, yj) + d(yj, yk) + d(y , xk), j T/= k; i.e. d(yj, yk) > d(xj, xk) -

d(xj, yj) -

d(yk, xk) > 5)~ - 2)~ - 2)~ -- ,~, j 7s k.

(9.17) Hence (yj) is a subset of Et - r which is not precompact. Since Et is of finite dimension, we know t h a t there exists a linear form u on Et such t h a t (u(yj)) is unbounded. The linear form q = uoCt is a holomorphic function on E with (q(xj)) is unbounded. Hence { x j } is not bounding in this case. Note t h a t here the family of linear mappings need not be equicontinuous. C a s e 2 . F o r e v e r y t E T t h e r e is a j

such that

the orbit

pt(xj) r B(O, 2A) " By the triangle inequality for the translation invariant metric d we have B(0, A) + B(0, A) C B(0, 2)~).

(9.18)

Choose 8 > 0 such t h a t pt(B(0,8) )C B(0,2A) for every t C T; this is possible in view of the equicontinuity of the family (Pt). Define

Vn - I.-jjn=lt~(Zj, (~) where {zj} is a countable dense set in T h e n for every t E T,

(9.19)

E; hence (Vn) is a covering of E.

flt(Vn) C ujn=lB(flt(zj), (~). Let now n be fixed. T h e n for some t e T we have p t ( z j ) C B(O,/~), j -- 1, ..., n. This is because Ct(zj) - z j - pt(zj) can be brought arbitrarly close to zj,j - 1, ..., n. Hence for this t we have

188CHAPTER

9 BOUNDING

AND WEAKLY-BOUNDING

SETS

p,(y,~) c B(0, ~) + B(0, ~) c B(0, 2~). But on the other hand we have by the assumption in this c a s e Pt(Xj) B(0, 2A) for some j. By the assumption B(0, 2A) is polynomially convex, so there is polynomial q w i t h l q ( p t ( x j ) l > 1, I q i< 1 on B(0,2A). The polynomial Q = qopt satisfies I Q ( z j ) I > 1 and l Q I< 1 on Vn.

(9.20)

We can now define inductively a covering (Wn) of E and a sequence (a,~) with am E Wrn+l/Wm, ?7"t- 1, ...,n as follows" Wm - V~(m), am - Xj(m)

(9.21)

where n ( m ) and j ( m ) are found by the procedure just described. If n n ( m ) has already been found we let j ( m ) - j be the index j found above and then we put n ( m + 1) as the smallest integer such that a m = xj E Vn(m+l) - Wm+l 9This is possible since Vn is a covering of E. We now have am E Wm+ l / Wm and polynomials Qm , m - 1, 2, ..., with

I Qm l< One can find C~rn c,~Qm, satisfies

Ct m E

1 on

Wn

and

IQm(~m)I> 1.

N and constants am such that the polynomial fm =

m-1

I f m l < 2 -m on Wm a n d l f ~ ( a m )

l>_m+l+

~ I A(~)I. (9.22) k--i

Since (W ~ is an increasing open covering of E, is holomorphic on E and

the function

~-~-m%1 f r n

(x)

m-1

I f(a~)[-I

fm(am)+ ~ m-1

>_ I fro(am) I-- ~

k=l

A(a.~) I

fk(am)+

k=l

fk(gm)-

k=m+ 1

oo

IIAIIwk -> "~k=m+l

f =

9.1 B O U N D I N G SETS

189

Hence { a m } {xj(m)} C A completes the proof. I

is not

bounding in this case.

This

R e m a r k 25 The proof in locally convex cases depends on a similar construction using a cover (Vn) consisting of convex sets. In our notation

g n - {ZI,...,Zn}~-B(0,6) is the vector sum of a finite set and a polynomially convex sets. I f B(0, 5) is convex we can use v: -

{Zl,

,

+ B(O,

where "cvx" denotes convex hull; as the sum of two convex sets V~ is then convex. However, the vector sum of convex set and a polynomially convex set need not be polynomially convex. This difficulty explains the difference between our proof and that in locally convex case and the role played by the mappings Pt ( see Schottenloher [196]).

9.1.3

B o u n d i n g Sets in Locally P s e u d o c o n v e x Spaces

Let us recall that a subset A of a topological vector space is b o u n d i n g if

II f IIA- sup If (x) l < oo xEA

for every holomorphic function f on E. D e f i n i t i o n 61 A topological vector space E is said to be locally p s e u d o c o n v e x if the origin has a fundamental system of pseudoconvex neighborhoods. E is locally p - c o n v e x if the fundamental system is absolutely pconvex, for some (0 < p < 1), Rolewicz [186]. We can see that a locally pseudoconvex topology is determined by a family of pseudo-norms while the locally p-convex topology is determined by a family of p-semi norms, 0 < p < 1. Let us recall that A p - s e m i n o r m on E i s a m a p p i n g I[-[Ip'E--~/~+ such that for t E ~ , x , y E E ,

(a) II tx IIp--I tiP. II x lip (b) I I x + y l l p < - I l X l l p + Ilyllp If II x IIpr 0 for x r 0, then II-lip is called a p - n o r m . If for some a _ > l , ( b ) is replaced by IIx+yllp-
190CHAPTER

9 BOUNDING

AND WEAKLY-BOUNDING

SETS

Finally II-lip will be a pseudo-semi norm if it is a p-semi norm for some p ( O < p < 1). The following theorem characterizes the bounding sets in a class of locally pseudoconvex topological vector spaces which may not be metrizable. T h e o r e m 91 [16](1990) Let E be a complete separable complex locally pseudoconvex space which has the bounded approximation property, and is such that the closed balls B ~ of E. dealed ~y it~ family of p~-~ni ~o~n~ ( II-I1~ ) - ~ / . (0 < p . < 1). are polynomially convex. Then the closed bounding subset of E are compact. P r o o f . Let

{z~} be dense in E and B ~ ( 0 , ~) - {x e E; II x I1~_< ~ }

where ( II x IIc~ )c,e/is the family of pc~-semi norms (0 < pa < 1), defining the topology of E. Let Pt - - I - Ct, t E T, where ( r is the given equicontinuous family of linear mappings. Assume A C_ E is closed and non-compact. Then it is not precompact; so there exists a sequence (xj) of elements of A such that

II ~y - x ~

II~> 5A > o,

when j 7~ k

for some suitable number A > 0 and some c~ E I. From now on the proof can be given along the same lines as in the above theorem. This completes the proof of the theorem. IH R e m a r k 26 For the quasi-complete locally convex spaces the result is due to Schottenloher [196] and for separable or reflexive Banach spaces it is due to Dineen[86]. Theorem

92 [16](1990)

A subset A of the pseudoconvex F-space

Zo - n~>o/~ -

x - (x~); II x Iio= sup~-~.lXn IP< oo p>O

1

,

(9.23)

9.1 B O U N D I N G

SETS

191

(0 < p < 1), of the intersection of all spaces lp of complex sequences x = (xn) with the F-norm II x Iio= supp>0 ~ - ~ I x~ IP< oc is bounding if and only if A is relatively compact. P r o o f . First the space lo is a complete metrizable pseudoconvex space with respect to the F-norm, oo

if x i]o= s u p ~-~lXn ]P. p>O

(9.24)

1

Hence by Theorem 90, the bounding subset of lo are relatively compact. In fact Io has polynomially convex balls, for the function

f(xl, ...x~) =ll ~,---, ~

JIo

is convex for x = (x], ...x~) C ( ~ n n E N, see Corollary 15. This completes the proof, i 9.1.4

Bounding

Sets in Non Locally Pseudoconvex

Spaces

In infinite dimensional locally convex spaces E the size of the bounding subsets has been investigated by Dineen [88] and Josefson[122], who answered the following questions negatively for l ~ : "Does there exist an infinite dimensional normed space E such t h a t t h e u n i t ball BE is b o u n d i n g . ? " In some non locally convex spaces the situation is different as it will be asserted by the following theorem.

T h e o r e m 93 / / 6 ] ( 1 9 9 0 ) A subset A of the non locally pseudoconvex F-space

z(~/-

9

(x~); li x ii- ~ l ~

I~n< ~ , P~ -~ 0

(9.25)

1

(0 < p~ < 1), Pn ~ O, of all complex sequences x = (xn) with the Fnorm [I x I[- ~ ] xn ]pn< oo, is bounding if and only if A is bounded. Consequently, the unit ball B l(pn)(0, 1) is bounding. P r o o f . The s p a c e l(pn) , (O < Pn < 1),p~ --~ 0, is Schwartz and hence it is Montel space. It is metrizable but not locally pseudoconvex since there is no countable family of pa-semi norms defining its metric topology, see

1 9 2 C H A P T E R 9 B O U N D I N G A N D W E A K L Y - B O U N D I N G SETS nolewicz [[186],p.153]. Now as l(pn), Pn ~ O, is a Montel space then every closed bounded subset A of E is compact and consequently A is bounding. On the other hand the bounding subsets of this space are relatively compact since its balls are polynomially convex, see the theorem above. Since B z(p~)(0, 1) is topologically bounded with respect to the only topology define by the given F-norm, it is bounding, and the proof of the theorem is complete, m C o n j e c t u r e 1 Are the weakly bounded subsets of l(p~), (0 < p~ < 1),p~ -~ 0, weakly-bounding? We have a negative conjecture.

9.2

WEAKLY-BOUNDING

9.2.1

(LIMITED)

SETS

W e a k l y - B o u n d i n g Set in Locally B o u n d e d Spaces

Definition 62 A subset A of a topological vector space E, whose dual E ~ separates its points is called. " w e a k l y - b o u n d i n g " ( o r limited) if every sequence (On) C E', which converges pointwise to zero in E, converges uniformly to zero on A. That is,

lim Cn (x) -- 0, for every x E E

lim

n---~oo

II

n - - ~ oK)

II Cn II-o,

(9.26)

IIA= sup A Ir

Proposition 3 A subset A of a locally bounded p-normed space E is weaklybounding if and only if II f ]]=sup If ( x ) ] < oo, for all A

xcA

f-

~

q5~ E H (E)

(9.27)

1

where ( cpn) C E' . P r o o f . =~)" Assume A is weakly-bounding. Then ]] On IIA--' 0 whenever r (x) --* 0 for every x in E. Let

1

9.2 W E A K L Y - B O U N D I N G

(LIMITED)

SETS

193

Since f - ~ r is analytic, the boundedness of f on A is given from the analyticity of the polynomials c n n C N and the assumption that II Cn IIA--~ 0 to have ~ ] 1 II On ]]~< c~. Notice that II ~-~.~ Cn n IIA~ EF

II

IIA< oo.

r If ~ r n G H (E) then ~-~ (On (x)) n tn/p converges for every x E E and t E (/'. By Generalized Cauchy-Hadamard formula in one variable (see subsection 7.2.3, p.139), limsup 1On (x) n 88 Ip -- lim 1r (x)I p -- 0. n--~oe

Now if oO

II

r

11,4<

1

then II Cn IIA-+ 0 as n --+ oo and also as a consequence of the radius of convergence formula Oo

R I (x) - - s u p ( r < 0; ~

[I Cn n ]I rn/p < c~)

(9.29)

1

of a p-normed space. This completes the proof of the theorem. I Josefson [122] has introduced the notion of weakly-bounding sets in his study of bounding subsets of l~. He has proved that the bounding and weakly-bounding subsets of l~ are the same. Dineen's result [ Cor.4.19 p.175] showed that this is obviously true for lp, (1 ~< p < oo). However the situation is totally different when we deal with non locally convex topological vector spaces as it is shown in what follows. Before we deal with the weakly-bounding subsets of a locally bounded F-space E, let us first characterize the duals of certain quotient spaces of E. For Banach spaces, see Rudin [[187] Th. 4.9 (b), p. 92]. T h e o r e m 94 [16](1990) Let E be a p-Banach space with a Schauder basis. Let M C E be a closed subspace. Then the dual space of the quotient space E / M is isomorphic to the annihilator M • of M, provided that E / M has a Schauder basis, i.e.

( E / M ) ' ~_ M •

(9.30)

194CHAPTER

9 BOUNDING

Let 7r" E ---* E / M

Proof. define

AND WEAKLY-BOUNDING

be the quotient map.

T (r

SETS

For each 4) E ( E / M ) '

- r

(9.31)

T h u s T is an isomorphism of (E/M)' onto M • : * F i x x * E M • 9 T h e n x* E E I and the r e s t r i c t i o n x i M - O. Let N be the null space of x*, hence M C_ N. It follows t h a t there exists a linear functional v on E / M such t h a t prr - x* as a consequence of the existence of a Schauder basis of E / M . (For example choose ~ such t h a t vlTr(N) -- 0 and v (Tr (x)) - x* (x) outside 7r (N)). T h e null space of v is 7r ( N ) , a closed subspace of E / M

by definition of

q u o t i e n t - t o p o l o g y in E / M and hence v is continuous, t h a t is, v E ( E / M ) ' . T h u s T (v) - vTr - x*. On the other h a n d if x E E and y* E ( E / M ) ' , then 7r (x) E E/M, hence x ~ y* (Tr (x)) is continuous linear functional which vanishes for x E M. T h u s T ( y * ) E M • T h e linearity of T is obvious. This completes the proof. I

We give now the following interesting result which shows to what extend the geometry of the spaces may play a role in Holomorphy.

Remark

Theorem

27

95 [16](1990)

Let E be a separable complex p-Banach space, (0 < p < 1). Assume that E is isomorphic to a quotient space of lp which has a Schauder basis. Then a weakly-bounding subset of E is not necessarily bounding in E. Moreover it could be an unbounded subset of E. P r o o f . A b o u n d i n g set is obviously weakly-bounding. Since E is isomorphic to lp/M for some closed subspace M of lp,(O < p < 1), so it is enough to s t u d y the w e a k l y - b o u n d i n g subsets of lp/M. This can be done t h r o u g h s t u d y i n g first the w e a k l y - b o u n d i n g subsets of lp(O < p < 1). Let

A-

l)l/p'

x(n);x(~) - ( ( n

I

(1)n

nth-place 1/p

,0,. ) noN

(9.32) I

9.2 W E A K L Y - B O U N D I N G

(LIMITED) SETS

195

and

B --

x(n);

x (n) - -

(1, ( ~1) l / p , , , , ,

(n1 - -

)l/p

' 0,. ) n c N } o

o

~

9

(9.33)

The space lp (0 < p < 1) is dense as a subspace of 11 and it has the same dual space which is isomorphic to lcr The bounding subsets of lp are bounding when they are considered as a subset of ll. Notice that the restriction of every f E H (/1) to lp is in H(lp). The set A is bounded in lp since for every x (~) E A, n

~-~.[xj[ p j=l

- - - - n1- - 1 . n

Moreover A is a weakly-bounding subset of lp for it is relatively compact in 11 where the bounding and weakly-bounding sets are the same. Notice that the condition cx:)

lim (sup ~ n--,c~

xs A

[xj]) -- 0

(9.34)

j=n

is satisfied. However A is not a bounding subset of lp, for it is not relatively compact. In fact the elements of A lie in different directions on the surfaces of the unit ball Bzp. Consequently A C lp, (0 < p < 1), is an example of a weakly-bounding set which is not bounding. The set B is a weakly-bounding subset of lp (0 < p < 1) since B is bounding as a subset of/1; it is clear that B is a relatively compact subset of ll. On the other hand, B is not bounded in lp for d (x (n) O) - ~ -j1 ~ oc. Therefore the weakly-bounding subsets of lp could be unbounded sets. Finally let us study the weakly-bounding subsets of a quotient space

lp/M where M C_ lp is a closed subspace and lp/M has a basis. Since the dual space (lp/M)' is isomorphic to the annihilator M • of the subspace M by the preceding theorem, t hen every sequence (r (lp/M)' can be assumed to be in (lp)/and vanish on M. Let

7~ " lp --> lp/M

in

196CHAPTER 9 BOUNDING

AND WEAKLY-BOUNDING

SETS

be the quotient mapping which maps the Fr6chet space lp onto the Fr6chet space lp/M with respect to the F-norms IJ 7r (x)I[= inf IJ x - m lip, where meM

II x lip= E F Ixjl p, for x -

(xj) C lp.

For the weakly-bounding subset B of lp given above, we claim that the subset 7r (B) of lp/M will satisfy the desired property, that is, it is non bounding but weakly-bounding in lp/M" Let (On)C (lp/M)' with (~n (X)~ 0 for every x in lp/M. Then oo

II ~ r 1

(x)

II ~ ~'~ II r 7r(B)

~-

(9.35)

1

This is because d/)nOTr is a sequence of continuous linear functionals on E which vanishes on M and B is a weakly-bounding subset of E. On the other hand 7r (B) is not bounding in lp/M. Notice that 7r(B) is not bounded in lp/M. This completes the proof. II

9.2.2

T h r e e Different Classes of H o l o m o r p h i c

Functions

The above theorem can be formulated using holomorphic functions as follows.

T h e o r e m 96 [16] Let E be a complex p-Banach space with a Schauder basis, (0 < p < 1). Then there exists a holomorphic function f in H (E) and a non-relatively compact set A in E such that

II f IIA= ~ pointwise in E where (r bounded set in E.

a~d

II On IIA~ 0

if On(X) ~ 0

(9.36)

C_ E'. Moreover A could be an unbounded weakly

P r o o f . Every bounding subset A of E is weakly-bounding but the converse is not always true; Remark the sets A , B of the preceding Theorem 95. Hence, not every set A in E satisfy II r ]IA~ 0, whenever Cn(x) ~ 0 pointwise for every (On) C_ E', is relatively compact. Therefore a function f E H (E) may exist and satisfies II / IIA= c~. Of course f will not of the form ~ r with II r IIA~ 0 wheneverr n ~ 0 pointwise since A it is not weakly- bounding, in accordance with Proposition 3, given at the begining of this part. Such a set A may be a proper weakly bounded set in E. This completes the proof. II

9.2 W E A K L Y - B O U N D I N G

( L I M I T E D ) SETS

197

R e m a r k 28 This last situation is almost similar to the finite dimensional

case(T n in which every f r H ((Tn) is unbounded on unbounded sets o f t ~. Theorem 95 shows the remarkable relevance of the concepts of bounding and weakly-bounding sets. We may consider them as basic concepts in holomorphy. To show this, let E be a locally bounded F-space and let By (E) and WBg (E) denote the classes of bounding subsets and weakly-bounding subsets of E respectively. According toTheorem 95 if E is not locally convex space with basis, one may obtain

Bg (E) ~ WBg (E) .

(9.37)

Moreover the class WBg (E) consists of two subclasses: (1) The bounded weakly-bounding subsets of E, (WbBg (E)), and (2) The unbounded weakly-bounding subsets of E, (W~Bg (E)). Therefore we have precisely the following relations

Bg(E) WbBg (E

(E)

Bg (E) C WbBg (E) C WWBg (E) . Figure 5: R e l a t i o n b e t w e e n t h e d i f f e r e n t classes of b o u n d i n g sets Consequently we will have the following result.

198CHAPTER

9 BOUNDING

AND WEAKLY-BOUNDING

SETS

Corollary 17 [16](1990).Let E be a locally bounded space with a Schauder basis. Let us denote by H b (E) the class of holomorphic functions of type Cn n, (r C E' which are bounded on all bounded sets in E. i.e.

H(E) = I

f=Er

nell(E);

(r

V bounded sets B in E

J C_

and by H b (E) the class of holomorphic functions of type E r E', which are bounded on all weakly bounded sets in E, i.e.

H (E) = I

f - ~ Cn n e H (E) ; (r

C E', II f lie< ~ ,

V weakly bounded set C in E

J

Then

H b (E) C H b (E) C H (E).

(9.38)

That is, the class of holomorphic functions H (E) on E contains properly two other different classes.

Proof.

See the proof of Theorem 95. m

(E)

H(E)

H b(E) C H D(E) C H ( E ) Figure6" Relation between the different classes H b ( E ) , H b (E)

and H (E) of holomorphic functions

199

9.2 W E A K L Y - B O U N D I N G (LIMITED) SETS

Remark 29 We notice that these above three classes of holomorphic functions coincide for many locally convex spaces, e.g. if E shown by Josefson [122] and Dineen [82].

lp (c~ >~p >~ 1) as

Remark 30 As we have seen,

the theorems of the two preceding parts explore a new class of bounding sets which appear in the study of holomorphy in the non locally convex spaces E. It is the class of bounded weakly-bounding sets ( B - W B g ( E ) or bWBg (E) for short).So recall that we have so far the following relation between the three classes of bounding sets in some non locally convex spaces : Bg(E) ~ B-

9.2.3

WBg(E) ~ WBg(E).

(9.39)

E x a m p l e s of H o l o m o r p h i c Functions

We introduce some of the remarkable topological vector spaces which satisfy the hypotheses of the theorems of the preceding subsections 9.2.1 and 9.2.2. We will also construct certain holomorphic functions and non bounding weakly- bounding sets in some spaces to explain the performance and effects of these concepts in Complex and Functional analysis.

Construction of Certain Holomorphic Functions ( O n L o c a l l y B o u n d e d Sequence Spaces) Example 31 Let E - lp, (0 < p < 1). According to the above comment the relation

means that not every A C_ E satisfying lim II r

IIA-- 0 i f

lim

Cn (X) -- 0 , e v e r y

x e E

(9.40)

is relatively compact. Equivalently, there is A C_ E which may be unbounded satisfying II r IIA--~ 0 i r e s ( x ) --~ 0, x E E, (r C_ E', and a function f = E r f C H (E) exists with II f IIA< c~. This is because weakly- bounding subsets are not necessarily relatively compact by Theorem 96.

200CHAPTER

9 BOUNDING

AND WEAKLY-BOUNDING

SETS

To show this, consider the following set given in the proof of Theorem 95.

B -- { x(n);

x (n) - -

1 ( 1 ) :/p (1, (~)l/p,..., ,0,...),

n:g

}

.

We have proved that B e WBg (E) /Bg (E) and B is unbounded in E. Now the following function (:x:)

1

:<x>

+niX>

~

~

o

n

I

will be the required one, i.e. II f liB< oo. Obviously (r

(1)

1)1//) __

1{~n {{B-- ( 1 ) 1( 1-1 )

"(n a s Tt---~ cx).

1

(9.41)

X n

c E' and hence

1

~(1-;)

1

--,0

Now oo

1 )l_~.Xnn e H (E)

(9.42)

1 and it is bounded on B"

m(1)

II fllB=sup ~ xmgB

m

1

!n P

n

=

1+~

-E


1 But the function g(X) g(X)

--

-- E1

Er~

( n1) 1 p: "xn

--

~ n ( X ) n i8 a n o t h e r

Er~

n1 ---+ (X) a8 m

having

ful~ction

%(x)- (~1)•

n

1

_l/n

.:15 n

and does not satisfy (9.40). It is not lin'ear, at

--~ (x).

This is because

9.2 W E A K L Y - B O U N D I N G (LIMITED) SETS

201

Construction of Certain Holomorphic Functions(On Locally bounded F-space) Example 32 A weakly-bounding subset may be unbounded by. Theorem 95. That is an unbounded set D may exist such that all holomorphic functions of type O0

1

will satisfy II E ~ r liD< ~, (r c_ E* To show this, let E - l(pn), infpn -~ 0, be a non locally pseudoconvex space, see Rolewicz [[186],p.153]. We now give an example of a function g (x) - ~

r

E H (E)

and a non-bounded weakly-bounding subset D of bounded. Set

E

such that gilD

is

Cn (X) -- X n

and let D -

1

x ( n ) ; x (n) - (0, ( 2 1 o g 2 2 ) 1 / P , . . . ,

withp=~ p ( 1 -

log log n ]~ and 0 < p < 1

(

1 )l/p 0 . . . ) , n e N } n log2n ' '

Obviously p--~ p ~ 0, i.e.p~ -~ O. n

Now D C E is unbounded: J

II ~(J)II- d(x(J), o) - ~

I ~(n'~)l~"

n--1

1 l'p]p(1- loglog. ) 1

-~

(nl~g2n) ~e n---1

a8 j --~(x).

--

2

n

lo~n -21oglogn)

e ( log log n

C~

202CHAPTER 9 BOUNDING

AND WEAKLY-BOUNDING

But

J cn (X)I~(~)

11

n=l

n--1

SETS

n/p < c~

(9.43)

nlog 2 n

D. Thus g = E cn < cx~ on the unbounded set D. On the other hand D is a weakly-bounding subset of l(p,~) since it is O0 compact in 11 " lim sup ~-~j=~ Izjl - o.A for every x--

(Xn) e

n--~C~x~ D

P r o b l e m 2. Let E be a topological vector space whose continuous dual space E ' separates the points of E. W h a t are the geometric and linear properties of E so t h a t all its bounding sets are relatively compact? P r o b l e m 3. Does there exist a non locally convex space E with a rich dual space E* whose bounding and weakly-bounding sets are the same? ( 0 b vious t h a t the space LP[0, 1] (0 < p < 1) satisfies this property but (LP) ' = {0} is a trivial dual). Could we construct such a space?

Image of B a n a c h Spaces Example

33 / o - S p a c e .

The space l o - A lp

(9.44)

p>o

is a complete locally pseudoconvex F-space with respect to the sup F - n o r m

II x

IIo=supll

x

p>o

lip

wh~ II x I1~- E j =n ~ Ixjl p x - (xj) e lp. The bounding subsets of lo are relatively compact since lo has polynomially convex balls. Notice that the funct i o n f ( x l , . . . , x n ) -[I ( e ~ , . . - , e n ) I[lo is convex f o r x - ( X l , . . . , x ~ ) e r n , n e N, see the author[J15], p. 17] or Corollary 15, p. 185. The weakly-bounding subsets of lo could be unbounded. For example the set A - {x (n)} where, x (~)

-

(1 ,

~

'

...

'

-n

' 0,...)

'

is weakly-bounding in lo for it is weakly-bounding in each lp. Notice that lo is dense in each lp (0 < p < 1). Moreover A is not bounded in lo " n ]l x(n) I l l o - S U p E ~=1

(1)I/pP -m

~ -

1 -- +oo

m,--1

m

203

9.2 WEAKLY-BOUNDING (LIMITED) SETS

as n -~ oo. Consequently a weakly-bounding set in lo may not be bounding. The importance of this F-space lo is that every separable Banach space E is isomorphic to a quotient space of lo; i.e. for a subspace M of lo,we have

(9.45)

E ~_ l o / M . A

Mapping with Rapidly Decreasing Sequences(F-Spaces) Example 34 Consider the space L ( E ) of all continuous linear mappings of a Banach space E. Let L ( E ) have the linear mappings norm. On To E L ( E ) defines r -th approximation numbers by (~(To) -

inf {fiT - Toll;T e L ( E ) with d i m T ( E ) <_ r } .

ff lP(E) -

{

To C

L(E); ~ [c~(To)] p < oo r=l

then on the intersection S(E)

-

(9.46)

}

e >ol (E)

(9.47)

a metrizable non locally convex vector topology is generated by the sets

Bp(To, ~) -

{

T C

S(E); ~ [ar(T- To)] p < c r=l

}

,p>O,E>O.

S ( E ) is complete and it has the bounded approximation property if E is complete, as it is shown by Pietsch[[18~],p.139]. If the balls defined by the metrics oo

dp(T, O) - ~

[C~r(T)]p ,

(9.48)

r:l

are polynomially convex then the bounding subsets of S ( E ) are relatively compact by Theorem 89. If S ( E ) has a basis for a normed space E we can apply Theorem 95 which considers weakly-bounding sets. Notice that L ( E ) is not a Fr~chet space if E is not Banach.

204CHAPTER

9 BOUNDING

AND WEAKLY-BOUNDING

SETS

R e m a r k 31 We could have another types of non locally convex Fr~chet spaces with mappings with rapidly decreasing approximation number as follows: Let Eq be a q-Banach space, (0 < q < 1) say with a Shauder basis and construct L(Eq) as above with the linear mappings q-norm:

IITI[ = sup IIT(x)II. x6BE

(9.49)

We shall then obtain different values for c~ and hence different spaces lP(Eq) and S(Eq).

Locally Pseudoconvex

F-spaces

E x a m p l e 35 Let E be a complete locally pseudoconvex space whose topology is defined by a sequence of pn-homogenous pseudonorms {llxll~;n e N } . Then, without loss of generality we can assume that IIxll, <_ llxll: <_ ...

Let E ~ = {x e E; IlxJl~ = 0}. Then ( E ~

(9.50)

is a decreasing sequence of vector subspaces of E; i.e. E~ n E~ n ....

Let E~

be the quotient space E~ E ~ i.e. En - E l E ~

(9.51)

The pseudo-no,ms, IlXlln, n e N induces pn-homogeneous pseudonorms Ilxll'n, n e N on the spaces En and consequently En with Ilxll'n, n e N are locally bounded spaces. Now if E has a Schauder basis, then the sequence of the cosets

{[ei]n} -~ {el + E~

i E g}n

(9.52)

generated by the basis {en} in the completion E of E, i8 a basis of norm o~,(~h~ ~ o,~it all t h o ~ {[~]n} ~ h ~ h a ~ ~q~al to z ~ o ) to th~ ~ ~o,~plete locally pesudoconvex spaces whose topology is defined by the sequence of pn-ho,~ogenous pseudonor,~s lixll'n, n e N , see R o Z ~ ~ z [[186],p.239.].

9.2 W E A K L Y - B O U N D I N G

(LIMITED)

205

SETS

In fact this can be metrized by the metric

i

]lxll'n

(9.53)

d(x, y) - ~ 1 ~n 1 + Ilxll'n " If the balls

un(o, r ) - {x C E; I[x]l'n_~ r},

(9.54)

are polynomially convex in En, n E N , then Theorem 91 can be applied and bounding sets turns out to be relatively compact.A

Product of Fr6chet Spaces. Example

36

Let E-

be the product of Fr~chet spaces E j by the metric d.j(x,

y) -

(9.55)

II~Ej such that the topology of E j

is given

ylJ-j

Iix -

where x j , y j C E j , and II.IIEj, J C N is a homogeneous pseudonorm and such that the balls defined by them are polynomially convex. It is easy to see that E has the bounded approximation property if each E j, j E N , has the bounded approximation property, see the author[J16], Example ~]. Let

r RN __. [0, §

(9.56)

be convex and homogeneous of degree 1; then an F-norm II'llE can be defined on E by II-IIE - r IIx2IIE~,...), x- (Xl,X2, ...). For example we can take

O(IIxlIIE1, IIx211E~, ...) - (~-~

IIxjIIEj) l/q,

(9.57)

j=l

wh~e

1 <_ q < ~ , t o h a v ~ O

< 1/q <_ 1 ,

a~d

IIxjllEj, j e N,

i~

p j - h o m o g e n e o u s with py --~ oo as j ~ c~. Now Theorem 90, p.186, can be applied and hence the bounding subsets of E are relatively compact. For example take

E-11

• lll2 •

• ....

(9.58)

206CHAPTER 9 BOUNDING AND WEAKLY-BOUNDING

9.3

PROPERTIES OF BOUNDING LIMITED SETS

SETS

AND

In this part we apply the results of the preceding two parts of this chapter to the study of the properties of the classes of bounding and weakly-bounding sets in some non locally convex topological vector spaces. 9.3.1

Bounding Sets and Compact Linear Maps

In this section we obtain some fundamental and interesting properties of the bounding subsets of certain non locally Convex topological vector spaces. They fit well with such spaces although some of them do not exist for locally convex spaces.

Definition 63 Let T:

Ep--~ Eq

be a continuous linear mapping between two locally bounded F-spaces Ep and Eq (0 < p, q < 1). If for every bounded set B of Ep, the image T ( B ) is relatively compact, i.e. T ( B ) - is compact. (9.59) then T is said to be compact.

T h e o r e m 97 [17](1989) If E is a normed space and T : E ~ lp,

(9.60)

(0 < p < 1), is a continuous linear mapping, then T ( B E ) , the image of the unit ball of E, is bounding in lp. Consequently, a domain of holomorphy of a normed space E, may have continuous linear image by T which is not a domain of holomorphy in lp.

Proof. We notice that every continuous linear mapping of a normed space E into lp is compact. This is proved by Stile[[201],Theorem 4]. Hence T ( B E ) - is compact and consequently is bounding. Now if U is a pseudoconvex domain of a Levi space E, then it is not bounding. That is, it is a natural domain of existence of a holomorphic function. However T(U) may not have this property since it may be bounding as a compact subset of lp. This completes the proof, n

9.3 P R O P E R T I E S

OF BOUNDING

AND

LIMITED

SETS

207

R e m a r k 32 ( 1 ) T h e property given by the preceding theorem means that : " T h e r e is n o c o n t i n u o u s l i n e a r m a p p i n g o f a B a n a c h space E i n t o lp, (0 < p < 1), t h a t c a n h a v e a n i n f i n i t e d i m e n s i o n a l b o u n d i n g closed range". (2) Also this theorem implies that : "A c o n t i n u o u s l i n e a r m a p p i n g T : Eq ~ lp o f a q - n o r m e d s p a c e Eq i n t o lp h a s a c l o s e d b o u n d i n g r a n g e i f O < p < q < 1 ". We have

of St l [[202], The linear completion (or hull) of a p-normed space E is called the B a n a c h e n v e l o p e / ~ . The following theorem studies the bounding sets in

T h e o r e m 98 [17](1989) If E is a p-Banach space with a Schauder basis, then a bounding subset of its Banach envelope E need not be bounding in E.

P r o o f . The space E is locally bounded. Thus the bounding subsets of E are relatively compact and are not the the same as the weakly-bounding subsets as we have shown in the preceding part. However on the linear completion /~ of E, the bounding and the weakly-bounding sets are the same because E is a separable Banach space. Therefore we choose a subset A of E which is bounding (or equivalently weakly-bounding)as a subset of /~ but not relatively compact in E. This completes the proof, m In what follows we give another property of bounding sets in some topological vector spaces. L e m m a 23 Let E be either metrizable or locally pseudoconvex space, with bounded approximation property, and polynomially convex balls. Then the vector sum of finitely many bounding subsets of E is bounding in E.

Proof. The sum of two compact subsets of a topological vector space is compact, Rudin [[187],p.37], and the bounding sets in such spaces E are relatively compact,by the author's result[16], (see section 8.1 on bounding sets). So the lemma is proved, m

208CHAPTER

9 BOUNDING

AND WEAKLY-BOUNDING

SETS

R e m a r k 33 The property given by the above lernma implies that: if U1 +U2 is not bounding then either U1 or U2 is not bounding. This means that if U =U~ +U2, is the vector sum of two domains, then U may be a domain of existence of a function f C H(U') for an open domain U' containing U, U1, U2 then it may happen that either U1 or U2 is a domain of existence of the function

f. For metrizable locally convex spaces, the result is due to Dineen[[82], p176]. The following theorem deals with some holomorphic properties of the p-convex hull of bounding sets, see Chapter 3, section 3.1 T h e o r e m 99 [17](1989) If E is a locally bounded p-normed space with a Schauder basis, then the absolutely p-convex hull of a bounding set in E is bounding. P r o o f . Assume that A C E is bounding. Then it is relatively compact by Corollary 16, p.181. Since the absolutely p-convex hull CpA is contained in

)~A + ... + )~2A provided that ~-]~' I Ai [P_< 1, then CpA is also compact. Notice that B(0,

= B(0,

(9.61)

Of course the convex hull of a bounding set in a non locally convex space may be unbounded and hence it may not be bounding. This completes the proof. II R e m a r k 34 For the non-separable normed space l~o, the preceding theorem is proved by Josefson [122]. It is proved for a balanced hull of a bounding subset A of a balanced open set U in a locally convex space by D ~ ~ [ [ 8 2 ] , 174]. Of course the property given in this theorem has its significance when we deal with analytic continuation and some topological problems. For example, it implies the analytic continuation froin a bounding set to its absolutely pconvex hull.

9.3 P R O P E R T I E S

OF BOUNDING

9.3.2

of The Different Weakly-Bounding

Properties

AND LIMITED SETS

209

Sets

The following theorem will show some of the properties of weakly-bounding subsets of locally bounded sequence spaces. It explores also the appearance of a n e w weakly-bounding class (denoted by B - W B g or bWBg) lies entirely between the class of bounding sets (denoted by Bg) and the class of weakly-bounding sets(denoted by W B g ) , This class may be also called Bayoumi weakly-bounding, B - WBg; see figure 7 drawn after the following theorem. T h e o r e m 100 [19](1989) Let E be a complete locally bounded F-spaces with a Schauder basis. Then we have

(1) The vector sum of finitely many bounded weakly-bounding subsets of E is bounded weakly-bounding. That is, AI,..., An E B - W B g ~ A1 q-... + An C B - W B g (2) The classes of weakly-bounding subsets and bounded subsets of E are different and their intersection contains properly the class of bounding subsets of E. That is, Bg Bdd n w s g wBg (9.62) where Bdd denotes the class of topologically bounded subsets of E. (3) Every weakly-bounding subsets of E is bounded with respect to the weak topology of E, but the converse is not true. That is a weakly-bounded set may not be weakly-bounding, i.e. WBg ~ WBdd.

(9.63)

P r o o f . P r o p e r t y (1)" follows directly from the fact that the sum of two bounded set E is bounded, by Rudin[187] and the properties of weaklybounding sets. Notice that sup xcAI,x2CA2

for any r E

Ir

-~-X2)l< sup I r xcA1

I -+- sup [ r

[

x2cA2

El and any sets A1, A2 in E.

To s h o w (2)" it is sumces to deal with it for the space lp, (0 < p < 1), since E is a continuous image of lp, by a continuous mapping T, i.e.E = T(lp), see Theorem 88, p.183.

210CHAPTER

9 BOUNDING

AND WEAKLY-BOUNDING

SETS

The set

{

x(n);

X(n)

-

-

1 (1, (~)l/p

~'''~(n

1) 1/p 0,.

.

o

)

neN}

is weakly-bounding for it is bounding in ll, but B is not bounded: n

cl(x (n), O) -- ~

1

--+OO as 1 fn

n--+ oo.

On the other hand the set D-{ei;

icN}

(9.64)

of unit vectors in lp is bounded but not a weakly-bounding subset of lp, for D is non relatively compact in 11. Thus the classes of bounded and weaklybounding sets are not the same. Now the set A -

I X(n);x(n) -- ((n1 )l/p '

1/p, O, nth-place (1) n

.) noN

I

is lying in the intersection of these two classes, for it is bounded and weaklybounding set in lp at the same time. However A is not bounding. So the class of intersection is larger than the class Bg of bounding subsets of lp. P r o p e r t y (3)" is obvious since the elements of the continuous dual E' are in H(E). In fact we have

E'c H (E)

(9.65)

where Hb(E) is the class of holomorphic (continuous and G~teaux analytic )H(E), whose elements have the form ~ r and bounded on weakly bounded sets. Finally the set D of unit vectors is weakly bounded in lp, for it is bounded in l l, where boundedness are the same for the original and the weak topologies. But D is not weakly-bounding of lp for it is not relatively compact in 11. This completes the proof of theorem. II

9.3 P R O P E R T I E S

OF BOUNDING

AND

LIMITED

SETS

211

Figure '7: R e l a t i o n b e t w e e n different classes of b o u n d i n g sets a n d classes of b o u n d e d a n d weakly b o u n d e d sets. Here, the given symbols mean : Bg: bounding sets in lp; WBg: the weakly-bounding sets in lp. B - WBg: bounded weakly-bounding sets in/p.(Or Bayoumi limited sets). Bdd: bounded sets in lp; WBdd:weakly bounded sets in lp. 9.3.3

W e a k * C o n v e r g e n c e in t h e D u a l of a p - B a n a c h S p a c e As Different From Norm Convergence

For infinite dimensional Banach spaces E the weak* topology and norm topology in E ~ differ. This follows from Alaoglu's and F.Riez theorem ( see Rudin[187]).Can they have the same convergent sequences? The answer is "no!". This was proved independently by Josefson [121] and A.Nissenzweig. They proved that : "Weak sequential convergence in the dual of a Banach space does not imply norm convergence . Therefore we recall one of our concepts used in [[15], [16]], that is, the weakly-bounding sets as follows : A subset A of a p-Banach space E is said to be limited if every weak null sequence in E ~ tends to zero uniformly on A. That is, "limn Cn(X) -- 0, uniformly for x E A if (r is weak*null sequence in E ~'' So limited sets may be regarded as the linear analogy of bounding sets, as we have treated them before. For this reason we have called them in [[15], [16]] weakly-bounding sets. By a bounding set A of E we mean a set which is bounded by all holomorphic (G~teaux-analytic and continuous) functions. That is, f(A) is a bounded subset of C for every C-valued holomorphic f on E.

212CHAPTER

9 BOUNDING

AND WEAKLY-BOUNDING

SETS

Generalized Josefson- Nissenzeig theorem

The fact t h a t the unit ball of an infinite dimensional Banach space is not limited (not weakly-bounding) is known also as Josefson- Nissenzeig theorem: Weak sequential convergence in the dual of a Banach space does not imply norm convergence. In what follows we extend their result to sequence non locally convex p-Banach spaces.That is to prove: "If E is an i n f i n i t e d i m e n s i o n a l p - B a n a c h s p a c e w i t h a S c h a u d e r b a s i s (0 < p < 1), t h e n t h e r e e x i s t s a w e a k * n u l l s e q u e n c e o f n o n z e r o p - n o r m v e c t o r s in E' ". 101 ( G e n e r a l i z e d J o s e f s o n - N i s s e n z e i g t h e o r e m ) Weak sequential convergence in the dual of a sequence p-Banach space (0 < p < 1)

Theorem

does not imply norm convergence. P r o o f . It is enough to consider the space E = lp. This is because E is a continuous image of lp, by a continuous linear mapping T, t h a t is, E - T(lp), and E' ~_ ( l p / M ) ' ~_ M • (seeTheorem 96) In fact, E _~ lp/M isomorphic to a quotient space of lp, which has also Schauder basis, see[[186],p.65]. T h a t is (lp/M)' are the continuous functionals on lp which vanish on M . We claim that there exists a set D in E such that lim Cn(x) = 0 pointwise in E

but

n

lim r

:fi 0 uniformly for x E D

n

Since bounded set may not be limited (i.e. not weakly-bounding ), [[13], Theorem3.2] the proof can also be given by constructing a such type of set. Consider the following set of unit vector in lp: D =

j X}.

(9.66)

D is bounded in lp. But it is not weakly-bounding in lp because D is not bounding in l l. Remember that bounding and weakly- bounding sets in 11 are the same. See[J13], Theorem3.2 and the diagram p.333]. Hence weak* convergence does not imply norm convergence in E'. This completes the proof. II Gelfand-Phillips spaces A space in which limited sets are relatively compact is known as GelfandPhillips space, see[126]. So far, it seems that such type of spaces d o n o t

9.3 P R O P E R T I E S

OF BOUNDING

AND

LIMITED

SETS

213

e x i s t in our study of the theory of bounding sets in topological vector spaces without convexity condition. The reason for this might be that the concept of bounded and weakly bounded sets are different, contrary to the cases of locally convex spaces. Theorem

102

Let E be a separable p-Banach space isomorphic to a quotient space with a Schauder basis. Then E is not a Gelfand-Phillips space. P r o o f . In [[16],Theorem 3.2], we have proved that, a weakly-bounding (limited) set of E may be unbounded. Consequently, our limited sets are not necessarily relatively compact. Thus E is not Gelfand-Phillips space. I C o r o l l a r y 18 ( I m a g e

of Banach

S p a c e s ) T h e locally pseudoconvex F-

space l o - N lp

(9.67)

p>o

is not Gelfand-Phillips spaces with respect to the sup F - n o r m oo

II x IIo:Supll x lip=sup ~

p>o

where

p>o j=l

Ixjl p

x -- ( xj ) E lp.

Proof. The bounding subsets of lo are relatively compact since lo has polynomially convex balls. Notice that the function f ( X l , . . . ,xn) =11 ( e ~ , --- , e ~ ) I1~ois convex for x - ( X l , . . . ,xn) E ~ " , n E N, see the author[[11],

p. 17]. The limited (weakly-bounding) subsets of lo could be unbounded. example the set A-

{

x(~); x ( ~ ) - ( 1 ,

(1) 1/p ~

(1) lip ,...,

For

} ,0,...), nEN

is weakly-bounding in lo for it is weakly-bounding in each lp. Notice that lo is dense in each lp(O < p < 1). Moreover A is not bounded in lo :

n(_.~)l/p.p

II x(n) Illo--sup ~

m-1

n 1

= ~

- - + oo

rn-1 m

as n ~ cx~. Consequently a weakly-bounding set in lo may not be compact.

214CHAPTER

9 BOUNDING

AND WEAKLY-BOUNDING

SETS

35 The importance of this F-space lo is that every separable Banach space E is isomorphic to a quotient space of lo; i.e. for a subspace M of lo, we have E ~_ lo/M. (9.68)

Remark

In what follows we prove that the F-spaces lp~ which may be locally bounded or not (this depends on (Pn)) are not Gelfand-Phillips spaces. Theorem

103 (Sequence F - S p a c e s ) T h e spaces oo

< ~ } , (1 > pn > 0)

lpn - - { X - - (Xn); X n e (~, ~

1

are not Gelfand-Phillips spaces Proof. F i r s t : I f pn /---~0, then each of lpn is metrized by a p-norm and lp~ become sequence locally bounded F-spaces. Consequently limited (weakly-bounding) sets in lp~ are not necessarily relatively compact, see [[12], Theorem3. 2] where we have proved that limited set could be unbounded S e c o n d : If pn ~ 0, the spaces lp~ are non locally pseudoconvex F-spaces. We claim that they are not Gelfand-Phillips spaces neither. We have to choose an unbounded set which is bounding or weaklybounding. In fact, the unit ball as we have seen [12] are bounding. Remember that these spaces are Schwartz and hence Montel spaces. Therefore every bounded closed subset of them is bounding. Let us consider the following set

A -

1

x(~); x ('~) - (1, (~)

1/pl

_1)1/pn, 0,. . .) , n o N } . '""(n

(9.69)

This set is unbounded; notice that

f;x

n

1 ----+ (:~

II1

m

aS Tt----+ 00.

However, A is weakly-bounding, i.e.

lim On (x) -- 0, for every x e lp,~ :=~ lim II r n----~

n----~(x)

IIA-- o,

(9.70)

9.3 P R O P E R T I E S

OF BOUNDING

AND

LIMITED

SETS

215

This is because the dual of lp~ is loo and A is weakly b o u n d e d in /1; we note t h a t O(3

lim sup ~ Ixjl-- O. n x C A .J - - n This completes the proof. II R e m a r k 36 Referring to the preceding theorems 102 and 103, we have seen a limited set may not be bounded( see [[12], Theorern3.2] ). Recall that the intersection between the classes of limited sets and of bounded sets is bigger than the class of bounding sets; consider for example the set of

B-

{

x(~) ; x ( n ) - (

/1) 1/p (1) 1/p -n

' ... '

-n

' 0,...),

}

of lp . Hence our space E - lp will have b o u n d e d and u n b o u n d e d limited sets which are not relatively c o m p a c t with the priority t h a t all its b o u n d i n g sets are relatively compact(see[16]). T h e following example explains the preceding theorems. E x a m p l e 3 7 A limited set may be unbounded by Theorem 103 and [[12],Th.3.2]. That is an unbounded set D may exist such that all holomorphic functions of type 1

will satisfy II r liD< C E*. {Sm}) To show this, let E - l(p~), inf Pn --~ O, be a locally bounded space. We now give an example of a function g (x) -- ~ r

C H (E)

such that g lD is bounded.

and a non-bounded Limited subset D of E Set (x)

-

and let 1 D - { x (~) ;x (n) - (0, (21og22)I/p

~---~

(

1 )liP 0 , . . . ) , n o N } n log2n '

216CHAPTER

with P=n p (1 -

9 BOUNDING 1 ~ log log n ]

AND

andO
WEAKLY-BOUNDING

SETS

Obviously p--~ p r 0

1

n

i.e.pn -~ O. N o w D C E is unbounded:

J II

x(J) II- d(x(J), O)-

l x(nn)lP rt--1

1 p( 1 - log log n )

nlog 2 n

n=l

~e 2 ---

--

n--1

as j ~ oo.

log n e ( log log n

--2 log log n)

n

--+00

But

J

cn

n= 1

--

1 ] ,~/p n log 2 n

< c<~

(9.71)

f o r every x - (x,~) e D . Thus g - y~ r < (x~ on the unbounded set D. On the other hand D is a weakly-bounding subset of l(p~) since it is compact in

11" n--<X~ lim sup x ~D

~j~n

[xj[ -- O.A

Notes and Remarks The following properties of Banach spaces are not similar to those we have already obtained in this paper and in [[15], [16], [17]], for at least, the non locally convex sequence p-Banach spaces. It is known in Banach spaces E that, if E is a separable Banach space then limited sets are relatively c o m p a c t . T h i s is contrary to([12],Theorem2.3). Further limited sets in E are b o u n d e d which is contrary to ([13]Theorem2). Now the set {en; n >_ 1} of unit vectors is limited if E = loo, but not if E=co. This set is in fact is limited in lpn only if Pn ~ 0,because it is closed and b o u n d e d in a Montel space and hence it is compact; but not in/p~, p= / - , 0 (0 < Pn < 1)because they are p-normed spaces for some (0 < p < 1), see T h e o r e m 103. Therefore, some m a t h e m a t i c i a n s paid a t t e n t i o n to the nonseparable Banach spaces. For example Josefson [124] has constructed a Banach space

9.4 H O L O M O R P H I C C O M P L E T I O N

217

in which every bounding set is relatively compact but which contains a closed non compact limited set. Independently, T.Schlumprecht[188] has also written a paper entitled: A limited set which is not bounding. Thus it is noteworthy to say that, in general, the theory of infinite dimensional holomorphy in locally and non locally convex spaces is once more different. We may need to invent and discover new criterion and terminology to solve the new problems that may appear during our study of Holomorphy in the absence of convexity condition and non linearity.

9.4

HOLOMORPHIC

9.4.1

COMPLETION

H o l o m o r p h i c C o m p l e t i o n in F - S p a c e s

Let E be a F-space whose dual be its linear completion (It is also the maximal space t o w h i c h all e x t e n d u n i q u e l y as c o n t i n u o u s

E' separates the points of E. Let /~ called the Banach envelope). That is, continuous linear functionals on E linear functionals.

Kalton [131] has studied the linear completion of some quasi-Banach spaces( i.e. for complete quasi-normed spaces). In analogy with the case of locally convex spaces we define the holomorphic completion.

Definition 64 Let F, be the linear completion of a topological space E whose dual separates points. The maximal subspace E5 of the linear completion to which all holomorphic functions f E H(E) (continuous and GOteaux analytic) are extended analytically is said to be h o l o m o r p h i c completion. E5 is well defined since any f E H(E) can be extended analytically to an open neighborhood ~ C_ E as can be shown by using Taylor series expansion of f at the points of E, see, Ch.. 7, section 7.2 It follows that each f has a unique m a x i m a l d o m a i n o f e x i s t e n c e ~fC_E. ^

Hence

E~ - n { f~:; f E H ( E ) }

(9.72)

and N { f~f; f E H ( E ) } is a vector subspace of E. E is called holomorphically complete if E-

E~.

(9.73)

218CHAPTER

9 BOUNDING

Now the linear span F

AND WEAKLY-BOUNDING

SETS

of the basis {e~; n C N}

of a Banach space E is proved to be holomorphically complete, by Noveraz

[~80]. In what follows we extend this result to some metrizable spaces to obtain examples of non complete holomorphically complete non locally convex spaces. T h e o r e m 104 Let E be metrizable space with a Schauder basis (en) and plurisubharmonic logarithmic metric. Then the linear span F of ( en ) , i.e.

F - span { en ; n C N }

(9.74)

is holomorphically complete. P r o o f . Assume t h a t log d(x, O) is a plurisubharmonic function on E where d is the metric defining E. Let

7r,~ " E---* En the projection of E into the first n basis vectors. Let y E E / F and choose a sequence (en), Cn > 0 such that (x)

E

en log d(Tcny, y) > - c o .

n=l

The function (:x:)

n(~) - ~ ~ log d(..x,

x).

xCE

(9.75)

n=l

is plurisubharmonic and hence

- {x e E; R(x) < R(y)} is a pseudoconvex domain of E. We notice that F C_ ~, since R ( x ) ~ - - o o , for x ~ U ~ ( E ) - F.

9.4 H O L O M O R P H I C C O M P L E T I O N

219

Now apply the Levi problem solution given by the author[12], [13],and [14], (see also Ch.10 ,subsection 10.1.1, p.231). We note ft is a domain of existence of a holomorphic function f E H(E). Hence the holomorphic completion F5 is contained in ft to have in particular y ~ Fs. Since y E E / F is arbitrary we conclude that

F-F~. This completes the proof of the theorem. I

Examples of Holomorphically Complete F-spaces Example 38

The complex spaces

l(p~)- { x--(Xn);~--~.lxn IP~< cx~ ,

0 < Pn <_ 1,

(9.76)

1

l o - np>olp-

x-

(Xn); sup p>0

I xn IP< c~

,

(9.77)

1

lp -- Uq<_plq -- { x E lp; x E lq for some q _< p},

(9.78)

where (X)

I xj Iq , q _ 1

d(x, 0) -- inf dq(x, 0) -- inf E q

q

(9.79)

1

and

l+p - nq>_plq -- { x e l ;suP lXj

,0 < p < _ 1

(9.80)

q>-P 1

are non locally convex spaces which satisfy the hypotheses of the above theorem. Notice that lp C l+ C lp+e.

(9.81)

220CHAPTER

9 BOUNDING

AND WEAKLY-BOUNDING

SETS

Relation Between Bounding Sets and Holomorphic Completion In what follows we give the relation between the union of the bounding subsets of a locally convex pseudoconvex F-space E and its holomorphic completion E.

T h e o r e m 105 [17](1989) If E is a locally pseudoconvex F-space with the bounded approximation property, then E D -- U A E B g ( E ) A -

where A orE.

is the closure of A, and Bg(E)

is the class of bounding subsets

For the proof of the theorem we need the following lemma.

L e m m a 24 Let E be a locally pseudoconvex F-space, with the bounded approximation property, f E H ( E ) , and A is a bounding subset of E. Then there exists a zero neighborhood of V in E such that

flfllA§ P r o o f . For f E H(E), and x E E

fz(Y) :

(9.82)

< we

define f~ : ~ - ~

by

f (x + y).

(9.83)

It is clear that f~ E H(E), for every x in E. Similar to the method of locally convex spaces, see Dineen [[82], proposition 4.22], we can show that the family (fx)~eA is t o - b o u n d e d subset of H(E), where ro is the compact open topology on H(E). In fact, if K is an arbitrary compact in E, then sup IlfyllA =

yEK

sup

yEK, xEA

IIf(x + Y)II = sup I]LIIK < xEA

Since E is metrizable, (fz)zEA is a locally bounded subset of H(E) and hence there exists a neighborhood V of zero, for example a ball B(0,e), such that sup IIfzIiv < oo.

xEA

Hence sup xEA,yEV,

] f ( x + y)I--]lfiiA+y < o o . I

9.4 H O L O M O R P H I C

COMPLETION

221

P r o o f o f T h e o r e m . If x C E5 then there exists (xn)~ C E such that x as n ~ oc. Since x C E5, Xn lim f(Xn) exists for every f E H(E), n----~ o o

and hence sup I f ( x ~ ) I < oo. n

Thus {xn} is a bounding subset of E

and x E UAEBg(E)A, that is

E5 C UACBg(E ) A.

On the other hand if A is a bounding subset of E and f E H(E), then by the above lemma a pseudovonvex balanced neighborhood V of zero exists in E such that LI/IIA+v < o o . By using the Taylor series expansion about points of A we find that there exists a holomorphic function f ~ on A + W such t h a t f A+W -- f A+V

where W is the interior of the closure of V in E. As A C A + W, we will have A C_ E5 and hence E D - UACBg(E)A-.

This completes the proof of the theorem. I

9.4.2

Holomorphic Extension Problem

In this section we study the holomorphic extension problem which is purely an infinite dimensional problem in some F-spaces : Let F be a subspace of a locally pseudoconvex metrizable space E, whose dual E' separates the points of E. " W h e n can every h o l o m o r p h i c function on F be e x t e n d e d to a h o l o m o r p h i c f u n c t i o n o n E ?"

222CHAPTER

9 BOUNDING

AND WEAKLY-BOUNDING

SETS

Holomorphic Extensions of Dense Subspaces The following theorem gives a counterexample to holomorphic extension problem. More precisely, it supplies us with a dense subspace which cannot be extended to the whole space all holomorphic functions.

Theorem 106 [19](1989) Not every holomorphic function on lo

-

Np>olp

with the sup-topology, that is, the one defined by the F-norm OG

jixll- sup p>0

,xo ,p 1

can be extended to a holomorphic function

on

lp

(1 > p > 0).

P r o o f . lo is a dense subspace of each lp (1 > p > 0), by Stile[[201],p.117]. Hence every bounding subsets of lo is bounding as a subset of lp. Since bounding subsets of lo and of lp are relatively compact, by Theorem 90, p.186, it suffices to pick up a non compact subset of lo which is compact as a subset of lp. For example the set

D-

~0,x(1),x(2),...~,

where

(0,. ,0,( n1 )I-l/p, 0,...), n E N

(9.84) (9.85)

is compact in lp if (1 > p > 0), but it is not compact in lo. Notice that oo

]lx(n)llo

-

-

s u p ~ - ~ . l x n Ip p>O 1

= sup(-1 )(1-1lp)p

----

n 1-p --+ oo

p>0 ?%

as n ~ c~. This completes the proof of the theorem, m As another counter example where the holomorphic extension fails to occur from a non locally convex subspace to a locally convex one we obtain the following result.

9.4 H O L O M O R P H I C

COMPLETION

223

T h e o r e m 107 If f is a holomorphic function on lp (1 > p > 0), then f may not be extended to a holomorphic function on 11. That is, not every holomorphic function on lp can be extended holomorphically to its Banach-envelope 11. P r o o f . The set A -

1 )i/p

x(~); x (~) - ( ( n

is not bounding as a subset of lp, for it is not relatively compact. However it is bounding as a subset of 11. See also proof of T h e o r e m 95. This completes the proof of the theorem, m

R e m a r k 37 It follows from the preceding theorem and the fact that the linear completion l~ - 1 1 that

lp C lpt . , also that ( lp)5 = lp. Holomorphic Hahn-Banach Extension Theorem We consider the second type of holomorphic extension problem : " W h e t h e r eve r y h o l o m o r p h i c f u n c t i o n d e f i n e d o n a c l o s e d s u b s p a c e F of a c e r t a i n F - s p a c e E can be e x t e n d e d analytically to E". Since there exists a continuous linear functional on a subspace of lp (1 > p > O) which cannot be linearly extended to lp, by Stiles [201], we will deduce the following theorem.

T h e o r e m 108 Let M be a closed subspace of lp (1 > p > 0) whose unit ball is ll-precompact. Then not every holomorphic function f in H ( M ) can be extended analytically to lp. P r o o f . By the result of Stile[[201], p . l l l ] , where under this assumption one can find a continuous linear functional on a subspace of lp (1 > p > 0) which cannot be linearly extended to lp. m Consider now the following normalized sequence (u~) which has disjoint supports:

224CHAPTER

9 BOUNDING

=

u2 -

AND WEAKLY-BOUNDING

SETS

(1,0,..),

1 1 (0, (~)I/P, (~)l/p, 0, ...), ..

Un--(O'""(n

1 )l/p

'

ooo~

(1)l/p, 0,. ) n ~

oo

(9.86)

nth-place Then we obtain the following theorem. T h e o r e m 109 Let M be a closed subspace of ~ which is spanned by the above sequence {u~; n E N } . Then not every holomorphic function on M can be extended analytically to 11. P r o o f . The set {Un;nEN}

is not bounding in the subspace M, for it is not relatively compact. On the other hand it is compact in 11. Hence it is bounding in 11. m R e m a r k 38 The above theorem implies that M~ = M r lp. The locally convex space l~ has the property given by this theorem. This is because the set D = {ej; j C N } o f unit vectors in l ~ is bounding in l~ but not as a subset of Co. That is the holomorphic completion satisfies

(Co)5 5r lc~

(9.87)

Discussion We have studied some problems which arise in infinite dimensional complex analysis. In fact, the study of bounding and weakly-bounding subset of non locally convex spaces provides information which is of fundamental interest in Holomorphy. There are essential relations between this study of this chapter and the radius of convergence problem suggested by Kisleman [134]. That is,

9.4 H O L O M O R P H I C

COMPLETION

225

"To c o n s t r u c t a h o l o m o r p h i c f u n c t i o n w i t h p r e s c r i b e d r a d i u s of c o n v e r g e n c e " and the Levi problem, which will be discussed in the next chapter, that is, to answer " W h e t h e r e v e r y p s e u d o c o n v e x d o m a i n is a d o m a i n of h o l o m o r phy". For example we can easily deduce that there are entire functions of H(E) with finite radius of convergence after we have characterized the bounding subsets to be relatively compact, see the examples of this chapter. In addition we can check : " W h e t h e r t h e Levi p r o b l e m will n o t h a v e a s o l u t i o n " if we can find a certain type of bounding subsets domain of E as Josefson [121] has done for the space loo. This chapter indeed explores an important point that : " N o t e v e r y h o l o m o r p h i c p r o p e r t y in locally c o n v e x s p a c e s c a n b e i n h e r i t e d b y n o n locally c o n v e x s p a c e s " . One of the reasons behind that is, for example, that the boundedness with respect to the original and the weak topologies are not the same. This has been explained via the study which has been held for the weaklybounding sets. It is hoped that the present study in this chapter will be helpful in claiming some achievments in pure and applied mathematics. The next chapter discusses the different approaches which are used to solve the Levi problem in seprable topological vector spaces. The following figure explains the Levi spaces among this class of the separable toplogical vector spaces.

Levi Problem in Separable t.v.s. Separable t.v.s.

F-Spaces with plurisubharmonic

tp(1 > p > 0)

Locally pseudoconvex spaces(Lps) with b.a.p.

~logarithmic metric with f.d.Schauder decomposition

/.C.8.

Banach spaces with b.a.p.

9

t~

" I p n ~ p r t "-"~ 0~

Ii x II=

Inductive spaces Up>tip ,

II 9 II= ~up~>0E~

~,~%~ Ix,, ITM Lps N F-spaces

loo 9is not a Levi space. Lp :(1 > p > 0) : is not aLevi soace. Figure 8 : S o m e L e v i s p a c e s a m o n g s e p a r a b l e t . v . s .

I~

I~

Chapter 10

LEVI PROBLEM IN TOPLOGICAL

SPACES

One of the important and interesting problems in the field of holomorphy is the Levi problem. To solve the Levi problem means to prove that the class of holomorphic functions has a certain type of richness. In one complex variable, given a domain ~ there exists a holomorphic function f in ~ which cannot be continued beyond the boundary of ~ as a holomorphic function. For instance we may prescribe the values f(zk) arbitrarily if (zk) is a discrete sequence in ~. This means that ~ is the natural domain of existence of f . In two complex variables, we have already domains like

no - {z ~r

1 <1 z I< 2},

which are not the natural domain of existence of any holomorphic function. That is, every holomorphic function in ~o can be extended holomorphically to

{z e~,~ I z I< 2

}.

E.E.Levi discovered in 1911 that the boundary of the natural domain of existence of every holomorphic function should satisfy a c e r t a i n c o n v e x i t y condition. He wondered:

W h e t h e r a n y d o m a i n w i t h this c o n v e x i t y p r o p e r t y is a n a t u r a l d o m a i n o f existence. In today's terminology he asked : 227

228CHAPTER

10 L E V I P R O B L E M

IN T O P L O G I C A L

SPACES

W h e t h e r a p s e u d o c o n v e x d o m a i n is a d o m a i n o f e x i s t e n c e ( o r a domain of holomorphy). Oka [182] proved the answer is affirmative in (T2; He, Norguet and B r e m e r m a n n [179] extended the result to (T~. For Hilbert spaces the Levi problem was solved by Gruman [113]; for Banach spaces with basis by Gruman ~Kiselman [114]. Dineen, Noverraz and Schottenloher [83] solved the problem in certain locally convex spaces. The problem has been solved for R i e m a n n domains spread over locally convex spaces with Schauder decomposition by Schottenloher [197]. The levi problem in locally convex nuclear spaces was solved by Coloumbeau ~ Mujica [78], thus exhibiting the first solution in a space which does not have a Schauder decomposition or is a direct subspace of a space with this propery. As for the negative results, Josefson [121] gave a c o u n t e r e x a m p l e : a pseudoconvex domain in the non-separable Banach space lc~(A) which is not a domain of holomorphy. In this chapter we explain our extensions of these results to many classes of topological vector spaces which are not necessarily locally convex. It is worthy to point out here that : The author has obtained the corresponding property in 1979(or the geometric condition ), corresponding to what Levi has discovered, but for domains of non locally convex F-spaces. He has called it the P B - p r o p e r t y and asked : W h e t h e r a p s e u d o c o n v e x d o m a i n w i t h the P B - p r o p e r t y is a domain of existence (or a domain of holomorphy). These spaces are called P B - s p a c e s cussed here in detail.

(or Bayoumi

spaces). It is dis-

Therefore in this chapter, we focus our attention on one of the central and important problems in this field of holomorphy without convexity condition, that is, the L e v i p r o b l e m . This problem will be studied in different classes of non locally convex spaces having finite dimensional Schauder decomposition (f . d. decomposition) or bounded approximation property (b. a.p.). In particular for locally bounded sequence spaces. We also discuss the different approaches that are used to solve it in those cases. That is, "To p r o v e t h a t e v e r y p s e u d o c o n v e x d o m a i n is a d o m a i n o f holomorphy".

10.1 LEVI

10.1

PROBLEM

AND

RADIUS

LEVI PROBLEM CONVERGENCE

OF CONVERGENCE

AND

RADIUS

229

OF

In this section we study the solution of the Levi problem using the radius of convergence approach. If E is a complex metric space and ~t is open in E, the radius of convergence Rf(x) of a holomorphic function f: ~ --~ (T is a positive number which measures the growth of f near x, to be precise

R f ( x ) - s u p {r > O; Taylor series of f at x converges uniformly in B(x, r) } If

E

is of finite dimension,

i.e. E - ( ~ ,

the radius of convergence

Rf(x) of any holomorphic function f" ~ --~ ~T is greater t h a n or equal to the distance to the boundary dn, i.e.

R: >_da where inf I I x - Y l l x e Ft. da(x) - ycOa

In particular Rf = c~ for every entire function on ~Tn. If E is of infinite dimension, this is not always the case. Dineen [88] has noticed the existence of entire functions with finite radius of convergence. This fact also follows easily from the study which we have undertaken on bounding and weakly-bounding sets in some separable F-spaces E, which are not necessarily locally convex, see the preceding chapter 9. In fact, there always exists an entire function f

on

E such t h a t

sup I f ( x ) I - c~ xEA

for every closed non-compact set A in E, provided that E is of the kind we considered in Ch.9. As an example is the function (X)

f(x) - ~-~ x~ 1

which is an entire function and has finite radius of convergence

Rf(x) - 1, every x E lp

230CHAPTER

10 L E V I P R O B L E M

IN TOPLOGICAL

SPACES

not only for 1 <__p _< c~, but also for 0 < p < 1. It can even happen t h a t inf R S (x) - 0

xEA

for some f E H ( E ) , and some bounded A in E, see Theorem 112,p.238 and examples 41,p.233. In this P a r t we start with the radius of convergence approach to solve the Levi problem. It is then natural to get as a consequence of having a solution to the Levi problem, several other results in Complex and Functional Analysis. Let E denote either a locally pseudoconvex space or an F-space D e f i n i t i o n 65 .A function f " U --~(T on an open subset U of E is said to be h o l o m o r p h i c if it is continuous and Gdteaux analytic i.e. f is analytic on every complex line of U. D e f i n i t i o n 66 An open subset U of E is said to be p s e u d o c o n v e x if - log 5u (x, a) is plurisubharmonic in U x E,

where 5v(x, a) -

inf{[ A [; x + Aa E OU}, x E U, a E E, A E ~T

(10.1)

= sup {r; x + rDa C U } , x e U, a e E, D unit disc in (/(10.2)

i.e. for every finite dimensional subspace F of E, the open set (U M F) pseudoconvex in F, see Noverraz[181]. The class of these subsets of E denoted by P C X ( E ) .

is is

D e f i n i t i o n 67 A function u " U --. [-oc, oc], where U is open in E, is said to be p l u r i s u b h a r m o n i c if it is upper semi-continuous and subharmonic on every complex line. The class of these functions is denoted by P S H ( U ) . D e f i n i t i o n 68 An open set U of E is called a d o m a i n o f existence, if there exists an f E H(U) which cannot be continued analytically beyond the boundary OU. D e f i n i t i o n 69 A space E is said to b a L e v i space if the Levi problem has a solution for E.

10.1 L E V I P R O B L E M A N D R A D I U S OF C O N V E R G E N C E 10.1.1

231

PB-Spaces

A metric d on a complex metric vector space E is not always h o m o g e n e o u s or even p-homogeneous, i.e. we do not necessarily have

d(tx, O) - ItlPd(x, O) (0 < p _~ 1 ) , t C ( ~ , x E E. So we found in 1979 t h a t it was useful to introduce the following concept in h o l o m o r p h y during the s t u d y of the Levi problem 9 70 A complex metric vector space E with a translation invariant metric d is said to have the p s e u d o c o n v e x b o u n d a r y d i s t a n c e p r o p e r t y ( P B - p r o p e r t y for short) if for every pseudoconvex domain f2 in E, it follows that the function

Definition

x ~ - l o g dn(x) is plurisubharmonic in ~.

Such a space E is also called a P B - s p a c e , where dn(x) -

inf d(x, y) yEOf2

(10.4)

is the distance to the boundary O~ of ~. Note t h a t - log da(x) is not necessarily plurisubharmonic whenever ~ is pseudoconvex. Example

39 The space ~Tn is a P B - s p a c e for any norm,

see e.g. Hormander [118], where it is only assumed that the distance is measured by a 1-homogeneous function which is positive except at the origin. Moreover a p-normed space E is a P B - s p a c e . This is easily seen from the definition of pseudoconvexity using the direction boundary distance defined by d(x, y ) = ] I x - YlIB, where Iltxll = ItlPllxlI.A Example

40 The space

lp - Uq<_plq - {x C lp;x C lq for some q <_p} with the metric (2<3

d(x, 0) - inf dq(x, 0) - inf q

q

Ixl q (q j=l

p

1)

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is a PB-space. Notice that if U is pseudoconvex in lp,

SPACES

then

- logd u ( x ) -- sup - 1Ogqday (x) q

oO

= sup s u p ( - l o g ~ Ixj q

Y

yjl q)

-

j=l

is plurisubharmonic as supremum of plurisubharmonic functions, of course d is not p-homogeneous for any p, (0 < p <_ 1). A

The following theorem gives a sufficient condition for a metric d to have the PB-property. T h e o r e m 110 [12](1979) Let d be a translation invariant metric on a complex metric vector space E such that each of the functions (10.5)

f~ " s --+ - l o g d ( e * x , 0), s E r

is convex in (F for each fixed x E E. Then (E, d) is a PB-space. In particular, this is the case if d is given by a p- homogenous norm ]1. ]]p, i.e. d ( x , v) - IIx - vll .

P r o o f . Assume that f~ is an arbitrary pseudoconvex domain in E. We claim that - l o g d a is plurisubharmonic in f~. Since the function (9.5) is convex and decreasing, it has the Legendre transform satisfying f:r,(s) -- s u p ( r s -

~x(r)),

x E E, s E/~,

(10.6)

fz(s)),

x e E, r e

(10.7)

r<0

tgz(r) -- s u p ( r s 8

T h a t is,

- l o g d ( e S x , O) -

s u p ( 7 - s - ~(~-,x)),

x E E , s E/~,

r<0

~ ( r , x) -- s u p ( r s + log d(eSx, 0)),

x E E , ' r E 1~.

8

Changing the notation by putting s - log Itl, we have

10.1 LEVI

PROBLEM

AND

d(tx 0) -

RADIUS

OF CONVERGENCE

inf Itl'~(~ - x) T>0

x e E , t e(T,

233

(10.8)

~

~(7, x) = sup Itl-'d(tx, 0),

x e E , T e ~.

(10.9)

tc--r

Note t h a t if d(x, 0) is p-homogeneous, i.e. d(tx, O) = ItlPd(x, 0), t e r x E E, 0 < p _ < 1, then we get ~ ( p , x ) = d ( x , 0) w h e n T - = p o r o t h e r w i s e

~(~,

x) = +oo.

Now

da(x) = =

inf d(x,y) =

y60~2

inf

x+tz60~

inf

x+yEOf~

d(y,O)

d(tz, 0) - i n f inf inf It] ~ (~- z) z

t

T>_O

~D

= inf inf inf Irish(w, z) - inf inf ~(~-, z)Sa(x z) ~ z

T>0

t

Z

T>0

for 5a(x, y) = inf(ltl; x + ty e Oft). Hence - log dfl(x) = sup s u p [ - log ~(~-, z) - 7 log 5a(x, z)] z

(10.10)

T>0

and since this function is either - - c o or else is continuous, it is plurisubharmonic as the s u p r e m u m of a family of plurisubharmonic functions. I R e m a r k 39 In the above theorem we have used the following property of a pseudoconvex domain f~ C E : if ~ C E and E is a topological vector space, then f~ is pseudoconvex if and only if the function

(x,y) H - l o g S a ( x , y )

is plurisubharmonic on ~ x E

w h e ?~e

5a(x, y) = sup(r; x + r D y C ~ ) , x E t~, y E E,

(lo.11)

D = {t C ~ ; Itl _< 1}. In fact, this is equivalent to our definition of pseudoconvexity, i.e. ft N F is pseudoconvex in F for every finite-dimensional subspace F of E, see Noverraz [ [181], Lemma 2.1.5]. I As an application of the above theorem we get Example

41

A complex vector space E

d(x, y)

with the metric given by

llx-yll - 1 + IIx-

YlI'

(10.12)

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where

I1 II i~ p-homogeneous norm, (0 < p < 1) is a PB-space. that the function

-

log d(etx, O) - log(1 +

Iletxll)

- log

is convex in t ~T for every fixed homogenous.A

Iletxll

- log(1 + oze-ltlp),

x C E. Notice that

d

We note 1 OZ-'-

IIxll

is not p-

40 According to the definition of a PB-space E: for every pseudoconvex domain f~ in E, - l o g dn is plurisubharmonic in ft. The converse is also true: Indeed, assuming that - l o g d~ is plurisubharmonic in ft and F is an arbitrary finite-dimensional subspace of E, f~ N F is pseudoconvex in view of the classical properties of plurisubharmonic functions of finitely many variables. Hence f~ is pseudoconvex. Consequently if E is a PBspace, then f~ is pseudo convex if and only if - log da is plurisubharmonic in f~.

Remark

10.1.2

Properties

of the Radius

of Convergence

Let us recall that the r a d i u s o f c o n v e r g e n c e R s ( x ) of a function f E H(f~) at a point x E f~ is the least upper bound of all numbers r > 0 such that the Taylor series of f at x converges uniformly in B ( x , r ) , the closed ball of centre x and radius r in E. Also the r a d i u s o f b o u n d e d n e s s Rb(x) of a numerical function

at a point x C f~ is the least upper of all numbers r > 0 bounded above in B ( x , r ) with B ( x , r ) C f~.

such that

u is

For a function f 6 H(ft),

-

sup(r > O; IlflIB( , )

is finite, B ( x , r ) C ~),

x E ~.

(10.14)

By these definitions of the radius of convergence R I and the radius of boundedness R} for a function f C H(ft), it is easy to prove t h a t

R} - inf(Rs, da)

(10.15)

where da(x) - infyeE/a d(x,y), x C f~, is the distance function on f~ defined by the metric d of E.

10.1 L E V I P R O B L E M

AND

RADIUS

OF CONVERGENCE

235

It may happen that f can be continued analytically beyond the boundary of Ft and hence R} will be less than R I (i.e. RbI <_ RI). It is clear that the radius of boundedness R b for any numerical function u is globally Lipschitzian continuous. However, the radius of convergence is only locally Lipschitz continuous. We formulate these results in what follows. L e m m a 25 If u

is any numerical function defined in a subset f~ o f a metric space E we define u(x) - +oo, Rb(x) - 0 for x e E / a . With this convention we have

I n~(x) - n~(y) ,< d(x, y) P r o o f . If R b ( x ) -

Rb(y)

for all x, y C E .

(10.16)

the conclusion is valid; let us assume that

n~(x) > n~(y) > 0. Then u is bounded above in B ( x , r - d ( x , y ) ) C B ( x , r ) for all r < Rb(x). Hence Rb(y) >_ r - d(x, y) and letting r tends to Rb(x) we get

R b (x) > R b (y) > R b (x) - d(x, y). This proves (10.16) in this case, and by s y m m e t r y the estimate holds everywhere, m L e m m a 26 If f

vector space E

is a holomorphic function on a subset f~ of a metric with a translation invariant metric d satisfying

d(tx, O) then

<_ d(x, O)

for all x C E , t C(T, I t I<_ 1

I n~(x)- n~(y)I<_ d(x, y), fo~ all x,y e a,

(~o.~7)

and d(x, y) < sup(df~(x), dry(y)). P r o o f . By s y m m e r t r y it is again enough to consider the case

hi(x) > h i ( y ) > o. Let w be the open ball B(x, R f ( x ) ) ~ and denote by g the extension of f l B ( x , da(x)) ~ to w. Then R g ( x ) = R l ( x ) and g is bounded in

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B(x, r) for all r < R l ( x ). Now, if R l ( x ) <_ d(x,y) we are done; assume t h a t R l ( x ) > d(x, y). T h e n y C w and if d(x, y) < d~ (x) or d(x, y) < d~ (y) the whole segment between x and y is contained in w A ~ must have Rl(Y ) - Rg(y). By the preceding lemma,

so we

R ,b( y ) > R ,b( x ) - d(x, y) - R z ( x ) - d(x, y). In conclusion we have

> R f ( y ) - Rg(y) > R gb ( y ) - d(x,y) = Rf(x) - d(x, y)

Rf(x)

t h a t is, (9.16) is proved. This completes the prove of the lemma, m The following result shows t h a t R~ for f E H(f~) admits a certain geometric property which is a consequence of the PB-property. Theorem

111

Let ~ be a pseudoconvex domain in a PB-space. Then - l o g R b is plurisubharmonic in

(10.18)

if u is plurisubharmonic in Consequently -logRbf is plurisubharmonic in

(10.19)

for every f E H(gt). P r o o f . Let u be a plurisubharmonic function in f~. Let

~

- {x e ~; ~(x) < k}, k e N.

(~0.20)

For every k E N, the set f~k is pseudoconvex. This follows from the definition of pseudoconvexity and the fact u is plurisubharmonic in f~. Let

d~k(x ) -- inf d(x,y),

x e f2k, k C N.

YEO~ k

Hence

R b - supd~ k k

lira d~k, k-~

(10.21)

10.1 L E V I P R O B L E M

AND

RADIUS

OF CONVERGENCE

237

that is - log R b - ir~f( - log d~ k).

(10.22)

The functions - l o g d~ k are plurisubharmonic in f~k for every k C N, since ftk is pseudoconvex and E is PB-space. Thus - l o g R b is plurisubharmonic as a decreasing limit of a sequence of plurisubharmonic functions. Since log l f ] is plurisubharmonic in ft for every function f E H(ft), we also get b ] ---- - log R} - log/i~loglf

(10.23)

is plurisubharmonic in Ft. I Let

Ep

be a complex vector space with the p-homogeneous norm

II.llp d(x,y) -II x - y l l p ,

such that

IItxllp -I t lp d(x, 0),

xEE, tE~. The following Lemma gives a formula for the radius of convergence R I for I E H(ft), ft C Ep open, which is well known for normed spaces, i.e. the 1-homogeneous spaces (see chapter 7). The given formula is called the pCauchy-Hadamard formula(or the Generalized Cauchy-Hadamard formula).

(p-Cauchy-Hadamard formula)The radius of convergence R S for f c H ( f ~ ) , ft is open inEp (O
Ri(x ) - l i m i n f IIP ll- , x e a

(10.24)

where Ilrnll-- sup

Ilxll
I rn(x) l

and Pn is defined by the formula f(x + y ) - ~ Pn(y), Proof.

p.139). I

y E Ep, Ilyll is small.

(10.25)

Similar to the one given in Theorem 68 (Ch.7, subsection 7.2.3,

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SPACES

The following theorem solves the radius of convergence problem stated by Kisleman [135], [136] 9 "Can one construct a holomorphic function with a prescribed radius of convergence?". D e f i n i t i o n 71 A topological vector space E is said to have a f i n i t e dim e n s i o n a l S c h a u d e r d e c o m p o s i t i o n (rid. d e c o m p o s i t i o n ) if there is a sequence of equicontinuous linear projections " E --,

n(E)

such that 7Tn ( E )

~ 00,

7~nOT~m ( x )

--

71"min(n,m)(x)

and x

--

lim ~n (x).

(10.26)

T h e o r e m 112 [12](1979) Let E be an infinite-dimensional complex metric vector space, with a translation invariant metric d and with a finite dimensional Schauder decomposition (~n ) , such that x ---, - log d(x, O) is plurisubharmonic on E. Let R " ft --~ ~+

be defined on a domain f~ c E - logR

with

p lurisub ha rmonic

and R <_ da.

Then, there exists a holomorphic function f E H ( ~ ) convergence

(10.27) with the radius of

(10.28)

R S _< R.

P r o o f . We shall construct, under the hypotheses of the theorem" a holomorphic function f on ~ such t h a t for a suitably chosen sequence (x~), which depends on the metric d, we have

I f ( x n ) I>_n,

nEN,

and R S < R

Here a covering (Vn) on ~ will be defined such t h a t

I[flly~ - sup [ f ( X n ) I < xC Vn

for

n e N.

(10.29)

10.1 L E V I P R O B L E M

AND RADIUS

OF CONVERGENCE

239

The proof can be completed as in the forthcoming Theorem 117, p.247, (see also the author [12land Schottenloher [198]). T h e o r e m 113 [12](1979) Let ~ be a domain in an infinite-dimensional metric vector space E with a monotone Schauder decomposition (~n) and with a translation invariant metric d such that log d(x,O) is plurisubharmonic function in E. Let R: ~ - - ~ + be such that

- l o g R plurisubharmonic in E

and R <_ d~

and I R(x)-

R(y)]<_ cd(x,y),

for x , y e ~, d ( x , y ) < d~(x)

and some c e ]0,1]. Then there exists f e H ( ~ ) K(c) > 0 depending on c such that

and a constant

m

(10.30)

K R <_R f <_ R. Without assuming (~n)

K

to be monotone we get

r

(10.31)

_< Rf < R,

where r is continuous with r depends on only d and (~n).

r

> 0 , for t > O which

Proof. See the author[[12],Proposition], m (For similar results in Banach spaces, see Schottenloher [198], Coeure [73] and Kiselman [134],[135] ). E x a m p l e 42 Consider the function f ( x ) - ~-~ x~. 1

Recall that it is entire and has finite radius of convergence R f ( x ) - 1 for every x E lp

not only for l <_p <_ oo, but also for O < p < l. We show that it can even happen that

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SPACES

inf R S (x) - 0

xEA

for some f E H ( E ) and some bounded set A in E (See Theorem 113, p.239 and Aron [~]). In fact, the problem of constructing holomorphic functions with this or similar properties is reduced to the much simpler problem of constructing plurisubharmonic functions with a certain growth by means of the foregoing result, Theorem 110. To illustrate how this result implies Aron's result [4]. We just take e > 0, and define r

= supn(I xn ] -e) +

x

=

e

n

Then r is convex, hence plurisubharmonic, on every lp, 0 < p < cx~,and still unbounded in every ball B(x, r) of radius r > e. An entire function f E H(lp) with radius of of convergence R I <_ e-r has the desired properties. A In a particular case, we have the following interesting result for all Fspaces /(p~)which they are either locally bounded when Pn /-+0 or non locally pseudoconvex metric spaces when Pn --+ O, see examples, p.85, 126. C o r o l l a r y 19 Let ~

and R

E-

be as in the above theorem, and

l(p~), (1 > Pn > 0).

Then there always exists an f E H ( ~ )

with

RS = R P r o o f . See the author[[12],Proposition 3.b.1,p.20].

(10.32) I

Regarding the richness of the space of holomorphic functions in some metric space we obtain the following result where To denotes the compact open topology on H(E).

10.1 L E V I P R O B L E M

AND

RADIUS

OF CONVERGENCE

241

T h e o r e m 114 [12](1979) Let E, f~, and R as in Theorem 111. Then the set

TI(p) - { f e H ( E ) ; n I < R.}

(10.33)

is sequentially dense in (H(E), T0). Moreover, if f in addition is locally Lipschitz continuous and the Schauder decomposition is monotone, then

T2(p) - { f e H ( E ) ; K R < R/ < R}

(10.34)

is sequentially dense in (H(E), To) for a constant K. P r o o f . See the author [[12] , proposition 3.c.l,p.29]. II

10.1.3

The Levi Problem

in PB-Spaces

In the preceding section we have discussed the problem of constructing a holomorphic function with a given radius of convergence. It has a relation with the Levi problem for some metric vector spaces E which contain in particular locally bounded sequence F-spaces. We have proved, in the preceding subsection 10.1.2, that for an open set U of E the radius of convergence, has the following properties: (i) -logRf is plurisubharmonic (if d is a PB-metric) and R/ <_ du. (ii) I R / ( x ) - R / ( y ) I <- d(x, y) if x, y e U and d(x, y) < sup(du(x), dv(y)). In fact, it is worth to note that condition (i) implies the pseudconvexity of U provided that E is a PB-space. Hence the problem of finding a holomorphic function with a prescribed radius of convergence is closely related to the Levi problem which can be formulated as follows: " G i v e n a p s e u d o c o n v e x d o m a i n U, d o e s t h e r e e x i s t a f u n c t i o n f E H(U) w i t h R / < dv, t h a t is, w i t h U b e i n g t h e d o m a i n of e x i s t e n c e of f ?" As a matter of fact this close relation was one of the reasons to tackle the radius of convergence problem in non locally convex spaces and to obtain the results of subsection 10.1.2, mainly Theorem 112, p.238. Now we give the solution to the Levi problem in PB-spaces as a consequence of Theorem 112.

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T h e o r e m 115 (Levi P r o b l e m i n PB-spaces)[12](1979) Let E be an infinite dimensional PB-space with f.d. decomposition such that x --, logd(x,O) is plurisubharmonic in E. Then every pseudoconvex domain U in E is a domain of holomorphy. P r o o f . Take R = du. Since U is pseudoconvex, - l o g R is plurisubharmonic. Applying Theorem 112, p.238, we find a holomorphic function f C H(U)with R I < R = du. This implies that U is a domain of existence of f, consequently a domain of holomorphy. This completes the proof, m C o r o l l a r y 20 (Levi P r o b l e m i n Locally B o u n d e d Spaces)J13]. Let E be a locally bounded space with f.d. decomposition. Then every pseudoconvex domain U in E is a domain of holomorphy. P r o o f . The topology of any locally bounded space can be defined by a phomogeneous norm for some p (0 < p < 1),see Rolewicz [186]. We can apply the above Theorem 115 noticing that E is a PB-space and x --~ log{{xll is plurisubharmonic by Lelong [169]. This completes the proof of the corollary. E x a m p l e 43 The spaces (X)

1

}

, (0<,o

_<1)

(~o 35)

with the metric d(x, y) - E l lxn [P~ has a Schauder basis. We recall that the linear and topological properties of l(p~) depend on the sequence (p~) we choose. For example, l(p~) is locally bounded if pn /--+0 and is not even locally pseudoconvex if p~ ~ O, Rolewicz [[186], p. 155o]. All such spaces have the interesting holomorphic property that the radius of convergence problem has an exact solution, see Corollary 19, p.2~0. When pn /--,0, l(p~) are locally bounded spaces which are Levi spaces. E x a m p l e 44 C o u n t e r e x a m p l e to the L e v i space: The space

L~ -

{f; j/~lI f I~< ~ }

(10.36)

of all measurable functions metrized by the metric, d(f, O) - f lo { F: Ip< cc ha~ th~ t~v~al d~al ~p~c~ L~ - {r H ~ c ~ a d o ~ n ~ U # L~ ( p ~ d o c o ~ ~ ]

10.2 LEVI P R O B L E M ( G R U M A N - K I S E L M A N A P P R O A C H ) 243 or not) can never be a domain of holomorphy. That is, Lp is not a Levi space. For a counterexample of a locally convex non Levi space, Josefson [121] has given the space lcr P r o b l e m 1: Does there exist a non locally convex space E with a rich dual E' which is not a Levi space ? R e m a r k 41 The class of the P B - s p a c e s E has been discovered in1978 while the author was writing his thesis at Uppsala University. He found the solution of the Levi problem for a big subclass of it to explore the fact that : " T h e r e is a t h e o r y o f i n f i n i t e d i m e n s i o n a l h o l o m o r p h y w i t h o u t any convexity condition". This fact was unbelievable to many mathematicians at that time. The reason for that might be the fact that the dual space of Lp is trivial, that is np - {0}, (0 < p < 1); this was discovered by Day in19~1. Although we have given in this chapter several of the geometric and holomorphic properties of the class of the PB-spaces, we still believe that there are many properties of it which have not been discovered yet. The PB-spaces are sometimes called Bayoumi spaces by some mathematicians. !

10.2

LEVI PROBLEM (GRUMAN-KISELMAN APPROACH)

In this part we shall focus our attention on the basic approach given by Grumman and Kisleman [114] which they used to solve the Levi problem in Banach spaces with bases. In fact, this represents the general common method that has been used by several authors, see [113], [114],[100],[101],[197] and [78] to solve the Levi problem in different classes of locally convex spaces. The technique is : To c o n s t r u c t a h o l o m o r p h i c f u n c t i o n on a p s e u d o c o n v e x dom a i n U w h i c h is u n b o u n d e d on s u f f i c i e n t l y m a n y s u b s e q u e n c e s of a s e q u e n c e (x~) in U w i t h t h e p r o p e r t y t h a t t h e a c c u m u l a t i o n p o i n t s (Xn) a r e d e n s e in t h e b o u n d a r y OU of U. We use this method to solve the Levi problem for domains either in metrizable spaces with f.d. Schauder decomposition or in locally pseudoconvex F-spaces with the bounded approximative property (b.a.p.). We start with extending Pelczynski's theorem which deals with locally convex Fr6chet space, to locally pseudoconvex F-space. This needs the following two lemmas which we state after introducing the following definitions.

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D e f i n i t i o n 72 A closed subspace M of a metrizable topological vector space E is said to be c o m p l e m e n t e d if there exists a continuous linear projection 7~ from E onto M ; hence E - M

| 71--1(0).

(10.37)

D e f i n i t i o n 73 A topological vector space E is said to be a Fr~chet space if it is a complete metrizable space. If moreover E is locally pseudoconvex, that is, E has a fundamental basis of pseudoconvex neighborhoods of zero (Un), then E can be determined by a sequence of pn-homogeneous pseudonorms, see Rolewicz [186]. Notice that not every metric vector space is locally pseudoconvex, e.g. the spaces E =/(p~), p,~ /-,0, with the metric d(x, 0) - ~ - ~ I xj IPr - (xj) E E, are not locally pseudoconvex, see Rolewicz [[186], p.153]. On the other hand a locally pseudoconvex space is not necessarily metrizable. For example, if its family (U~) of pseudoconvex neighborhoods of zero is not countable, see also Example 3, p.20, of the inductive spaces Ul>_p~>olp, with the q-topology is non-metrizable locally pseudoconvex space. Pelczynski [[183] ] has succeeded in proving t h a t : A locally c o n v e x F r ~ c h e t s p a c e has t h e b . a . p , if a n d o n l y if it is a c o m p l e m e n t e d s u b s p a c e of a m e t r i z a b l e locally c o n v e x s p a c e w i t h basis. In what follows we extend this fundamental result to all locally pseudoconvex Fr6chet spaces having the b.a.p. We need the following two lemmas: L e m m a 28 ( A u r b a c h ) L e t E be a p-normed space with d i m E - n and with the p-norm IlXllp. Then there exist n points (xi)? e BE(O, 1) of the unit ball of E, and n linear functionals (x*)'~ C E* such that

x~(xy) - 5ij

and

IIx~llp- sup I x*(x) I< 1

(10.38)

xCBE

where 1 < i , j < n, and with equality when p -

1.

P r o o f . Let V ( y l , ..., Yn) be the determinant of n vectors yi - (a/l, ..., a~) in BE. Assume that V takes its maximum at the points x l , . . . , Xn. Set

10.2 LEVI PROBLEM(GRUMAN-KISELMAN

x ~, ( x )

-

V(xl

,...,

,...,x~ x, Xi+l

APPROACH)

245

Xn)

Y ( X l , ...,Xn)

x e E.

,

(10.39)

Since V ( X l , . . . , X , . . . , X n ) ~ V ( X l , ...,Xn) for every x e B, then (x;)~ and (xj)~ satisfy the required properties. This completes the proof of the lemma. i The n points and the n linear functionals determined by the preceding lemma are called the A u r b a c h s y s t e m of E. L e m m a 29 Let E be a p - n o r m e d space with d i m E - n (0 < p <_ 1). Then there exist n 2 continuous linear mappings f r o m E into E with dim u i ( E ) - 1 such that

n2

x-

~

u~(x)

for every

x e E

(10.40)

1

and k I

ui__2

for every

k, (l__k_
(10.41)

P r o o f . Let (xi)~ and (x~)~ be the Aurbach system of E. For i - r n + j , (0 <_ r
.~(x)

x;(x) ~j ,

--

x C E.

(10.42)

n

Then for k-rn+j

we obtain, k

rn

1

1

j 1 rn

_< sup (11~ Ilxll <1

1

k

u~(x)ll + II ~ ~+~(x)ll) 1

_< II III § sup ii -. 1 Ilzl]
]

k

/.e. i l ~ u i i l 1

_< 1 +

sup i l x i l - < 2 - I I iixi]~ 1

n

9 xj(x).~jll,

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R e m a r k 42 Lemma 28 can be extended to any finite dimensional invariant metric space E. In fact, (xi)~ represents the basis of E and (x*)~ represents its orthogonal dual in E' the dual space of E. The proof of Lernma 29 shows that the lemma is valid for any invariant metric vector space E of dimension n with a nondecreasing metric d. This is obvious from the inequality (9.~1). d is called a non-decreasing metric if d(tx, O) <_ d(x, O) for all t C (~ with It ]<_ 1. Note that d' (x, 0) = sup d(tx, 0) (10.43) 0
is a non-decreasing metric if d is invariant, Rolewicz [186]. Let us now state one of the main results of this section:

T h e o r e m 116 [13](1989) Let E be a locally pseudoconvex Fr~chet space. Then E has the bounded approximation property if and only if it is a complemented subspace of a rnetrizable locally pseudoconvex space with basis. P r o o f . Since the topology of E can be determined by a sequence of Pnpseudonorms, ]]xiip~ , it is enough to give the proof in the case of a complete p-normed space. This can be achieved immediately by using the above lemmas and the method used in Pleczynski [18] for a Banach space. I

D e f i n i t i o n 74 A domain U C_ E is said to be d o m a i n of h o l o m o r p h y if there are no domains U1 and U2 such that

r162

uf g ,

(10.44)

with the property that for every f E H(U) there exists f~ E H(U1) such that f IU2-- fl IU2" D e f i n i t i o n 75 U is said to be a n a t u r a l d o m a i n o f e x i s t e n c e if there is a function f C H(U) which cannot be continued beyond the boundary OU of U, i.e. there are no domains U1 and U2 with

r and g E H(U1) such that

C_UNU1, U~U1,

10.2 LEVI P R O B L E M ( G R U M A N - K I S E L M A N A P P R O A C H ) 247

f ] u 2 - g [u2.

The following theorem extends the results of [84],[114], [197], and [80] for locally convex spaces. T h e o r e m 117 ( L e v i p r o b l e m f o r m e t r i z a b l e spaces)J12](1989) Let E be an infinite dimensional complex metric vector space with a f. d. decomposition (~n). Then every pseudoconvex domain U in E is a domain of holomorphy Proof. We first assume that d is a translation invariant and (Trn) is monotone with respect to d. Since U is pseudoconvex then Un - U N PEn and (10.45)

O'n - Un n Trn-l (Un-1)

are pseudoconvex in En - 7~n(E) for every n C N. Let (zn) be dense in U, z , E Un and such that d(zn, O) + du(zn) ~_ n.

(10.46)

Xn -- Zn + )~nen+l

(10.47)

Let

(e, e E , , n , _ l ( e , )

-- 0 and d(en, 0) - 1) and An e (T is chosen such

that d(xn, zn) -- du(zn)

tin

2'

n e N

where the sequence (e~) is given by Cl -- d u ( Z l ) / 2 , Cn -- m i n ( d u ( z n ) , En_l, d u ( x n - 1 ) ) / 2 ,

n > 1.

We define a cover (Vn) of the domain U to have 9 Xn C Yn+l/Yn, Xn E

Kn+l

(10.48)

where K~+I is some compact holomorphically convex set in a~+l. In fact the cover is taken as in the author[[12], Th. 3.a.1]. Let Xn-{xEU;zrm(X)

EU,

for all m ~ n } ,

(10.49)

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IN TOPLOGICAL

Ln+l - {x Ca_~Tn+l; d(x, O) <_ n, do.n+ 1 (x) >__ en+l },

SPACES

(10.50)

Vn - N~>k>n {x E Xn; du(7rkx) > ~n -]" d("KkX, "KmX), 7Tin(X) E LAIn}

(10.51)

and V1 -- {Zl}, where L A denotes the holomorphically convex hull of a set L in U and da is the distance to the b o u n d a r y Oa of an open set a in U. Now we have the following properties 9 (1) x . c_ X . + l . ~ m ( x . ) - vm a Xn and V. C_ V.+I. r.~ (V.) - Vm a Vnforall m>_n (2) For all x E V and r <_ du(x), there exists n E N with B(x, r) C_ Xn and 7rm(B(x, r)) C_ LAm for m > n. (3) For a l l x E U , t h e r e e x i s t h > 0 andnEN w i t h B ( x , 5) CVn. (4) The sequence (x,,) has the properties t h a t x , C Vn+l/Vn and hence Xn ~ 71"n+l (Yn). The set,

7rn+l(Yn)- {x e LnA+I nTrnl(LA);dv(vr,x) >_ s --]--d(TrnX,X)}

(10.52)

is not necessarily holomorphically convex since -logdu(x) and logd(x, O) m a y not be plurisubharmonic functions in general. Thus we may have xn in 7rn+l(Vn) A although Xn ~ 7rn+l(Vn) as in (4). Fortunately we have the following lemma. L e m m a 30 The previous cover (Vn) of U has the property that, for every fixed n E N, there is a natural number mn <_ n with 7rn+l (Vmn) A C 7rn+l (Vn).

Proof.

(10.53)

S i n c e ~ c_ V2 C_ ... c_ U, it follows for a fixed n, 7rn+l(V1) ~ 7rn+l(V2)---~ 7Tn+l(Vn).

Every set 71"n+l(Vn), m < n lies in the element A 1 n "Kn 1(LnA) n ... n 7r~n1 (LAm) Ln+

of the strictly increasing sequence {Zl} C (Ln+IA n 7rn 1 (LnA) n ... n r ~ 1 (L~))C ... C Ln+ 1 A nTrn I (LnA)

10.2 LEVI P R O B L E M ( G R U M A N - K I S E L M A N A P P R O A C H ) 249 of compact holomorphically convex subsets of O'n+l. This sequence covers Zrn+l (Vn) because A 1 A 7rn+l(Vn) C L n + 1 N "zn (Ln+l).

Hence there is an mn <_ n such t h a t 7rn+l(Vm,~)A C_ 7rn+l(Vn). Indeed this l e m m a deals, with the finite dimensional case where the different topologies ( locally and non locally convex) are equivalent. Note t h a t the sets 7rn+l(Vrn) lie in the ball B ( O , m - 1) M E n + l , m _< n + 1. I To complete our proof of the t h e o r e m let us put now Kn+l -

7rn+l ( V m ) A

(10.54)

where rn is the biggest mn given by L e m m a 30. T h e n we can deduce from the previous properties and our l e m m a t h a t 9 (5)K~+I is compact holomorphically convex set in a n + l , Xn

~ Kn+l

and 7rn(Kn+l) C_ On.

By (4) and (5),

Xn E Vn+l/gn and xn

r Kn+l, n E N.

(10.55)

Therefore there exists a holomorphic function f~ E H(Vn+I) with

[ f~(Xn) I> 1 > IlfnllKn+~

(10.56)

where Ill.IlK = s u p g I fn(X). Now we apply the extension of G r u m a n and Kiselman's l e m m a [114], which says t h a t " F o r a n y h o l o m o r p h i c f u n c t i o n f o n cr~ = Un N 7r~l(u) a n d a n y e > 0 there exists a holomorphic function g on U with

Ilg-

foTrnllv <

e and

[Igllvm < oo for all rn E N "

see Schottenloher [197]. We approximate the function fnOTr~ by a function g,, on V,,+I. Thus there exists a function gn E H(V) with [[giivm < oo, m E N and Ilgn- fnOTrn+l[[V,-,+l < min(I f , ~ ( x ~ ) I - - 1 , 1 -- Ilfnl[K~+l)Since 7r,+l(Vm,) C_ K , + I by (5), we have

Ilgllvmn

IIg' - fnO~~+lllv..~ + IlAoTr~+lllVm

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10 L E V I P R O B L E M

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SPACES

< 1 -IIf~lIK~+l + IIf~lIK~+l ~ 1. Since Xn E V~+I we have

[ gn(X~) I>1 f ~ ( x ~ ) I -

I g~(Xn) -- fn(Xn) l> 1.

Thus

I gn(Xn) l> 1 > IIg llv .,

n e N.

(10.57)

Let

hn - kn. g ~ ,

kn > O, O~n C N.

Then the function f = y~, hn is holomorphic in Vm~and by (3) it is holomorphic in U with ] f(xn)I>_ n. Thus U is a natural domain of existence of the function. Consequently U is a domain of holomorphy. Finally, for any metric d there is an equivalent metric which is translation invariant, Rolewicz [[186]. Th.l.l.1]. Also the metric d*(x, 0) Supnd(Trnx, 0) is equivalent to d and (Trn) is monotone with respect to d,. Thus the complete solution is established, m R e m a r k 43 Observe how the Lemma 30 has been used to overcome the absence of the convexity condition in the above Theorem 117, thus departing from the proof in the locally convex case where there is no need of proving it. The next corollary generalizes Schottenloher's results [[197], Cor. 3.4], for locally convex cases. C o r o l l a r y 21 (Levi p r o b l e m in locally p s e u d o c o n v e x F-spaces)J13] Every pseudoconvex domain in a locally pseudoconvex Frdchet space E with the b.a.p, is a domain of holomorphy. P r o o f . According to Theorem 116, E is isomorphic to a complemented subspace of a metrizable space with basis. Hence the corollary follows from Theorem 117. m

10.2 LEVI PROBLEM(GRUMAN-KISELMAN

APPROACH)

251

E x a m p l e 45 ( M a p p i n g s with R a p i d l y D e c r e a s i n g A p p r o x i m a t i o n Numbers ) Consider the space L(E) of all continuous linear mappings of a Banach space E. Let L(E) have the linear mappings norm. For To E L(E) we define the r th a p p r o x i m a t i o n n u m b e r s by a~(To) = inf {]IT -Toll; T C L(E)

(10.58)

with dim T ( E ) < r}.

If lP(E)-

{

[a,(To)] p < co

To C L(E); ~ r--1

}

(10.59)

,

then on the intersection

(10.60)

S(E) = Mp>olP(E)

a metrizable non locally convex vector topology is generated by the sets

Bp(To, e)

-

-

T e S(E); ~

[ar(T

-

To)]p < e

,p > 0,~ > 0

r--1

S ( E ) is complete and it has the bounded approximation property if E is complete, as it is shown by Pietsch[[184],p.139]. It is worthy to point out that the function

log dp(T, 0), T e S(E), is a plurisubharmonic function where oo

0) :

(10.61)

r:l

This is also the case for the function Oo

- log dn (T) -

sup - log ~ ToCO~

[a~ (T - To)]P, T e ~t

r=l

It is plurisubharmonic if ~ is any pseudoconvex domain of S(E). That is, S(E) is a PB-spaces. So by Theorems 113, 115, p.,239, 242, it is a Levi space. Note that L(E) is not a Fr~chet space if E is not Banach. A

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SPACES

E x a m p l e 46 ( T h e I n f i n i t e P r o d u c t o f Fr~chet Spaces) We show that the Levi problem can be solved for the product H ~ E j of Fr~chet spaces (complete metric spaces) Ej, when the topology of Ej is given by the metric d(x, y)E~ -- IIx --

yll~j,

II.ll~j

where is a homogeneous pseudonorm, j E N, and each Ej has the bounded approximation property (b.a.p.), j E N. Let us first show that: the product

E-

II~~

(10.62)

has the b.a.p, if each Ej, j C N, has the b.a.p. By Theorem 114, the Fr~chet space Ej, j E N, is a complemented subspace of a space Fj with basis; hence it can be written as Ej - Fj |

(10.63)

Gj.

and there exists a continuous projection IIj

nj.Ej--,Fj with H~-1 (0) - Gj. Then E - nFEj

- nFFj

@ nF

Gi

Let r 9 ~ N ~ [0,--~oo] be convex and homogeneous of degree 1, F-normllxl] E can be defined for x - (Xl, x2, ...) e E by

]l.llE - r

(10.64) t h e n an

IIx211E=,---),

For example we can take oo

r

IIx211E~., ...) -

(~-~ IlxjllqEj)1/q j=l

where 1 <_ q <_ co, and II.IIE~, is pj-homogeneous with, e.g. p j ~ O a s j ---+ (X).

Since Hj is continuous,

Ilnj(xjDIIEj ~ Cj IlxjlIEj, Now for x -

(Xl, X2, ...) C E we define

Cj e R,j e N.

10.3 L E V I P R O B L E M ( S U R J E C T I V E

LIMIT

APPROACH)

n(x) - (nl(Xl), n2(x2), )

253

(10.65)

Then [[II(x)[tE-

][(IIl (Xl),II2(x2), ...)]lE

-- r

IlIh(x2)ll,-.-) oo

= ( y ~ l l i I j ( x j ) l [ E~ q)l/q j=l oo

j=l (:x)

< (supCj)(y~llxjllq)l/q Ej -

-

J

'

j=l

i.e. IIn(x)llE ~ CIIxlIE. T h ~ II i~ co~tin~o~ on E - n ~ Ej. Consequently E is a complemented subspace of the space I I ~ F j | H ~ Gj where I I ~ F j has basis. Applying Corollary 21, p.251 we will have that E has a solution to the Levi problem, e.g. take

E -

-

ll •

• 11/3 •

H~ll/n

(10.66) (10.67)

Of course E is neither locally bounded nor locally convex A

10.3

LEVI PROBLEM (SURJECTIVE LIMIT APPROACH)

In this section we deal with : " T h e p r e s e r v a t i o n of t h e p r o p e r t y of b e i n g a L e v i s p a c e u n d e r surjective limits". Our setting here is the L o c a l l y p s e u d o c o n v e x ( L p s ) which may not be metrizable or locally convex. This will include the special class, the p-Banach spaces with the bounded approximation property, (1 > p > 0). We shall first describe some basic properties of surjective limits which can be obtained conveniently as for locally convex spaces.

254CHAPTER

10 L E V I P R O B L E M

D e f i n i t i o n 7 6 A Lps E write

IN TOPLOGICAL

is a s u r j e c t i v e l i m i t of Lps

SPACES

(El)lEA

E = lim Ei

(10.68)

icA

if there exists a continuous linear mapping ui from E ~i : E --~ Ei,

each i E A

and we

onto Ei, i.e.

(10.69)

and the inverse images of the neighborhoods of 0 in Ei , as i ranges over A, form a basis for the neighborhood system at 0 in E. That is, for each neighborhood W of 0 in E, there exists an i C A and a neighborhood V of 0 in Ei such that 7r71(V) C W. E

is called o p e n s u r j e c t i v e l i m i t if 7ri is an open map for each i C A.

The surjective limit approach was used by Hirschowitz [119] in studying the Cartesian product of the complex planes, and by Dineen [84] in characterizing the collection of locally convex spaces (l.c.s) in which certain holomorphic properties are true. The following example has useful consequences. E x a m p l e 47 C P ( E ) denotes the set of all continuous pseudonorms, i.e. p , - s e m i n o r m s for some p,, (0 < p , < 1) on a Lps E. For each q C C P ( E ) , let Eq denote the vector space E equipped with the topology generated by q and let 7rq denote the canonical map from E onto E / q - l ( o ) . Then E has the surjective representation ( E / q -1 (0), nq) which will be called p s e u d o n o r m

(10.70)

s u r j e c t i v e r e p r e s e n t a t i o n of E. A

In what follows we introduce a big collection of Lps spaces which are closed under the operation of surjective limit. L e m m a 31 [14] The collection of locally pseudoconvex spaces (Ei)icA with the approximation property (a.p.) is closed under the operation of surjective limit. That is, lira Ei has the a.p. if each Ei, i C A, has the a.p.

10.3 LEVI P R O B L E M ( S U R J E C T I V E LIMIT A P P R O A C H )

255

P r o o f . For the proof notice that the a.p. of a topological vector space E does not depend on whether or not E is locally convex. But it does depend on the existence of a family (Tri)icA of finite-rank of continuous linear functionals 7li : E--~ E which can be used to approximate the identity mapping I. So we may follow here Dineen's method [[84] Example 2.31 for a locally convex space. This completes the proof of lemma, m The following proposition will be helpful to reduce the study of the Levi problem for Lps to spaces which have continuous p~ -norms (0 < pa < 1). P r o p o s i t i o n 4 [14]Every Hausdorff Lps E with f.d. decomposition (~i) is a surjective limit of spaces Ei, i E A, where Ei has a continuous p i - n o r m and f. d. decomposition. P r o o f . This can be established in a manner similar to that of Dineed [3, Example 2.4] taking into account Example 47, p.254, m The next result gives a sufficient condition on a pseudoconvex domain U of a Lps E in order to have U - 7F-1(71-(U))

(10.71)

7r" E --~ E / q - l ( o )

(10.72)

where

is the quotient map and q is a continuous p-seminorm on E. L e m m a 32 Let U be a pseudoconvex domain of a Lps E. Let q be a continuous p-seminorrn on E, (0 < p < i) such that Bq(O, 5) C_ V for 5 > O. Then u = u + {y; q(y) = 0 } . (10.73) P r o o f . The proof is a finite dimensional one which does not depend on whether or not E is Lps. So we can follow Dineen's method [[84], Lemma 1.1] taking into account the result of Hirschowitz's [119]. m The following lemma is of special interest when we deal with the solution of the Levi problem. The proof is analogous to that given by Dineen ([84], Lemma 1.8) using Lemma 31 instead of the corresponding result in normed spaces.

256CHAPTER Lemma

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SPACES

33 Let U be an open pseudoconvex subset of open surjective limit lim(Ei, 7ri)

(10.74)

lEA

of locally pseudoconvex paces Ei, i C A. Then U is ~i-open for some i C A. That is,for every x E U, there exists Vz open in Ei such that x C 7ri-l(Yx) C U.

(10.75)

I We are now in a position to introduce a basic theorem, which will enable us to obtain the solution of the Levi problem for Lps spaces. T h e o r e m 118 [1~{1(1989) The collection of Hausdorff Lps spaces with the b.a.p, in which pseudoconvex domains are domains of holomorphy is closed under open surjective limit.

That is, the open surjective limit lim(Ei, ~i) iEA

of the Levi spaces Ei, i C A with the b.a.p, is a Levi space. P r o o f . To establish the proof, we assume that U is a pseudoconvex domain of the surjective limit

lim(Ei, 7ri). lEA

Then there exists i E A such that v

-

with ~i(U) a pseudoconvex domain, see Lemma 33. By the assumption, ~i(U) is a domain of holomorphy. Now if there exist open connected sets U1, U2 in E such that

u2 c_ u, u2 c_ u n u 1 and for each f E H ( U ) , there is an fl E H(U1) with f [u2-- fl Iu2-

(10.76)

10.3 LEVI P R O B L E M ( S U R J E C T I V E LIMIT A P P R O A C H )

257

Then, as a consequence of 7ri being open, we obtain 7ri (U1) C_ 7ri(U2). Note that :ri(U) is a domain of holomorphy. Hence U1 C_ 7r~-l(Tr(iV)) - U, and consequently U is a domain of holomorphy. This completes the proof of the theorem. II We are now in a position to give the solution of the Levi problem in Lps having the b.a.p. (The case of L.c.s.. was considered by Dineen [84] and Schottenloher [197] ). 119 (Levi problem in Lps)[14(1989 ) Let E be a Hausdorff Lps with the b.a.p. Then every pseudoconvex U in E is a domain of holomorphy.

Theorem

P r o o f . Note that every Lps E can be expressed as a surjective limit of p~-normed spaces with continuous p~-norms q~ (see Example 48, p.254). Hence applying Theorem 116 and the solution of the Levi problem for pnormed spaces with b.a.p. (see Corollary 20, p.242) we obtain the required result and the proof is complete. II The following is an example of a non-metrizable Lps. E x a m p l e 48 ( T h e inductive space ). Ul>_p>0lp

(10.77)

We recall that we have defined the q-topology on Ul>_p>olp to be the strongest vector topology such that each injection

ip " lp --~ Ulp

(10.78)

is continuous. The space Ulp with this q-topology is a complete separable non locally convex Lps and has the unit vectors (e~) as its symmetric Schauder basis. A sequence converges in Ulp if and only if it is contained in and converges in some lp. A set is compact in Ulp if and only if the set is contained and compact in some lp. Note that no closed infinite dimensional subspace of Ulp is metrizable. Hence Ulp itself is not metrizable, see Stiles

[203]. Since E is Lps with a Schauder basis, Theorem 117 implies that E is a Levi space. A

258CHAPTER

10.4

10 L E V I P R O B L E M

IN TOPLOGICAL

SPACES

LEVI P R O B L E M (QUOTIENT MAP APPROACH)

In this P a r t we solve the Levi problem in some s e p a r a b l e p - B a n a c h s p a c e s , (0 < p _< 1). T h a t is, in a complete locally b o u n d e d space with p-homogeneous norm. (The case p = 1 corresponds to Banach spaces). Since every separable Banach space E is isomorphic to a quotient space of

lo = N p > l p , t h a t is,

E ~_ lo/M for a closed subspace M of E, see Stiles [203], and since E is a Levi space, see Corollary 20, p.240, we can obtain the following interesting partial result for the Levi problem. 120 (Levi p r o b l e m in certain separable B a n a c h space)[l~] Let U be a pseudoconvex domain in a Banach space E. Suppose that E is isomorphic to a quotient space of lo, that is,

Theorem

E ~_ lo/M,

(10.79)

and lo/M has the bounded approximation property. Then U is a domain of holomorphy. P r o o f . Since l0 is a Fr~chet space, its quotient space lo/M is a Fr6chet space. Now the space lo/M is assumed to have the b.a.p. Hence a direct application of Corollary 20, p.242, will imply the required result.This completes the proof of the theorem, m Let Ep be a p-Banach space. The fact t h a t every separable locally bounded space is isomorphic to a quotient space of lp(O < p <_ 1) will help us to obtain the following consequence which is of special interest. 121 (Levi p r o b l e m in separable p - B a n a c h space)[l~] Let Ep be a separable p-Banach space which is isomorphic to a quotient space of lp , that is E "-" lp/M (10.80)

Theorem

and lp/M has the bounded approximation property. convex domain in Ep is a domain of holomorphy.

Then every pseudo-

10.4 LEVI P R O B L E M ( Q U O T I E N T M A P A P P R O A C H )

259

P r o o f . Ep is isomorphic to lp/M for some closed subspace M of lp. lp/M is a Fr~chet space, and by assumption it has the b.a.p. Then by applying Corollary 20, p.242, we achieve the solution and the proof is established. II R e m a r k 44 The assumption that lp/M

has the b.a.p, cannot be dropped out of the above theorem. We note that Lp[0, 1], (0 < p < 1) does not admit holomorphic functions other than 0, and so each domain (pseudoconvex or not) is not a domain of holomorphy.

P r o b l e m 2.What is the class of separable p-Banach spaces each of its elements is isomorphic to a quotient space lp/M with the b.a.p.? C o u l d H a r d y s p a c e Hp(0 < p < 1) be a n e l e m e n t of t h i s class? We note that lp(0 < p < 1) is an element of this class since it has a basis and hence this class is nonempty. We give now more examples of non-locally convex spaces which have either a Schauder basis or the bounded approximation property. In fact, by the results of this chapter, all pseudoconvex domains in these spaces are domains of holomorphy, that is, they are Levi spaces. E x a m p l e 49 The H a r d y spaces

Hp(1 > p > 0).

The Hardy space Hp of all analytic functions f on the unit disc of (~ is separable, locally bounded, non locally convex space with respect to the pnorm

Ilfll-

~-~llim/02~ I f @ d ~ Ip dO,

f

H,(r

The Banach envelope of Hp is isomorphic to 11, that is H p ,.o ll

(10.81)

by Kalton [130]. This implies that H p has the b.a.p. Moreover, Hp contains a non locally convex closed subspace Mp of E isomorphic to lp (0 < p < 1), that is

Mp ~_ lp,

(10.82)

see Shapiro [189]. Now every pseudoconvex domain in this complemented subspaces M of

Hp, in H ~ is a domain of holomorphy, by corollary 20, p.242. That is, H p and Mp are Levi spaces. A

260CHAPTER

10 L E V I P R O B L E M

IN TOPLOGICAL

SPACES

R e m a r k 45 ( I m p o r t a n t ) T h e previous results of this chapter can be generalized to non-schlicht domains over a suitable space E. Let us first recall the following concepts which are analogues of those considered by Schottenloher [19~ for locally convex spaces : A Riemann

d o m a i n s p r e a d o v e r a metric vector space E

(ft, q) where ft

is a pair

(10.83)

is a connected Hausdorff space and

q" ~ --~ E

(10.84)

is a local homeomorphism. That is, for every x E f~ there exists a neighborhood w of x such that qw " W --~ E (10.85) is a homeomorphism of w onto q(w). If q is injective, the domain (ft, q) is called s c h l i c h t d o m a i n , and can be identified, via q, with a domain in E. The boundary distance function dn on a fixed domain (ft, q) over E is defined by: dn(x) -

sup(r; there exists a neighborhood U of x s.t.

q I u " U ~ B ( q ( x ) , r ) is a homeomorphism), x E f t

(lo.86) (lO.87)

The ball B ( x , r ) for x E ~, r > dn(x) is just the component of q - l ( B ( q ( x ) , r ) ) which contains x. The plurisubharmonic, the holomorphic or for that matter any locally defined class of functions, can now be defined on (ft, q) using restrictions q[~ of the projection q, see the author[12], [13], and [1~].

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spaces.

278

Notations

~T: Space of complex numbers. K: Scalar field. (Tn: Space of n complex numbers. C~' 9Space of continuousely m-quasi-differentiable maps. ~ : Space of n real numbers. A" Polynomial corresponding to a multilinear map A. a: Constant > 1. E ' : Continuous dual of E. [[T[[ : Norm of T. ElM" Quotient space. PB-space : Bayoumi space. 0U : Boundary of U. Ilxll: Norm(or p-Norm ) o f x. FD: Fr~chet Differential. GD: Gateaux Differential. QD: Quasi-differentiable (or Bayoumi differentiable) maps. Dr(a): Quasi-differential (or Bayoumi differential)of f at a. D 2f: Second Quasi-differential. D m f : m th Quasi-differential. dr(a) = D f ( a ) : Bayoumi quasi-differential of at a. d2f(a) e L(2E; F)" Isomorphic to D2f, a bilinear map. deg T : Degree of a mapping T. a~(T): r th approximation numbers. (ELM) ~_ M • 9Annihilator of A. RI (x) : Radius of convergence of f. R} (x). Radius of boundedness of f. Hp(1 > p > 0) : Hardy spaces. I I ~ E j " Infinite product of Ej. NLP: Nonlinear programming. mrnq /P m! : New universal constant. qu(x): Minkowsky functional. E - limieA Ei 9Surjective limit of Ei. d(x, y): Distance between x, y. H ( E ) : Entire functions(continuous and Gateaux analytic). Hbb(E) 9Entire functions of type ~ CX, bounded on bounded sets of E. /)" Linear completion. !

279 Ea : Holomorphic completion.

HBEP: Hahn Banach extension property. HBSP: Hahn Banach separation property. CP(E) : The set of all continuous pseudonorms. (f t, q) : Schlichet domain.

La(E, F): Space of linear mappings. L(E, F): Space of continuous linear mappings. La(mE, F): Algebraic m-linear maps of E into F. L~s(mE, F): Algebraic symmertric m-linear maps of E into F. L(E m, F): Space of continuous m-linear maps of E into F. L~(mE) = Ls(mE,K) : Continuous symmertric m-linear maps. L(E1, .., Em; F): Continuous multilinear maps of E1 x ... x Em into F. P~(mE; F): Algebraic m-homogenous polynomials of E into F. P(mE; F): Continuous m-homogenous polynomials of E into F. p(mE): Continuous m-homogeneous polynomials of E into K. A b - { (1 - t)l/pa + tl/pb}, 0 _< t _< 1 "Arc segment between a and b. BE(0, 1): Unit ball of E of center at the origin. Cp(A)" Closed p-convex hull Of a set A. Ep(K) : p - E x t r e m e points of K. d(x, O) =11 x lip: p-norm defined by a p-homogeneous metric d. L p ( I ) - {f; f : l f

Ip< oo}" Lebsgue integrable functions on [0, 1].

A(x) = f (x) - f (a) - T(x - a): Affine linear functional. QA (U,F): Quasi-analytic (Bayoumi analytic)maps from U to F. QH (U; F): Quasi-holomorphic(Bayoumi holomorphic)maps fromUto F. QA (U) = QA (U; IK): Quasi-analytic( Bayoumi analytic)maps. QH(U) = QH(U,~) : Quasi-holomorphic(Bayoumi holomorphic)maps. Qc~ (U, F) : Quasi-C~Analytic maps from U to F. ~(a~_l ~) 9Characteristic function on (c~i-1, ai). deg(f, M, x ) : Degree of a mapping f On M at x C f(M). det(A) : Determinant of A.

II I llA= sup A JI (x)IT0: Compact open topology. Bdd(E) : Bounded susets of E. WBdd(E) : Weakly bounded subsets of E. Bg (E) : Bounding sets in E. WBg (E) 9Weakly-bounding sets(Limited sets) in E. WbBg (E) :Bounded Weakly-bounding sets(Bayoumi limited sets) in E. P C X ( E ) : O p e n pseudoconvex sets of E. PSH(U) : Plurisubharmonic functions on U.

280

a.p." Approximation property. b.a.p: Bounded approximation property. [ei]~ 9Cosets of the basis ei. H~ 9Banach envelope of Hp, l~ - {x - ( ~ ) ; x~ e r E ~ I~nP < ~ } , ( 0 < p < 1). l o - n ; > 0 1 ; - { x - (x~); sup;>0 E ~ I ~ I;< o o } . l~-Uq<_plq-{xElp;xElq for some q__p}, where d(x, 0 ) - infq dq(x, 0 ) - infq E l lXj I~, q s 1 l+ - Nq>_plq - {x E/q; supq>__p~--}7 I xj Iq< c~ }, (0 < p _< 1). f ~ ( s ) - sup~<0(Ts- ~ ( T ) ) " Legendre transformation of ~ where, ~ ( ~ ) - s u p ~ ( ~ - f~(~)), x e S , ~ e ~, ,~ e ~. dmf (a) =re!Am "Isomorphic to D m f (a),using m-linear map Am. dmf (a) =m!P~" Isomorphic to Dmf(a) by m-homogeneous polynomial Pm. L(mE, F) ~- L ( E , L ( m - I E ; F ) " Space of m- times repeateadly continuous linear maps. lp(E)" {T E L(E); ~-}~1 [a~(T)]P < c~}. Space of rapidly decreasing approximation numbers. 5u(x, a)= inf(]tl; x + ta E OU), x c U, a E E, U C_ E. Distance function to the boundary of U in t.v.s. l ( p n ) - {x = (Xn); Xn E(~, ~-~c~[xnlP,~ < oo}" Locally bounded F-spaces, if Pn /-*0, and non locally pseudoconvex Montel space if P n ~ O. Hb(E) 9Entire functions of type ~ r r e E' bounded on weakly bounded sets of E.

Index Affine maps, 63 Annhilator of subspace of Metric space, 193 Applications to Non-Convex analysis to Operations research, 58 Arc segment in Vector space in t.v.s., 50 Baire Category Theorem, 20 Banach Open mapping p-Banach space, 23 Banach-Steinhaus type Theorem for Polynomials, 46 Banach-Steinhaus Theorem for Linear maps Continuous polynomials, 25 Bayoumi Space PB-Space, 231 Bayoumi-differentiability Quasi-differentiability Interchange order with limit, 96 Bolzano Intermediate-Value theorem Finite and Infinite spaces, 163

Bounded Linear map Norm, 3 Bounded approximation property (b.a.p.) Approximation property, 186 Bounding sest in Lbs Bounding sest in F-spaces Bounding sets in Lps and t.v.s., 186 Bounding sets (Bg) Weakly bounding sets (WBg),Limited sets Bounded-Weakly bounding sets (B-WBg),Bayoumi Limited set, 181 Bounding sets and Holomorphic Completions, 220 Bounding sets in Non Lps in t.v.s., 189 Brouwer Fixed point Generalized, 63 9 Cauchy Sequence, 5 Cauchy-Hadamard formula Generalized Radius of Uniform convergece, 139 Cauchy-Hadamard formula Radius of convergence 281

282 Holomorphic function, 237 Compact p-Exteme set p-Convex set, 56 Complemented subspace, 244 Completion Linear Holomorphic, 207 Complex Mean-Value theorems Mean-Value Inequality, 94 Construction of holomorphic maps on locally bounded spaces on Lps F-spaces, 199 Continuity of Linear Map, 3 Continuous Mapping Derivative, 3 Continuous dual space of E ( E' ), 194 Convex Function Set, 182 Counterexample to Levi space Non Locally convex space, 242 Criterion for continuity, 40 Curvilinear integral of Maps of Vector-Valued functions, 160 Decreasing Approximation numbers Sequence, 19 Degree of Polynomials

INDEX Degree theory Degree of maps, 124 Derivative Directional, G~teaux, 111 Differential pq-Differential Quasi-Differential (Bayoumi -Differential), 77 Dineen, 14 Domain Pseudoconvex of Holomorphy, 228 Rienhardt, Logarithmically convex, 72 Entire function Continuous and Gateaux analytic, 229 Equicontinuous Maps, Family Group, 63 Equivalence relation Vector space of maps of Quasi-tangency, 78 Examples Non Metrizable space F-spaces, 18 Extension Property Problem, 7 Extension Theorem in Lbs and Lps in t.v.s., 11 A

Fixed points theorem of Brouwer, or of Kakutanis Generalized, 63 Fr~chet spaces Product of

INDEX

Complete metric spaces, 205 Function Regular Bounded, 158 Functional Analysis on p-Normed space, 1 Fundamental Theorems in Functional Analysis, 1 Fundamental Theorem of Calculus Riemann Integration Step functions, 157 Hahn-Banach Extension property(HBEP) Separation property(HBSP), 7 Hardy space Banach envelope of Closed subspace of, 259 Hausdorff Lps, 255 Holomorphic G~teaux Analytic and continuous Quasi, 124 Holomorphic extension problem Dense subspaces Closed subspace, 221 Holomorphic functions Three different classes of, 196 Holomorphic mappings Fixed points of, n non convex domins in ~ , 63 Holomorphically complete Spaces Example of, 217 Homeomorphism between Lbs, 22

283

Homotopy Continuous maps Invariant, 167 HSrmander, 72 Image of separable Banach spaces

(Zo) Image of separable p-Banach spaces (Lp), 202 Inductive space Locally p-convex space Locally pseudoconvex spaces, 257 Infinite dimensional Complex analysis Discussion on, 224 Holomorphy case, 1 Infinitely Quasi- diffe re nti als Continuousely, 152 Integral Mean-Value theorem on Arc segment, 176 Integral domain of Quasi-Analytic maps, 155 Internal point of Arc segment of p-Extreme set, 52 Inverse mapping theorem of Quasi-differentiable map, 85 Isometry Natural Isomorphic, 34 Josefson Counterexample to Levi problem, 228 Limited sets, 212 K-space

284 Three-space problem, 11 Kakutani Fixed point theorem Generalized, 66 Kalton, 14 Theorem Failure of linear extension, 10 Kisleman Approach Radius of convergence, 239 Krein-Milman Theorem p-Convex hull of p-Extreme points, 68 Lemma

Approximation Schwartz, 96 Zorn Aurbach, 10 Levi problem and Surjective Limit Approach in Lps, 257 Grumman and Kisleman Approach, 243 in Metrizable spaces in Locally pseudoconvex spaces, 247 in PB-spaces (Bayoumi spaces) in F-space, 241 in Separable p-Banch spaces and Quotient Mapping Approach, 258 Radius of convergence Approach Bayoumi PB-spaces, 242 Linear Mapping Functional, 2 Space, 1, 8

INDEX Lipschitzian property Generalized, 80 Local Property Constant map, 153 Locally Bounded space Pseudoconvex space, 8 Locally bounded Space F-space, 2 m Quasi-differential, 107 m-Expansion of maps Finite Uniquness of, 125 m-Linear map 2-Linear map(Bilinear) Contiuous ...L{,m E,F), 30 m-Quasi tangency m-quasi differential, 124 Mapping into Product spaces Quasi-differentiable, 82 Martin's theorem, of p-Normed spaces, 41 Mathematical Programming Nonlinear, Constraints, 61 Mean-Value General Theorem of Integral, 131 Mean-Value theorem Lagrange Real, or Complex, 90 Metric Non decreasing by F-norm, 246 Metric space Translation invariant

INDEX Complete, 229 Milman Generalized theorem of p-Extreme points, 50 Multilinear maps, of p-Banach space Continuous,, 29 Multinomial Formula Binomial Formula, 36 Natural domin of Existance Levi space, 230 Normalized sequence of disjoint supports, 223 Open mapping theorem of Locally bounded space, 20 p- homogeneous norm, quasi-norm Polynomial, 24 p-Convex combination p-Convex hull of Compact set, 58 p-Convex hull Absolutely, 208 p-Convex set Absolutely, 51 p-Extreme Points sets, 50 Path of type Arc segment in r 160 PB-property Pseudoconvex boundery distance, PB-Space (Bayoumi Space), 231 Pelczynski theorem Fr~chet space with b.a.p.

285 Complemented L.c.s. with basis, 244 Permutation, 35 Physics Thermodynamics Application, 8 Plurisubharmonic Functions Logarithmically...metric, 230 Point-wise Separation property, 16 Polarization Formula of Symmetric m-Linear maps, 38 Polynomially convex of Ball of Set, 181 Polynomials m-Homogenuous, Continuous, 35 Power series Properties between p-Normed spaces, 136 Uniqueness of, Quasi-differential of, 146 Principle of uniform boundedness Equicontinuous family Bounded pointwise, 25 Product of p-Normed spaces Cartesian, 29 Properties Radius of convergence Radius of boundedness, 234 Properties of Bounding and Weakly bounding Compact linear maps, 206 Property Finite intersection,

286 PB-property, 70 Property (p) Extension property Separation property, 11 q-Banach Space Separable, 5 q-normed space, 5 Quasi tangent pq-Tangent of Two maps, 78 QuasiDifferential norm, 2 Quasi-Analytic Principle of, Maps, 151 Quasi-Analytic Continuation Strong form Weak form, 153 Quasi-Differential Higher, G~teaux, 106 Second , Higher, 101 Quasi-differential Properties of, Bayoumi, 79 Quasi-differential (or Bayoumi differential) of Multilinear maps of Polynomials, 83 Quasi-Holomorphic (or Bayoumi Holmorphic) Maps Function, 123 Quotient space Quotient map

INDEX Complemented space, 185 Radius of Absolute Convergence Normal Convergence, 144 Radius of convergence Radius of boundedness of Holomorphic function, 229 Relationship between Bayoumi and Frdchet differentials Bayoumi (Quasi) and G ~teaux differentials, 115 Relatively compact sequentially compact, 185 Riemann domain Spread over Metric vector spaces, 260 Rolewicz, 183 Metric Linear Spaces, 2 Schauder Basis Decomposition, 185 Schwartz Symmetric theorem Generalized, 103 Separable Banach spaces Levi Problem in some Separable p-Banach spaces, 258 Separation Property Problem, 7 Separation Theorem in Lbs (Locally bounded sp.) in Lps (Locally pseudoconvex sp.), 12 Sequence spaces with Logarithmically convex balls

INDEX with p-Convex ball, 73 Space Locally p-convex Locally pseudoconvex (Lpc, 52 of Continuous linear maps L(E,F) of Continuous Polynomials P(E,F), 4 of Convex type Sequence,, 73 Orlicz Montel or Schwartz, 68 Sublinear Functional Absolutely~ 8 Surjective limit of Lps Preservation of Pseudonorm Surjective representation, 254 Taylor formula with Lagrange Remainder with Assymptotic property, 133 Taylor Polynomial Differentiation of Taylor formula, 129 Theorem Generalized Milman, 56 Theorems Hahn-Banach Mean value, 8 Topology on L(E,F)

Compact open ...on H(U), 5

Uniform Convergence Normal Convergence Absolute Convergence, 138 Universal constant,

287 of m-Linear map and Polynomial Generalized, 42 Vector spaces Hahn-Banach Theorem in, Sublinear functional in,. 8 Vector sum of Bounding sets Weakly bounding ( or Liraited)set, 207 Weakly Bounding Closed, 16 Weakly bounded sets Bounded weakly-bounding sets Bounded sets, 210 Weakly bounding sets in Lbs Weakly bounding sets in F-spaces Limited sets in Lbs and Fspaces, 192

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