C* -Algebras : Banach Spaces (North-Holland Mathematical Library)

C*-ALGEBRAS

V O L U M E 1" B A N A C H SPACES

North-Holland Mathematical Library Board of Honorary Editors: M. Artin, H. Bass, J. Eells, W. Feit, P.J. Freyd, F.W. Gehring, H. Halberstam, L.V. H6rmander, J.H.B. Kemperman, W.A.J.Luxemburg, F. Peterson, I.M. Singer and A.C. Zaanen

Board of Advisory Editors." A. Bj6mer, R.H. Dijkgraaf, A. Dimca, A.S. Dow, J.J. Duistermaat, E. Looijenga, J.P. May, I. Moerdijk, S.M. Mori, J.P. Palis, A. Schrijver, J. Sj6strand, J.H.M. Steenbrink, F. Takens and J. van Mill

VOLUME 58

ELSEVIER Amsterdam- London- New York- Oxford- Paris - Shannon- Tokyo

C*-Algebras Volume l: Banach Spaces

Corneliu Constantinescu Departement Mathematik, ETH Ziirich CH-8092 Ziirich Switzerland

2001 ELSEVIER A m s t e r d a m - L o n d o n - New Y o r k - O x f o r d - Paris - Shannon- Tokyo

ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands

9 2001 Elsevier Science B.V. All rights reserved.

This work is protected under copyright by Elsevier Science, 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 Science Global Rights Department, PO Box 800, Oxford OX5 I DX, UK; phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-mail: [email protected]. You may also contact Global Rights directly through Elsevier's home page (http://www.elsevier.ni), by selecting '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 internal circulation, but permission of Elsevier Science 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 Science Global Rights Department, at the mail, 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 2001 Library of Congress Cataloging in Publication Data A catalog record from the Library of Congress has been applied for.

ISBN (this volume): ISBN (5 volume set): Series ISSN:

0 444 50749 3 0 444 50758 2 0924 6509

O The paper used in this publication meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Pape Printed in The Netherlands.

Preface

Functional analysis plays an important role in the program of studies at the Swiss Federal Institute of Technology. At present, courses entitled Functional Analysis I and II are taken during the fifth and sixth semester~ respectively. I have taught these courses several times and after a while typewritten lecture notes resulted that were distributed to the students. During the academic year 1987/88, I was fortunate enough to have an eager enthusiastic group of students t h a t I had already encountered previously in other lecture courses. These students wanted to learn more in the area and asked me to design a continuation of the courses. Accordlingly, I proceeded during the academic year, following, with a series of special lectures, Functional Analysis III and IV, for which I again distributed typewritten lecture notes. At the end I found that there had accumulated a mass of textual material, and I asked myself if I should not publish it in the form of a book. Unfortunately, I realized that the two special lecture series (they had been given only once) had been badly organized and contained material that should have been included in the first two portions. And so I came to the conclusion that I should write everything anew - and if at a l l - then preferably in English. Little did I realize what I was letting myself in for! The number of pages grew almost impercepetibly and at the end it had more than doubled. Aslo, the English language turned out to be a stumbling block for me; I would like to take this opportunity to thank Prof. Imre Bokor and Prof. Edgar Reich for their help in this regard. Above all I must thank Mrs. Barbara Aquilino, who wrote, first a WordMARC TM., and then a IbTF~ TM version with great competence, angelic patience, and utter devotion, in spite of illness. My thanks also go to the Swiss Federal Institute of Technology that generously provided the infrastructure for this extensive enterprise and to my colleagues who showed their understanding for it.

Corneliu Constantinescu

This Page Intentionally Left Blank

vii

Table of Contents of Volume 1

Introduction

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

Some Notation and Terminology 1

Banach Spaces 1.1

1.2

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

N o r m e d Spaces

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

1.1.1

General Results . . . . . . . . . . . . . . . . . . . . . . .

1.1.2

Some Standard Examples

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

1 7 7 7 12

1.1.3

Minkowski's Theorem . . . . . . . . . . . . . . . . . . . .

31

1.1.4

L o c a l l y C o m p a c t N o r m e d Spaces

35

1.1.5

P r o d u c t s of N o r m e d Spaces

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

37

1.1.6

S u m m a b l e Families . . . . . . . . . . . . . . . . . . . . .

40

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

Operators 1.2.1

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

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

General Results

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

61 61

1.2.2

Standard Examples . . . . . . . . . . . . . . . . . . . . .

74

1.2.3

Infinite M a t r i c e s

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

92

1.2.4

Q u o t i e n t Spaces . . . . . . . . . . . . . . . . . . . . . . .

113

1.2.5

Complemented Subspaces

1.2.6

T h e T o p o l o g y of Pointwise C o n v e r g e n c e

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

123 134

1.2.7

C o n v e x Sets . . . . . . . . . . . . . . . . . . . . . . . . .

138

1.2.8

The Alaoglu-Bourbaki Theorem . . . . . . . . . . . . . .

148

Bilinear Maps . . . . . . . . . . . . . . . . . . . . . . . .

150

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .

153

1.2.9

1.3

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

xix

The Hahn-Banach Theorem 1.3.1

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

159

The Banach Theorem . . . . . . . . . . . . . . . . . . . .

159

1.3.2

E x a m p l e s in M e a s u r e T h e o r y

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

171

1.3.3

The Hahn-Banach Theorem

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

180

1.3.4

T h e T r a n s p o s e of an O p e r a t o r . . . . . . . . . . . . . . .

191

viii

1.4

1.5

1.6

Table of Contents

1.3.5

P o l a r Sets

1.3.6

The Bidual

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

199

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

211

1.3.7

The Krein-Smulian Theorem

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

228

1.3.8

Reflexive S p a c e s . . . . . . . . . . . . . . . . . . . . . . .

240

1.3.9

C o m p l e t i o n of N o r m e d Spaces . . . . . . . . . . . . . . .

245

1.3.10

Analytic Functions

246

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

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .

254

A p p l i c a t i o n s of B a i r e ' s T h e o r e m . . . . . . . . . . . . . . . . . .

256

1.4.1

The Banach-Steinhaus Theorem . . . . . . . . . . . . . .

256

1.4.2

Open Mapping Principle

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

264

E x ercises . . . . . . . . . . . . . . . . . . . . . . . . . . .

280

Banach Categories

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

281

1.5.1

Definitions . . . . . . . . . . . . . . . . . . . . . . . . . .

281

1.5.2

Functors . . . . . . . . . . . . . . . . . . . . . . . . . . .

288

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

308

1.6.1

General Results . . . . . . . . . . . . . . . . . . . . . . .

308

1.6.2

Examples

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

322

O r d e r e d B a n a c h spaces . . . . . . . . . . . . . . . . . . . . . . .

334

1.7.1

O r d e r e d n o r m e d spaces . . . . . . . . . . . . . . . . . . .

334

1.7.2

Order Continuity

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

340

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

357

Subject Index

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

359

Symbol Index

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

371

1.7

Nuclear Maps

Name Index

ix

C o n t e n t s of All V o l u m e s

T a b l e o f C o n t e n t s of V o l u m e 1

Introduction

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

Some Notation and Terminology B a n a c h Spaces 1.1

1.2

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

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

N o r m e d Spaces

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

1.1.1

General Results . . . . . . . . . . . . . . . . . . . . . . .

1.1.2

Some S t a n d a r d Examples

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

1 7 7 7 12

1.1.3

Minkowski's Theorem . . . . . . . . . . . . . . . . . . . .

31

1.1.4

L o c a l l y C o m p a c t N o r m e d Spaces

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

35

1.1.5

P r o d u c t s of N o r m e d Spaces

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

37

1.1.6

S u m m a b l e Families . . . . . . . . . . . . . . . . . . . . .

40

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

Operators 1.2.1

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

General Results

61

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

61

1.2.2

Standard Examples . . . . . . . . . . . . . . . . . . . . .

74

1.2.3

Infinite M a t r i c e s

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

92

1.2.4

Q u o t i e n t Spaces . . . . . . . . . . . . . . . . . . . . . . .

113

1.2.5

Complemented Subspaces

123

1.2.6

T h e T o p o l o g y of P o i n t w i s e C o n v e r g e n c e

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

134

1.2.7

C o n v e x Sets . . . . . . . . . . . . . . . . . . . . . . . . .

1.2.8

The Alaoglu-Bourbaki Theorem . . . . . . . . . . . . . .

148

1.2.9

Bilinear Maps . . . . . . . . . . . . . . . . . . . . . . . .

150

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3

xix

The Hahn-Banach Theorem

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

138

153 159

1.3.1

The Banach Theorem . . . . . . . . . . . . . . . . . . . .

159

1.3.2

E x a m p l e s in M e a s u r e T h e o r y

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

171

1.3.3

The Hahn-Banach Theorem

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

180

1.3.4

T h e T r a n s p o s e of a n O p e r a t o r . . . . . . . . . . . . . . .

191

x

1.4

Table of Contents

1.3.5

Polar Sets

1.3.6

The Bidual

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

199

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

211

1.3.7

The Krein-Smulian

1.3.8

Reflexive Spaces . . . . . . . . . . . . . . . . . . . . . . .

Theorem

1.3.9

C o m p l e t i o n of N o r m e d Spaces

1.3.10

Analytic Functions

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

1.6

1.7

240

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

245

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

246

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .

254

A p p l i c a t i o n s of Baire's T h e o r e m

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

256

Theorem . . . . . . . . . . . . . .

256

1.4.1

The Banach-Steinhaus

1.4.2

Open Mapping Principle

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

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5

228

Banach Categories

264 280

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

281

1.5.1

Definitions . . . . . . . . . . . . . . . . . . . . . . . . . .

281

1.5.2

Functors

Nuclear Maps

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

288

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

308

1.6.1

General Results

1.6.2

Examples

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

308

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

322

Ordered Banach spaces . . . . . . . . . . . . . . . . . . . . . . .

334

1.7.1

Ordered normed spaces . . . . . . . . . . . . . . . . . . .

334

1.7.2

Order Continuity

340

Name Index

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

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

357

Subject Index

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

359

Symbol Index

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

371

xi

Table of Contents of Volume 2

Introduction 2

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

Banach Algebras 2.1

2.2

2.3

2.4

Algebras

xix

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

3

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

3

2.1.1

General Results

2.1.2

Invertible Elements

2.1.3

The Spectrum

2.1.4

Standard

2.1.5

C o m p l e x i f i c a t i o n of A l g e b r a s . . . . . . . . . . . . . . . .

51

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

Normed Algebras

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

3

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

13

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

17

Examples

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

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

2.2.1

General Results

2.2.2

The Standard

2.2.3

The Exponential

2.2.4

Invertible E l e m e n t s of U n i t a l B a n a c h A l g e b r a s . . . . . .

2.2.5

The Theorems

2.2.6

Poles of R e s o l v e n t s

2.2.7

Modules

69

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

Examples

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

Function and the Neumann

Series

. . .

of R i e s z a n d G e l f a n d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

32

69 82 114 125 153 161

.....

174

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .

197

Involutive Banach Algebras

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

201

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

201

2.3.1

Involutive Algebras

2.3.2

Involutive Banach Algebras

2.3.3

Sesquilinear Forms

2.3.4

Positive Linear Forms

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

287

2.3.5

The State Space . . . . . . . . . . . . . . . . . . . . . . .

305

2.3.6

Involutive Modules

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

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

241 275

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

322

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .

328

Gelfand Algebras

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

331

2.4.1

The Gelfand Transform . . . . . . . . . . . . . . . . . . .

331

2.4.2

Involutive Gelfand Algebras

343

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

xii

Table o] Contents

2.4.3

Examples

2.4.4

Locally Compact Additive Groups .............

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

365

2.4.5

Examples

378

2.4.6

The Fourier Transform

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

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Compact Operators 3.1

3.2

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

The General Theory

358

390 396 399

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

399

3.1.1

General Results . . . . . . . . . . . . . . . . . . . . . . .

399

3.1.2

Examples

419

3.1.3

Fredholm Operators

3.1.4

Point Spectrum

3.1.5

S p e c t r u m of a C o m p a c t O p e r a t o r

3.1.6

Integral Operators

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

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

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

437 468 477 489

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .

517

L i n e a r Differential E q u a t i o n s . . . . . . . . . . . . . . . . . . . .

518

3.2.1

B o u n d a r y Value P r o b l e m s for Differential E q u a t i o n s . . .

518

3.2.2

Supplementary Results . . . . . . . . . . . . . . . . . . .

530

3.2.3

L i n e a r P a r t i a l Differential E q u a t i o n s

...........

549

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .

563

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

565

Subject Index

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

568

Symbol Index

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

588

Name Index

xiii

T a b l e o f C o n t e n t s of V o l u m e 3

Introduction

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

xix

C*-Algebras

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

3

4.1

4.2

4.3

4.4

The General Theory

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

3

4.1.1

General Results . . . . . . . . . . . . . . . . . . . . . . .

4

4.1.2

T h e S y m m e t r y of C * - A l g e b r a . . . . . . . . . . . . . . .

30

4.1.3

F u n c t i o n a l c a l c u l u s in C * - A l g e b r a s

56

4.1.4

T h e T h e o r e m of F u g l e d e - P u t n a m

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

75

The Order Relation . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1

Definition and General Properties

4.2.3

Examples

4.2.4

P o w e r s of P o s i t i v e E l e m e n t s

4.2.5

The Modulus

4.2.6

Ideals a n d Q u o t i e n t s of C * - A l g e b r a s

4.2.7

T h e O r d e r e d Set of O r t h o g o n a l P r o j e c t i o n s

4.2.8

Approximate Unit

92

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

92

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

116

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

123

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

143

...........

150

.......

162

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

178

S u p p l e m e n t a r y R e s u l t s on C * - A l g e b r a s . . . . . . . . . . . . . .

208

4.3.1

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

208

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

215

The Exterior Multiplication

4.3.2

Order Complete C*-Algebras

4.3.3

The Carrier

4.3.4

Hereditary C*-Subalgebras

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

263

4.3.5

Simple C*-algebras . . . . . . . . . . . . . . . . . . . . .

276

4.3.6

Supplementary Results Concerning Complexification

W*-Algebras 4.4.1

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

243

. .

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

General Properties

297

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

297

4.4.2

F as an E - s u b m o d u l e

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

309

4.4.3

Polar Representation

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

335

4.4.4

W*-Homomorphisms

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

Name Index

of E '

286

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

.

361 385

xiv

Table of Contents

Subject Index

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

388

Symbol Index

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

411

xv

Table of Contents of Volume 4

Introduction

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

5 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Pre-Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 General Results . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Hilbert sums . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Orthogonal Projections of Hilbert Space . . . . . . . . . . . . . 5.2.1 Projections onto Convex Sets . . . . . . . . . . . . . . . 5.2.2 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Orthogonal Projections . . . . . . . . . . . . . . . . . . . 5.2.4 Mean Ergodic Theorems . . . . . . . . . . . . . . . . . . 5.2.5 The Frkchet-Riesz Theorem . . . . . . . . . . . . . . . . 5.3 Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 General Results . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Supplementary Results . . . . . . . . . . . . . . . . . . . 5.3.3 Selfadjoint Operators . . . . . . . . . . . . . . . . . . . . 5.3.4 Normal Operators . . . . . . . . . . . . . . . . . . . . . . 5.4 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Cyclic Representation . . . . . . . . . . . . . . . . . . . 5.4.2 General Representations . . . . . . . . . . . . . . . . . . 5.4.3 Example of Representations . . . . . . . . . . . . . . . . 5.5 Orthonormal Bases . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 General Results . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Hilbert Dimension . . . . . . . . . . . . . . . . . . . . . 5.5.3 Standard Examples . . . . . . . . . . . . . . . . . . . . . 5.5.4 The Fourier-Plancherel Operator . . . . . . . . . . . . . 5.5.5 Operators and Orthonormal Bases . . . . . . . . . . . . 5.5.6 Self-normal Compact Operators . . . . . . . . . . . . . . 5.5.7 Examples of Real C-Algebras . . . . . . . . . . . . . .

XiX

3 3 3 14 19 24 24 29 33 54 63 72 72 86 108 123 130 130 146 156 166 166 191 206 218 223 243 258

Table o] Contents

xvi

5.6

Hilbert right C*-Modules 5.6.1

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

Some General Results

286

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

286

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

310

5.6.2

Self-duality

5.6.3

Von Neumann

5.6.4

Examples

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

373

5.6.5

JCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

430

5.6.6

Matrices over C * - a l g e b r a s

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

477

5.6.7

Type I W*-algebras

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

515

Name Index

right W*-modules

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

341

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

535

Subject Index

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

539

Symbol Index

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

567

xvii

Table of Contents of Volume 5

Introduction 6

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

Selected Chapters of C'-Algebras . . . . . . . . . . . . . . . . . . . 6.1 LP.Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Characteristic Families of Eigenvalues . . . . . . . . . . . 6.1.2 Characteristic Sequences . . . . . . . . . . . . . . . . . . 6.1.3 Properties of the CP-spaces . . . . . . . . . . . . . . . . 6.1.4 Hilbert-Schmidt Operators . . . . . . . . . . . . . . . . . 6.1.5 TheTrace . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.6 Duals of Cp-spaces . . . . . . . . . . . . . . . . . . . . . 6.1.7 Exterior Multiplication and Cp-Spaces . . . . . . . . . . 6.1.8 The Canonical Projection of the Tridual of K . . . . . . 6.1.9 Integral Operators on Hilbert Spaces . . . . . . . . . . . 6.2 Selfadjoint Linear Differential Equations . . . . . . . . . . . . . 6.2.1 Selfadjoint Boundary Value Problems . . . . . . . . . . . 6.2.2 The Regular Sturm-Liouville Theory . . . . . . . . . . . 6.2.3 Selfadjoint Linear Differential Equations on T . . . . . . 6.2.4 Associated Parabolic and Hyperbolic Evolution Equations 6.2.5 Selfadjoint Linear Partial Differential Equations . . . . . 6.2.6 Associated Parabolic and Hyperbolic Evolution Equations 6.3 Von Neumann Algebras . . . . . . . . . . . . . . . . . . . . . . 6.3.1 6.3.2 6.3.3 6.3.4 6.3.5 6.3.6 6.3.7 6.3.8 6.3.9

The Strong Topology . . . . . . . . . . . . . . . . . . . . Bidual of a C*-algebra . . . . . . . . . . . . . . . . . . . Extension of the Functional Calculus . . . . . . . . . . . Von Neumann- Algebras . . . . . . . . . . . . . . . . . . The Commutants . . . . . . . . . . . . . . . . . . . . . . Irreducible Representations . . . . . . . . . . . . . . . . Commutative von Neumann Algebras . . . . . . . . . . . Representations of W*-Algebras . . . . . . . . . . . . . . Finite-dimensional C*-algebras . . . . . . . . . . . . . .

xix

3 3 3 10 21 46 56 72 79 102 116 124 125 139 150 153 184 192 202 203 218 263 283 293 299 320 325 334

xviii

Table of Contents

6.3.10 A generalization . . . . . . . . . . . . . . . . . . . . . . . 7

C * - a l g e b r a s G e n e r a t e d by Groups 7.1

7.2

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

P r o j e c t i v e Representations of Groups

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

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

355 369 369

7.1.1

Schur functions

7.1.2

Projective Representations . . . . . . . . . . . . . . . . .

7.1.3

S u p p l e m e n t a r y Results . . . . . . . . . . . . . . . . . . .

431

7.1.4

Examples

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

466

Clifford Algebras

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

492

404

7.2.1

G e n e r a l Clifford Algebras

7.2.2

C~.p,q . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

518

7.2.3

C~(IN)

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

538

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

559

Subject Index

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

563

Symbol Index

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

592

N a m e Index

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

369

492

xix

Introduction This book has evolved from the lecture course on Functional Analysis I had given several times at the ETH. The text has a strict logical order, in the style of "Definiton- Theorem - P r o o f - E x a m p l e - Exercises". The proofs are rather thorough and there are many examples. The first part of the book (the first three chapters, resp. the first two volumes) is devoted to the theory of Banach spaces in the most general sense of the term. The purpose of the first chapter (resp. first volume) is to introduce those results on Banach spaces which are used later or which are closely connected with the book. It therefore only contains a small part of the theory, and several results are stated (and proved) in a diluted form. The second chapter (which together with Chapter 3 makes the second volume) deals with Banach algebras (and involutive Banach algebras), which constitute the main topic of the first part of the book. The third chapter deals with compact operators on Banach spaces and linear (ordinary and partial) differential equations- applications of the theory of Banach algebras. The second part of the book (the last four chapters, resp. the last three volumes) is devoted to the theory of Hilbert spaces, once again in the general sense of the term. It begins with a chapter (Chapter 4, resp. Volume 3) on the theory of C*-algebras and W*-algebras which are essentially the focus of the book. Chapter 5 (resp. Volume 4) treats Hilbert spaces for which we had no need earlier. It contains the representation theorems, i.e. the theorems on isometries between abstract C*-algebras and the concrete C*-algebras of operators on Hilbert spaces. Chapter 6 (which together with Chapter 7 makes Volume 5) presents the theory of/:P-spaces of operators, its application to the self-adjoint linear (ordinary and partial) differential equations, and the von Neumann algebras. Finally, Chapter 7 presents examples of C*-algebras defined with the aid of groups, in particular the Clifford algebras. Many important domains of C*-algebras are ignored in the present book. It should be emphasized that the whole theory is constructed in parallel for the real and for the complex numbers, i.e. the C*-algebras are real or complex. In addition to the above (vertical) structure of the book, there is also a second (horizontal) division. It consists of a main strand, eight branches, and additional material. The results belonging to the main strand are marked with (0). Logically speaking, a reader could restrict himself/herself to these and ignore the rest. Results on the eight subsidiary branches are marked with (1), (2), (3), (4), (5), (6), (7), and (8). The key is

xx

Introduction

1. 2. 3. 4. 5. 6. 7. 8.

Infinite Matrices Banach Categories Nuclear Maps Locally Compact Groups Differential Equations Laurent Series Clifford Algebras Hilbert C*-Modules

These are (logically) independent of each other, but all depend on the main strand. Finally, the results which belong to the additional material have no marking and - from a logical perspective - may be ignored. So the reader can shorten for himself/herself this very long book using the above marks. Also, since the proofs are given with almost all references, it is possible to get into the book at any level and not to read it linearly. We assume that the reader is familiar with classical analysis and has rudimentary knowledge of set theory, linear algebra, point-set topology, and integration theory. The book addresses itself mainly to mathematicians, or to physicists interested in C*-algebras. I would like to apologize for any omissions in citations occasioned by the fact that my acquaintance with the history of functional analysis is~ unfortunately, very restricted. For this history we recommand the following texts. BIRKHOFF, G. and KREYSZIG, E., The Establishment of Functional Analysis, Historia Mathematica 11 (1984), 258-321. 2. BOURBAKI, N., Elements of the History of Mathematics. (21. Topological Vector Spaces), Springer-Verlag (1994). 3. DIEUDONNt~, J., History of Functional Analysis, North-Holland (1981). 4. DIEUDONNI~, J., A Panorama of Pure Mathematics (Chapter C III: Spectral Theory of Operators), Academic Press (1982). HEUSER, H., Funktionalanalysis, 2. Auflage (Kapitel XIX: Ein Blick auf die werdende Functionalanalysis), Teubner (1986), 3. Auflage (1992). KADISON, R.V., Operator Algebras, the First Forty Years, in: Proceedings of Symposia in Pure Mathematics 38 I (1982), 1-18. MONNA, A.F., Functional Analysis in Historical Perspective~ John Whiley & Sons (1973).

Introduction

xxi

8. STEEN, L.A., Highlights in the History of Spectral Theory, Amer. Math. Monthly 80 (1973), 359-382. There is no shortage of excellent books on C*-algebras. Nevertheless, we hope that this book will be also of some utility to the mathematics commutity.

This Page Intentionally Left Blank

Some N o t a t i o n and Terminology We use in this book the notation and terminology which are usual in the current m a t h e m a t i c a l literature. In the following list we present some of those for which we felt that difficulties in interpretation may arise. Any set theory (with the axiom of choise or equivalently with Zorn's Lemma) will do for the present book. 3 and V denote "there exists" and "for all", respectively; 3! means "there exists uniquely". We write iff for "if and only if". On special occasions (appearing very seldom) we choose the axiomatic setting of von Neumann: we call class a collection of sets (which need not be itself a set) and we define an ordinal number ~ as the set of ordinal numbers 7/ strictly smaller than ~, i.e.

(0 := 0). A cardinal number is the smallest ordinal number having a given cardinality; we denote for every set T by C a r d T its cardinal number, i.e. the cardinal number with the same cardinality as T . If P is a proposition and x a variable (which may occur in P ) , then

{x IF(x)} denotes the class of x for which P(x) holds. If in addition X is a set, then we put

{x e x lP(x)} : - {x Ix e x and P ( x ) } . If A and B are sets, then

A\B := {x e d lx ~ B}, A A B := (A\B)U (B\A), dxB:={(x,y)

lxed

and y e B } .

A partition of a set X is a set of pairwise disjoint nonempty subsets of X the union of which is equal to X . A function or a map is a triple f := (X, Y, F) (denoted also by f : X -+ Y ), where X , Y are sets and F is a subset of X x Y such that

x E X ==~ 3!y~ C Y, (x, yz) C F. X, Y, and F are called the domain, the range of values (or codomain), and the graph of f , respectively. We set then f(x) := y~

for all x E X and

f ( A ) := {y E Y I ::Ix E A , f ( x ) = y}, -1

) E B},

f(B):={xEXIf(x -1

-1

f (Y):= f ({Y}) for all A c X , B

C Y , and y E Y. We call f a m a p o f

X into Y. If f , g

are maps of X into Y, then we set {f = g} := {x E

X lf(x) = g ( x ) } ,

{f -~ g} : : {x e X I f ( x ) ~ g ( x ) } . If T is a term and x a variable (which may occur in T ) and X, Y are sets such that

x EX ~

T(x) E Y ,

then we denote the map f := (X,Y, {(x,y) E X x Y IY = T(x)}) by

f : X -~ Y,

x,

>T ( x ) .

If f : X --+ Y is a map and Z is a subset of X , then the restriction of f to Z (denoted f i Z ) i s

the map

z--~y,

x,

~f(z).

The map f : X --+ Y is called injective (surjective), if (~, y e X ,

f(x) = I(Y)) ~

x = y

( / ( x ) = Y). The expression "f is a map of X onto Y" means f is a surjective map of X into Y. f is called bijective if it is simultaneously injective and surjective, in which case we set f-l

:y

; X,

f (x) ,

>z

and call f - 1 the inverse of f . If Y is a set and X is a subset of Y , then the inclusion map X --+ Y is the map X

>Y,

x~

>x;

if X = Y then we may call the inclusion map X -+ Y the identity map of Y . If X, Y, Z are sets and

f :X

~Y,

g:Y

>Z ,

x.

>g ( f ( x ) )

then we put

go f : X - - + Z ,

and call this map the composition of f and g. If X, Y, Z are sets and

f:XxY

>Z,

then we put

f ( a , . ) : Y----+ Z ,

y,

> f(a,y),

>Z,

x,

~f(x,b)

f(-,b):X for all a E X

and b E Y .

A family (X)~Er (indexed by I ) is in fact the m a p c ~ x~ defined on the set I for which the range of values (codomain) is not specified. Any set X defines the canonical family (x)xEx. If (X~),Er is a family of sets, then we put

1-I x~ .= ((x,),,~ !~ E t ~

x, E x~}

and call 11 X~ the product of the family (X~)~ei. tEI

Let X be a set. An equivalence relation on X is a binary relation ~ on X such t h a t we have for all x, y, z E X " Xt,,.,X~

x~y::~y,',-,x, ( x , ' ~ y and y ~ z )

~x~z.

An equivalence class of the equivalence relation ~ is a nonmepty subset A of X such t h a t

x , y E A ==~ x ~ y , (x c A , y C X , x ~ y) :==v y E A . For every x C X , the set {y C X I x ~ y} is an equivalence classe of ,-~ called the equivalence class of x (with respect to ~ ). The set of equivalence classes of is a partition of X which is denoted by X / ~ .

The m a p X --+ X / ~

which

sends every x C X into its equivalence class is called the quotient map. A free ultrafilter on a set X is an ultrafilter on X possessing no one-point sets. If X is an infinite set then the filter on X , {X\AIA

finite set}

is called the filter of cofinite subsets of A. A totally ordered set is an ordered set X such that for all x , y E X either x O},

K'-{~EI

la~ IK,

tD

> limxn(t). u-+co

Then for all n E IN, n > n~, sup ixn(t) - x(t)l < c. tET

Thus x n - x E t~co(T) and I1~ - ~II~ -< c

for every c > 0 and n C IN, n > n~. Hence x C fco(T) and lim Xn -- X. n--+ oo

gco (T) is thus a Banach space. It is obvious t h a t gco(T) is separable whenever T is finite. Given two distinct subsets A, B of T , we have t h a t i]~ - ~]i-

1.

Hence the distance between any two distinct elements of the set {e A I A E (T)} is 1. If T is infinite, then this set is uncountable and gco(T) is not separable.

Remark.

I The assertion t h a t gco(T) is not separable whenever T is infinite will

be generalized in Corollary 6.3.6.16 b). Example

1.1.2.3

( 0 ) Let T be a set. If T is finite then put

~~

:: c(T)-- c0(T):: ~(T).

I f T is infinite, let ~ denote the filter on T consisting of the cofinite subsets ofT,

i.e. 9= {A C T O T \ A is finite},

and define c ( T ) - = {x e gco(T) ] x ( ~ ) converges},

14

1. Banach Spaces

e~

:= c0(T) := {x e e~(T) I lim x ( ~ ) = 0}.

Then c(T) an co(T) are closed vector subspaces of e~176 and therefore Banach spaces with respect to the induced norm. Given x 9 co(T), if x 7~ O, then the set

{t e T I Ix(t)i = Ilxllt is finite and nonempty, c(T) and co(T) are separable iff T is countable. The norm on co(T) defined by the restriction of the norm on f ~ ( T ) to co(T) is

~o,n~ti,~ e ~ o t ~ by

li-llo we ~t e ~ := ~o := c o ( ~ ) ,

~ := ~ ( ~ ) .

Only the last assertion needs proof. Assume that T is countable. The set of linear combinations of the vectors et (t 9 T) and eT with coefficients in (resp. ~ + i ~ ) is countable and dense in c(T). Hence c(T) and co(T) are separable. Given distinct s, t 9 T ,

Hence co(T) and c(T) are not separable if T is uncountable. E x a m p l e 1.1.2.4

i

( 0 ) Let T be a topological (measurable) space. Define C(T) := {x 9 e~(T) l x is continuous},

(B(T) := {x 9 e~(T) i 9 i~ measurable}). Then C(T) (B(T)) is a closed vector subspace of e~(T) and consequently a Banach space with respect to the induced norm. i E x a m p l e 1.1.2.5

( 0 ) Let T be a set and p a real number, p > 1. Put

f

eP(T) := / z c IKr

, ~ Ix(t)l' < ~ }

O'(T) is a vector subspace of IK T and [].[[p: gP(T) --+ IR+,

x,

defines a norm on gP(T), called the p - n o r m , gP(T) with this norm is a Banaeh space. If T is infinite then Card T is the topological cardinality of gP(T). In particular, gP(T) is separable iff T is countable. We put

e~ := e~(~)

1.1 Normed Spaces

15

By the Minkowski inequality,

(

I~(t) + y ( t ) l ~

0 there is an n E E IN with IlZm -- x~ll~
n ~ , n

> n . . We deduce t h a t for every t E T ,

(x~(t))~E~ is a Cauchy sequence. Define x'T

>IK,

t,

> limx=(t). n - - + (x)

Then 1 p

_ Card T . Remark.

m

The spaces gP(T) (p C [1, c~]) are special cases of the LP-spaces of

integration theory. They are precisely the LP-spaces with respect to counting measure on T .

Proposition 1.1.2.6

( 0 )

Let T be a set and take p,q 6 [1,c~[, with

P b, (tn)ne~N has a Catchy subsequence (s,~)neIN. Since T is complete, (Sn)ne~ converges in T. Lemma 1.1.2.11 c ~ b shows that A is relatively compact. I P r o p o s i t i o n 1.1.2.13

( 0 ) (Fr6chet, 1907) Take a set T . Take p E [1, c~[.

Let A be a subset of g~(T) (resp. co(T), resp. c ( T ) ) . Then the following are equivalent:

a)

A is relatively compact.

b)

Given c > O, there is a finite subset S o f T

with

1__ p

Iz(t)l ~


a. Take ~ > 0 and let S be a finite subset of T with 1

I*(t)l"


I i x ( s t ) - y(s,)ll


nq(x,~). Given n E IN, put 1 y n : = nq'x,~'[ ) x n

Then, given n C IN

P(Yn) ~- p(Xn) > 1, nq(xn) q(v~)

=

q(~)

_

1

Hence lim q(Yn) = 0 n---+oo

and so, by c), lim P(Yn) = O,

n--+oo

which is a contradiction. Hence there is some c~ > 0 with p _< a q . Interchanging the roles of p and q, it follows t h a t p and q are equivalent.

I

32

1. Banach Spaces

Remark.

For a substantial part of the theory of normed spaces (which includes

the first four chapters of this book) only the topology generated by the norm, and not the norm itself is significant. (Thus our interest is not really in normed spaces but, in "normable" spaces). In this "topological case" the norm may be replaced by any other equivalent norm. The theory of locally convex spaces is the best context for this theory. For more precise studies, the "geometrical aspect" is very important and we cannot exchange the norm without damaging the results. This is the case, e.g., with Hilbert spaces and C*-algebras, which are treated in the last four chapters of this book. C o r o l l a r y 1.1.3.2

I"

(

0

\

)

Let p,q be equivalent norms on the vector space

E . I f E is complete with respect to p , then it is also complete with respect to q.

II

D e f i n i t i o n 1.1.3.3

Theorem

1.1.3.4

( 0 )

Given n E IN U {O, o c } , we define

( 0 )

(Mi,~ow~i'~ Theorem, ~Sg~).

Norm~ of

finite-dimensional vector space are equivalent. Given n c IN, let p be an arbitrary norm on IK n and I1" II the Euclidean norm on IKn . Furthermore, let el, e 2 , . . . , en be the standard basis vectors of IK n and put

a :=

p(ek) 2

> O.

k--1

Given x := (xk)ke~. C IK n , we have, by Schwartz's Inequality, that 1

p(x) = p

xkek k=a


__

for distinct elements m, n E IN.

We construct the sequence recursively. Take n E IN and suppose that the sequence has been constructed up to n -

1. Let F be the vector subspace of

E generated by X l , X 2 , . . . , x n - 1 . Then F is finite-dimensional and so E ~ F .

36

1. Banach Spaces

By Corollary 1.1.3.6, F is closed, so, by Proposition 1.1.4.2, there is an x~ E E

IIx~ll

with

= 1 and

1

d~(x~) > -~ Then

Ix n - x m l l >

1

for every m E IN, m < n, which completes the recursive construction. Remark.

I

The above corollary can be improved (see Exercise 1.3.5).

T h e o r e m 1.1.4.4

( 0 )

(F. Riesz, 1 9 1 8 ) E v e r y locally compact normed

space is finite-dimensional.

Let E be a locally compact normed space. We assume that E is infinitedimensional. By Corollary 1.1.4.3 there is a sequence (xn)ne~ in E such that

[[xnI[-1,

Ixm-xni[_

1

for distinct elements m, n E IN. Since

{x E E I Ilxll = 1} is compact (Corollary 1.1.1.4), the sequence (x,)ne~ contains a convergent subsequence, which is obviously a contradiction. P r o p o s i t i o n 1.1.4.5

I

( 3 ) Let A be a nonempty subset of the normed space

E . Then, given x , y E E , IdA(x) -- dA (Y)I ~ IIx - yll.

We have

dA(x) ~ I I x - zll ~ I I x - Yll + IlY- zll for any z E A , so that

dA(~) ~ IIx - yll + dA(y), dA (X) - dA (y) ~ IIx - yll, IdA(x) - dA(y)I ~ IIx - yll.

I

1.1 Normed Spaces

37

1.1.5 P r o d u c t s of N o r m e d S p a c e s

Proposition 1.1.5.1

( 0 ) Let ( E L ) t e l be a finite family of normed spaces and p be a norm on IK I such that given (c~L),ei, (~L)LeI 9 IR~ with c~L

IR+,

(x~)Le,'

>

suP lxL I LEI

(Example 1.1.2.5, Proposition 1.1.5.1) is called the supremum norm

of the pro-

duct 11 EL and for every p 9 [1, oo[ the norm LEI

IIE

--+ m ,

>

LEI

is called the p - n o r m

IIx ll LEI

of the product

11 EL (Example 1.1.2.5, Proposition tEI

1.1.5.1). These norms are denoten sometimes by I1"

lb.

Th~ 2-~o~m i~ also

called the Euclidean n o r m of the product 11 EL. Unless otherwise specified, LEI

we take the Euclidean norm on the product of normed space.

All these norms generate the product topology (Proposition 1.1.5.1) and they are therefore equivalent (Proposition 1.1.3.1 c =~ a). When our interest

38

1. Banach Spaces

is restricted to "topological aspects" of the theory (i.e. if we are concentrating on "normable" and not on normed spaces) any of these norms on products will do. But as soon as the "geometric aspects" are important, we have to choose a specific one of these norms. For example, the Euclidean norm will be the appropriate one in the case of Hilbert spaces while the supremum norm will be needed in the case of C*-algebras. P r o p o s i t i o n 1.1.5.3

( 0 )

Let E be a normed space and take (~,~ c IK.

Then the map

ExE

>E,

(x,y),

>c~x+~y

is uniformly continuous.

We have

II(~x~ + ,~yl) -(o~x2 + ~y~)ll = IIc~(Xl

z2) + ,/~(yl

-

-

-

y~)ll

E,

(x,y),

~ax+C~y. --1

- -

By Proposition 1.1.5.3, ~ is continuous and so ~ (F) is closed. Since

_1

FxFc~(F)

c

~(

T),

it follows that - 1

FxF=FxFc

m

cp(F).

Hence m

c~x + f l y - ~(x, y) C F for every x, y E F , i.e. F is a vector subspace of E .

1.1 Normed Spaces

C o r o l l a r y 1.1.5.5 ( 0 )

39

Let E be a normed space, A a subset of E , and

F the vector subspace of E generated by A . Then F is the smallest closed vector subspace of E constaining A . It is called the closed vector subspace o f E generated by A . If A is countable, then F , endowed with the induced norm, is separable.

By Corollary 1.1.5.4, F is a vector subspace of E and it is clear that it is the smallest closed vector subspace of E containing A. Assume now that A is countable and let B be the set of linear combinations of elements of A with coefficients in Q (resp. ~ + iQ). Then B is countable and m

BcFcB.

Thus m

m

m

BcFcB,

so that F -

m

B , i.e. F is separable.

Example 1.1.5.6

Let E be a normed (Banach) space and c~, fl, 7, 6 scalars

such that

~-

f17 # o.

Then E x E

> IR+,

(z,y) ,

>

sup{ II~x + ~YlI, 117x + @IF}

is a (complete) norm.

The map

ExE

>ExE,

(x, y),

,

(~x + &, 7~ + @)

is linear and bijective and the map

ExE

>Ia+,

(x,y),

is a (complete) norm (Proposition 1.1.5.1).

>

sup{ll*ll, Ilvll} n

40

1. Banach Spaces

1.1.6 Summable Families

Definition 1.1.6.1

( 0 )

An ordered set T is called upward (downward)

directed if for every s, t 9 T , there is an r 9 T such that s 0 there is a J E ~S(I) such that

O the set

is finite and the set

is countable.

44

1. Banach Spaces

There is a J

9~ 1 ( I )

with

g

X~

< C

for every K e q3S(I\J ) (Proposition 1.1.6.6). We get

{ ~ Z lllx~ll >_e} c J, i.e.

is a finite set. From

Ilz~ll> _~1} we deduce that

is countable. C o r o l l a r y 1.1.6.8

I Let s be a Banach space. If (xt)~E, is a summable family

in E , then (atxt)tE, is summable in E for every (at)rE, E g ~ Take c > 0. Put := sup ilatl.i tel

There is a J C q3y(I ) such that

4(1 + 5) for every K C q3S(I\J ) (Proposition 1.1.6.6). By Proposition 1.1.1.5, b & last assertion. Let c > O. There exists a J E ~J.f(I) such t h a t

~EK

for every K E ~ 3 y ( I \ J ) . We get

for every K E g l S ( I \ J ) . By Proposition 1.1.6.6 the family (x~)~ei is summable. We have

for every J E ~ S ( I ) and therefore

b =:> c is trivial.

46

1. Banach Spaces

c =~ a. Let (xn)ne~ be a Cauchy sequence in E . There is a strictly increasing family (kp)pe~u{o} in IN such that

II~.-

1

~,,II

< 2-;

whenever n ~ IN, n > kp. Then 1

pEIN

pEIN

By c), (xkp - xkp_l)pelN is summable. Given n E IN zk. - Zko + ~ - ~ ( z k , - Zko) p=l

so that (xk.)ne~ converges. It follows that (x.).e~ converges. E is thus complete, m P r o p o s i t i o n 1.1.6.11

( 0 ) Let E be a normed space, (xt)tei,(yt)tci be

summable families in E and c~, ~ C IK. Then (c~x~+ j3yt)~ei is summable and

tel

tel

tel

We have (Proposition 1.1.5.3) that aZxt

+ ~EyL

tel

=alimZxt J,~ l

tel

+ ~lim y'~y~ J,q~l

tEJ

tEJ

= lim ( ~ ~-~ xt +/3 ~-'~ y ~ ) = lim y ~ (c~x~ +/3yt) . J,~l

LEJ

P r o p o s i t i o n 1.1.6.12

tEJ

( 0 ) Let

.

J'TdI tEJ (Xt)te I

be a family in the normed space E

and Io -

{t e l J x, # O}.

Then (x,),et is summable iff (x,)te,o is summable and in this case Z

X t -- ~

tel

Xt .

tEIo

We set :-~3/(I)

> q3:(Io),

J,

> IonJ.

Then P ( ~ t ) - ~Io and ExL=Ext tEJ tE~(J)

for every J E q3l(I ) . The proposition now follows.

m

I. 1 Normed Spaces

47

C o r o l l a r y 1.1.6.13 ( 0 ) Let (x~)~e/, (Y~)~eL be summable families in the normed space E such that I A L = O. Define Ztt :---

f xu

[

if # e l

Yu

if p e L .

Then

E zt'--Ex~+EY~"

#EIUL

tel

AEL

For each A e L and each ~ e I , define xa:=0,

y~:=0.

The corollary now follows from Proposition 1.1.6.12 and Proposition 1.1.6.11.

I Proposition 1.1.6.14

( 0 )

The following assertions are equivalent for

every family (c~)~i in IK :

a)

(oQ)~E I is sumrnable.

b)

(oLt)~e I is absolutely summable.

c) For every injective map ~" IN --+ I the sequence (~'~ c~v(k))ne~ convcrk=l ges.

d)

sup Je~s(I)

In particular, the summable and absolutely summable families coincide in finitedimensional normed spaces.

a =:~ b. Assume first IK = IR. We set

Then (OLt)tEI+ , (Ct~)~EI_ , and (--OQ)~EI are summable. By Corollary 1.1.6.13, (la~i)~ei is summable, hence (a~)~e/ is absolutely summable. Assume now IK = (!?. Then (rea~)~ci , (ima~)~ei are summable (Corollary 1.1.6.5), so by the above considerations they are absolutely summable. We deduce sup

~

la~l _< sup E Ireatl + sup E ]ima,] < oc, JEq3f(I) ,EJ J ~ s (I) ~cJ

48

1. Banach Spaces

hence (c~,),ct is absolutely summable (Proposition 1.1.6.3). b => c is trivial. c ~ d. Assume

JEg~I(I)

~EJ

Then there is an increasing sequence (J~),c~ in

liml~o~l

q3f(I) such that

~

n---~ o o

Let qa : IN --+ I be an injective map for which there is an increasing sequence (Pn)ne~ in IN with

for every n E IN. Then lim

~--~a~(k)

n--+ oo

_-~,~limIZo~I

~,

tE Jn

k=l

which contradicts c). d :::> a. Assume first IK = JR. We set I+ := {t E I I o~, _> 0},

I - := {~ E I J c~, < 0}.

Then sup ? . a , JE~I (I+) ~cJ

< cx~,

sup

).(-c~)

< oo,

JEgls(l- )

(OQ)~EI+ and (-c~)~c/_ are summable (Proposition 1.1.6.3). Hence (oQ)tEI is summable (Proposition 1.1.6.11, Corollary 1.1.6.13).

SO

Assume now IK = r

sup i zeEJoo l

JEq3i(I)

Then

sup Iro Co l

JEq3y(I)

sup I E i m c ~ L [ - s u p limEc~Ll< Jcq3I(I) ~cJ Jcq3f(t) ~cJ

JEq3I(I)

sup I E c ~ ] < o o . JEq3f(I) ~cJ

By the above considerations (re a+)~ct, (ima+)+c/ are summable, hence (a+)+ci is summable (Corollary 1.1.6.5).

Remark. a) Let

(OQ)LEI be a family in IK,

I

1.1 Normed Spaces

f "I

> IK,

~t

49

>a~,

and p be the counting measure on I, i.e. p ' ~ S ( I ) ----+ 1R,

J,

> CardJ.

Then (a~)~ei is summable iff f is #-integrable and in this case Ea~=lfdP.

b) A. Dworetzky and C.A. Rogers proved (1950) that if every summable family in E is absolutely summable, then E is finite-dimensional. Example 1.1.6.15 ( 1 ) ( 3 ) Let T be aset, p 9 [1, oc[, a n d x 9 T. Then (x(t)et)tET is summable in gP(T) (in ~ ( T ) ) iff x 9 tP(T) (x 9 co(T)) and in this case the sum is x. Assume first x 9 fP(T) (x 9 co(T)). Then 1

E teA

x(t)et -- x

-

Ix(t)l . t

p

,

(ll

~(t)~,tea

-

c~

sup tET\A

Ix(t)l)

for every finite subset A of T. Hence (x(t)et)tET is summable in gP(T) (in g~(T) ) and its sum is x. Assume now (x(t)et) summable in gP(T) (in t~~ Let c > 0. Then there is a finite subset A of T such that

V x(t)~ EB

P

for every B 9 ~ s ( T \ A ) (Proposition 1.1.6.6). We get EtEWIx(t)lP -- EteAIx(t)lP + sup {EtEB Iz(t)IPlB C ~ / ( T \ A ) } _


'

2

-

for every n E IN. We set

U " ~cr

~E,

x,

~y~z4(-~an

nEIN

(Corollary 1.1.6.10 a =:v c). u is linear (Proposition 1.1.6.11). Let x E t~~ so that ux E U En . Then there is a p E ] N so that ux E Ep . Let q E ] N , q > p ,

nEIN

such that x(q) # O. Then

4q( q

x(q)

u x - ~-~--~-an

)

E Eq,

n--1

so that 1

-
0 and ro O(3

n=p+l

nr'd-lllx~ll

c < 5

Suppose that /5' E Uro_l~l(a). ~ Then ]/31 < r0, and so n-1 E

n-1 ~m(:~n-m-I

n--1

IR+,

a,

> Irea I + l i m c ~ I

is a norm when ~ is endowed with its canonical structure as real vector space, but it is not a norm if r E 1.1.3

is taken to be complex vector space.

Let n C I N . Show t h a t there is a norm p on ]K 2 such that

(

) 1

p (1,0) < -I1(1,0)11, n p((0,1)) > nil(o, 1)11. E 1.1.4

Let

E

be the vector space of continuously differentiable maps

] - 1, 1[--, IK with bounded derivatives. Show that E E

>

IR+,

9,

sup ]dx

tCl-l,ll ~(t)

]

> IR+,

te ]- 1,1[ are equivalent norms. E 1.1.5

Show that a normed space E is separable iff it possesses a dense

vector subspace with a countable algebraic basis. E 1.1.6

Show that IK ~ admits no norm which generates the product topo-

logy. (Hint: Given a sequence (an)ne~ in IK, the sequence (ane~)neiN converges to 0 with respect to the product topology.) E 1.1.7

Let T be an infinite set and given f E ]0, oc[ T , define

qs'IK(T)

>IR+,

x,

>E f(t)ix(t)l. tET

Prove the following: a) For each f C ]0, oc[ T , q/ defines a norm. We write IK(T) for the vector space IK (T) endowed with this norm. b) If f, g C ]0, co[ T then:

1.1 Normed Spaces

59

b 1) IK~T) and IK~T) are isometric iff f = g.

b2) qI and qg are equivalent iff inf f (t)

tGT- ~ ~

O,

g(t)

tETinff - ~

~: 0.

c) ]0, c IR+,

y,

> sup I x ( t ) y ( t ) l . tE[0,1]

Show that: a) p is a norm iff {z :/- 0} is dense. b) If p is a norm then C([0, 1]) is separable with respect to p. c) If p is a norm then C([0, 1]) is complete with respect to p iff there exists an a > 0 such that {Ix[ > a} is dense. E 1.1.12

Let b~ be a cardinal number such that C a r d { A C R [ C a r d A < R} = R.

Show: a) There exists a set 9.1 of subsets of R such that C a r d 9 . 1 = 2 ~,

CardA=R,

Card(ANB)
0. Show that there is an x E E with IIx[I = 1 and dE(X) > 1-- C.

E 1.1.14 Let (xt)tei be a summable family in the normed space E and the filter of cofinite subsets of I . Show that lim xt = 0. E 1.1.15

Let

(Xt)~)(t,)~)EI•

be a summable family in the Banach space E .

Prove that

E tEI

XtA 9

AEL

(t,~)ElxL

E 1.1.16 Let (xt)~ei be a summable family in the normed space E and f " L --~ I a bijective map. Prove that -- 2___,xt. ~EL

tel

E 1.1.17 Let ( x ~ ) n ~ be a sequence in the Banach space E . Show that the following are equivalent: a) (X,~)n~ is summable. b) The sequence ( ~

XS(k))

k--1

is convergent for every bijective map nEIN

f :IN~IN. Show further that if these conditions are fulfilled, then for every bijective map f : IN--+ IN, E nEIN

xn -- lim ~ x l ( k ) . n---~oo

k-1

1.2 Operators

1.2

61

Operators

In the language of category theory, normed spaces are the objects and operators the morphisms of the category of normed spaces, where an operator is a continuous linear map between normed spaces. It is the operators which are the subject of this section. An i m p o r t a n t and useful feature of the set of operators between two normed spaces is t h a t it too admits a natural normed space structure.

1.2.1

General Results

Proposition

1.2.1.1

( 0 )

(F. Riesz, 1911) Let u " E --+ F

be a linear

map from the normed space E to the n o r m e d space F . Then the following are equivalent:

a) u is continuous. b) u is continuous at O. c) u is continuous at a point of E . d) There is an c~ c lR+ with

f o r every x 6 E . e) u ( E #) is bounded.

f) u is uniformly continuous. f =~ a ==~ b => c and

d r

e are trivial.

c ::v d. Suppose u is continuous at x0 c E . Then there is a 5 > 0 such that

II~x- ~011 _< 1 for every x e E with I I x - ~011 _~ a. We put Ot

"--

1 5

--.

Take x E E\{O} and let 6

Y "-- X o + v~x-~X.

Then

62

1. Banach Spaces

lly - ~011 = and so

mlluxll Ilxll

( )11

=

9

-II~y-~xoll

_< 1,

1

II~xll < ~llx I = ~1 x l l d ::, f. Since

for every x, y E E ,

u is uniformly continuous.

C o r o l l a r y 1.2.1.2 Every infinite-dimensional normed space admits a discontinuous linear form. Let A be an algebraic base of the infinite-dimensional normed space E . Let f : A ~ IK be unbounded and let x' be the linear form on E defined by

x'(x) := f(x)[[x[[ 1 for x E A. Since i-~x c E # whenever x E A,

f(A) C x'(E#). Hence x'(E #) is not bounded and x' is not continuous (Proposition 1.2.1.1 a~e). D e f i n i t i o n 1.2.1.3

( 0 ) Let E and F be normed spaces. We define

L(E, F ) : = {u: E

~ Flu is linear and continuous},

~:(E, F):-

F) I u(E) is finite-dimensional},

{u: s

and

ilull

:-inf{c~ C IR+ J x C E ==, Iluxll < c~llxll }

(the n o r m o$ u)

/or u e L ( E , F ) (Proposition 1.2.1.1 a ::~ d). The elements of s called operators f r o m E to F . We further define

z' := L(E,~:),

c(z):= L(E,E),

C:(E) = C:(E,E),

are

1.2 Operators

1 :=IE:E----+E,

x~

~x

63

(the identity operator on E )

and

(~, x') := (~', ~/:= z'(~) for (x, x') C E x E ' . E' is called the dual of E (H. Hahn, 1927). The elements of E ( E ) are called operators on E . Remark.

a) Because of the equivalence a r

e in Proposition 1.2.1.1, the ope-

rators from E to F are often referred to in the literature as bounded operators. b) It will be proved in Theorem 1.2.1.9 a) that the map >]R+,

E(E,F)

u,

>llull

is a norm. Proposition

ue s

1.2.1.4

( 0 ) Let E and F be normed spaces and take

F).

~) 9 e E ~ II~xll < il~ll IIxil. b) II~ll = sup II~ll = sup Iluxll = s u p , xCUI(O)

xEE#

xEE

Ilxll=l where in the last equality it is assumed that E # {0}. c) If G is a subspace of E , then li~ l all _< II~li.

a) We put A := {~ c IR+ I x E E ==~ lluxll

lixll.

Then 1

~x

1

= p-~llxll . . . . < 1,

64

1. Banach Spaces

so that we deduce successively

fill~xll-

u(z)

_F,

x~

>limutx.

u is linear and if

lim inf ilu, li < o~, then u is continuous and

[u[[ _< lim inf [lut[I 9 t,;~

66

1. Banach @aces

We have that u(o~x + ~y) = limt,;~ut(o~x + j3y) = l~,~(oLutx + j3uty)) = ozux + ~uy for x, y C E and a,/~ C IK (Proposition 1.1.5.3), i.e. u is linear. Moreover

Iluxll : 1~,~ II~xll ~ lim lmf II~[I IIxll - IIxll limte;~inf IiutI[ for every x E E (Corollary 1.1.1.4, Proposition 1.2.1.4 a)), which proves the last assertion.

I

Lemma 1.2.1.8

( 0 )

Let E and F be vector spaces. For every x C E ,

the map FE

>F ,

u:

> ux

is linear (it is called the e v a l u a t i o n o f x ) and {u E FE l u is linear} is a vector subspace of F E . The first assertion is a consequence of the definition of the vector space structure on F E . Let u , v be linear maps of E into F and put w:--uWv. Then w ( a x +/~y) = u ( a x + fly) + v ( a x +/3y) = a u x + 3 u y + a v x +/~vy = = ~ ( ~ x + v~) + ~(~y + vy) = ~ x

+ ~wy

for x , y E E and a, 3 c IK, i.e. w is linear. Let u" E --~ F be linear. Take 7 C ]K, and put v "= 7 u . Then

~(~x + Z~) - 7 ~ ( ~ x + / ~ )

- 7(~

+ 9~)

- ~

+ 9vy

for x , y C E and a, fl C ]K, i.e. v is linear. It follows from the above considerations that {u E F E ] u is linear} is a vector subspace of F E .

I

1.2 Operators

Theorem a) s

1.2.1.9

67

( 0 ) Let E and F be normed spaces.

F) is a vector subspace of F E and the map s

>JR+,

u,

>llull

is a norm; it is called the canonical n o r m oJ' /:(E, F ) .

b) If F is complete, then s a) Take u, v C s

II(u + v)~ll-

F) is also complete.

F ) . Then

Ilux + vxll ~ I1~11 + Ilvxll ~ Ilull Ilxll + Ilvll Ilxll = = (11~[I + Ilvll)llxll

for every x C E (Proposition 1.2.1.4 a)), so that u + v c /:(E, F ) (Lemma 1.2.1.8, Proposition 1.2.1.1 d =~ a) and

II~ + ~ll -< II-ll + Jl~Jl. Take c~ E IK. Then

sup I(~u)xl-

I~1 sup Iluxll = I~l Ilull

xEE#

xEE#

(Proposition 1.2.1.4 b)), so that c~u E F_.(E,F) (Proposition 1.2.1.1 d =:> a, Lemma 1.2.1.8) and

II~tl-

I~l Ilull

(Proposition 1.2.1.4 b)). Finally, it is easy to see that u - 0 whenever

Iluil

= o.

b) Let (Un)ne~ be a Cauchy sequence in E ( E , F ) . For every c > 0 there is an m~ such that Ilun-upl I <e for every n , p E IN, with n , p > m~. Take x E E . Then

for every c > 0 and each n , p C IN with n , p >_ m~ (Proposition 1.2.1.4 a)). Hence

(UnX)nelN is

a Cauchy sequence and therefore a convergent sequence in

F . Define u'E

.....> F ,

x,

> limunx. n -----~(:x3

68

1. Banach Spaces

Then _(u~ - u)x. = u n x -

ux = u ~ x -

limupx p--~ o o

= lim (unx - UpX) p-+ oo

lim (un - Up)X p----> c ~

whenever n E IN and x E E . By Proposition 1.2.1.7, un - u E s

F) and

II~.- ~ll < for c > 0 and n E IN, n > m~. It follows t h a t u E s

F) and

lim Un - u .

n---> ~

s

I

F ) is thus complete.

C o r o l l a r y 1.2.1.10

( 0 )

The dual space of a normed space is a Banach

I

space.

C o r o l l a r y 1.2.1.11 D e f i n i t i o n 1.2.1.12

( 0 ) ( 0 )

If E is a Banach space, then so is f~(E). Let E , F

I

be n o r m e d s p a c e s and u " E - + F

a

bijective linear map. u is called an i s o m e t r y o f n o r m e d spaces if

II~xll- IIxll for ever'9 x E E . u is called an i s o m o r p h i s m o f n o r m e d spaces if u and u -~ are continuous. E and F are called i s o m e t r i c ( i s o m o r p h i c ) if there exists an isometry (isomorphism) E - + F .

If u : E --+ F is an isometry (isomorphism), then so is u -1 . Every isometry is an isomorphism and every isomorphism is a homeomorphism. Isometry is equivalence in the category of normed spaces (geometrical aspect), whereas isomorphism is the equivalence relation in the category of normable spaces (topological aspect). We may replace "bijective" by "surjective" in the definition of isometry, since the main property of isometry implies that the injectivity follows. Let T be a finite set with at least two elements and take p,q E [1, oo] with p ~= q. Then gP(T) and gq(T) are isomorphic (Minkowski Theorem), but they are not isometric. This can be seen by considering their unit balls. If T is infinite then ~P(T) and ~q(T) are not even isomorphic. There exists an infinite-dimensional Banach space E not isomorphic to any of its proper subspaces (W.T. Gowers, Bull. London Math. Soc. 26 (1994), 523530). Then E is not isomorphic to E • ]K.

1.2 Operators

Proposition

1.2.1.13

1

\

0

(

69

) Let F be a dense subspace of the normed space

E and G a Banach space. Then the map

~(E, a)

>~(F, a) ,

> uIF

u,

is an isometry. It is obvious that the above map is linear. By Proposition 1.2.1.4 b), IlulYll =

for every u C s

sup

x6F#

Iluxll :

sup

x6E#

II~xll :

I1~11

since F # is dense in E # . Take v C s

By

Proposition 1.2.1.1 a :=v f, v is uniformly continuous. Since G is complete, there exists a continuous extension u : E --+ G of v. u is then linear, i.e. u E s

and uiF = v. Hence the m a p

~(E, G)

>~(F, G) ,

> ulF

u,

I

is surjective and so it is an isometry. Proposition

1.2.1.14

Let E, F be isomorphic normed spaces. If E is com-

plete then so is F . Let u : E -+ F be an isomorphism and let (Yn)ne~ be a Cauchy sequence f

%

in F . Then ( u - l y n ) \

/ nCIN

is a Cauchy sequence (Proposition 1.2.1.1 a ==v f ) a n d

so a convergent sequence in E . Hence u ( n--+cx)limu - l y n ) - n~lim U(u-lyn)

- - n---~oclimYn.

I

Thus (Yn)n~lN converges and F is complete. Proposition

1.2.1.15

Let p and q be norms on the vector space E . Let

Ep (resp. Eq ) be the vector space E endowed with the norm p (resp. q ), and define u:Ep--+ Then the following are equivalent: a)

u is an isomorphism.

b)

p and q are equivalent.

Eq,

x~

>x .

70

1. Banach Spaces

a :=~ b. Given x c E ,

q(x) = q(ux) < Ilullp(x), p(x) = p(u-~x) < Ilu-~llq(x) (Proposition 1.2.1.4 a) ), i.e. p and q are equivalent. b ==~ a. For some a > 0 1

- p < q gP(T)',

x,

~~

(resp. el(T)

~ co(T)',

x,

~ ~[c0(T))

is surjective. By b) (resp. c)) it is an isometry. f) It is obvious that x' E g~ x'-

x'(eT) -- 1, and x ' = 0 on IK (T) . Hence

0 on co(T) (Proposition 1.1.2.6 c)). Assume that there is an x E tl(T)

with ~ = x'. Then ~ = 0 on co(T) and so, by c), Ilxlll - II~ l co(T)ll - O,

x=O.

This leads to the contradiction l=x'(eT)=~(eT)=O.

g) follows from e) and f). Remark.

ll

A representation of t ~ ( T ) ' is given in Corollary 1.2.2.14.

E x a m p l e 1.2.2.4

Let T be an infinite set and ~ the filter of cofinite subsets

o f T (i.~. 9= { A C T I T \ A

x " c(T)

~ IK,

x,

is f i n i t e } ) ,

~ limx(~),

to C T . Let ~ " T \ {to} ~ T be a bqection. We set

_

x "T

~IK

'

t~

f x'(x)

if t - t o

~ ~(~(t)) - ~'(~)

if t # to

for every x C c ( T ) .

a)

x' C c ( T ) ' ,

b)

~ E co(T) for every x C c(T) ; define

IIx'll - 1.

c(T) ~ c o ( T ) , c)

~,

u is an isomorphism and

JJ~ I = IJ~-~ll = 2. d)

co(T) and c(T) are not isometric.

~~

1.2 Operators

79

a) and b) are obvious. c) It is clear t h a t u is linear and injective. We have t h a t

I~(to)l = Ix'(x)l ~ Ilxll~, sup I~(t)l-

sup

tET\ { to }

Ix(~(t))-x'(x)l _~ Ix'(x)l +

tET\ { to }

sup

Ix(~(t))l ~ 211xll~,

tET\ {to }

so t h a t

I1~11~ ~ 211xllo~ for every x 9 c ( T ) . Hence u is continuous (Proposition 1.2.1.1 d =~ a ) and

I1~11 ~ 2. From e T - 2eto 9 c ( T ) ,

IleT- 2etoll~ --- 1,

Ilu(eT- 2~,o)I~ = 2,

we see t h a t ]lull = 2. Let x C co(T). Define y "-- X ( t o ) e T nt- X o (t9-1 e c ( T ) .

Then x'(v)

x(to),

-

so t h a t y--x.

Hence u is surjective and U-I(x)

-- X ( t o ) e T -t- X 0 ~ - 1

We deduce t h a t

I1~-~11~ _< 211xll~ for every x C co(T). Thus

Ilu-lll

U -1

is continuous (Proposition 1.2.1.1 d =~ a ) and

_< 2. From eto + e~-l(to) E co(T),

I1~o + %-1(to)11~ --- 1, it follows t h a t Iiu-ll] = 2. d) Assume t h a t

I1~-1(~o + ~-,(~0))1 ~ -

2

80

1. Banach Spaces

v "c(T) -----+ co(T) is an isometry. Put

A "- { IVeTI - 1} and

At "- {Iv(eT + et)l = 2} for every t C T . Since v is an isometry, A and At are finite, nonempty, and

At C A for every t 9 T (Example 1.1.2.3). Hence there are distinct t', t" 9 T with At, N At,, :/: O. Take t 9 At, N At,,. Then

(vet,)(t) = (VeT)(t) = (vet,,)(t), so t h a t

2-

I(vet, + vet,,)(t)l ~ Iv(et,-+ et,,)Ioo = I1~,' +

e,,,ll~

- 1,

which is a contradiction. Hence co(T) and c(T) are not isometric,

Remark. Example

d) will be generalized in Corollary 1.2.7.18.

1.2.2.5

( 0 ) (F. Riesz, 1910)

Let # be a positive measure and

p, q conjugate exponents. a)

xy 9 L l(#) and

Ilxylll ~ Ilxllqllyltp

(I-I~tder

inequality)

for every x 9 Lq(p) and y 9 LP(p). For each x 9 Lq(#), put ~" LP(#)

b)

m

~ 9 LP(#) ' and [[~[] =

c) ~f p ell, ~[

Ilxllq for

) IK,

y,

) f xyd#. J

~v~y x e Lq(#).

th~n th~ map

L q(p) is an isometry.

> L p(p)',

x,

~

1.2 Operators

d)

81

The map

>LI(#) ',

L~(#)

x~

>~

is an isometry iff L ~ ( # ) is order complete.

The proofs can be found in most books on integration theory.

I

If T is a set and p counting measure on T, then Example 1.2.2.3 a),b),d) becomes a special case of the above example. Remark.

Definition 1.2.2.6

( 0 )

5(s, t) "- (~st "=

We set

S 1 if s = t

I

(the K r o n e c k e r ' s symbol)

0 if s # t

for all s, t .

E x a m p l e 1.2.2.7

( 0 ) Let [Olij]iEINm,jEINn be a matrix over IK. Let

u 9IK n --+ IK TM be the associated linear map and A the set of the roots of the equation

=0. If we endow IK TM and IK n with the Euclidean norm, then:

a) A C JR+. b)

Given A C A and a solution x := ( x j ) j c ~ tions

j=l

i=1

we have that II~xll 2 = Allxll 2 m

c)

I/nil2 - sup A _< E ~ [aijl 2. )~cA

i=1 j = l

of the system of linear equa-

82

d)

1. Banach Spaces

If n = 2 , then

ilull2

1

12

m i=1

-4

i=1

]OZ/1]2 -]- E ]OL/2 2 i=1

i=1

-~-

m

]c~i1[ 2 i=1

[~

-

i=1

-2

--"

E ~ i=1

m

E (J~,~l~ + J~,,J') +

1

i=1

-1-

(IOdil

--1-IOLi2I2)

i=1

O~i20Ljl

-- 2 E [a/10/j2i,j=l

a & b. We have that //1

r

j,k=l i=1

k=l

c) Take x = ( x k ) k e ~ such that Iluzll ~- - s u p { l l u y l l 2 1 y By Lagrange's Theorem, there is a A E IR such that

c II( n, Ilyll = 1}.

Oll~yll ~ = ~ o lyll 2 0~k 0~k for every k E INn when y = z . Thus

j=l

i=1

for every k C INn. By a), b), and Proposition 1.2.1.4 b), ]]u]]2 = supA. ,~EA

Take ,~ ~ A with ) ~ - I1~112 and for every k C INn,

9 c IK ~ such that ( , ) i s

fulfilled. Then

1.2 Operators

83

m

E

aij~ik

j=l i=1

j=l

i=1

-< i=1

Ixjl

i=1

I~,~l ~

~

j=l i=l

I~jl ~

Ilxll.

It follows

11~l1411xll~ =

k=l

II~ll~lx~l ~ _


gP,

x,

>

x~

(the left s h i f t on gP).

u~, u~ e s

IlUrl] - 1, llU~]] = 1, U~ is injective, u~ is surjective.

c) u~ o ur is the identity operator on gP. d)

If a e ]K\{O} and x, y E gP such that

( a l - ur)x = y then x(n)-

1

~-~aky(k ) k=l

for all n E IN.

86

1. Banach Spaces

e)

al-

ur is injective for all a G IK.

All the above assertions hold for c in place of gP.

a), b), and c) are easy to see. d) We prove the assertion by complete induction. The assertion is obvious for n = 1. Let n C IN and assume the assertion holds for n. Then a x ( n + 1) - x(n) = y(n + 1),

so that x(n + 1) =

1 1 x(n) + - y ( n + 1) - ~ O~

n 1 n+l y ~ aky(k) + - g - ~ a y(n + 1) = k=l

1

n+l

= a.+2 Z

akY(k)"

k--1

e) follows from d).

m

E x a m p l e 1.2.2.10 ( 0 ) Let T be a locally compact space and let Co(T) be the set consisting of those elements of C(T) which vanish at the infinity (if T is compact then Co(T):= C(T)). Then Co(T) is a closed vector subspace of C(T) and Co(T)' can be identified with the Banach space .Mb(T) of bounded Radon measures on T . The proof can be found, for example, in N. Bourbaki, Integration, Ch. II (1952), w II Remark.

If T is endowed with the discrete topology then ~o(T) - C o ( T )

E x a m p l e 1.2.2.11 ( 1 ) (2) (3) If T is a completely regular space, then C(T)' can be identified with the Banach space of Radon measures on the Stone-Cech compactification of T . This is an immediate consequence of Example 1.2.2.10. E x a m p l e 1.2.2.12 set T .

m

( 0 ) gl(T), co(T)', and c(T)' are isometric for any

1.2 Operators

87

By Example 1.2.2.3 e) e I(T) and co(T)' are isometric. To show that el(T) and c(T)' are isometric, we may assume T to be infinite. Consider T whith the discrete topology and let T* be the Alexandroff compactification of T . Then c(T) can be identified with C(T*). Hence c(T)' can be identified with the Banach space Mb(T*) of bounded Radon measures on T* (Example 1.2.2.10). But Mb(T*) is isometric to gl(T*) and gl(T*) is isometric to t~l(T), since T and T* have the same cardinality. Therefore c(T) and t~l(T) are isometric. I

Remark.

co(T)' and c(T)' are always isometric even though co(T) and c(T)

are not isometric when T is infinite (Example 1.2.2.4 d) ). ( 0 ) Let (Et)tET be a family of normed spaces. Take

P r o p o s i t i o n 1.2.2.13

p and q weakly conjugate exponents. Define

tET

g-- {xtE

HE~ tET

I (IIx'tII)tETE ~q(T) },

and endow E and F with the norms

E

>XR+,

x,

>ll(Ix, ll):~rllp,

F

>n%,

x',

>ll(llx'~ll)~TIIq,

(Proposition 1.1.2.7 a)). Given x' E F , define x'' E

>]K,

x,

,

tET

a)

x' E E' and

I1~'11- IIx'll for

b)

If p r oc and q r O or if T is finite, then the map

~w~y x' E F .

>

F

E'

x'~

N ~ x'

is an isometry. a) Given x E E and x' E F ,

y ~ I<xt, x~>l < ~ tET

tET

IIx~ll Ilx~ll < Ilxll IIx'll

88

1. Banach Spaces

N

(Proposition 1.2.1.4 a), Example 1.2.2.3 a)), so that x' E E' and IIx'll < IIx'll.

Take x' E F and a E ]0, Ilx'l][. By Proposition 1.2.2.3 b),c), there is a z E gP(T) # such that z > 0 and

II~'~llz(t) >

~.

~-:~ I ~'~llz(t) >

~.

tET

Let S E ~3s(T ) with

tES

By Proposition 1.2.1.4 b), there is an (Xt)tET E l - l E t # tET

such that Xt - - O

for every t E T \ S

and

E(xt, x~t)z(t) :> a . tES

Put y := ( z ( t ) ~ ) t ~

E E.

Then 1

1

_< ( E z ( t ) P ) ~ tET

tET

if p E [1, oc[ and IlyI - supz(t)ilxt I I1~'11,

IIx'll = IIx'll.

b) Take v~ C E ' . Let s E T . Given x C E s , let Y denote the element of E given by Y

t,

~

x

L 0

ift=s ift=fis

and define X s' " E~ !

> IK

x,

> O(Y).

!

Then x s C E s . P u t

x' :=

9 I-[ E;. tCT

Take y C IK (T) . Then

tET

tET

for every x C E with

IIx~ll < 1 for every t C T . Hence

tET

(Proposition 1.2.1.4 b)). Since y is arbitrary, x ' E F and

IIx' I < IlOll (Proposition 1.2.2.2), and x ' = 0 on

G := {x c E l{t z T Ix~ # 0} is finite}. Since G is dense in E (Proposition 1.1.2.7 c) ), we deduce that x' = 0 and

IIx'll < IlOll- IIx'll < IIx'll,

IIx'll- Ilx'll.

Hence the m a p F is an isometry,

)E',

x'.~

;x' i

90

1. Banach Spaces

C o r o l l a r y 1.2.2.14 Let T be an infinite set endowed with the discrete topology and let fiT denote the Stone-Cech compactification of T . Put E "- co(T) x C ( f l T \ T ) ,

F

:=

e'(T)

x Mb(flT\T)

and endow E and F with the norms

E

~ JR+,

(x,y),

F ---+ IR+,

(z,#),

sup{l]x[[~, []Y [}, ~1 zll, + ll,II

(Proposition 1.2.2.13 a)).

a) For (z,#) E F , define (z, ~) . E

~~ ,

(~. y) .

E

x(t)z(t)+

fvd,.

tET N

Then (z,#) e E' for every (z,#) e F and the map u . F

> E' ,

(z. . ) .

. (~. . )

is an isometry.

b)

For (z,#) e F , define

(z, #) 9C(flT)

) IK ,

X

E

x(t)z(t) + f x

[(flT\T)d#.

tET

Then (~,~) E Adb(flT) for every (z,#) e F (Example 1.2.2.10) and the map

is an isometry.

c) For y E g~176 denote by ~ its continuous extension to fiT and for # C M b ( Z T ) , put

Th~. ~ c ( e ~ ( T ) ) ' fo~ ~v~y ~ ~ M ~ ( g T ) and th~ map ~

is an isometry.

M~(gT)

> (e~

~,

~

1.2 Operators

d)

91

w o v o u - l : E' --+ ( t ~ ( T ) ) ' is an isometry. a) follows from Example 1.2.2.3 e), Example 1.2.2.10, and Proposition

1.2.2.13. b) is easy to see. c) follows from Example 1.2.2.11. d) follows from a), b), and c).

m

92

1. Banach Spaces

1.2.3 I n f i n i t e M a t r i c e s

D e f i n i t i o n 1.2.3.1 ( 0 ) An infinite matrix is a function S x T -+ IK, where S and T are sets. Let k : S • T -+ IK be an infinite matrix and take p C [1, oc] U {0}. Let q be the conjugate exponent of p. If k(s, .) c gP(T) for every s C S , then define

M

kx" S

~ IK,

s,

)E

k(s,t)x(t)

tET for x C gq(T) (Example 1.2.2.3 a)). If k(., t) C gP(S) for every t E T , then we define

u sES for x e gq(s) (Example 1.2.2.3

a) ).

If S and T are finite, then k is a matrix and kx is the usual multiplication of the matrix with a vector. D e f i n i t i o n 1.2.3.2

(0)

Let S , T be sets and take p,q e [1,oc] U {0}.

Let gP'q(s, T) denote the set offunctions k: S x T ~ ]K such that k(s, .) E ~.q(T) for every s c S and that

(Ilk(s, .)llq)scSE ~P(S) endowed with the norm gP'q(S,T)

>]R+,

k,

~ll(llk(s,.)[lq)sesllp

(Proposition 1.1.2. 7 a) ). We define ~o'q(s, T)

D

~

{k E ~~

It E T ~

k ( . , t ) C co(S)}

and endow fo'q(s, T) with the restriction of the norm of f~'q(s, T ) . By Proposition 1.1.2.7, gP'q(S,T) is a Banach space. ]K (sxT) is dense in fP'q(S,T) whenever p :/: oo and q :/: oo. It is easy to see that fo'q(S,T) is a closed subspace of t~'q(s, T), so that it is a Banach space, f~ T) is a closed subspace of to'q(s, T). Let k C IK SxT. Then k C t!2'2(S,T) iff

Z

(s,t)ESxT

Ik( ,t)l
kx

belongs to f_.(gP(T), ~7"(S)) and N

Ilkll ~ Ikll w h e n e v e r k e ~r,q(s, T ) . I f r E {0, (3o} (resp. if r e {0, (3o} and p -- cxD)

then N

Ilkl = I kll

N

N

(reap ]ik I c0(T)ll- Ilkl]- Ilkl])

whenever k C gr,q( s, T) .

c)

For S ~ 0 and r - oc, the map A

is an isometry iff p or T is finite.

94

1. Banach Spaces

d) If p ~ oo, then the following are equivalent for every k 9 gr'q(s, T)-

dl)

N

k(e'(T)) c co(S)

d~.) k(., t) 9 co(S) e)

(~esp. c(S) ); (rasp.

c(S) ) for every t 9 T .

The map ~.o'I(S,T)

A

> E.(co(T),co(S)),

k, ~ ~ k

is an isometry.

f) (Woeplitz, 1911) Given k 9 go'l(S,T), the following are equivalent: A

fl) kz e c(S) fo~ ~w~y x e c(T), f~) E k(., t) e c(S). tET

g) If R is a set, u 9 s

and k 9 t~2'2(S,T), then there is a

unique h 9 ~2,2( R, T) such that N

N

h =uok; we have

ilhii < ii~li iiklla) Given s 9 S,

tET A

(Example 1.2.2.3 b), Proposition 1.2.1.4 a)). Thus kx 9 gr(s) and N

ilk~ilr < iikli il~ii~. b) By a), Proposition 1.1.6.11, and Proposition 1.2.1.1 d =v a, N

k 9 L(~,(T), e r(s)) and N

ilk i_< ilkll. Assume now that r 9 {0, oo}. By Example 1.2.2.3 b), c) and Proposition 1.2.1.4 b),

1.2 Operators

{n

_< ~up IIk~ll~

95

I 9 9 e(T)#) = Ilklln

for every s E S , and so n

Ikll_< Ilkll,

n

Ilkll = Ilkl.

Finally, assume t h a t r E {0, cx)} and p = cx~. T h e n

< sup

{nI]kx]]o~ Ix E co(T) # } =

]]k ]co(T)]]

for every s E S (Example 1.2.2.3 c), Proposition 1.2.1.4 b) ). Thus N

N

Ilkll d2), the map M

~ o " ( s , T ) --+ ~(~0(T), c0(S)),

k,

~k

1.2 Operators

97

is an isometry. fl ==~ f2. We have t h a t n

T

k(., t ) : k ~ c c(S). tET

f2 ::a fl. We may assume that T is infinite. Let ~ denote the filter of cofinite subsets of T , i.e. I e g3s(T)},

~ "= { T \ B

and put a " - lim x ( ~ ) . Then x -

ae T 6 c o ( T ) , so that by b) and d2 ~ d l , N

k(~- ~)c

~0(s).

Hence n

n

n

T

N

k~ = k(x - ~ 4 ) + ~ k ~ = k(~ - ~ 4 ) + ~ ~

k(., t) c ~(z).

tET

g) We set

h-R•

~,

N

(r,t), "(~k~T)(~).

Then N

2

E

2

t E T rE R

(r,t)EIRxT

tET

~

12--

tET

tET

sES

(Proposition 1.2.1.4 a)), so that h e tF'2(R, T) and

Ilhll __ I1~II Ilkll. We have N

N T

T

for all (r, t) C R x T , so that

n

2

98

1. Banach Spaces

n

T

n

T

he t = uke t

for all t E T. By Proposition 1.1.2.6 c), n

n

h=uok.

The uniqueness of h is obvious. Remark. Some of the results in Proposition 1.2.3.4 are actual special cases of the theory of integral operators, which will be treated in Section 3.1.6.

P r o p o s i t i o n 1.2.3.5

( "l )

( 2 ) Let R , S , T be sets and p,p' weakly conjugate exponents. Take n, q E [1, cr U {0}. Let n' be the conjugate exponent of n . Take k C gP'n'(S, T ) , h e ~.q'P'(R, S) , and let h o k. R • T --+ ~,

(~,

t),

~ ~ h(~, ~)k(~, t) sES

(Example 1.2.2.3 a)). Then h o k e gq,n'(R, T),

lib o kll _< llhll Ilkll, and n n n

h o kx = h ( k x ) whenever x e gn(T) . If k e ~ , 1 ( S , T) and h e Go" (R, S) , then

h o k e e~ ,~(n, s). Take x c t~n(T) and r C R. Then

(s,t)eSxT

sES

IK,

t,

if

nEIN

if t 6 Bn.

an e)

t6T\ U

)

k 6 g~'~(S,T).

If these conditions are fulfilled, then the map Cl

N

k e ~ ( T ) --+ e~176 is in s

~(S))

x:

~,kx

and M

M

[Ikl] = Ilk I c o ( T ) [ I - sup IIk(s,.)[I,. sES

a :=> b and a :=> c are trivial. 1 b ==~ d. Given n E IN, put cn .._-_ ~.

c ::=> d. Given n r IN, put cn - 1. d => e. Given t c T , N T

k(, t) = kr Define

e ~(s)

Bn

where x is

104

1. Banach Spaces

kl:SxT

>IR,

(s,t),

;sup{rek(s,t),0}

k2:SxT

>IR,

(s,t),

>sup{-rek(s,t),O}

k3:SxT

>IR,

(s,t),

>sup{imk(s,t),0}

k4: S x T

> IR,

(s,t),

>sup{-imk(s,t),0}

Assume that (Ilk(s, ")lll)s~S is not bounded. Then there is a j 9 {1, 2, 3, 4} such that (llkj(s,.)ll~)ses is not bounded. P u t B0 -

0 and construct a sequence

(sn)ne~ in S and an increasing sequence (Bn)ne~ in 913I(T) inductively such that n

Sn r {Sk I k 9 I N n - l } ,

E kj(sn, t) > - En tEBn\Bn-1

for every n 9 IN. Take n 9 IN and assume that the sequences have been constructed up to n -

1. Then there is an Sn 9 S~k{Sk I k 9 I N n - l }

such that

Ilkj(s~,.)llx >

n

Cn

+ 1 + (CardBn_l)

sup IIk(.,t)ll~.

tE B,~-1

There is a finite subset B~ of T with B~-I C Bn and

E

kj(sn, t)< 1.

tET\Bn Then

E kj(sn, t) tCBn\Bn-1

Ilkj(Sn, .)Ill -

kj(sn' t) -

E

teBn- 1

n

Z kj(sn' t) > Cn tCT\Bn

This completes the inductive construction. We put

cn . - {t e B.\B~_~ I kj(~,~, t) > 0}. Then (Cn)nc~ is a disjoint sequence in q3l(T ) . By d), there is a sequence N

(an)nelN in IR such that an > ~,~ for every n 9 IN and kz 9 g~176 where

x'T

>IK,

t,

0

if t c T \

)

U cn nEIN

an

if t c C n ;

1.2 Operators

105

Given n E IN,

n

i

i

I(k~)(~.)l = ~

k ( ~ , t)x(t) >_ E

tET

E

~ O~n

kj(sn, t)x(t) >_

tET

tE Bn \ B n - 1

k j ( s n , t ) > oLn - n ~ n , gn

n

which contradicts kx C g~ ( S) . e ===>a, and the last assertion follow from Proposition 1.2.3.4 b). Theorem

1.2.3.11

(

1)

I

(Kojima-Schur) Let S, T be sets and k" S • T -+ IK

a ]unction with k(s, .) C gl(T) for every s E S . Then the following are equivalent: M

~) k~ e c(S) /o~ ~ y b)

x e c(T).

k(-,t) E c(S) for every t c T , E k(-,t) C c ( S ) , and k C g~c'I(S,T). tET

If these conditions are fulfilled, then the map

~(T) is in s

>~(S),

M

~,

>kx

Ilkll.

c(S)) and has norm

a =~ b. Given t E T , e T and e T belong to c(T) and N T

k(., t ) - k~ ~ c(S),

N T

Z k(., t)= k~ c ~(s). lET

By Proposition 1.2.3.10 b =~ e, k E f ~ ' I ( S , T ) . M

b =~ a. By Proposition 1.2.3.4 a),b), kx E g~(S) whenever x c g~(T) and the map

e~(T)

~e~(s),

x,

is continuous. It is obvious that

By continuity, n

k(co(T)) c c(S).

M

~ kx

106

1. Banach Spaces

There is an a 9 IK with x -

s e T 9 co(T). Since

k4 - Z

k(., t) c c ( S ) ,

tET M

kz c c ( S ) . The final assertion follows from the last assertion of Proposition 1.2.3.10.

I Remark.

The above theorem still holds for co(S) in place of c ( S ) .

T h e o r e m 1.2.3.12

1)

(

(I. Schur) Let S , T be sets and k" S • T -+ IK a

function with k(s, .) E gl(T) for every s E S . Then the following are equivalent: N

a)

kx e c(S) for every x 9 t ~ 1 7 6

b)

ke s 9 c(S) for every countable subset B of T .

M T

c) k(.,t) c c(S) .for every t O T ,

and for any e > O there is a finite subset

B of T such that

teT\B

for every s E S . Assume that k(., t) e c(S) for every t e T . Define e ~S(S)},

a "- { S \ A I A

a'T

~IK,

t,

~limk(-,t).

If the above conditions are fulfilled, then a C gl ( T ) ,

lim Ilk(s,- ) -all~ = 0 , s,~ the map N

M

k . e~ ( T ) ~

N

c(s),

~,

) k~

A

kll = Ilk I co(T)ll = sup IIk(~, )Ill sCS

1.2 Operators

107

and N

a(t)x(t),

limkx = ~

tcT for every x C g ~ ( T ) . a =~ b is trivial. n b =~ a. Let B be a subset of T . Assume that ke T ~ c(S). Then there is a sequence (s~)ne~ in S with

n T n T lim inf I (keB)(Sn) --(keB)(Sn+l ) [ > O. Given n c IN, put

Ca "= { k ( ~ , ") # 0}

n

B

and

c:=Uc,. ncIN

Then C is countable and n T

n T

(keB)(sn)-(kec)(Sn)

for every n E IN. Thus

I

NT lim inf I (kec)(Sn) - ( k eNcT) ( S n + l ) > 0 n T

n T

which contradicts the fact t h a t ke c C c(S). Hence ke B c c(S). n By Proposition 1.2.3.10, kx c g~(S) whenever x c g~(T) and the map

e~(T)

~e~(s),

~~

n

k~

is continuous. Since the vector subspace of e~(T) generated by {e T I B C T} n is dense in g~(T) (Proposition 1.1.2.6 d)), kx C c(S) for every x c g~(T) (Example 1.1.2.3). a =~ c. Given t c T , eT E ~ ( T )

and n T

k(., t) - k ~ e ~(S). Take c > 0. Assume t h a t for every finite subset B of T there is an s E S with

108

1. Banach @aces

Ik(,, t)l > c.

tET\B P u t Bo "= 0 and construct a sequence (Sn)neIN in S and an increasing sequence (Bn)nelN in ~I3s(T) inductively such t h a t s

8n ~ {Sk ] k C I N n - l } ,

E Ik(sn't) - a(t)l < -6' tCBn-1 5s

~ Ik(sn, teT\B,~

Ik(,~,t)l > 6 '

teB,~\Bn-1

t)l




tET

k(~2n,~lxl'll-I ~ (k(s2n,'/- k(s2n ,,,~)x(,~ I-

tCT\ B2n

tEB2n\B2n-1

and

I~(s2~,,~l - ~

~

tCB2n-2

-~ tET\B2n

k(s2~

tcT\B2n

Ik/s2n,,t-o~,/I - ~

I~(s2n,~/I-

tEB2n-2

~ tcT\B2n-1

Ik~2n 1,'~-a/'/I -

I~(s~o-l,'~l>

110

I. Banach @aces

5g"

E

> ~--2g-2g=g

E

E

M

for every n 9 IN, which contradicts the fact that kx 9 c(S). c ==> a & the last assertion. Take e > 0. By c) there is a finite subset B of T with

Ik(~, t)l < E tET\B

whenever s 9 S . We obtain successively that

Z

la(t)l _< lim

inf

tET\B

Z Ik(~,t)l
limxs(t).

xlll

=

s,~

Then x E ~1 (T) and

limllxs

-

0.

s,N

We define k" S x T

>IK,

(s,t),

>xs(t).

n T

By hypothesis, ke B E c(S) for every countable subset B of T and the conclusion follows from Theorem 1.2.3.12. C o r o l l a r y 1.2.3.14

(

1)

I

Let S , T

be sets and k a function S x T --+ IK

for which k(s, .) C gl(T) for every s C S . Then the following are equivalent:

a)

k e e~

b)

kx c co(S) fo~ ~ y

c)

k(.,t) e Co(S) for every t e T , and given c > O, there is a finite subset

M

x c e~(T)

B of T such that

tET\B

whenever s C S .

a =v b follows from Proposition 1.2.3.4 a). b ::v c. Given t E T , etT C g~

so that M T

k(., t ) = k~ e c0(S) By Theorem 1.2.3.12 a =a c, given c > 0, there is a finite subset B of T such that

112

1. Banach Spaces

Ik(s,t)l

< c

tET\B

whenever s E S. c =~ a. Let c > 0. By c) there is a finite subset B of T with

Z

Ik(~, t)l
0 such t h a t

p(x) ~ ~llxll for every x E E . Thus

Iluxll = p(x) ~ ~llxll for every x C E , i.e. u is continuous (Proposition 1.2.1.1 d :=~ a). Now let u be arbitrary. Ker u is a finite-dimensional vector subspace of E and hence closed (Corollary 1.1.3.6). Let q:E

> E/Ker u

be the quotient m a p and v the factorization of u t h r o u g h E / K e r u .

Then

v is injective ( L e m m a 1.2.4.6). Since E / K e r u is finite-dimensional, the above considerations show that v is continuous. Thus u = v o q is also continuous.

I

120

1. Banach Spaces

C o r o l l a r y 1.2.4.9

( 0 ) If E, F are finite-dimensional normed spaces, then

every bijective linear map E ~ F is an isomorphism, C o r o l l a r y 1.2.4.10

m

The dual and the algebraic dual of the finite-

( 0 )

dimensional normed space E coincide. In particular, the dimension of E and E' coincide,

m

C o r o l l a r y 1.2.4.11

Let E , F

( 0 )

be normed spaces and u" E --~ F be

a linear map. If K e r u is closed (e.g. finite-dimensional) and I m u is finitedimensional, then u is continuous. Let q : E --+ E / K e r u be the quotient map and v the factorization of u through E / K e r u . Then v is injective (Lemma 1.2.4.6). Since

v ( E / K e r u) = Im u, E / K e r u is finite-dimensional. Hence v is continuous (Proposition 1.2.4.8) and so u = v o q is also continuous. If Ker u is finite-dimensional, then it is closed by Corollary 1.1.3.6.

Corollary 1.2.4.12 ( 0 ) The following are equivalent: a)

x' is continuous.

b)

Ker x' is closed.

c) x ' = 0

a

m

E

or Kerx' is not dense.

a => b =:> c is trivial. b => a follows from Corollary 1.2.4.11. c => b. Assume that Kerx' is not closed. Then Kerx' has a point of adherence x which does not belong to Ker x'. Then given y E E , 9'

( xy '-t x )-X='-(- Y~ ) )

-

x' ( y ) -

z , - ~ x C Ker

~'(y) ~

x-~X'(Y)x' ( ~ ) - o ,

C Ker x',

z'(y)

(Corollary 1.1.5.4), E C Ker x', which is a contradiction. Hence Ker x' is closed.

1.2 Operators

121

Every infinite-dimensional normed space contains a den-

C o r o l l a r y 1.2.4.13

se proper vector subspace. Let E be an infinite-dimensional normed space. By Corollary 1.2.1.2, E admits a discontinuous linear form x'. By Corollary 1.2.4.12 c ==~ a,

Ker x' is

a dense proper vector subspace of E . Example

II

Let /3T be the Stone-Cech compactification of the dis-

1.2.4.14

crete space T . We put A := ~ T \ T

and consider every x C g~

extended

continuously on ~ T . Put u : e~(T)

~ C(a),

~ ,

~ xla.

Then the factorization e~176

~ 6(A)

of u through e~(T)/co(t) is an isometry. By L e m m a 1.2.4.6 and Tietze's theorem, the factorization

e~(T)/co(T)

~ C(A)

of u is bijective and it is easy to see t h a t it is also n o r m - preserving. Example

1.2.4.15

I

Take E:--~ 1 XCo,

F := {(~,~) I 9 c e~}, G:={(x,0) u :E

~ co,

lxefl}, (z,y)

,

)y--x,

and endow E with the 1-norm.

b)

F = Keru.

c)

The factorization of u through E / F

d)

is an isometry.

G is a closed vector subspace of E but q(G) is not closed in E / F . q : E -+ E / F

is the quotient map.)

(Here

122

1. Banach Spaces

-1

e)

F + G = q (q(G)) = { ( x , y ) l x, y 9 gl} is not closed.

a) u is linear and

I1~(~, y ) l l -

I l y - xllo ~ Ilyllo + Ilxllo ~ Ilxll~ + Ilyllo - I I ( x , y)ll~.

b) is obvious. c) By L e m m a 1.2.4.6, the factorization of u is bijective. Take z C co and (x, y) E E with

z = ~(~, y) = y - ~.

By a), Ilzllo ~ II(y,x)ll~. It follows from z = u(O, z) and tlzllo = I1(o, z)ll~ that the factorization is an isometry. d) By c), we may identify E / F

with co. W i t h this identification,

q(C) = e 1 .

e) is easy to see.

I

1.2 Operators

1.2.5 C o m p l e m e n t e d

Subspaces

Proposition 1.2.5.1 E.

123

( 0 ) Let F and G be subspaces of the normed space

Then the map F x G

~, E ,

( x , y ) ~-+ z + y

is linear and continuous.

It is obvious t h a t the above m a p is linear. If we endow F • G with the 1-norm of the product then

ll~ + yll _< Ilxll + Ilyll = ll(x, y)ll~ proving continuity (Proposition 1.2.1.1 d =~ a).

Proposition 1.2.5.2 E,

q: E -+ E / F

( 0 )

Let E

I

be a normed space, F , G

subspaces of

the quotient map, and put u:F•

(x,y)

>E,

v :G

~, E / F ,

y~

>x+y,

> qy.

Then the following are equivalent:

a)

u is an isomorphism.

b)

F and G are closed in E and v is an isomorphism.

c)

F is closed in E and v is an isomorphism.

a ::v b. F x {0} and {0) x a

are closed sets of F x G . Since u is a

homeornorphism, F and G are closed sets of E .

q is continuous (Theorem

1.2.4.2 b)) and so v is also continuous. Take Z E E / F

and z C Z . There is a

pair (x,y) C F x G with

~(x, y) = z. It follows that vy = qy = q(x + y) = q ( u ( x , y ) ) = qz = Z .

Hence v is surjective. We show that v is lower bounded. Take y E G . Then

124

1. Banach Spaces

Ilyll ~< Ilxll + Ilyll = II(x, y)ll~ = Ilu-~(x + y)ll ~< Ilu-~ll IIx + ytl for every x E F (Proposition 1.2.1.4 a) ), so t h a t IlYll < Ilu-lil inf I]x + Yll - Ilu-lll IlqYl]

IvYll = IlqYll >

1

By Proposition 1.2.1.18 a), b), v is bijective and v -1 is continuous, i.e. v is an isomorphism. b =~ c is trivial. c :=> a. Let z E E . There is a y E G with qy = q z .

Hence z - y E F

and u(z - y,y) = z-

y + y = z,

i.e. u is surjective. We show t h a t u is lower bounded. Take (x, y) E F x G . Then

q(z + y) = qx + q~ = qy, so t h a t (Proposition 1.2.1.4 a), Theorem 1.2.4.2 b))

Ilyll =

v-~q(~ " + y)ll ~< IIv-'ll Ifq(x -4- y)fl ~< IIv-'ll I1~ + yll,

I1~11 = I x + y - y l l

~< I I ~ + y l l +

Ilyll ~< ( 1 + IIv-~lI)ll x + y l l ,

II(x, y)ll~ = sup{llxll, Ilyll) ~< (1 + IIv-~ll)llx + yll 1

II~(x, y)ll - I1~ + yll > - - I I ( x ,

y)ll

9

By Proposition 1.2.1.18 a), b), u is injective and u -~ is continuous. Since u is continuous (Proposition 1.2.5.1), it is an isomorphism. /

D e f i n i t i o n 1.2.5.3 orE.

[

0

)

Let E

be a n o r m e d space and F, G be subspaces

We say that E is the d i r e c t s u m o f F E=F|

if the m a p

1

\

a n d G , and we denote this by

1.2 Operators

F x G

>E ,

125

(z,y) ~ , '~ z + y

is an isomorphism. Let E be a normed space. A c o m p l e m e n t e d subspace of E is a subspace F of E for which there i s a s u b s p a c e G of E with E=F| In this case G is also a complemented subspace of E ; it is called a c o m p l e m e n t of F i n E .

By Proposition 1.2.5.2 a =~ b, complemented subspaces of a normed space are closed. But there are closed subspaces of Banach spaces which are not complemented subspaces (see Corollary 1.2.5.14). Let (E~)~ci be a finite family ofnormed space. For each A C I , the space E~ (canonically identified with a subspace of 1~ E~ ) is a complemented subspace of 1-I E~ 9

C o r o l l a r y 1.2.5.4 E , and G , H

Let E be a normed space, F a complemented subspace of

complements of F in E . Then G and H are isomorphic.

By Proposition 1.2.5.2 a =~ c, F is closed and G and H are isomorphic to E / F . Hence F and G are isomorphic to each other, C o r o l l a r y 1.2.5.5

m

Let E be a normed space and F be a complemented sub-

space of E . E is complete iff F and E / F

are both complete.

By Proposition 1.2.5.2 a =~ c, F is closed. Thus the assertion follows from Theorem 1.2.4.2 e),f), C o r o l l a r y 1.2.5.6

I

~

m

0

) Let F be a closed subspace of the normed space E

which is of finite codimension in E . Then E has a finite-dimensional subspace G such that E=FoG. As G we may take any algebraic complement of F in E . In particular, E is complete iff F is complete.

126

1. Banach Spaces

Let q : E -+ E / F for E l F .

be the quotient map and (XL)tE I a n algebraic basis

Given ~ E I , take x~ E X~ and let G be the vector subspace of E

generated by (x~)~Er. Then G is finite-dimensional and qIG is an algebraic isomorphism. By Corollary 1.2.4.9, qlG is an isomorphism, so by Proposition 1.2.5.2 c ::~ a E=FGG.

If H is an algebraic complement of F in E , then there is an isomorphism u : G ~ H . The map F•

>F•

(x,y),

>(x, uy)

is then an isomorphism, so E=F|

The last assertion follows from Proposition 1.1.5.1, since every finitedimensional normed space is complete (Corollary 1.1.3.5). [

D e f i n i t i o n 1.2.5.7

(

0

)

I

Let F be a vector subspace of the normed space

E . A projection in E is an operator p on E such that p o p = p. If F = I m p then we say p is a projection o f E onto F . In this case xEFc---~px=x for every x E E , so F is closed.

For every u E s

with u ( E ) C F , u is a projection of E on F iff

ux = x for every x E F .

Theorem

1.2.5.8 ( 0 )

(Murray,1937)Let F

and G be subspaces of the

space E . Then the following are equivalent:

~) E = F e a . b)

there is a projection p of E onto F such that G = K e r p .

If these assertions hold, then 1 - p is a projection of E onto G , F = Ker (1 - p), and E / F

is isomorphic to G.

1.2 Operators

a ==~ b. Let q 9 E -+ E / G

127

be the quotient map. By Proposition 1.2.5.2

a =~ c, G is closed and the map u " F ---+ E / G ,

x,

) qx

is an isomorphism. P u t j" F

) E,

p'-jou

x ~

>x ,

-1 o q .

Then p is an operator on E and I m p C F . Given x c F , p x = u - l (qx) = x .

Thus I m p = F and p o p = p. T h a t G = K e r p is obvious. b ==> a & the last assertion. Since (1-p) 1 -p

o(1-p)=l-p-p+p=l-p,

is a projection in E . Take x E E . T h e n (x e F ) ~

(x--px)

r

( ( 1 - p ) x = O) ~

(x e K e r ( 1 - p))

(x e G) r

(px = 0) ~

((1 - p ) x = z ) r

(x e Ira(1 - p)).

Thus F-Ker(1-p),

G-Im(1-p).

Define u "F x G v :E

~ E,

~ F x C,

( x , y ) ~-+ x + y , z ~

(pz, ( 1 - p ) z ) .

u and v are linear and continuous (Proposition 1.2.5.1, Proposition 1.1.5.1). Since uvz = pz + ( 1 - p ) z

= z,

for every z E E and v u ( x , y) = (p(x + y ) , (1 - p ) ( x

+ y)) = (x, y)

for every (x, y) C F • G , it follows t h a t u is an isomorphism. Hence E=F@G.

By Proposition 1.2.5.2 a =v c, E / F

and G are isomorphic.

I

128

1. Banach Spaces

C o r o l l a r y 1.2.5.9

Let E be a normed space, F a subspace of E ,

( 0 )

and G a subspace of F . a)

If F is a complemented subspace of E and G is a complemented subspace of F , then G is a complemented subspace of E .

b) If G is a complemented subspace of E , then it is a complemented subspace ofF. a) By Theorem 1.2.5.8 a =~ b, there are projections u of E onto F and v of F onto G. Then E

>

E,

x ~,

>

vux

is a projection of E onto G. By Theorem 1.2.5.8 b ::~ a, G is a complemented subspace of E . b) By Theorem 1.2.5.8 a ::v b, there is a projection u of E onto G. Then F

;,F,

X~

-)Ux

is a projection of F onto G. By Theorem 1.2.5.8 b =~ a, G is a complemented subspace of F .

II

Corollary 1.2.5.10

( 0 )

Let E , F

be normedspaces. Take u 6 s

such that the map E~Imu,

x,

) ux

is an isomorphism. Let G, H be vector subspaces of E and F respectively, such that G is not a complemented subspace of E . If u(G) c H and there is a projection of H onto u ( G ) , then H is not a complemented subspace of F . Assume H to be a complemented subspace of F . By Murray's Theorem,

u(G) is a complemented subspace of H . By Corollary 1.2.5.9 a), u(G) is a complemented subspace of F and by the same corollary b), it is a complemented subspace of Im u. It follows that G is a complemented subspace of E and this is a contradiction. E x a m p l e 1.2.5.11 of ~(T)

II

( 0 ) If T is an infinite set, then there exists a projection

o~ ~o(T) ~ h o ~ no~,~ i~ 2 and ~ y

~h

p ~ o j ~ t i o ~ ha~ a ~o~.~ at

least 2. co(T) is a complemented subspace of c(T) with codimension 1.

1.2 Operators

129

Let ~ denote {A C T I T \ A

is finite},

the filter on T consisting of the cofinite subsets of T . It is easy to see that c(T)

~ c(T),

x,

~x-

(lim x(~))eT

is a projection of c(T) onto co(T) with norm 2 and with one-dimensional kernel. By Murray's theorem, co(T) is a complemented subspace of c(T) with codimension 1. Let u be a projection of c(T) onto co(T). Set X

:=

?_re T .

Then, given t C T , et e co(T) ,

IleT-2etll=l,

I1~11 ~ II~(~T- 2e,)ll- IIx- 2e, II ~ Ix(t) -- 21 ~ 2 - Ix(t)l, and so Ilull _> l i m ( 2 - I x ( t ) l ) - 2. Remark.

m

This result will be generalized in Proposition 4.2.8.23.

Example 1.2.5.12 Let E be a normed space, T a topological space, S a closed set of T , and u:C(S,E)

~C(T,E)

an operator with (~)

IS = x

for every x E C ( S , E ) . Then {x C C ( T , E ) ] x = 0 on S} is a complemented subspace of C(T, E) . The map C(T, E)

~ C(T, E) ,

x.

~ x - u(xlS )

is a projection of C(T, E) onto {x C C(T, E) Ix = 0 on S} and the conclusion follows from Murray's Theorem. Remark.

m

Exercises 1.2.14, 1.2.16 (resp. Corollary 1.2.5.15) present examples

where such an operator u exists (resp. does not exist).

130

1. Banach Spaces

Example

1.2.5.13 ( 0 )

If a complemented subspace of go contains co,

then it is not separable. Let E be a complemented subspace of g~ containing Co and let u be a projection in go with E -

K e r u (Theorem 1.2.5.8 a =, b). Put F'-

Imu

and define

z~'F

)IK,

x,

)z(n)

for every n E IN. By Lemma 1.1.2.17, there is an uncountable set 91 of infinite subsets of IN such that A N B is finite for distinct A, B r 91. Let ff~ be a finite subset of 91. Given A c if3, put

C(A) "- A \ O B BE~3 B#A

and x "-- E

eC(A) "

AE~

Then I xll < 1. Since E contains Co, Ue A -- UeC( A )

for every A C 91. Hence U X --- ~

?-teA .

AE~

Take x' C F t . By the above,

ff3Eg3f (!21) AE!B

AE~

sup Ix'(ux)l ~ IIx'o ~ll ~ IIx'll Ilull xcE#

(Proposition 1.2.1.4 b), Corollary 1.2.1.5). Hence (x'(ueA))A~ is a summable family (Proposition 1.1.6.14 d =v a) and

{A e ~ I ~ ' ( ~ ) # O}

1.2 Operators

131

is a countable set (Corollary 1.1.6.7). Thus

:-- U

i

o}

nEIN

is also countable. Take A E 9.1\9./o. Then (~)(~)

= x'(~)

= o

for every n E IN, so that U e A = O.

Thus eA E Ker u -

E.

{CA I A E 92\9.10} is thus an uncountable subset of E such that

for distinct A, B E 9.1\920. Hence E is not separable. Remark.

I

Lindenstrauss (1967) proved that every infinite-dimensional com-

plemented subspace of t ~ is isomorphic to t ~ , and therefore not separable (Example 1.1.2.2). C o r o l l a r y 1.2.5.14

( 0 )

(Phillips, 1940) For no infinite set T are c(T)

and co(T) complemented subspaces of t ~ ( T ) .

Let S be a countable infinite subset of T. Given x E t ~ "x "T~

IK

'

t~

! x(t)

if t E S

/

if t E T \ S

0

Define 9e ~ ( s )

~e~(T),

x,

~.

Then t c~(S)

>Imu,

is an isometry, U(Co(S)) C co(T), and

x,

>ux

define

132

1. Banach Spaces

co(T)

>co(T),

y,

>esy

is a projection of co(T) onto U(Co(S)). By Example 1.1.2.3 and Example 1.2.5.13, Co(S) is not a complemented subspace of t ~ ( S ) , so that by Corollary 1.2.5.10, co(T) is not a complemented subspace of t ~ ( T ) . By Example 1.2.5.11, co(T) is a complemented subspace of c(T) so that by the above considerations and Corollary 1.2.5.9 a), c(T) is not a complemented subspace of ~ ( T ) . m C o r o l l a r y 1.2.5.15 Let T be an infinite set endowed with the discrete topology and let T* be its Stone-Cech compactification. Then there is no operator u "C(T*\T)

~ C(T*)

with the property that x = ux[(T*\T) for every x e C ( T * \ T ) .

By Example 1.2.5.12, the existence of such an operator would imply that {x C C ( T * ) I x - 0 on T * \ T } is a complemented subspace of C(T*), i.e. that co(T) is a complemented subspace of t ~ ( T ) , contradicting Corollary 1.2.5.14. m Proposition 1.2.5.16 Given x E ~ ( T )

Let T be a set and take p E [1,c)c]U{0}.

( 0 )

, define 5" ~P(T)

and for u c s

~ IK,

t,

Take x e g ~ ( T ) . Then 5 e s

p-t~~ b)

y,

>x y ,

define it" T

a)

> ~P(T),

~ (uet)(t). I]511- [Ixll~. We define

>s

x,

~ is linear and the map

g~(T) is an isometry.

>Imp,

x,

>~x

.~ 5.

1.2 Operators

c)

u E s

=~/~ E e ~ ( T ) ,

d)

9 ~ e~(T)~

~ =

e)

The map

Itall~ -< II~*ll-

~.

s is a projection of s f)

133

,~ s

u , ~ ~, u

onto I m p of norm 1.

I m p is a complemented subspace of E(lP(T)). a), b), c), and d) are easy to see. e) By c) and a) the map is a well-defined operator of norm at most 1. By

d), it is a projection of F_.(gP(T)) onto I m p . Hence it has norm 1. f) follows from e) and Murray's Theorem.

i

134

1. Banach Spaces

1.2.6 T h e T o p o l o g y of P o i n t w i s e C o n v e r g e n c e /

D e f i n i t i o n 1.2.6.1

(

0

\

)

Let S be a set, A a subset of S ,

space, and J~ a set of maps of S into T .

T a topological

.~A denotes the set 2c endowed with

the topology of pointwise convergence in A .

9rs denotes the set ~ endowed with the topology of pointwise convergence, i.e. with the topology on .%- induced by the product topology on T s . Let E , F be normed spaces. By Proposition 1.2.1.4 a), the topology of pointwise convergence on L:(E, F) is coarser than the norm topology of L:(E, F ) . In particular, the topology of E~ is coarser than the norm topology of E ' . /

P r o p o s i t i o n 1.2.6.2

(

0

) Let E be a vector space, F a vector space of

linear forms on E , and V' a O-neighbourhood in FE. Then there is a finite subset A of E such that

{x' C F i x

c A ~

Ix'(x)l_

1} C V'.

Given x' c F , there is an a E IR+ with x' c a V '

By the definition of the topology of pointwise convergence, there is a finite family (x~)~e, in E and a family (c~)Lei in IR+\{0} such that

{x' c E ' I ~ e I ~

I~'(x~)l _< ({t})l=l

(Proposition 1.2.7.11). T h e converse is easy to see. The last assertion follows from the first one. I

( 0 ) Let T be a completely regular space. Then x E C(T) is an extreme point of C(T) # iff Ixl = eT. Example

1.2.7.15

Take x E C(T) # and assume t h a t there is a t E T with Ix(t)l r 1. Let U be a neighbourhood of t such that := sup I~(~)l < 1. sEU

Take y E C(T) with {y r O} C U , Then

[lYi[oo= 1 - a .

146

1. Banach Spaces

x+yEC(T)

l(x+y)

+

1

# ,

(x-y)-x

Thus x is not an extremal element of C(T) # . Now assume that Ix I = aT. Take y , z E C(T) # and a E ]0, 1[ with a y + (1 - a ) z = x . Then

ay(t) + (1 - a ) z ( t ) = x ( t ) , so t h a t ~(t) = z(t)

- ~(t)

for every t E T . Hence y~-z

--X

and x is thus an extreme point of C(T) # . Example e~(T)

1.2.7.16

Let T be a set. Then x E g~

I

is an extreme point oj"

~/y Ixl = aT.

This assertion follows immediately from Example 1.2.7.15. Example

1.2.7.17

I

Let T be a locally compact space. Then x of Co(T) is an

extreme point of Co(T) # iff x ] -

aT. In particular, Co(T) # admits extreme

points iff T is compact. Take first x E C0(T) # and assume t h a t there is a t E T with Ix(t)I # 1. Let K be a compact neighbourhood of T , so that a := sup Ix(~)l < 1. sEK

Take y E Co(T) # with {y#O}cK, Then

Iiyil~=l-a.

1.2 Operators

14 7

x :t= y C Co(T) # ,

l(x+y) +

1

(x-y)=x

Thus x is not an extreme point of Co(T) # . Now suppose that Ixl = eT. Then T is compact and, by Example 1.2.7.15, x is an extreme point of Co(T) # . C o r o l l a r y 1.2.7.18

9

Let T be a compact space and S a locally compact n o n -

compact space. Then C(T) and Co(S) are not isometric,

m

148

1. Banach Spaces

1.2.8 T h e A l a o g l u - B o u r b a k i T h e o r e m T h e o r e m 1.2.8.1

( 0 ) (Alaoglu 1940, Bourbaki 1938) Let E be a normed

space. Then E~E# (i.e. the unit ball of E' endowed with the topology of pointwise convergence) is compact. If E is separable then E~E# is metrizable. Let ~ be an ultrafilter on E '#

y':E

Define

>IK, x,

>limx'(x); x ! ,~

y' is linear and

ly'(x)l- lim Ix'(x)l ~ II~ll x I~ for every x e E (Proposition 1.2.1.4 a)). Hence y' is continuous and Ily'll ~ 1 (Proposition 1.2.1.1 d =~ a), i.e. y' E E '# . Since ~ converges to y' in the topology of pointwise convergence, E~ # is compact. Now suppose that E is separable. Let A be a countable dense set of E . Since E '# is equicontinuous, the topology of pointwise convergence in A coincides with the topology of pointwise convergence in E , i.e. E~ # - E~ # (Proposition 1.1.2.15). But E~ # is metrizable (since A is countable). Hence E~ # is metrizable.

1

T h e o r e m 1.2.8.2

( 0 )

(Banach-Dieudonn~)Let E be a normed space

and 92 the set of subsets A of E , such that

{x 9 A I Ilxll ~ c} is finite for every e > O. Then of the topologies on E ' , the topology of uniform convergence on the sets of 92 is the finest one to induce the topology of pointwise convergence on the equicontinuous sets of E ' . Put ~s

~ U I C E'

[

given an equicontinuous set A' of E ' , A' N U' is an open subset of A~

D

J

It is easy to see that T' is the finest tology on E' inducing the topology of pointwise convergence on the equicontinuous sets of E ' . Let |

be the topology

(on E ' ) of uniform convergence on the sets of 9.1. Since the sets of 9,1 are relatively compact, |

induces the topology of pointwise convergence on the

equicontinuous sets of E (Proposition 1.1.2.15). Hence |

C T'.

1.2 Operators

149

Take U' E ~s and x' E U'. Let n E IN. Given A c E , put

ft. "- {y' E n E ' # I x E A ~

I ( x , y ' - x'l] < 1}.

Then

n

is a downward directed set of closed sets of nE'#E, the intersection of which is {x'} n n E '#. Since U ' N n E '# is an open set of nE'#E and since nE'#E is compact (Theorem 1.2.8.1), there is an An E ~ I ( 1 E #) with

An C U'. Define A-

UAn nE IN

.

Then A E 91 and

x' E {y' E E' I x E A

~

Il ~ 1} C U'.

Hence U' is a neighbourhood of x' with respect to | Thus G ' -

%'.

We deduce that ~' C | I

C o r o l l a r y 1.2.8.3

( 0 ) Let E be a Banach space and ~ the set of compact

convex sets of E . Then the topology on E' of uniform convergence on the sets in .~ is the finest one to induce the topology of pointwise convergence on the equicontinuous sets of E ' . Endow E' with |

the topology of uniform convergence on the sets of

and let ~s be the finest topology on E' inducing the topology of pointwise convergence on the equicontinuous sets of E' (Theorem 1.2.8.2). We have to prove that |

= ~'. |

C ~s follows from Proposition 1.1.2.15. By Theorem

1.2.8.2, to prove the reverse inclusion it suffices to show that, given a sequence (Xn)nEIN in E converging to 0, there is a K E ~. containing {Xn t n E IN}. But this was proved in Proposition 1.2.7.8.

I

150

1. Banach Spaces

1.2.9 B i l i n e a r M a p s Definition 1.2.9.1

( 0 ) Let E, F, G be vector spaces. A map u" E x F --+ G

is called bilinear if it is linear in each variable (i.e. u(x, .) and u(., y) are linear for every x E E and y E F ).

Proposition 1.2.9.2

(0 )

Let E, F, G be normed spaces and u" E x F--+ G

a bilinear map. Then the following are equivalent:

a) u is continuous. b) u is continuous at (0,0). c)

There is an a E IR+ such that

Ilu(~, y)il < -Ilxll Ilyll for every ( x , y ) E E

d)

x

F.

The restriction of u to E # • F # is uniformly continuous.

d~a~b

is trivial. b ==> c. There is a 5 > 0 so that liu(x,y)]l 0. There is an rn E IN such that bp - b m
_ m . Take p C IN, with p _> 2m and let A

:-- {(i,y)e (~,, u {o})x (~,, u {o})I,~ < sup{i,j}},

B 9=

,~(~'J)~ A l~ + ; >

~}.

Then p

(i,j)EB

n=O

-
~'(~)+~'(y)

Show that z' E (E • F ) ' and

llz, ll (ll ,ll § ii 'll )

1

where E x F is endowed with the 2-norm. E 1.2.7'

Let T be a set. Take x E f ~ ( T ) , and define x'el(T)

>]Z,

y,

>Ex(t)y(t) tET

Show that the following are equivalent (Example 1.2.2.3 b))" a)

There is a y E t~I(T) with [[Y[I = 1 and 5(y) - I ] ~ [ ] .

b)

There is a t 9 T with Ix(t)I = IIx ]oo.

E 1.2.8

Let T b e a s e t . Take x E t ~

Xco(T)

(resp. c(T) )

>IK,

Y'

>E x ( t ) y ( t ) " tET

Show that the following are equivalent (Example 1.2.2.3 b) ): a)

There is a y E co(T)

(resp. y E c(T)) with i l y l I - 1 and

I~(y)i = li~ll. b)

x E IK (T)

(resp. the map

T\~'(0)---+~,

~,

>

i~(~)l

is in c(S)). E 1.2.9

Take p,q, r E [ 1 , c r

with ~1 + q _1 > 1 and k" IN • IN --+ ]K such that

k(m, .) E g' for every m E IN. Given x E gq, define 9IN

~IK,

m,

~ Ek(m'n)xn" nE IN

Show that if ~ E fr whenever x E t~q, then the map X,

is continuous.

)X

1.2 Operators

E 1.2.10

155

Take k e C([0, 1] x [0, 1]). Given x 9 C([0, 1]), define 1

kx" [0,1]

> IK,

/ k(s, t)x(t)dt.

s,

0

Show the following: a) kx C C([0, 1]) whenever x e C([0, 1]). b) The map ,.,.,

k -c([o, 1])

x,

> C([0, 1]),

>kx

is linear and continuous and 1

Ilk I -

E 1.2.11

sup

~e[o,q

/

Ik(s, t)ldt.

0

Take p C [1, cx~]. Given x c l K 2z , define

Xr " 2Z

~ It(,

n,

> x( n-

l ),

xe" ~

> IK,

n,

> x(n + l).

Show that"

a) x e ~p(2z) ~ Xr, Xe e ~P(~), I x~llp --Ilxell~ --Ilxll~. b) x c c(2Z)

(resp. Co(2Z)) =:> xr,xe C c(~)

(resp. Co(2Z)).

c) The maps eP(~) --+ eP(~), c(2Z) -+ c(2Z), Co(2Z) -+ Co(2Z) defined by x ~-~ Zr (resp. xe) are isometries. They are called the right (resp. left) shift of t P ( ~ ) , E 1.2.12

Let

c(2~), and Co(~), respectively.

(OLn)nEiN

1 1 be a sequence in lK. Take p, q C ]1, c~[ with p+~ - 1,

and put

F'-{xcgPl

lirn ~-~akXk--O }. k--1

Show that the following are equivalent: ~3

a)

Y ~ l a n l q = O or c~. n=l

156

b)

1. Banach Spaces

F is dense in fP.

E 1.2.13

Let F be a closed vector subspace of the Banach space E ,

q : E --+ E l F

the quotient map, and T a set. Take u 9 ~_.(fl(T),E/F)

> 0. Show that there is a v 9

E) such that

u = q o v,

E 1.2.14

and

Ilvll ~< (1 + c)ll~ll,

Let T be a metrizable topological space, S a closed set of T , and

E a normed space. Prove the following: a)

There exists an operator

u:C(S,E)

>C(T,E)

of norm 1 such that (uz) I X = z

for every x 9 C(S, E) (see e.g. Jun-Iti Nagata, Modern general topology, Theorem VII.14). b)

{x 9 C(T, E) I x = 0 on S} is a complemented subspace of C(T, E ) .

E 1.2.15

Let E be a normed space, T a completely regular space, S a

closed set of T , A a countable subset of T such that A \ U is finite for every neighbourhood U of S and put f := {~ e

C(T, E)

Ix = 0 o. St.

Show t h a t there is an operator

U : co(A, E)

~ .T"

of norm 1 such that

(uz) i A = x for every x C co(A, E ) . E 1.2.16

Let E be a normed space, T a completely regular space, and K a

compact set of T . For each closed set S of T let S' denote the set of t c T such that t is not a point of adherence of S \ { t } , and define Kr for each ordinal number { by means of transfinite induction as follows:

1.2 Operators

157

Ko--K, nE~

Assume that there is a countable ordinal number ~ with K~ - O. Show that there is an operator

u" C(K, E)

~ C(T, E)

such that

(ux) I / ( -

x

for every x C C(K, E) and that {x E C(T, E) ] x = 0 on K} is a complemented subspace of C(T,E). (Hint" Use the preceding exercise.) E 1.2.17

Let E be a vector space,

A a convex (resp. absolutely convex)

set of E , (x~)~ci a finite nonempty family in A, and (a~)~i a family in IR+ (resp. in IK), such that Ea~-I

(resp.

~EI

Zla~]_~l). LCI

Prove the following:

a) ~a~x~CA. LCI

b)

There is an x C A such that E

E 1.2.18

x~ = (Card I) x.

Let E be a (separable) normed space. Show that there exists a

(metrizable) compact space T such that E is isometric to a subspace of C(T). E 1.2.19

Let E be a separable normed space. Show that E is isometric to

a subspace of t~~ E 1.2.20

Let E, F be normed space with F finite-dimensional. Show that

the closed unit ball in s

F) endowed with the topology of pointwise con-

vergence is compact (and metrizable if E is separable) (generalization of the Theorem of Alaoglu-Bourbaki).

158

1. Banach Spaces

E 1.2.21

Let E, F , G be normed spaces. Define

B(E, F; G):= {u " E x F

~ G lu is bilinear and continuous}

and

Ilull :-inf{o~ e JR+ [(x, y) e E x F ~ ~. E ~

~(F, C ) ,

9~

II~(x, Y)II ~ o~llxll Ilyll},

u(~, .)

for every u E B(E, F; G). Prove the following: a)

B(E, F; G) is a vector subspace of B(E,F; G)

G ExF and the map

~]R+,

u,

>llull

is a norm. b)

u is linear and continuous and

c)

The map

I1~1-lull for every u e B(E,F; G).

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

u:

~u

is an isometry. E 1.2.22

Let F be a closed vector subspace of a Banach space. Show that

the set of projections of E on F is a convex set of / : ( E ) , closed with respect to the topology of pointwise convergence.

1.3 The Hahn-Banach Theorem

1.3 T h e H a h n - B a n a c h

159

Theorem

The H a h n - B a n a c h Theorem is the most i m p o r t a n t result in the theory of normed spaces, without which the theory would lose all interest. It ensures t h a t the dual space of a normed space contains sufficiently many vectors to allow every normed space to be isometrically imbedded - - by means of the evaluation map into its bidual. It also allows us to associate to each operator its transpose and bitranspose. The evaluation map enables us to define the most important class of Banach spaces: the reflexive ones.

1.3.1 T h e B a n a c h T h e o r e m Lemma

1.3.1.1

( 0 )

Let E be a vector space and F , G

vector subspaces

of E with F A G -- { O} . Put H:=F+G. Let x I and yt be linear forms on F and G , respectively. Then there is a unique linear f o r m z ~ on H with

z ' l F = x',

z'lG = y '

The uniqueness is obvious. Take (x~, Yl), (x2, y2) E F • G with x l + yl = x2 + y2. Then Xl-X2--y2-Yl

C FNG,

so t h a t Xl = x2,

Yl --- Y2 9

It follows that the map z':H

>IK,

x+y,

>x'(x)+y'(y)

is well-defined, z' has the required properties. Theorem

1.3.1.2

( 0 )

(Banach, 1929) Let F be a vector subspace of the

real vector space E . Let p be a real f u n c t i o n on E F such that

I

and y' be linear f o r m on

160

1. Banach Spaces

a)

p(x + y) < p(x) + p(y) for every x , y 9 E .

b)

p(ax) = ap(x) for every x 9 E and a 9 IR+.

c) y'(y) IR,

x~

> lim ~1 n-+cr

Xm

.

7Z

rn=l

Then F is a vector subspace of g ~ ,

y' is a linear form on F with y' < P l F ,

and

p(x + y) < p(x) + p(y) ,

p(.z) - .p(~)

for every x, y C gor and c~ E 1R+. By Banach's Theorem, there is a linear form x' on g~ with

x ' l F = y' ,

x' < p.

Thus

x' (x)

-x' (-x) > -p(-x)

__

__

lim inf ~1 n-+ cr

Xm

?'t

m=l

for every x E g ~ . The inqualities lim infxn _< lim inf -1 ~ n--+oo

n---+ cx~

T~

xm < x'(x) < lim s u p -1 ~ n - + cx:~

rn=l

rt

xm < lim sup xn n--+oo

m--1

are trivial for x C g ~ . We deduce that x' is continuous with norm 1 and vanishes on co. Take x E g~ and put y := utx

(resp. y := urx).

Then

1~ n

(xm

Ym) -- xn

(resp.

X 1 --

m=l

Xn+

1 ]

n

n

for every n C IN. Hence lim

-l~(xm-ym)-0.

n - + o o ?Z

m=l

Thus x - y E F

and * ' ( * - v) = v ' ( * - v) - 0 ,

Remark.

.'(x)

= x'(v),

tt

Let ~ be a free ultrafilter on IN. Then x"g~

x~

~liml n , 8 ~ rZ

Xrn m--1

has the properties required in the above example. This is another proof for the existence of x' which does not use Banach's Theorem. Of course, not all x' are of the above form.

1.3 The Hahn-Banach Theorem

P r o p o s i t i o n 1.3.1.4

( 0 ) Let E be a vector space and A a convex (resp.

absolutely convex) set of E with x E a A .

163

such that f o r every x C E

there is an a E JR+

Define

>IR, x ,

p:E

>inf{aEIR+ixcc~A}.

Then

p(x + y) < p(~) + p(y), f o r every x, y E E

p(.x) = I-Ip(x)

and (~ E ]R+ (resp. a C IK ).

Take /3, 3' e IR+\{0} with x C fl A ,

y c "),A .

1 1, Then ?x, ~y E A, so that

1

~(x fl+7

+ y) =

fl 1 3/ 1 -yEA. ~f l +-T f l x + fl + 7,, /

Hence x + y E (fl + 7 ) A ,

p ( x + y) < fl +'),.

Since fl and -), are arbitrary,

p(x + y) < p(x) + p(y). The other assertion is trivial. L e m m a 1.3.1.5

( 0 ) Let E , F

I be complex vector spaces and u

E -+ F

an I R - l i n e a r map. T h e n E

>f,

>ux-iu(ix)

x,

is C-linear.

Define v :E

) F,

x ,

>ux-

iu(ix).

v is obviously IR-linear and v(ix) = ~(i~) -i~(-~)

= ~(~

- i~(ix)) = ~x

for every x C E . Hence v ( ( ~ + i g ) x ) = v ( ~ x ) + v ( i g x ) = ~ v x + i f l v x = (~ + i g ) v x

for every x E E and a,/3 C IR, i.e. v is C-linear.

I

I 63

1. Banach Spaces

P r o p o s i t i o n 1.3.1.6 space E .

Let B

( 0 ) Let A be a n o n e m p t y convex set of the vector

be an absolutely convex set of E

there is an ~ C IR+ with y E a B .

such that f o r each y C E

Take x C E \ ( A + B ) .

Then there is a linear

f o r m x' on E , bounded on B , such that

sup re x' (y) < re x'(x). yCA

Now 0 E B. We may assume that 0 c A (otherwise we replace x and A by x - a

and A - a

for an a c A ) . W e p u t 1

C:=A+-~B,

p:E

~IR,

y,

~inf{a~IR+lyeaC},

y' : F .... ~ IR,

c~x ~-+ c~p(x) .

By Proposition 1.2.7.3, C is convex and by Proposition 1.3.1.4,

p(y + ~) < p(y) + p(~),

p(~v) = ~p(y)

for every y, z C IR+ and c~ E IR+. Take c~, fl E IR+ with

x e ( ~ B ) n (ZC). Then c~ > 1, fl > 1. There is a pair (a, b) E A x B, with 1

1

~-

a + ~b.

Then x-

( (11- ~)

x+

1-

(

b

1

=~x-

2 -aeA,

1

x+~bCB,

~) 1B B . 1-~ z~ 2 ' x~ ~2(/3-1)

1.3 The Hahn-Banach Theorem

165

2a 2a-

2 ( / 3 - 1)

1

Since x ~ C , it follows that 2cg p(x) > -2a-1

> 1.

Thus we have y' < p F . By Theorem 1.3.1.2, there is an N - l i n e a r map z " E ~ ]R such that z' < p and z' (x) = y'(=) = v ( * ) > 1.

Put X I .--- Z I

if I K - I R

and x' . E - - + r

v.

", z' (v) - iz' (iy)

if IK - C .

By Lemma 1.3.1.5, x' is a linear form on E . Since p _< 2 on B and

-B-B,

x' is bounded on B . T h e n s u p r e x ' ( y ) = supz'(y) < supp(y') 0 with V f (x) c E \ A .

Then x E E \ ( A + U E ( 0 ) ) . By Proposition 1.3.1.6 (and Proposition 1.2.7.2), there is a linear form x' on E , bounded on UE(0), such that sup re x'(y) < re x'(x). yEA

By Proposition 1.2.1.1 e => a, x' E E ' .

166

1. Banach Spaces

Now suppose that A is absolutely convex. Then x'(x) :/= O. Define

Y "

~'(~) ~,. Ix'(x)l

Take y 9 A with y'(y) :/= O. Then

x,(z) y,(y) Ix'(x)l ly'(y)l so that

ly'(y)l

(x,

y 9 A,

(x) y'(y) Ix'(x)l ly'(y)l y

= rex'

)

x'(z)



xdA. *

g) Take w "- (t~)tEi C ,(2. The sets - t t + Aa are dense Gs-sets of T for every (~, A) C I x L. Since T has the Baire property, we can choose

(N tel

+

(N tel

AEL

AEL

Then p~(Ea;~eAx ) =

1

sup(Eut~(Ea),(eAx

2(#(W)

)~EL

1

sup

ate(W)

1

sET

(

seT

E

Eax

tel

ACL

(

teI

+ U e A x ) ) ) ( s ) --

AEL

(

e-t~+A~ + e-t~-A~

)

1

Since w is arbitrary,

(

))

(s) >_

)

I AEL

ACL

174

1. Banach Spaces

E x a m p l e 1.3.2.2

Using the notation of L e m m a 1.3.2.1, let .T be a vector

subspace of e~176 and y' a linear f o r m on .7c with y' < p l Y .

Then there is

an x' 9 t~176 ' such that

1) IIx'll ~< 1; 2)

x' o ut - x' o u = x' for all t 9 T ;

3)

x'(x) 9 IR+ for every positive function x of e~176 ;

4) ~ ' l T = y ' . By Lemma 1.3.2.1 b), c), and the Banach Theorem (Theorem 1.3.1.2), there is a linear extention x' of y' on t?~(T) such that x' < p. Then

x'(x) ~ p(x)~ Ilxll,

- x ' ( x ) = x'(-x) ~ p(-x) ~ II- xll = Ilxll

(Lemma 1.3.2.1 a)). Hence

I~'(x)l < I1~11 for every x 9 ~?~176 Therefore x' is continuous and I1~'11 _ o for every positive function x of g~176

I

1.3 The Hahn-Banach Theorem

L e m m a 1.3.2.3

Let S be an infinite subgroup of the additive group T , ~

ring of subsets of T . Take C 9 ~ on ~

175

a

and let # be a bounded positive real function

such that for every A, B 9 9~ with A M B = O and every s 9 S :

1) #(AU B)

=

2) s + A 9 1 6 2

#(A)

+

#(B),

#(s+A)=p(A),

3) V(c) c ~ , 4) for every t c T there is an r c C with t - r r S . Then there is a subset B of C such that (s + B)ses is a partition of T and

,(~ + B ) = o for every s C S .

We define s ~ t : v=:~ s -

t c S

for s, t C T. ~ is an equivalence relation in T. By 4), there is a subset B of C containing exactly one point in each equivalence class of ~ . Then (s + B)scs is a partition of T. By 2) and 3), s + B E g~,

#(s + B) = p ( B )

for every s E S. Thus, by 1), #(B) CardD

=

Ep(s

+ B)=

sED

#( U(s

+

sED

B)) _< sup p(A) AC~

for every finite subset D of S. Hence #(s+B)=p(B)=O

I

for every s C S. Lemma 1.3.2.4

Let T be an infinite additive group and p : ~ ( T ) -+ JR+ a

map such that # ( A U B) = It(A) + It(B) for disjoint A, B C ~3(T) and

1"[6

1. Banach Spaces

#(t + A) = #(A) for every A 9 g3(T) and t 9 T . Then there is a partition (An)ne~ of T , such that

#(An) = 0 for every n 9 IN.

Let S be a countable infinite subgroup of T . The assertion follows from Lemma 1.3.2.3 by setting 9l : = g l ( T ) , E x a m p l e 1.3.2.5

Let T

C := T .

be a compact additive group and )~ be its Haar

measure normed by )~(T) = 1. Given x 9 ~ ( T ) utx : T

) IK,

ux : T Then there is an x' C g ~

1)

I

~ IK,

s,

and t 9 T , define

~ x ( s + t) ,

s,

>z ( - s ) .

with the following properties:

IIx'll - 1,

2) x' out = x' o u = x' for every t E T , 3) x'(x) r IF[+ for every positive real function in e~ 4) x'(x) = f xd)~ for every x E s176 If T is infinite, then for every such x' there is a partition (An)ne~ of T such that

9' ( ~ )

=o

for every n E IN. (Hence the Lebesgue Convergence Theorem does not hold for such an x' .)

First consider the case IK = IR. Let p be the function defined in Lemma 1.3.2.1. By Lemma 1.3.2.1 f),

I

x d~ x(s + t),

~ x(-s).

with the following properties:

1) IIx'll- 1, 2) x' o u t = x' o u = x' for every t c T , 3) x'(x) e

IR+ for every

positive real function x in g ~ ( T ) ,

4) x'(em) = 1 for every dense G~-set A of T . If T is infinite, then for every such x' there is a partition (A,~)n~ of T such that

9'(~o) = o for every n C IN. (Hence the Lebesgue Convergence Theorem does not hold for such an x' .)

First consider the case IK = JR. Let p be the function defined in Lemma 1.3.2.1, 91 the set of dense Gs-sets of T, ~ the vector subspace of g~176 generated by {eA I A C 92}, and y'" ~

>JR,

E ~ A691

'

(In the above sums,

{A ~ ~ I o~,4--/:0}

> E OZA" A691

178

1. Banach Spaces

is obviously finite and MA:~0 AE~I ~A~0

so the value of y' does not depend on the representation.) By the last assertion of Lemma 1.3.2.1, y' < p I ~ - The existence of x' with the desired properties now follows from Example 1.3.2.2. If IK = ~ , then /~~

~r

x,

) x'(rex)+ix'(imx)

has the required properties. The final assertion follows from Lemma 1.3.2.4. E x a m p l e 1.3.2.7

Let n 9 IN, ~ the Lebesgue measure on lR n, and

.7" := {x 9 g~(]Rn) I {x :/- 0}

is bounded}.

Define u t x : IR n

) IK,

ux : lR ~

s,

) IK,

) x ( s + t) ,

s,

~ x(-s)

for x 9 .T and t 9 1Rn . Then .T is a vector subspace of f~(]l=['~), u t x , ux 9 .T for every x 9 .T and t 9 IR n, and there is a linear f o r m x' on .T with the following properties:

1) x' o u t = x' o u = x' for every t 9 T , 2)

x'(x) 9 IR+ for every positive real function x in 3c,

We may replace 3) by

3')

There is a A-null set A with x'(eA) = 1.

For every such x' there is a dosjoint sequence (An)nelN Of subsets of ]R '~ such that

U An is bounded, n c IN

X'(en A ) 0

1.3 The Hahn-Banach Theorem

179

and

z'(~o) = 0 for every n E IN. (Hence the Lebesgue Convergence T h e o r e m does not hold for such an x' .)

Let T be the compact additive group IR~/2Z n , ~o : ]R ~ --4 T the quotient map, and x' the linear form from Example 1.3.2.5 (resp. Example 1.3.2.6). For every p

=

(Pk)kelNn E 2Zn the map n

I-[[pk,1 + pk[

>T ,

t ,

) ~o(t)

k=l

is bijective. Let ~op denote its inverse. The m a p

7

~IK, z, ~ ~

~'(xo~,)

pCTZ n

has the required properties (for 3', it is sufficient to take as A a Gs-set of ]R ~ which is dense in [0, 1[~ and which is a A-null set). In order to prove the last assertion, put

s : = Q ~, := {A C #:~

]Pu n I

> IR+,

c = [ 0 , 1 [ ~, A is bounded }, A ~ " x'(eA).

By L e m m a 1.3.2.3, there is a B C C such t h a t (s + B ) , e s is a partition of T and

~(~ + B ) = 0 for every s c S . Given s c S , we put As:=(s+B)

A [ 0 , 1 [ ~.

Then x'(As) = 0 for every s C S and

X'(enAn) x'(eC01E)

I

180

1. Banach Spaces

1.3.3 T h e H a h n - B a n a c h Theorem

1.3.3.1

Theorem

( 0 ) (Hahn 1927, Banach 1929) Let F be a subspace of

the normed space E . Take y' E F ' . Then there is a continuous linear extention x' of y' to E with IIx'll- Ily'll.

Case 1

]K = IR

Define

p:E

>~,

x,

>lly'llllxll,

Then

p(x + y) = Ily'll IIx + yll ~ Ily'll(llxll + Ilyll) ~ p(x) + p(y), p(~x) = Ily'll II~xll- Ily'llallxll - ~p(x) for every x, y E E and a E IR+. Moreover

y'(x) ~ Ily'll Ilxll = p(x) for every x C F . (Proposition 1.2.1.4 a)). By the Banach Theorem, there is a continuous linear extension x' of y' to E with

x'(x) < p(~) for every x C E . Then

x'(x) ~ Ily'll IIxll and

-x'(x) : x ' ( - x ) ~ Ily'll II- xll = Ily'll IIxll and so

Ix'(x)l ~ Ily'll IIxll for every x e E . Hence x' is continuous and IIx'll _~ Ily'II. By Proposition 1.2.1.4 c), IIx'[I- Ily'll. Case 2

IK = ~

1.3 The Hahn-Banach Theorem

rey'

is a c o n t i n u o u s real l i n e a r f o r m on F .

a continuous

] R - l i n e a r f o r m z' on E

181

B y t h e a b o v e proof, t h e r e is

extending

re y' w i t h

IIz'[[ = [ire o Y'll-

Define

x' : E Then

>r

x p is l i n e a r ( L e m m a

x,

>z'(~)-iz'(iz).

1.3.1.5) a n d c o n t i n u o u s . B y C o r o l l a r y 1.2.1.6,

IIx'll = I1~ o x'll = I1~'11 = I1~ o y'll = Ily'll. Then re o x' (x) = z' (x) = re o y ' ( x ) ,

i m o x'(x) = - z ' ( i x ) = - r e

o y'(ix) = re o ( - i y ' ) ( x ' ) = i m o y ' ( x ) ,

x'(x) = re o x'(x) + i i m o x'(x) = re o y'(x) + i ira o y'(x) = y'(x)

I

for e v e r y x C F , i.e. x ~ is a n e x t e n t i o n of yl. Corollary

1.3.3.2

( 0 )

Let F be a finite-dimensional vector subspace of

the normed space E . Let G be a closed vector subspace of E with F N G = { 0 } . Let (x~)~I be an algebraic basis of F .

a)

There is a family (xt~)~, in E' such that x'~ vanishes on G for every c I and

x'~(~) - M for every L,A E I . b)

The map p "E

>E ,

x,

>

E 'x~(x)x~ L6I

is a projection of E onto F , vanishing on G. c)

F is a complemented subspace of E .

d)

If the map FxG

>E,

(x,y)~

>x+y

is surjective, then E is the direct sum of F and G and 1 - p projection of E onto G , vanishing on F .

is the

182

1: Banach Spaces

a) Put H:=F+G. Take ~ E I

and

x ~'F

)IK,

Ea)'x~'

>a,.

.XEI

By Lemma 1.3.1.1, there is a linear form y' on H such that y'IF = ~',

y ' l a = o.

K e r y ' is the vector subspace of H generated by C U {x~ I A E I\{~}}. It is thus a closed vector subspace of H (Corollary 1.2.4.3). Hence y' is continuous (Corollary 1.2.4.12 b =~ a). By the Hahn-Banach Theorem, there is a continuous linear extention x~' of y' to E . x ' vanishes on G and

z~(x~) = ~ !

for every A E I . b) p is linear and continuous. It vanishes on G and Imp c F . Moreover, given ~ E I ,

AEI

Hence I m p = F . Thus

tEI

tEI

~EI

for every x E E , so that pop=p. Hence p is a projection of E onto F . c) follows from b) and Murray's Theorem (Theorem 1.2.5.8). d) Since Kerp-

G,

E is the direct sum of F and G , and 1 - p vanishing on F (Murray's Theorem).

is the projection of E onto G, I

1.3 The Hahn-Banach Theorem

D e f i n i t i o n 1.3.3.3

( 0 )

183

Let E, F be normed spaces. Set

O. By Corollary 1.3.3.5, there is an x' E E ' , vanishing on F , such that x'(x) = dF(x).

Then x' -~ 0.

Corollary 1.3.3.7 dual E' is.

I ( 0 )

The norrned space E is separable whenever its

1.3 The Hahn-Banach Theorem

185

Let (x'~)ne~ be a dense sequence in E ' . For each n E IN, there is an xn C E with

llx,~ll ___1,

1

l='(x.)l _> ~llx'nll

(Proposition 1.2.1.4 b)). Let F be the closed vector subspace of E generated by (Xn)neIN and let x' be a continuous linear form on E vanishing on F . (x~)~e~ contains a subsequence (x~,)nc~ which converges to x'. Then I

I

I

-2 Ilxk. II _< Ixko (~k~

I(xk~ xko)-(xk~ , x'>l- I(xk. , x'k~ x')l < II~k.' -*' II,

IIx'll ~ IIx'- x%~II-4-Ilxg= II ~

311x'k=

-

x'll

for every n E IN. Thus

IIx'll _< lim 31Ix'~o - m ' I I - 0 . Thus x ' = 0 and F = E (Corollary 1.3.3.6). Hence E is separable (Corollary

1.1.5.5).

1

Remark. The separability of E does not imply that of E' as is shown by taking E := ~1 (Examples 1.1.2.5, 1.2.2.3 d), and 1.1.2.2). C o r o l l a r y 1.3.3.8

( 0 ) Let E be a normed space and take x E E \ { 0 } .

a)

There is an x' e E' with IIx'[I = 1, x'(x) - lixll.

b)

Ilxl] =

sup Ix'(x)l = sup Ix'(x)l. XI E E l #

x IEE t

lix'll=l a) follows from Corollary 1.3.3.5. b) Given x' c E ' # ,

i~'(x)i < llx'll iizli < iixli (Proposition 1.2.1.4 a)) and so sup Ix'(x)l ( I l x l l . x, E E , #

By a), there is an y' C E' with

ily'll =1,

r

Hence Ilxll _< sup ]x'(x)] < x' E E'

IIx'ii=l

sup ]x'(x)] < ]]x]]. x, E E , #

i

186

1. Banach Spaces

C o r o l l a r y 1.3.3.9 y in E .

0

I

~

\

) Let E be a normed space and take distinct x and

Then there is an x t E E '

with

9'(~) # ~'(y) Since x - y ~ 0, the assertion follows from Corollary 1.3.3.8 a). C o r o l l a r y 1.3.3.10

(0 )

I

oo

Let E be a Banach space and ~ tnxn a power n--O

series in E . If there is an r > 0 smaller than the radius of convergence of the above power series such that E

olnxn -- 0

n--O

for every c~ E U~(O), then Xn --~ 0

for every n = IN u {O} .

Take n E I N U { 0 }

and x ' E E ' . T h e n

for every c~ E U~(0) (Proposition 1.2.1.16), so that

::,:'(~n) = o by a classical result of function theory. Since x' is arbitrary, Xn = 0

(Corollary 1.3.3.8 b) ). C o r o l l a r y 1.3.3.11

I Let E , F

be Banach spaces. Take u E s

Let U

be a domain of IK and take f : U--4 E .

a)

If f

is differentiable at (~ E U then u o f is differentiable at (~ and ( u o f ) ' ( o l ) = uf'(c~) .

b)

f

is constant iff f

is differentiable and f ' = O.

1.3 The Hahn-Banach Theorem

187

a) is trivial. b) The necessitiy is trivial. Take a, fl C V and assume that f(c 0 ~ f ( f l ) . By Corollary 1.3.3.9, there is an x' c F ~ with x ' ( f (ol)) :/: x ' ( f (,3)) .

By a), x' o f is differentiable, and by the above, its derivative does not vanish identically. Hence, by a), the derivative of f does not vanish identically.

I C o r o l l a r y 1.3.3.12

Let E be a normed space and p a norm on E' equivalent

to the canonical norm on E ' . Put

q(x) := sup{Ix'(x)l lx' e E',p(x') < 1}. Then

E

~ JR+, x',

~ q(x)

is a norm of E equivalent to the initial norm of E . Let Eq (resp. E p ) denote the vector space E

(resp. E ' ) endowed with the norm q (resp. p ). If p is

lower semicontinuous on E'E, then E~ is the dual of Eq.

Since p is equivalent to the canonical norm on E t , there is an a > 0 such that 1

- p ~ II-II ~ ~p. Hence

1

--q --< II" II _< t~q. ol

(Corollary 1.3.3.8 b)). In particular, q is finite and x E E , q(x) = 0 ~

x = O.

Take x, y E E . Then ]x'(x + y) = Ix'(x) + x'(y)] < Ix'(x)] + Ix'(y)l < q(x) + q(y)

for every x' E E ' , p(x') < 1, so t h a t

q(x + ~) < q(~) + q ( y )

188

1. Banach Spaces

Moreover,

q(j3x) = sup{]x'(~x)] ] x' E E ' , p(x') < 1} = = [/3[sup{[x'(x)[ ] x' E E', p(x') a.

Take y' E E ' \ { 0 } and let c~ E ]0,p(y')[. Then !y, E E ' \ A ' so

by the above considerations. It follows successively that p'(y') > . .

p'(y') ___ p ( y ' ) ,

p - p,.

and E~ is the dual of Eq.

Remark. The final assertion does not hold if p is not lower semicontinuous (see Example 1.3.1.16 c)). (E. Helly) Let E be a normed space, (xP~)~e, a finite linearly independent family in E t, and ~ > O. Given a E ]K l , the following are equivalent:

C o r o l l a r y 1.3.3.13

1.3 The Hahn-Banach Theorem

a)

189

For every c > O, there is an x E E such that I1~11 ~ ~ + ~ and !

x,(x)-

a~

for every t E I . b)

] E atat] < a]] E atx'tl] tel

for every (at)tEl E IK I .

tel

a ~ b. Take ~ > 0 and x satisfying a). Then

, tel

= I tel

I

t6I

tel

I inf{flE]R+lbE/3A

}.

Since A is absolutely convex, p is a norm. Endow IK I with this norm. By Corollary 1.3.3.8 a), there is an x' E (IKI) ' such that x'(a)

II~'ll = 1,

= p(a)

.

Take (at)rE1 E IK I such that

tEI

for every b E IK'. By b) (and Proposition 1.2.1.4 b)),

p(a) - x'(a) = E a t a t

yll = ~up I = 1.

By a), ((',x')yn)ne~ is a Cauchy sequence in L ( E , F ) . Denote by u its limit. Then

ux = lim <x, x')yn = lim yn. n ----~oo

Hence (Yn)ne~ converges and F is complete. The converse implication follows from Theorem 1.2.1.9 b).

I

1.3 The Hahn-Banach Theorem

191

1.3.4 T h e T r a n s p o s e of an O p e r a t o r

Definition 1.3.4.1 take u C s

( 0 )

(Banach 1 9 2 9 ) L e t E , F

be normed spaces and

F ) . The map u " F'

> E',

y' J

> y' o u

is called the transpose (operator) o f u .

T h e o r e m 1.3.4.2 a)

( 0 ) Let E, F be normed spaces and take u C s

F).

u' is the unique map F' --+ E ' , such that

for every (z, y') e E x F'

b)

u' e s

E'),

I1~'11 =

Ilull.

c) u is determined by u'. a) Given (x,y') c E x F ' , (~, ~'y') = (x, y' o ~) = y' o ~(x) = y'(~x) = ( ~ , y'). Take f " F' --+ E ' , such that (~, f(y')) - ( ~ , y') for every (x, y') E E x F ' . Then (x, f(y')) = (x, ~'~') for every (x, y') e E x F ' , s o that f(y') = u'y' for all y' e F'. Thus f = u'. b) Take x', y' C F' and c ~ , / ~ C I K . B y a ) (x, ~'(~x' +/~y')) - ( ~ , ~x' +/~y') - ~(~x, ~') + Z ( ~ , y') -

= . ( z , ~'~') +/~(~, ~'y') - (~, .~'x' + Z~'~') for every x C E . In other words,

Hence u' is linear.

192

1. Banach Spaces

Given y' E F ' ,

II~'y'll = Iiy'o ~11 < Ily'll ii~ll (Corollary 1.2.1.5), and so u' is continuous (Proposition 1.2.1.1 d ==> a) and

il~'il < il~il. Take x E E with ux ~ O. By Corollary 1.3.3.8 a), there is a y' C F' such that

ily'll = 1,

y'(~x) - ii~ll 9

y'o ~(~)-

(x, ~'y') < i~ll il~'y'li < il~ll ii~'l[

Thus

li~xll = y'(~)-

(Proposition 1.2.1.4 b) ), so

II-II u'

(Theorem 1.3.4.2 b) ) is linear and continuous.

Take u, v c s

F ) and c~, fl C IK. Then

= ~(x, ~ ' y ' / + Z(~, ~'y') - (~, ~ ' y ' + Z~'y') = (~, ( ~ ' + Zv')y') for every (x, y') e E x F' (Theorem 1.3.4.2 a)). Thus

~u' + fl,'=

(~ + 9,)'

(Theorem 1.3.4.2 a) ). Hence the map s

>s

E') ,

u ~ ~ u'

is linear. By Theorem 1.3.4.2 b), it is continuous. C o r o l l a r y 1.3.4.4

( 0 ) If E is a normed space, then (1E)' -- 1E,.

m

1.3 The Hahn-Banach Theorem

C o r o l l a r y 1.3.4.5

193

( 0 ) Let E , F , C be normed spaces and take u e ~(E, F),

~ 9 L(P, a ) .

Then V 0 U) I ~

~t t 0 V ! .

Given (x, z') C E x G',

(vo~(~),z') = (wx, z')= (~,v'z')= (~,~'~'z')= (~,~' o~'(z')) (Theorem 1.3.4.2 a)). Thus u' o ~ ' =

(~ o ~)'

(Theorem 1.3.4.2 a) ). C o r o l l a r y 1.3.4.6

m The transpose of a projection is a projection.

This is an immediate consequence of Corollary 1.3.4.5. C o r o l l a r y 1.3.4.7

( 0 ) Let E , F

be normed spaces and take u E s

If u is an isomorphism (isometry), then u' is an isomorphism (isometry).

Let u be an isomorphism and let v := u -1 . Then uov=

1F,

you=

1E.

Thus v ~ou I = 1F,,

u ~ o v ~ - 1E,

(Corollary 1.3.4.5, Corollary 1.3.4.4), i.e. u' is an isomorphism. If u is an isometry, then for y' E F ' , ll~'y'll = sup I(~,~'y')l = s ~ p I ( ~ , y ' ) l xEE#

xcE#

= sup I(y,y')l = Ily'll yEE#

(Proposition 1.2.1.4 b), Theorem 1.3.4.2 a)), i.e. u' is an isometry. Remark.

The reverse implication, u' isomorphism (isometry) ==~ u isomorphism (isometry),

holds whenever E and F are complete (Corollary 1.4.2.5).

R

193

1. Banach Spaces

C o r o l l a r y 1.3.4.8

(2)

(3)

Let E , F , G , H

be normed spaces. Take

v E / : ( G , H ) and let ((y~,zt))te, be a finite family in F ' •

u E s

G.

Then

(E (, ~:>z~) o ~: E ( , ~,~:>z~, v o ( E (, ~:>z,) - E (, ~:>~z~ tEI

Given x E E

tel

tel

tel

and y E F ,

tel

tEI

tcl

",

-

Yt)

t)x

eEI

(Theorem 1.3.4.2 a) ), and

v( ~ < , ~:>z~)~ =

~( Ez,) = ~vz,

tel

tel

: ( E ~ , ~:>vz,)~

tel

tel

which proves the assertion, C o r o l l a r y 1.3.4.9

i

( 0 ) Let E and F be normed spaces. Take

9 L(F,E'). Then the following are equivalent:

a)

The map u" F~ --~ ErE is continuous.

b)

There is a v E / : ( E , F )

such that u - v ' .

The operator u of b) is unique and is called pretranspose o f u.

a =v b. Take x E E . By a), the linear map F~ ---+ IK,

y',

~ (x, uy')

is continuous. By Corollary 1.2.6.5, there is a vx E F such that

(x, ~y') - (~x, y') for every y' E F ' . It is obvious that v" E --+ F is linear. If x E E # , then

I(~, y'>l = I(x, ~y')l ___ Iluy'll _< I~11 Ily'll for every y' E F ' . By Corollary 1.3.3.8 b),

1.3 The Hahn-Banach Theorem

195

Hence v is continuous. By Theorem 1.3.4.2 a), v' = u. b ~ a. By Theorem 1.3.4.2 a),

(~, ~y') = (vx, y') for every (x, y') C E • F ' . a) now follows. The uniqueness of v follows from Theorem 1.3.4.2 c). Theorem

1.3.4.10

(Banach-Stone) Let S , T

be compact spaces. Given

u E f_,(C(S),C(T)), the following are equivalent:

a)

u is an isometry.

b)

There is a h o m e o m o r p h i s m f : T - + S and a y C C(T) such that ly(t)l = 1

for every t C T and

~ = y(x o / ) for every x C C ( S ) .

a => b. We identify C ( S ) ' , C(T)' with the Banach space of Radon measures on S and T , respectively. By Corollary 1.3.4.7, u' is an isometry. Given s c S (resp. t e T ) , let as (resp. at) denote the Dirac measure on S (resp. on T ) at s (resp. t). Take t c T . Then at is an extreme point of C(T) '# (Example 1.2.7.14) and so u'at is an extreme point of C(S) '# . By Example 1.2.7.14, there are f (t) e S and y(t) E IK such that

ly(t)l = 1,

u'at = y(t)a/(t) .

f is injective, for otherwise u' would not be injective, f is also surjective, since u' is surjective. Take x C C(S). Then y ( t ) x ( f ( t ) ) = (x, y(t)5/(t)) = (x, u'ht) = (ux, at) = u x ( t ) .

We see from this relation that y and f are continuous. Since f is bijective, it is a homeomorphism. Moreover, ~

b =~ a is easy to see.

= v(~ o f).

I

196

1. Banach Spaces

Example

Take p 9 [1, c~[, and let q be the conjugate exponent of

1.3.4.11

p. Let u be the right (left) shift in ~ . Then u' is the left (right) shift in gq, where tq is identified with (tP)' (Example 1.2.2.3 d) ). Let v be the left (resp. right) shift in t~q . Then (x)

(x)

y> =

= n=2

=

oo

(resp. (ux, y> - E

o(3

xn+,Yn = E

n=l

for every (x, y)

9 e~

vy>

n=l

xnYn-1 -- (X, vy))

n=2

• t~q and the assertion follows from Theorem 1.3.4.2 a). m

Let S, T be sets. Take p, q 9 [1, oo] t3 {0} and let p' and q' be the conjugate exponents of p and q, respectively. Take

E x a m p l e 1.3.4.12

k 6 gP'q'(S,T) and put N

n

k" gq(T)

~ gP(S),

x,

~ kx

(Proposition 1.2.3.~ b)). If either S or q is finite, then ~ ) x = kx for every x' 6 gP'(S), where s (Example 1.2.2.3 b) ).

has been identified with a subset of gP(S)'

Given x c eq(T),

tET

t6T

sES

tET

s6S

(Theorem 1.3.4.2 a) ).

m A

Remark.

If S and T are finite, then k is the matrix associated to k and the N

transpose of k is the matrix associated to (k)'.

1.3 The Hahn-Banach Theorem

197

E x a m p l e 1.3.4.13 Take n C IN, and let u 9IK n --+ IK 2 be a linear map, with associated matrix [aij]ie~2,je~,. If we endow IK ~ and IK 2 with the Euclidean norms, then

]alj +

]ltt]2--~ j=l

]a2j

+

j=l

n

n

,~2

Elalj]2+Ela2j]2~ j=l j=l / -4 j=l

lalJ]2} ( E

la2J12} - I E

/

/

\j=l

alj-~2jl2 j=l

= 1(~-~2 ( IOLljl2--]-]0/2j]2) -'~j--1

)2 j=l

) i,j=l

By Example 1.3.4.12, the matrix associated to u" IK'2 -+ I[4n is the transpose of the matrix [O~ij]ielNe,jelNn, and by Theorem 1.3.4.2 b), I~'11 = I1~11 m

The assertion follows now from Example 1.2.2.7 d).

E x a m p l e 1.3.4.14 ( 1 ) Let S , T be sets, p,p' weakly conjugate exponents, and q, q' be conjugate exponents. Take k C t~P"q(s, T) and put U

[J

(Proposition 1.2.3.8 b) ). If T or p is finite, then CI !

(;)'~' = kx f o r e v e r y x t C ~q' ( T ) , where ~q' ( T )

(Example 1.2.2.3 b) ).

has been identified with a s u b s e t of gq(T)'

198

1. Banach Spaces

Given x E tP(S),

sES

= Ex(s)( sES

tET

E

sES

k(s, t)x'(t)) = (x, kx'}

tET

(Theorem 1.3.4.2 a) ).

I

Example 1.3.4.15

Let S , T be locally compact spaces, f 9S ~ T a proper continuous map and put u ' g o ( T ) ---+go(S),

x,

~xo f .

Then for each p E .Mb(S), u'# is the image f ( # ) of p (Example 1.2.2.10).

For the proof see, for example, N. Bourbaki, Integration (1956), Ch. V, w I

1.3 The Hahn-Banach Theorem

199

1.3.5 P o l a r Sets D e f i n i t i o n 1.3.5.1

( 0 ) Let E be a normed space, F a vector subspace

of E , and G a vector subspace of E ' . Put

(polar of F),

F ~ := {x' e E' I x'lF = 0} ~

(prepolar of G).

:= ['~ Kerx' x'cG

F ~ is a closed vector subspace of E~ and ~ of E . We have

{0}~

',

E~

is a closed vector subspace

~ ~

(the last equality follows e.g. from Corollary 1.3.3.8 a) ). P r o p o s i t i o n 1.3.5.2

( 0 )

Let E be a normed space, F a closed vector

subspace of E , and q : E -+ E / F

the quotient map. Then

Im

q' = F ~

and the map (E/F)'

>F ~

y',

> q'y'

is an isometry. In particular, F ~ is a dual space.

The assertion follows from Proposition 1.2.4.7. Remark.

II

( E / F ) ' and F ~ are frequently identified using the above isometry.

P r o p o s i t i o n 1.3.5.3

( 0 ) Let E be a normed space, F a vector subspace of

E ' , and (x~)~ei a finite family in E such that no nontrivial linear combination of (x~)~i belongs to ~

Then there is a family (x'~)~i in F , with

x:(~) = ~ for every ~, A C I .

200

1. Banach Spaces

We prove the assertion by induction on the cardinality of I . Take A E I and put J "- I\{A}. By the inductive hypothesis, there is a family (Y~)~cz with

for every t,# C J . Given t C I , put

~-F

~IK,

x',

>x'(x~).

Assume that N Ker ~ C Ker ~ . tCJ

(If J - 0, replace the intersection by F . ) Then, by Lemma 1.2.6.3, there is a family (c~)~ej in IK with XA -- E

~

"

tEJ

Thus

tea

tCJ

for every x' C F and so E

OF.

LCJ

This contradicts the hypothesis of the proposition. Hence, we can find an x'E (NKer~)\Ker~. Now put 1

x~! .. _-_ ~x,(x~) x ! and !

.

f

I

!

for t E J . The family (x'L)~eI has the required properties.

I

1.3 The Hahn-Banach Theorem

P r o p o s i t i o n 1.3.5.4

( 0 ) Let E be a normed space and F ~ is the closure of F in ErE .

201

a vector

subspace of E ' . Then (~

Let F be the closure of F in E E . It follows from F C (~176 that F C (~176 Take x' C E ' \ F . We prove that x' r (~176 Assume the contrary. Let A be a finite subset of E with

CI {y' < E l

-x'(x)I < I}

xEA

(Proposition 1.2.6.2). Let G be the (finite-dimensional) vector subspace of E generated by A. Let (xL)~e~ be an algebraic basis of G such that {x~ [ C I , x~ E ~ is an algebraic base of G A ~ Put S := {~ C I Ix~ q~ ~ By Proposition 1.3.5.3, there is a family (x[)Lej in F with 9: ( ~ ) = 5 ~

for ~, A c J . Put

cEJ

Then y'(~) = x'(~)

for every ~ C J and y'(z~) = 0 = ~ ' ( ~ )

for every ~ E I \ J .

Hence y ' - x ' contradiction we sought. C o r o l l a r y 1.3.5.5

( 0 )

Let E be a normed space, F a subspace of E'

closed in ErE, and q " E ~ E / ~ (E/~

= 0 on G, so that y' r F , which is the I

the quotient map. Then the map '--+ F,

x'~

~ q'x'

is an isometry.

The assertion follows immediately from Proposition 1.3.5.2 and Proposition 1.3.5.4. I

202

1. Banach Spaces

1.3.5.6

Corollary

( 0 )

Let E be a normed space. Then a subspace of E'

is a dual space whenever it is closed in E~E .

The assertion follows immediately from Corollary 1.3.5.5. Proposition

1.3.5.7

I

( 0 ) Let F be a vector subspace of the normed space

E . Then

~176 = F. The inclusion FC~

~

is trivial, so F C ~176 Assume that F :/: ~176 Then there is a y' E E' with

y'l F - O ,

y' I ~176 ~ 0

(Corollary 1.3.3.5). But then y' E (F) ~ = which is a contradiction. Hence r =

Proposition 1.3.5.8 s

( 0 )

F ~

= (~176176 (Proposition 1.3.5.4),

o(po),

Let E , F

m be normed spaces and take u E

F) . Then

Keru'=(Imu) ~

Keru=~

Imu=~

Take (x, y') E E x F ' . Then (~x, y') - (~, ~'y')

(Theorem 1.3.4.2 a) ). It follows immediately from this, that y'EKeru'~y'E

(Imu) ~

1.3 The Hahn-Banach Theorem

203

so t h a t Ker u' = (Im u) ~ and Im u = o((im u) ~ = ~

u')

(Proposition 1.3.5.7). By the above equality and Corollary 1.3.3.8 a), it further follows t h a t x E Ker u r

x E ~(Im u').

Hence Ker u = o (Im u'). Corollary

1.3.5.9

( 0 ) Let E , F

I

be normed spaces and take u E f _ . ( E , F ) .

Then u' is injective iff Im u is dense.

u' is injective

r

K e r u ' = {0} r

~

= F ~

Im u = F

(Corollary 1.3.3.8 a), Proposition 1.3.5.8). Remark.

I

The injectivity of u does not imply t h a t u' i~ surjective, as the

inclusion m a p t~1 --+ co shows. Corollary

1.3.5.10

( 0 )

Let E, F be normed spaces. Take u E f_.(E, F)

such that I m u is closed. Let q: F ~ F / I m u be the quotient map. Then

Im q~ = Ker u ~ and the map

(F/Imu)'

> Keru',

x'. ~ ~ q'x'

is an isometry.

We have Im q' = (Im u) ~ = Ker u' (Proposition 1.3.5.2, Proposition 1.3.5.8), and the assertion follows from Proposition 1.3.5.2.

I

204

1. Banach Spaces

C o r o l l a r y 1.3.5.11

Let E be a normed space and p a projection in E . Then

p~ is a projection in E ~ with

Kerp' = (Imp)~

Imp' = (Kerp)~

By Corollary 1.3.4.6, p' is a projection, and by Proposition 1.3.5.8, Kerp' = (Imp) ~

Imp' C (~

~

(Kerp) ~

Take x' E (Kerp) ~ . Then C Kerp

x-px

so that

(x, p'x') = (px, x') = (x, ~') for every x C E (Theorem 1.3.4.2 a) ) and (Kerp)~

x'=p'x'EImp',

Theorel~ 1.3.5.12

I m p ' = (Kerp)~

( 0 ) L~t r b~ a ~ p a ~

m

4 th~ ~o~m~d ~pac~ E a~d

u : F --+ E the inclusion map.

a)

Imu'=F'.

b)

Keru' = F ~

c)

The factorization E ' / F ~ ~ F' of u' through E ' / F ~ is an isometry.

d)

I f G is a closed vector subspace of ErE and v : ~ map, then the f a c t o r i z a t i o n E ' / G isometry. In particular, E ' / G

--+ (~

-+ E is the inclusion

of v' through E ' / G

is a dual space.

a) follows from the Hahn-Banach Theorem. b) follows from Proposition 1.3.5.8. c) Let v be the factorization of u' through E ' / F ~ . Then

Ilvll = If~'ll = II~lf _< 1 (b), Proposition 1.2.4.7, Theorem 1.3.4.2 b)). Take X ' E E ' / F ~ . Then

IlvXll ~ Ilvll IIX'll ~ IIX'll (Proposition 1.2.1.4 a)). Take x' E X ' . Since

is an

1.3 The Hahn-Banach Theorem

205

x ' l F - x' o u = u'x' = v X ' ,

it follows that

<x,x'> = <x,vX'> ~ IlvX'll Ilxll for every x E F (Proposition 1.2.1.4 a) ). Hence

IIx' FII ~ IlvX'll. By the H a h n - B a n a c h theorem, there is a y~ C E ~ with

y'IF- x' F,

IlY'II- Ilx'lFII.

Then y ~ - x ~ C F ~ i.e. y~ C X ~. Hence IIX'll ~-IlY'II = IIx'IFII-~ IIvX'll,

IlvX'l = IIX'II 9

By a), v is surjective. Hence v is an isometry. I

d) follows from c) and Proposition 1.3.5.4. Proposition

1.3.5.13

( 0 )

Let T

be a compact space and let C(T)'

identified with the Banach space of Radon measures on T . Let ~ subspace of C ( T )

be

be a vector

such that x-2, x y c jc f o r every x , y E .T and let It be an

extreme point of jco A C(T) '#

Then the f u n c t i o n s of ~

are constant on the

support of It.

We may assume that # ~ 0. Let x be a positive real function in ~ . Assume x is not constant on Supp #. Put

1

y:=-x,

ct

u := y . # .

Then

~#o,

~-~#0,

and I L'II + I # - u l ' - / y d l # +

f(l-y)dlp= /d,#,-,,#ll-I

(Proposition 1.2.7.11). Then zy, z(1 - y) e 9r , so t h a t

206

1. Banach Spaces

(z, u) -- f

zy d# = O ,

(z,p-u>=fz(1-y)dp=O for every z E .T'. Hence u,#1

1

I1~11 v, I1~- vii

u E .T~

( I t - u) e .T ~ fq C(T) '#

Since 1

1

and since # is an extreme point of .To N C(T) '# , 1 I~11 ~ = ~ '

y -I1~11 on S u p p # ,

9 = ~11~11 on S u p p l .

Hence x is constant on S u p p # . Now let x be an arbitrary function in .T and take s, t E Supp # with

x(~) # ~(t). Since x~ is a positive real function in .T, it follows from the above that x~ takes the same values at s and t. P u t / z~

if x ( s ) z ( t ) = 0

Y

I x ~ - ~(~)~1 '

if ~(~)x(t) # 0.

Then y is a positive real function in .T taking different values at s and t, which contradicts the above result. Hence x is constant on S u p p # . Theorem

1.3.5.14

I

( 0 ) Let T be a compact space and .T a vector subspace

of C(T) such that x-2, xy E .T for every x, y E .T. Put

S "-- N xl(O) xE.~" and /

t" ~ for s, t G T \ S .

\

( x E .T ~

xIs ) -- x I t ) )

Let ~ denote the set of x E C(T) which vanish on S and for

which z(s) = z(t) whenever s, t E T \ S

satisfy s ~ t. Then .T = G.

1.3 The Hahn-Banach Theorem

207

Take x E g . Identify C(T)' with the Banach space of Radon measures on T . Let # be an extreme point of ~-~

C(T) ~#. By Proposition 1.3.5.13, the

functions in ~" are constant on S u p p # . Hence there is a y E $" with x = y on S u p p # and so (~, , ) = ( y , , )

= 0

9v~ is a closed vector subspace of C(T)~c(T). Hence by the Alaoglu-Bourbaki Theorem, yo N C(T) '# is a compact set of C(T)~c(T). By the K r e i n - M i l m a n Theorem~

c.. tT~,# ~- j "

'

o n t ~

the extreme points of ~'~ N C(T) '# . By the above, (x,#) = 0 for every # C 9v~ A C(T) '# . Hence

~ e~176 =7 (Proposition 1.3.5.7) and

gc.F.

I

The reverse inclusion is easy to see.

Remark.

The idea of using extreme points for such denseness problems is due

to de Branges (1959). C o r o l l a r y 1.3.5.15

Let T be a locally compact space and 3z be a

( 0 )

vector subspace of Co(T) such that: 1)

If x, y c ~ , then xS, xy c J~ ;

2)

Given distinct s, t e T there are x, y e .~ such that x(s)y(t) ~ x ( t ) y ( s ) .

Then :7z is a dense set of Co(T). Let T* be the Alexandroff compactification of T and extend each function in Co(T) by setting it equal to 0 at the Alexandroff point of T . By 2), S of Theorem 1.3.5.14 contains only the Alexandroff point of T , and the equivalence classes of ~ are one point sets. Corollary

1.3.5.16 ( 0 )

be a compact space and ~

(Weierstrass-Stone Theorem,

I

1885, 1937).

a vector subspace of C(T) such that:

Let T

208

1. Banach Spaces

1) x , y E iF => x h , xy c iF. 2)

Given distinct s, t e T , there are x, y e iF with x(s)y(t) =/=x ( t ) y ( s ) .

Then iF is a dense set of C ( T ) .

C o r o l l a r y 1.3.5.17

I

( 0 ) Let T be a set and iF a closed vector subspace of

g ~ ( T ) such that x-2, xy E iF for every x, y E jc. Take x C iF and f E C(x(T)) such that o e x ( T ) , ~ r J= ~

S(O) = o.

Then f o x C iF.

Let G be the smallest vector subspace of C ( x(T)) such that: 1) the function x(T)

> IK,

a,

>

is in G, 2) e~--(-~C G whenever eT C iF, 3) g-~, gh G ~ for every g, h C G. Then f ox C iF for every f C G .By Theorem 1.3.5.14, f belongs to the closure of ~ in C ( x ( T ) ) . Hence there is a sequence (f~)ne~ in ~ converging uniformly to f . Then (fn o X ) ~

converges uniformly to f o x. Since iF is closed and

fn o X C iF for every n C IN, it follows that f o x E iF.

C o r o l l a r y 1.3.5.18

1

Let T be a compact space and iF a vector subspace of

C(T) such that xy C iF for every (x, y) C C(T) x iF. Put s ::

Nx

(0) 9

xC.~

Then

7-

{~ e c(T) I ~lS - 0 }

Given distinct s, t c T \ S , there is an x c iF with

9(~) ~ ~(t). The assertion thus follows from Theorem 1.3.5.14.

I

1.3 The Hahn-Banach Theorem

Proposition

1.3.5.19

Let T

be a locally compact space and ~

209

a vector

subspace of C(T) such that: 1)

5 x , xy C .~ for every x , y E .T.

2)

For distinct s, t E T , there are x, y C .~ with

9(~)y(t) - ~(t)y(~) # 0 3)

e T C . ) E" .

Let 9 be the coarsest topology on C(T) for which the functions C(T)

> IK,

~, f x d #

x~

are continuous for every bounded Radon measure # on T . Then jz is dense in C (T) with respect to ~ . Let /~T be the Stone-(~ech compactification of T . We consider the functions in .~ to be extended continuously on f i t and put ~

/~T

~ IK 7 ,

t,

>, (x(t))xej:

Then ~ is continuous and, by 2), it is injective on T . Let x c C(T) have compact support. Let (#~)~e~ be a finite family in .s

and take c > 0. T has a compact set K such that Supp x c K

and I#~I(T\K)
~(t)

is a homeomorphism. Hence x l K - y o r for some y C C ( ~ ( K ) ) . By Tietze's Theorem, y can be extended to a continuous function on ~(/~T) such that

Ilyll- I1~11 Then

210

1. Banach Spaces

ly o ~ldl~l

=

q'x"

is an isometry. Hence Im j " = F ~176 , and the m a p F"

> F ~176 y" l

> j"y"

is an isometry, Proposition

i

Let E be a normed space, F a closed subspace of E ,

1.3.6.18

and q : E ~ E / F

the quotient map. Then Ker q" = F ~176

and the factorization of q" through E " / F ~176is an isometry. By P r o p o s i t i o n 1.3.5.2, I m q' = F ~

and the m a p

u : (E/F)'

>F ~

y',

> q'y'

is an isometry. Let j : F ~ ~ E ' be the inclusion map. Then

q'=jou so t h a t ql, ~ U' O

jl

(Corollary 1.3.4.5). Now Ker j ' = F ~176 and the factorization v of j' t h r o u g h E " / F ~176 is an isometry (Theorem 1.3.5.12 b), c)). Since u' is an isometry (Corollary 1.3.4.7), Ker q" = Ker j' = F ~176 . The factorization of q" t h r o u g h E " / F ~176is u' o v and therefore an isometry.

m

1.3 The Hahn-Banach Theorem

Proposition

1.3.6.19

(0)

217

Let E be a n o r m e d space and let .!

u := jE' o 3E" a)

fE o jE, = 1E, .

b)

u is the projection of E'"

onto I m j E ,

and

Ilull

~ 1; u is called the

c a n o n i c a l p r o j e c t i o n o f E'" (better: o f the t r i d u a l o f E ).

C) K e r u = (ImjE) ~ d)

E'" = (Im jE') @ (Im jE) ~

e)

If E is complete and q : E"

>E " / I m j E ,

r :Era

~ E'/ImjE,

are the quotient maps (Corollary 1.3.6.5), then r o q' is an isometry.

a) Given (x,x') c E x E ' , (x, j ~

o

jE,(X')) = (jEx, jE, X'} = {jEX, X') -- {X, X')

(Theorem 1.3.4.2 a) ), and so

j!E O j E ' -- 1E' b) By a),

.! u o jE' = jE' o 2E o jE' = j E ' ,

U O U _ _ U O j E , o j'E - - j E , O J"E - - U .

Hence u is a projection in E'" and it follows from Im jE' = Im (u o jE') C Im u = Im (jE' o J~E) C Im jE' ,

Im u = Im jE' that u is a projection of E"' onto I m j E , . By Theorem 1.3.6.3, IIJEI]-< 1,

so that

IIJE'II _< 1,

218

1. Banach Spaces

llull = IIJE' o J~ll ~ llJE'II llJ~ll ~

1

(Corollary 1.2.1.5, Theorem 1.3.4.2 b)). c) By a), jl

"I

"I

"I

E 0 U -- #E 0 jE' 0 #E "-- 3E"

Hence Ker j~ ~ Keru = Ker (jE' o J~E) ~ Ker j ~ ,

Ker u -- Ker j~ - (ImjE) ~

(Proposition 1.3.5.8). d) follows from b), c) and Murray's Theorem. e) The maps (E"/ImjE)'

(ImjE) ~

> (ImjE) ~

~ E ' " / I m jE, ,

x' ~ ~, q'x' ,

x'" ,

~ rx'"

are isometries (d), Proposition 1.3.5.2, Proposition 1.2.5.2 a =a c), and the assertion now follows, m

Corollary 1.3.6.20 ( 3 )

Let ~T be the Stone-Cech compactification of

the discrete space T . Put A :=

and take each x C e ~ ( T ) co(T)", and co(T)'"

ZT\T,

to be extended continuously to ~ T . Identify co(T)',

with g l ( T ) , / ~ ( T ) ,

and the Banach space M(flT)

Radon measures on ~ T , respectively (Example 1.3.6.2). Let i and j

of

denote

the evaluation of co(T) and /~I(T), respectively, and put u:=joi'. Then i and j

are the inclusion maps, u is the projection of M ( f l T )

gl(T) , Kern is the Banach space M ( A )

onto

of Radon measures on A ,

M(flT) = el(T)@ M(A), and e ~ ( T ) / c o ( T )

and ( e ~ ( T ) / c o ( T ) ) ' are canonically isometric to C(A) and

.hl ( A ) , respectively,

m

1.3 The Hahn-Banach Theorem

P r o p o s i t i o n 1.3.6.21

( 0 )

Let E

vector subspace of E ' . Put G := (~ map. Put H "= E l ~

be a Banach space and F

219

a closed

~ and let j " F --~ G be the inclusion

and let q" E ~ H be the quotient map. Finally let u

be the isometry H'

, ~ G,

~' ,

> q'~'

(Corollary 1.3.5.5). \ ~" ~ "11

b)

\ \ J .l-] " l " ~ ]

.

.

.

.

.

.

(jg~) o u -1 is continuous on GE whenever ~ E H .

c) If F # is dense in G#E, then II(JH~) o u - l l Fll = II~]l

for every ~ E H .

d)

If F # is dense in G#E and if every continuous linear f o r m on F

is

continuous on FE, then the map v" H

> F' ,

~ ~ } (jH~) o u -1 I F

is an isometry and uov' ojF = j.

e)

Under the hypothesies of d), if we identified F" with G via the isometry u o v' then j becomes the evaluation of F .

a) We have ((jHqx) 0 u - l , rl} = (jHqx, u-lrl) = (qx, u-lrl) = (x, q'u-lrl) = (x, r]) .

b) follows from a). c) Since F # is dense in GE~ , it follows from b) that I[(JH~) o u-1 I Fll = Ii(JH~)o u - l I =

IIJH(II-

I[~ll

(Theorem 1.3.6.3 a) ). d) Let ~' E F ' . By the hypothesis of d) and Corollary 1.2.6.5, there is an x E E such that

220

1. Banach Spaces

~'(~) = (x,~)

for every ~ E F . By a), (jHqx) o U-1 I F = ~'.

Hence v is surjective. By c), it is an isometry. Take ( E F and x E E . Bya), (x, uv'jF~) = (x, q'v'jF~} = (vqx, jF~) = (vqx, ~) = ( ( j s q x ) o U-1, ~) -- (X, ~) .

We deduce that uV'jF~ = ~ = j ( ,

u o v'ojF -- j .

e) follows from d) (and Corollary 1.3.4.7). P r o p o s i t i o n 1.3.6.22

I

( 1 ) Let T be a set and ~ S the Stone-Cech com-

pactification of the infinite discrete space S . Put

Take k E

~..o,1(S, T ) , and let k be the continuous extention of the map S

)~I(T),

Zs

s,

~ k(s,.)

~ e '(T)i',(~),,

where gl(T) is identified with a subspace of gl(T)" via the evaluation map.

~) k e e~

~?f k l ~ = O.

b) k E go'l(S, T) iff k ( A ) C co(T) ~ , where co(T) is identified with a subspace of gl(T)' via the evaluation map (and Example 1.2.2.3 e) ). The continuous extention k exists by the Alaoglu-Bourbaki Theorem. a) Let ~ be the filter on S consisting of all cofinite subsets of S, i.e. 9-- { A E ~ ( S ) ] S \ A

k lA -- 0 is equivalent to

finite}.

1.3 The Hahn-Banach Theorem

221

x' e el(T)' = : , lim(k(s,-) x ' ) = O. By Corollary 1.2.3.13, this is equivalent to lim Ilk(s, )Ill = 0 i.e. to k c t~~

T).

b) Take k e go'I(S,T) and s e A. Then (e T k ( s ) } =

lim (e T , k ( r , - ) ) -

lim k ( r , t ) - O

S ~ r --. s

S ~ r-+s

for every t C T , where t~I(T)' is canonically identified with t ~ ( T ) (Example 1.2.2.3 d)), so that k(s) e (IK(T)) ~ . Since ]K (T) is dense in co(T) (Proposition 1.1.2.6 c)), k(s) e co(T) ~ . Hence k(A) C co(T) ~ Now suppose that k(A) C co(T) ~ . Then lim k(r, t) =

lim (e T, k(r,-)}

S ~r--~ s

--

( e Tt ,

k(8))

--

0

S~r---~ s

for every s E A and t E T . Hence k(.,t) E co(S) for every t c T . Thus k e eo'~(S,T).

I

E x a m p l e 1.3.6.23 ( 1 ) Let S , T be sets and I~S the Stone-Cech compactifieation of S with respect to the discrete topology on S . Put A := ~ s \ s and for k C g~'~(S, T) let "k denote the continuous extension of the map

s

>e~(T),

s,

>k(s,.)

to

A-

~ ~o(T)~

with the identifications in Proposition 1.3.6.22. Put

M .- {klk ~ eT'I(S,T)},

U ' eo'~ ( S, T )

". M ,

k .

~~ ,

and endow .M with the norm

M.

> IR+,

k,

> supllk(s)ll. sCA

222

1. Banach Spaces

Then Ker u -- 60'1(S, T)

and the factorization

~.~"(S,T)/e~ of u through g o ' I ( S , T ) / g ~

>M

is an isometry.

By Proposition 1.3.6.22 a), Ker u = t'~ (S, T ) . By L e m m a 1.2.4.6, the factorization

v" e ~ ' I ( S , T ) / ~ ~ of u through g o ' I ( S , T ) / g ~

~ .A4

is bijective. Take K E g o ' I ( S , T ) / g ~

Then

il~Kli = ikl] = ~up ]lk(~)ii _< sup ilk(~,-)ii~ = i kii sEA sES for every k C K , so t h a t

II~gli < Ilgi]. Take a E IR with a < []vKl], and k E K . There is an s C A , such that IIk(~)ll > - .

Furthermore, there is an x' e (t'l(T)') # with

J<x', k(~))l

> ~.

Then lim I(x', k(r, .))] = I(x', k(s))] > a .

Sgr--~s

Hence there is an r E S such t h a t

< I<x',k(r, ")>I < Ilk(r, ")ll, < IlkllThus

< IIKII and JivKiJ ~ IIKII, since k and a are arbitrary. Hence ilvKiI = [IKII.

I

1.3 The Hahn-Banach Theorem

P r o p o s i t i o n 1.3.6.24

223

( 4 ) Let T be a compact space and # a positive

R a d o n measure on T

with support equal to T . D e n o t e by u and v the eva-

luation map of C(T)

and L ~ ( # ) ,

C(T)--+ L ~ 1 7 6

respectively, and by w the inclusion map

Then

(Im v) M (Im w") = Im(w" o u) = Im(v o w). Take a e ( I m v ) ~ (Imw"). There are x e L~176 and x" e C ( T ) " such that a - vx

wttx tt .

--

By the Vitali-Lusin Theorem, there is a disjoint sequence (Kn)ne~ of compact sets of T such that for every n C IN, x I K n is continuous and K,~ is the support of e g n ' # and such that

#(T\ U Kn)= o. n E IN

Let to E T and suppose x} U Kn has two distinct limits

OL1

and a2 in

nEIN

to. Put [0~ 1 - - 0/2[ C "-----

Take k C {1,2}. Put

s

"- {U (3 Ak [ U open neighbourhood of to},

denote by ~k an ultrafilter on Hk finer than the lower section filter of Hk (ordered by inclusion), and define x'k " L ~ ( # )

'

) IK

) lA,ak im ~ 1

y,

Then x'k E L ~ ( # ) ' and

(y, ~ ' x l ) - (~y, 4 ) = y(t0) for every y E C ( T ) . Hence I

W

I

X 1 ~

!

W

!

X 2

9

J(A yd#.

223

1. Banach Spaces

Since (x, x') = (vx, x'} - (w"x", x') = (x", w'x') for every x' E L ~ ( # ) ' , we get

(X, X'I) -- (X, X;). But

I(x,x'k)

~kl

-

= lim 1 A,~k ~

fA(X _ ak)d# < c

for every k c {1, 2} and we get the contradictory relation 3e = lal

-

a~l _ I~, - ( ~ , ~ i ) l + I ( x , G ) - ~ 1 _ 2c.

Hence x[ U Kn has a unique limit at to. nc1N Since to is arbitrary, we may extend x[ U Kn continuously on T . Thus nEIN xEImw, aCIm(vow),and (Im v) n (Im w") C Im (v o w). By Proposition 1.3.6.16, w" o u = v o w, so that Im(v o w) = Im(w" o u), and this implies Im(v o w) C (Im v) n (Im w"). Proposition

1.3.6.25

( 7 )

m

Let E , F be n o r m e d spaces and u C s

Then

(Im jE') N (Im u'") = Im(jE, o u') = Im (u'" o jF')Take x'" E I m ( j E , ) n (Im u'"). There are x' c E' and y'" C F " such that x m =jE,

X ~ =umy

m.

By Proposition 1.3.6.19 a) and Proposition 1.3.6.16 (and Corollary 1.3.4.5), x'

- - J "'" EJE, X

'

~- J "' E u

" 'y" '

--

(u"

0

jE)'Y"'

7_ ( j R

0

u)'y"

- - U ' 3"F Y

"'

,

xm -- jE 'xl = jE' u I'l 2FY ,t e jE' o u ' ( F ' ) = u ' " o j r , ( F ' ) .

Hence (Im jE') n (Im u") c Im(jE, o u') -- Im(u"' o j r , ) . The reverse inclusion is trivial.

II

1.3 The Hahn-Banach Theorem

Proposition

225

Let E be a complex Banach space and F' a vector

1.3.6.26

subspace of E' such that EF, is Hausdorff. We denote by E the underlying N

real Banach space of E and by G' the set of continuous linear forms on EF, and put y' " E

~,

x :

~ y'(x) - iy'(ix)

for every y' E G'. a)

G' is a vector subspace of E ' , y' c F' for every y' E G', the map G'

> F' ,

y' l

> y'

is an isometry of real normed spaces (with respect to the induced norms), and F'

> G'

x' t

>rex'

is its inverse. b)

If F ' = E' then G ' = E ' .

c)

/ f t h e map

E - - - + F",

z,

~ (z,.)lF'

is an isometry of complex Banach spaces then the map N

E

~a",

z,

>(z,.)la'

is an isometry of real Banach spaces. N

a) It is obvious t h a t G' is a vector subspace of E ' . By L e m m a 1.3.1.5, y' is a linear form on E and by L e m m a 1.2.6.4, y' C F ' . Moreover, N

r e y ' ( x ) = re (y'(x) - iy'(ix)) = y ' ( x ) , for every x C E , so t h a t re y' = y'. For x ' E F '

and x C E , N

r e x ' ( x ) -- (x, rex') - i(ix, rex') --

226

1. Banach Spaces

= re (x, x'} - i re i(x, x'> = re(x, x') + i ira<x, x') - (x, x'), so that re x' = x'. Hence the two given maps are bijective and everyone of them is the inverse of the other map. Since they are obviously IR-linear, it folllows from Corollary 1.2.1.6, that they are isometries of real normed spaces. b) follows from a) and Proposition 1.3.6.12. c) Let y" E G". Put x"'F'

>~,

x',

~y"(rex')-iy"(irex').

By a), Lemma 1.3.1.5, and Corollary 1.2.1.6, x" e F" and ]lx"ll = IlY"]I-By the hypothesis of c), there is an x E E such that

(x,.)lF' = x",

I xll -I1~"11- Ily"ll.

Take y' E G'. By a), y' C F ' and (x,y') = (x",y') = y " ( r e y ' ) - i y " ( i r e y ' ) .

Since (x, y') = (x, y ; - iy'(ix)

it follows

(z, y') - r Since y' is arbitrary

(x, .)IV' = r Hence the map

E---+

G",

z,

) (z,-)lG'

is an isometry of real Banach spaces. P r o p o s i t i o n 1.3.6.27 E' and

( 0 )

u-E

B

Let E be a Banach space, F a subspace of

)F',

z,

~(z,.)lF.

If EeF is compact, then u is an isometry of Banach spaces.

1.3 The Hahn-Banach Theorem

227

The map EF

} FtF,

xl

)'l,tx

being continuous, u ( E #) is a compact and therefore a closed set of F ~ . For every y E F , Ilyll = sup I ( y , x ) l = xEE#

~up I ( y , ~ x ) l

xEE#

By Corollary 1.3.1.9, u ( E #) = F '# . Since EF# is compact, EF is Hausdorffand so u is injective. Since EF# is compact, EF is Hausdorff and so u is injective. It follows that u is an isometry of Banach spaces.

I

228

1. Banach Spaces

1.3.7 T h e K r e i n - S m u l i a n T h e o r e m /

P r o p o s i t i o n 1.3.7.1

(

0

x

)

Let E

be a normed space and K

a weakly

compact convex set of E . Then

{ ~ I(,~, ~)e z # • K} is a weakly compact, absolute convex set of E containing K .

Put u : IK • EE,

~, EE, ,

(a,x),

~ax.

Then u is continuous, so that {ax I ( a , x ) 9 IK# x K} = u(IK # x K) is weakly compact. This set is obviously absolutely convex and contains K . I T h e o r e m 1.3.7.2

( 0 ) (Mackey) Let E be a normed space, ~ the set of

convex weakly compact sets of E , and A' a convex set of E' which is closed with respect to the topology on E I of uniform convergence on J~. Then A' is a closed set of ErE.

Take x' 9 E ' \ A ' . There is a K 9 ~l, such that {y' 9 E' [ x 9 K ==~ I ( x , x ' - Y'}I

rex"(y') > 5 c ~ -

o~

which is a contradiction. Hence ~1 x l t E jE(L) and there is an x E E with X" :

jE(X).

Then

sup re y' (x) -- sup re x" (y') < re x" (x') -- re x' (x). y~ E A ~

y~ E A ~

Therefore x' is not in the closure in E~ of A' and so A' is a closed set of E ~ . I Theorem

1.3.7.3

( 0 )

(Krein-Smulian, 1940) Let E be a Banach space

and A' a convex set of E ' . If A ' M n E '# is a closed set of ErE for every n E IN, then A' is a closed set of E~E. Let %' be the finest topology on E t inducing the topology of pointwise convergence on the equicontinuous sets of E ~ (Theorem 1.2.8.2). Since every equicontinuous set of E' is contained in a set of the form n E '# (n E IN), E ' \ A ' E ~s i.e. A' is closed with respect to %'. By Corollary 1.2.8.3, A' is closed with respect to the topology on E ~ of uniform convergence on the convex compact sets of E and the assertion follows from Theorem 1.3.7.2 since every compact set is weakly compact. I

Remark.

The theorem no longer holds if E is not complete.

230

1. Banach Spaces

D e f i n i t i o n 1.3.7.4

( 0 )

Let E

be a vector space. A cone of E

is a

nonempty subset A of E for which a A c A whenever a E JR+. The cone A is called s h a r p if

A M ( - A ) = {0}. 0 belongs to every cone. The cone A is convex iff A+ACA.

C o r o l l a r y 1.3.7.5

( 0 ) Let E be a Banach space and A' a convex cone

of E ' . Then A' is a closed set of ErE iff A' A E '# is a closed set of E~E .

First assume that A' N E ~# is a closed set of E ~ . Take n C IN and put

1,. ?2

Since u is continuous and A' A n E '# - u 1(A' A E ' # ) , A ' A n E '# is a closed set of E ~ . By the Krein-Smulian Theorem, A' is a closed

set of E ~ . The reverse implication follows from Proposition 1.2.6.6. Proposition

1.3.7.6

( 0 )

Let E

I

be a Banach space and (u~),e, a finite

family of projections in E' such that Im u, is a closed set of E~E and that UL o U x - - 0

for distinct ~, A E I . Then ~

Im u~

is a closed set of E~E .

tCI

First observe that ~ u~ is a projection in E' and that

Im E u ~

= EImu~.

LEI

~EI

Let x' be point of adherence of E '# A ~ Im u~ in E ~ . There is an ultrafilter LEI

;~ on E' converging to x' in E~ with E ~# N E I m u ~ LEI

E ~.

1.3 The Hahn-Banach Theorem

231

By the Alaoglu-Bourbaki Theorem, x' E E '# and ut(~) converges in E~ for every ~ E I . Moreover, lim u~(~) E I m u~, since Im ut is a closed set of E~. Then lim (~-'~ ut(~)) = ~ tEI

lim ut(~) E ~

tEI

tel

Imu~.

tEI

tel

x ' = lim ~ - l i m (:~-~ u ~ ) ( ~ ) = lim (~--~ ut(~')) E ~ tEI

tEI

Imu~.

tEI

Hence E '# N ~ Imut is a closed set of E~. By Corollaryl.3.7.5, ~ Im ut is tel

tel

a closed set of E~.

I

P r o p o s i t i o n 1.3.7.7

( 0 ) Let E be a normed space. Given a linear f o r m

x" on E ' , the following are equivalent:

a) x" E I m j E . b)

Kerx" is a closed set of E'E.

c) x" is continuous on E'E. a :=~ b and a ==~ c are trivial. b ==>a. We may assume that x" =/=0. Take x' E E'\Ker x". By Proposition 1.2.6.2, there is a finite subset A of E such that {y' E E' l x E A ~

](x,x'-

y')] < 1} C E ' \ K e r x " .

Then x" is bounded on {y' E E' i x E A ==~ I(x,y'>[ < 1}

as can be seen by factorizing x" through E'/Ker x". Hence, by Lemma 1.2.6.4, there is an x E E with x " ' - - (x,.)

and x" E ImjE. C ::=>a. Put

232

1. Banach Spaces

V' := {x' 9 E' I Ix"(x')l < 1}. By Proposition 1.2.6.2, there is a finite subset A of E such t h a t

Ix'(x)l
ux'

is continuous. c)

u'(ImjF) C ImjE. a =~ b. Given (x',y) c E' x F ,

c. Take y E F . Then (u'jFy, x') = (jpy, ux'} = (y, ux'> for every x' E E' (Theorem 1.3.4.2 a) ). Thus the map

Ek

~IK, z',

~(u3ry, z') !

9

is continuous. By Proposition 1.3.7.7 c =~ a, u'jFy belongs to I m j E . Hence u'(Im jR) C Im j E . c ~ a. Put v :F ~ (Corollary 1.3.6.5). Then v E s

E,

y,

) j~xlu'jry

E) and

(y, v'x'} = (vy, x'} = (u'jFy, x'} = (jpy, u x ' } -

(y, ux')

for every (x', y) C E' x F (Theorem 1.3.4.2 a)) and so u = v'.

I

1.3 The Hahn-Banach Theorem

/

Proposition

1.3.7.9

(

0

233

\

) Let E be a Banach space. Given a linear f o r m

x" on E ' , the following are equivalent: The restriction of x" to E '# is continuous at 0 with respect to the topology on E '# of uniform convergence on the weakly compact convex sets orE.

b)

E ' # N K e r x " is a closed subset of ErE .

c)

K e r x " is a closed set o.f E ~ .

d)

x" E I m j E . a ==>b. Let ~ be the topology on E ~ of uniform convergence on the weakly

compact convex sets of E and |

the topology induced on E '# by %7'. Take

x ~ E E '# and c > 0. There is a weakly compact convex set K of E such t h a t

I~"(y')l
ux'

is continuous.

b)

Themap

is continuous at O.

a :=> b is trivial. b =:v a. Take y C F . By b), the map EE#, ----+ IK ,

x' :

; (y, ux')

(resp. (y, ux'> )

is continuous at 0. By Proposition 1.3.7.9, a =:v d, there is an x C E such that (y, ux') = (x, x')

(resp. (y, ux') - (x, x') )

for every x' E E ' . a) now follows. [

C o r o l l a r y 1.3.7.12

~

0

) Let E be a Banach space. Given a projection u

in E ' , the following are equivalent:

a)

The map

is continuous.

b)

I

I m u and Keru are closed sets of E~E .

1.3 The Hahn-Banach Theorem

235

Put v := 1E --U. Then Im u = Ker v. a =~ b. Ker u is obviously a closed set of E ~ . The map

E'E

~, E'E,

x"~

~, vx'

is continuous, so that Ker v is a closed set of E ~ . Hence I m u is a closed set of EE,. b =~ a. Let ~ be an ultrafilter on EE#, converging to 0. Let x' denote the limit of u(~) in EE#, (Alaoglu-Bourbaki Theorem). Then v(~) converges to - x ' in EE#,. By b), x'CImu, and

-x'EImv=Keru

so

X~ -- ux ~ = O. Hence the map

E#E,

~ E'E,

x"

~, ux'

is continuous at 0. By Proposition 1.3.7.11 b :=> a, the map

is continuous. Lemma

1.3.7.13

I

Let T be a compact space. Take x E C(T) , and let (Xn)neIN

be a sequence in C(T) for which x is a point of adherence in the topology of pointwise convergence. If every subsequence of (x,~)ne~ has a point of adherence in C(T) with respect to the topology of pointwise convergence, then there is a subsequence of (xn)nc~

converging to x in the topology of pointwise conver-

gence. First assume that T is separable. Then, by the diagonal procedure, we may construct a subsequence (Yn)ne~ of (Xn)ne~ converging to x on a dense set

236

1. Banach Spaces

of T . Take t E T . Assume that (Yn(t))nC~ does not converge to x(t). Then there is an e > 0 together with a subsequence (z,~)ne~ of (Yn),~e~N, such that

Iz~(t) - x(t) l > for every n E IN. By assumption, (z,)nc~ has a point of adherence z in C(T) with respect to the topology of pointwise convergence. Then

Iz(t)-

~(t)l > ~,

and this is a contradiction, since z and x coincide on an dense set of T . Hence (Yn(t))nEIN convergens to x(t). Since t is arbitrary, (yn)ne~ converges to x in the topology of pointwise convergence. Now let T be arbitrary and let S:= H

xn(T),

nEIN

~'T

>S,

r

t,

~a(T),

> (x~(t))~e~,

t,

)~(t),

and 7rp'~a(T)

~IK,

(Sn)neIN'

;Sp

for every p E IN. Then p is continuous and ~a(T) is compact. Let y be a point of adherence of (Xn)nE~ in C(T) with respect to the topology of pointwise convergence, and take t', t" E T such that

r

- r

Then

xn(t')=xn(t") for every n E IN, so that

y(t') = y(t"). Hence there is a unique map y: S ~ IK with y=yor

1.3 The Hahn-Banach Theorem

Since p(T) is the quotient space of T with respect to ~ ,

237

y is continuous, x is

a point of adherence of (Tr~)ne~ and every subsequence of (Trn)ne~ has a point of adherence in C(~(T)) with respect to the topology of pointwise convergence. Since p(T) is separable, the first part of the proof implies the existence of a subsequence (Trk~)ne~ of (Trn)ne~ converging to x in the topology of pointwise convergence. It follows that the subsequence (xkn)~e~ of (x~)ne~ converges to x in the topology of pointwise convergence.

I

L e m m a 1.3.7.14 Let T be a compact space and let C(T)T denote the set C(T) endowed with the topology of pointwise convergence. Given 9c C C ( T ) ,

the following are equivalent: a)

Every sequence in Y has a point of adherence in C(T)T.

b)

Every sequence in T contains a sequence which converges in C(T)T.

c)

jc is a relatively compact set of C(T)T. b :=~ a and c :::> a are trivial. a =:~ b follows from L e m m a 1.3.7.13. a :=> c. Let ~ be an ultrafilter on C(T) containing IF. By a), {x(t)lx e ~ }

is bounded for every t E T , so t h a t the map x'T

>IK,

t,

> l i m y(t )

is well-defined. We show that x is continuous. Take t E T and c > 0. Assume t h a t every neighbourhood of t contains a point s such that

I x ( ~ ) - ~(t)l > ~ We construct inductively a sequence (tn)neli in T starting with tl "= t and a sequence (X~)ne~ in 9c such t h a t the following hold for every n C IN" 1) n--/: 1 ~

]X(tn)- x(t)] > e.

2)

I x ( t k ) - x~(tk)l < ~1 for every k C ]N~_I

3)

Ixk(tn)- xk(t)l < n1

for every k C INn

Choose Xl arbitrarily. Take n C IN, n > 1, and assume that the sequences have been constructed up to n -

1. By the definition of x , there is an xn C 9c

such that 2) is fulfilled. Since the functions in 9r are continuous,

238

1. Banach Spaces

n

~ T I I x k ( s ) - xk(t)l < 1 } k:l

is a neighbourhood of t. By hypothesis, there is a tn in this neighbourhood of T satisfying 1). This finishes the inductive construction. Let s be a point of adherence of (tn),~e~ and y a point of adherence of

(Xn)n~IN in C(T)T. Then

y(tk) - x(tk) for every k C IN, by 2). Thus

l y ( ~ ) - x(t)l > by 1). By 3),

xk(~) = x~(t) for every k E IN, so that

y(~) - y(t) = y(t~) = x(t~) - ~ ( t ) , which is a contradiction. Hence there is a neighbourhood V of t such that

for every s E V, and so x is continuous at t. It follows that x r C(T), converges to x in C(T)T, and Y is relatively compact. T h e o r e m 1.3.7.15

9

Let A be a subset of the Banach space E . Then the

following are equivalent: a)

Every sequence in A has a point of weak adherence in E .

b)

Every sequence in A has a weakly convergent subsequence in E .

c)

A is weakly relatively compact. By the Alaoglu-Bourbaki Theorem, EE#, is compact. Given x C E , define

~:'E~, and

,~IK,

x',

~(z,x')

1.3 The Hahn-Banach Theorem

~. E---~C(E~,), a ~

~-~

239

~.

b. Let (xn)ne~ be a sequence in A. By a), every subsequence of

(~(xn))ne~ has a point of adherence in C(E~#,) with respect to the topology of pointwise convergence. By Lemma 1.3.7.13, there is a strictly increasing sequence (k~)n~iN in 1N such that (~(Xkn)),~eIN converges to some y 9 C(E#E,) in the topology of pointwise convergence. Define x ' " E'

~ IK,

x', ~ " lim (x~n x').

Then x" is linear and Xll l E t #

-_ y .

Hence E '# N Ker x" is a closed set of E ~ . By Proposition 1.3.7.9 b =:~ d, there is an x C E , such that jEX = X".

Then (Xk.)neIN converges weakly to x. a =:v c. Let ~" be an ultrafilter on E containing d . By a), {x'(x) Ix e A} is a bounded set of IK for every x' c E ' . Thus the map x 'r" E' ---+ IK,

x' ~

limx'(x) x,~

is well-defined. It is obviously linear. By a), every sequence in ~p(A) has a point of adherence in C(E~#,) with respect to the topology of pointwise convergence. By Lemma 1.3.7.14 a ::~ c, ~(~) converges to some y e C(E#E,) with respect to the topology of pointwise convergence. We have x'r lE '# = y ,

so E '# N Ker x" is a closed set of E ~ . By Proposition 1.3.7.9 b ~ d, there is an x E E with jEX -- X".

Then ~ converges weakly to x , and A is weakly relatively compact. b :=> a and c ==> a are trivial.

I

Remark. a) The implication a ~ b was proved by Smulian (1940) and the implication a =~ c was proved by Eberlein (1947).

b) It is possible to prove a stronger form of Lemma 1.3.7.14 (for T a compact instead of compact) so that the above theorem can be proved without the use of Proposition 1.3.7.9.

240

1. Banach Spaces

1.3.8 Reflexive Spaces Definition 1.3.8.1 ( 0 ) (H. Hahn, 1927) A normed space is called reflexive if its evaluation map is surjective (in which case it is an isometry (Corollary 1.3.6.5)). It may happen that a Banach space is isometric to its bidual without being reflexive (R.C. James, A non-reflexive Banach space isometric with its second conjugate space, Proc. Nat. Acad. Sci. USA 37 (1958) 174-177).

Proposition 1.3.8.2

( 0 ) Every finite-dimensional normed space is retie-

xive.

This follows immediately from the fact that the dual and the algebraic dual of a finite dimensioual normed space coincide (Corollary 1.2.4.10). 9 P r o p o s i t i o n 1.3.8.3 ( 0 ) Every reflexive space is complete and its bounded sets are weakly relatively compact. Let E be a reflexive space. Then, using the evaluation map, we may identify E with E" and EE#, with ~"# By Corollary 1 2.1.10 E is complete and, by the Alaoglu-Bourbaki Theorem, EE#, is compact. Hence every bounded set of E is weakly relatively compact. 9 E

Proposition 1.3.8.4

I

9

.

,

( 0 ) (P.J. Pettis, 1938) A Banach space E is retie-

zive iff E' is reflexive.

If E is reflexive, then E' is obviously reflexive. Assume that E is not reflexive. Identify E with a subspace of E" via the evaluation map. Since E is complete, it is a closed subspace of E". By Corollary 1.3.3.6, there is an x'" E E'"\{0} vanishing on E . Then x'" does not belong to ImjE, and so E' is not reflexive. II P r o p o s i t i o n 1.3.8.5

Let E be a normed space and F a subspace of E .

a) F is reflexive iff j E ( F ) = F ~176 b)

If E is reflexive and F is closed, then F is reflexive (P.J. Pettis, 1938).

1.3 The H a h n - B a n a c h Theorem

241

a) Let j 9F ~ E be the inclusion map. T h e n Im j "

--

F ~176

(Proposition 1.3.6.17). If F is reflexive, then F ~176 = Im j " = Im (j" o j F ) = Im (jE o j ) -- j E ( F ) (Proposition 1.3.6.16). Now suppose t h a t j E ( F ) -- F ~176 . Take y" E F " . T h e n j " y " C F ~176 , and so there is an x C F with jEX =

j"y".

Then .,,. 2 3FX

__

jEjX =jEjx

__

j,,y,,

(Proposition 1.3.6.16) and y" C j F x

(Proposition 1.3.6.17). Hence jF is surjective and F is reflexive. b) Take x" E F ~176 . Since E is reflexive, there is an x C E with x" = jEX .

Given x ~ c F ~ ,

(x, x'> = (jEX,X'> = (X", X'> = 0 so t h a t m

xE~176 (Proposition 1.3.5.7). Hence F ~176 C jE(F).

The reverse inclusion is trivial. By a), F is reflexive. Proposition of E ,

1.3.8.6

Let E

be a n o r m e d space, F

and q" E " --~ E ' / F ~176the quotient map.

I a closed vector subspace

242

1. Banach Spaces

a)

ElF

b)

If E is reflexive, then so is E / F .

is reflexive iff q o jE is surjective.

a) Let r : E --+ E l F

be the quotient map. Then jE/F o r = r" o jE

(Proposition 1.3.6.16), so that jE/F is surjective iff r" o jE is surjective. By Proposition 1.3.6.18, the factorization of r" through E " / F ~176is an isometry. Thus r" o jE is surjective iff q o jE is surjective. b) If E is reflexive, then jE is surjective. Thus q o jE is surjective. By a), E/F

is reflexive.

C o r o l l a r y 1.3.8.7

I Let F be a closed subspace of the normed space E . Then

E is reflexive iff F and E l F

are reflexive.

The necessity follows from Proposition 1.3.8.5 b) and Proposition 1.3.8.6 b). For the converse, assume that F and E l F

are both reflexive and let

q: E" -+ E " / F ~176 be the quotient map. Take x" C E " . By Proposition 1.3.8.6

a), there is an x C E such that qjEX = qx".

Then q(x" - jEX) = O,

x" - jEX E F ~176 By Proposition 1.3.8.5 a), there is a y with jEY -- X tt -- jEX.

Hence x" = j ~ ( x + y),

i.e. jE is surjective and so E is reflexive. P r o p o s i t i o n 1.3.8.8

I

( 0 ) A Banach space, which is isomorphic to a refle-

xive Banach space, is itself reflexive.

1.3 The Hahn-Banach Theorem

243

Let u : E --+ F be an isomorphism of Banach spaces and assume E reflexive. Then u" is surjective (Corollary 1.3.4.7). Since

jF o u

-

-

~tt 0 jE

(Proposition 1.3.6.16), it follows that j F o u is surjective. Hence jF is surjective and F is reflexive. E x a m p l e 1.3.8.9

every set T .

I ( ]. ) ( 7 )

g P ( T ) i s reflexive for every p r

o0[ and

c0(T), c(T), gX(T) and g~(T) are reflexive iff T is finite.

For gP(T) (p e [1, c~] U {0}) this follows from Example 1.2.2.3 d),e) and Proposition 1.3.8.4. By Example 1.2.2.4 c), c(T) and co(T) are isomorphic, so that the assertion for c(T) follows from that for co(T) and from Proposition 1.3.8.8.

I

Example 1.3.8.10 Let S, T be sets and p, q E ]l, cx)[ be conjugate. Then gP,q(S, T) is reflexive. This is an immediate consequence of Example 1.3.8.9 and Proposition 1.2.3.6 b). Examp'r6')l.3.8.11

I

If It is a measure and p e ]1, oc[, then LP(It) is reflexive.

The assertion follows from Example 1.2.2.5 c).

I

E x a m p l e 1.3.8.12 If T is a completely regular space, then C(T) is reflexive

iff T is finite. If T is finite, then C(T) is reflexive py Proposition 1.3.8.2. Assume that T is infinite. Replacing T by its Stone-(~ech compactification, if necessary, we may assume that T is compact. There is a sequence (tn)ne~ in T for which

tn ~ {tin I m e ]N\{n}} for every n E IN. First suppose that there are two distinct ultrafilters ~ , ~ on IN with lim tn = lim tn. n,~ n,~ Take A r ~'\$ and put

B:={tnlneA}.

244

1. Banach Spaces

Then

eB

is a Borel function on T and

#J ) f eB d#

IK ,

x"" 2t4b(Z)

is a continuous linear form on Adb. Take x 9 C(T). Then limx(tn) - l i m x ( t n ) n,~

n,~

'

and so there is an n 9 IN with

9(t~) # ~ ( t ~ ) . Hence

jC(T)X r X". Thus jC(T) is not surjective (Example 1.2.2.10), and C(T) is not reflexive. Now suppose that for any two distinct ultrafilters ;~, ~ on IN lim t , r lim t , . n,;~

n,~

Then {tin I n 9 IN} is homeomorphic to the Stone-0ech compactification of IN. Define 9v := {x e C(T) in 9 IN ==, x(tn) = 0},

u" C(T) ~

~oo,

x,

> (x(tn))

nEIN

Then Ker u = 9c and the factorization of u through C(T)/5 c is an isometry (Tietze's Theorem). By Example 1.3.8.9, go~ is not reflexive, so that C(T)/.~ is not reflexive. By Proposition 1.3.8.6 b), C(T) is not reflexive.

I

Every bounded sequence in a reflexive Banach space has a weakly convergent subsequence. P r o p o s i t i o n 1.3.8.13

By Proposition 1.3.8.3, every bounded set of a reflexive space is weakly relatively compact and the assertion now follows from Theorem 1.3.7.15 c =:> b.

I

1.3 The Hahn-Banach Theorem

1.3.9 C o m p l e t i o n

of N o r m e d [

D e f i n i t i o n 1.3.9.1

(

0

245

Spaces

) Let E be a normed space. A c o m p l e t i o n o f E is

a Banach space F such that E is a dense subspace of F . Theorem ImjE

1.3.9.2 ( 0 )

Let E

be a normed space. I f E

(Corollary 1.3.6.5), then I m j E [

Theorem

1.3.9.3

(

0

is identified with

is a completion of E .

ll

\

) Let E be a normed space and let F , G be comple-

tions of E . Then there is a unique isometry u" F --+ G with U X z X

for every x E E . The uniqueness of u is trivial. Let jl " E -+ F ,

j2 " E ~

G be the

inclusion maps. Then, by Proposition 1.2.1.13, we can extend them to operators j-~ C E.(G, F ) , j~ E / : ( F , G ) , respectively, with IlJl I = IIJ21 = 1. We have jl o j2(x) - x ,

j2 o jl (x) = x

for every x C E . We deduce ~oj-~=lF,

j-2ojl= la.

u " - j 2 now has the required properties. Remark.

The above theorem allows us to identify all completions of E . This

justifies the use of the term the completion of E .

246

1. Banach Spaces

1.3.10 A n a l y t i c Functions Definition 1.3.10.1

( 0 ) Let E be a Banach space and U an open set

of IK. A function f

9U -+ E is called analytic if for given so C U there is

a power series ~ t n x n i n E

and an r > 0 such that r is smaller than the

n--O

radius of convergence of this power series,

U~(ao) c u, and oo

:(,) = E(,-,o)o,. n----0

for every s C U~IK(C~o).

By Proposition 1.1.6.11, if f " U ~ E and g" U ~ E are analytic, then s f + ~g" U ~ E is analytic for all s,/~ E IK. P r o p o s i t i o n 1.3.10.2

Analytic functions are differentiable and their deriva-

tives ~re analytic.

The proposition follows immediately from Proposition 1.1.6.25. oo

P r o p o s i t i o n 1.3.10.3

( 0 ) Let E be a Banach space, ~ n--O

series in E ,

r its radius of convergence, So C IK, and oo

f " Urn(SO)

>E ,

s,

> ~-~.(s - So)" X, . n--O

Then f is analytic.

Take ~0 E U~(s0) and put r'=~-I~0

- Z01

Take a E UT,(/~0) and let p : = I~ - ~ol + 19o - ~ol < ~' + I~o - ~ol = ~ .

Then

(:),o m--0

Oonm

/o

Oo, n

I

t'x. a power

1.3 The Hahn-Banach Theorem

247

for n E IN U { 0 } , and so

i[(n)

,o

for n C IN U {0} and m e IN. U { 0 } . Hence oo

oo,n

E

.

, ( cx:).

Given m C IN U { 0 } , put

ym := ~

(~o - ~o)"-mx.

n---m

(Corollary 1.1.6.10 a =~ c) and

(:) ::o for n C IN with m > n . Take p C IN. Then

I~- ~olmllYmll -< ~ m=0

: E

I~o -~ol~-mltz~l

m=0

[]Xn[[

]ZO- OzO[n-mloL- ~0[ m

n=0

I~- 9o:m =

n=0

~ E

m=0

]]Xn]lpn < 00.

n=0 [

(Corollary 1.1.6.10). Since p is arbitrary, the family

[(a-

\

~O)mym)meiNU{O } -

is absolutely summable. It follows that the radius of convergence of the power (DO

series ~

tmym is greater than r' (Theorem 1.1.6.23).

m--0

Take c > 0. There is a p C IN such that

f(~)-~-~(~-~o

x~ < 5 ,

~

n=0

I1~11 < 5

m=p+l

Then (Proposition 1.1.6.11, Corollary 1.1.6.10),

E

(OL -- /~O)mym -- E m=0

m=0

=E n--0

(OL -- /~0)m

(2~0 -- OLO)n-mxn n=0

( ~ - ~o/~(~0- ~0)o-~ ~~ m--0

--

248

1. Banach Spaces

=

(~ - ~ o ) - x . +

Z

n=O

n=p+l

(~ - ~o)m(9o - ~o) --m

Xn,

m=O

P

P

II ~ - ~ ( ~ - ~o)mym- ~-~(~- ~o)'x~ll _< m=0

_< ~ n=p+l

n=O

I~.11

I~-~o1~1~o-~ol "-~

< ~

m=0

I~.11o' 0, such

n=0

that r is smaller than the radius of convergence of the power series,

u~(ao) c u, and (2O

:(-)- E(---o)'-o n--O

1.3 The Hahn-Banach Theorem

249

for every a E Ur~(ao). By Corollary 1.2.1.17, r is smaller than the radius of oo

convergence of the power series ~

tnux~ and

n--0 oo

n--0

for every a E Ufi(ao). Hence u o f is analytic. C o r o l l a r y 1.3.10.5

I

Let E be a Banach space, U a domain in ]K, and

f : U -+ E an analytic function. If f vanishes on an open nonempty subset of U, then f vanishes identically.

Take x ~ E E ' . By Proposition 1.3.10.4, x' o f is analytic. Since it vanishes on an open nonempty subset of U, it vanishes identically. By Corollary 1.3.3.8 a), f vanishes identically. T h e o r e m 1.3.10.6

I

( 0 ) (Liouville's Theorem) Let E be a complex Banach

space. Every bounded analytic function (~ --+ E is constant.

Let f :~ -+ E be a bounded analytic function and take a,/~ E ff~. Assume that f (a) ~ f (/3). Then there is an x' E E' with X' o f (a) ~: X' o f (fl)

(Corollary 1.3.3.9). By Proposition 1.3.10.4, x t o f is analytic. Since it is bounded, it is constant by the classical form of Liouville's Theorem. Hence x' o f (a) = x' o f (j3)

which is a contradiction. Remark.

I

The above theorem was proved by Cauchy (1844) for E = ~ .

C o r o l l a r y 1.3.10.7

I

~

0

\

) Let E be a complex Banach space and f "r

an analytic function. If

lim f(o~) = 0 t~----~o o

then f is identically zero.

E

250

1. Banach Spaces

f is bounded, so it is constant (Theorem 1.3.10.6).

I

T h e o r e m 1.3.10.8 ( 6 ) (Laurent's Theorem, 1843) Let E be a complex Banach space. Take so E 9 and 0 < rl < r2. Put V : - {c~ E e l r l

< Ic~-sol < r2},

and let f " U -+ E be an analytic function. Then there is a unique family (xn)ne~ in E such that oo

n'----O0

for every a E U. The radius of convergence of the power series Orb

O0

Et"Xn

(resp.

n:0

Et~x_n) n:l

is greater than r2 (resp. ~1 ) . The expression oo

(t -

;

n'----O0

is called the Laurent series of f , -1 E

(t-

OLo)nXn

n:--O0

is called its principal part and x-1 is called its residue.

Take r E ]rl, r2[. Given n E IN, put

1/

2rr

xn := 27rrn

f(a0 + reit)e-intdt,

0

where the integral is defined (as in the classical case) with the help of the Riemann sums. Take n E IN. Then 27r

Xn , X I) - -

1 f 27crn

x ' o f(c~o + reit)e-i'adt

0

for every x' E E ' . By Cauchy's Theorem, (zn, x') does not depend on r. Hence, by Corollary 1.3.3.9, xn does not depend on r.

1.3 The Hahn-Banach Theorem

Set fl "= sup IIf(ao + ~e")ll < ~ . tEIR

Then

I(x,~, x')l C ( E ' # ) ,

x,

~ jEX

i E'#.

u preserves norms (Corollary 1.3.3.8 b), Theorem 1.3.6.3 a)) and so I m u is a closed set of C(E'#B) (since E is complete). In order to show that A is

1.4 Applications of Baire's Theorem

261

relatively compact, we must therefore prove that u(A) is a relatively compact set of C(E'#B). Since u(A) is a bounded set of C(E'#u), it is sufficient to show that u(A) is equicontinuous (Ascoli Theorem). Take x' E E'#B and c > 0. We show that there is a neighbourhood U of x t in E~B # such that

I(x, y')- (~, ~')l < for every yt E U and x E A. Assume the contrary and take n E IN. Then 1 n

is a neighbourhood of x' in E ' ~ . There are yn E A and x~ E Un such that !

I(y~, ~ -

x')l _> c.

Then lim X'n(X~ ) = X' (xk)

n--+ oo

for every k E IN and so (X~z)nEIN converges to x' in E'p#. By Proposition !

1.1.2.15, (x~)ns~ converges to x' in E~ and by b), (Xn)nE~ converges to x' uniformly on A. Hence e _< lim I(Yn, x'n

_

X r

n--+(x)

)1 = 0 I

and this is a contradicition. E x a m p l e 1.4.1.10

(4)

Let Sl "-- {OL E r

] [OL[- 1}

and 27r

XnA .__ ~1 f

e_in tx(t)dt

0

for

X E C(Sl)

and n E 2Z. There is an x E C(S1) such that

p E IN

does not converge, i.e. the Fourier series of x is not pointwise convergent.

269

1. Banach Spaces

Given p E IN, set P n=-p

and P X p! " C ( S 1 )

) ]Z ,

X l

) E

Xn 9

n---p

For p E IN and t El0, 27r[

fp(eit ) Xp'

= sin(p + sin t

1)t

is linear and 27r

x,,(~)

1 /

=

fp(eit)x(t)dt.

o

Thus

1/ [fp(dt)ldt ~ Ixll~ ~ 27r

14r

o

for every x E C(S1) 9Hence xp' is c o n t i n u o u s and

1/

27r

Ix;l[ a). Define

f "S1

) ]K

Take g > O. There is an x E

1/

'

a'

C(S,)

fv(e it)

1

>

-1

if f p ( a ) > 0 if f p ( a ) < 0.

such t h a t ]lxlloo _ 1 and

(xlet) - .lezt))dt < e .

o

Then

I

x'p(x) - ~1

2/

]fp (eit )[dt _[x~p(x)l>_-~1/ [fp(e~t)ldt- c 27r

0

(Proposition 1.2.1.4 b)). Since ~ is arbitrary, 27r

[IXpll > -

1 [ ]fp(eit)[dt > ~

27r

1/

,sin(P+ 89

"It

o

t

dt =

o

(2p+1)~ (k+l)~ 1 / ]sint[dt- 1~o / ]sin nt[ dt > t 7r t

7r

0

--

k~

(k+l)lr

1~-~.

- > -T"

k=O

1

(k + 1)7r

flsinntldt_2~-~l k~

~

k=O

k+l

Hence lim ][x;I I - oc.

p---~c~

By Corollary 1.4.1.3, there is an x e C(S1) for which converge,

(Xp(X))p~

does not m

264

1. Banach Spaces

1.4.2 Open Mapping Principle Proposition

1.4.2.1

( 0 )

Let E be a normed space, F a Banach space,

and u " E -+ F a linear surjective map. Then 0 is an interior point of u ( E # ) . u ( E # ) is absolutely convex (Propositions 1.2.7.2, 1.2.7.7, and 1.2.7.5), and U

nu(E#)D

n E IN

U

U

nu(E#)-

n E IN

u(nE#)-

n E IN

u(U

nE#):

By Proposition 1.4.1.1, 0 is an interior point of u ( E # ) . Proposition

1.4.2.2

and take u E s

( 0 )

u(E)-

F.

n E IN

I

Let E be a Banach space, F a normed space

F) . If 0 is an interior point of u ( E # ) , then 0 is an interior

point of u ( E # ) .

By hypothesis, there is an c > 0 with cF # C u(E#).

We prove t h a t F# C u(E#).

Take y E ~ F ~ . We construct inductively a sequence (xn)~e~ in E # such that

I

1 )

y - u

2--~xm kin-- 1

< mc

2n

for every n E IN. Take n E IN and suppose that x~,...,x,~_~ have been constructed. Then y-u

~ Xm

E

F # C 2--~

km = 1

Hence there is an x,~ E E # with

y - u

~xm

c - ~---~uxn < 2n+l '

\m=l

i.e.

y - u

2--;xm m--1

2n_}_ 1 -

1.4 Applications of Baire's Theorem

265

This completes the inductive construction. (~nXn) is an absolutely convergent sequence in E . Put nE1N

1 X-- Xn nE IN

(Corollary 1.1.6.10 a =~ c). Then Iixll _< E

lIIxni[ _< 1

nEIN

(Corollary 1.1.6.10) and y=

lim u

n --+c~

(~2~ 1 m=l

-~

xm

)

=uxEu.E#,()

Hence

F# C u(E #) and 0 is an interior point of u(E#).

I

T h e o r e m 1.4.2.3 ( 0 ) (Open Mapping Principle, Banach 1932) Every surjective operator between two Banach spaces is open, i.e. maps open sets into open sets. Let E, F be Banach spaces and u " E --+ F a surjective operator. Let U be an open set of E and y C u(U). Take x E U with

ux=y. Then x + e E e c U, for some e > 0. Hence

y + ~ ( E # ) = ~(z + ~E#) c ~(U). By Proposition 1.4.2.1 and 1.4.2.2, 0 is an interior point of u(E#). Hence y is an interior point of u(U) and u(U) is thus open. I C o r o l l a r y 1.4.2.4 ( 0 ) (Principle of Inverse Operators) Every bijective operator between Banach spaces is an isomorphism.

266

1. Banach Spaces

Let E, F be Banach spaces and u : E ~ F a bijective operator. Since u is open (Theorem 1.4.2.3), u -1 is continuous. Hence u is an isomorphism,

i

Let E , F be Banach spaces. An operator u : E --+ F is an isomorphism (isometry) iff u' is an isomorphism (isometry).

C o r o l l a r y 1.4.2.5

The necessity was proved in Corollary 1.3.4.7. So assume that u' is an isomorphism (isometry). By Corollary 1.3.4.7, u" is an isomorphism (isometry). Then

u"

o jE -- jF o u

(Proposition 1.3.6.16), so that jE

=

'U'/-I O j E o u .

Hence Ilxll = IIjExll = itu ' ' - ' o jF o u(x)l I q'x'

(Proposition 1.3.5.2) and let

w" F ' / ( I m u) ~

~ (Im u)'

(Proposition 1.3.5.8) be the factorization of i' through F ' / ( I m u ) ~ .

a)

Keru'-(Imu)

~

Imu'=(Keru) ~ Keru-~

b) ~ and

~'

Imu=~

are i s o m o r p h i s m s of B a n a c h spaces.

c) v and w are isometries. B

d)

u'-vog'

ow.

a) By Proposition 1.3.5.8 K e r u ' = (Imu) ~

Keru=

~

Imu-

~

and, by Proposition 1.4.2.9 b), Im u' - (Ker u) ~ . b) By Proposition 1.4.2.9 a), g is an isomorphism of Banach spaces. Hence, by Corollary 1.3.4.7, the same holds for g~. c) follows from Proposition 1.3.5.2 and Theorem 1.3.5.12 c). d) Given x E E and y~ E F ' , (x, j v ~ ' w r y ' } - (x, q'~'i'y') -- (i~qx, y'} = (ux, y'} = {x, u'y'} - ( x , j - ~ r y ' ) , SO

j

ovo~'

owor

-

j

ou' or.

Since j is injective and r is surjective, VO~lOW=U l .

I

1.4 Applications of Baire's Theorem

( 0 )

Proposition 1.4.2.11 s

Let E , F

271

be Banach spaces and take

u E

F ) . If I m u ' is closed, then Im u is also closed. Define G := I m u ,

v" E

~ G,

x'

~ ux,

w:G

~F,

y~

>y.

Then v' is injective (Corollary 1.3.5.9) and w' is surjective (Theorem 1.3.5.12 a)). Since

~'(a') = v ' ( ~ ' ( F ' ) ) = ~'(F') (Corollary 1.3.4.5), v'(G') is closed. By Corollary 1.4.2.8 a ~

b, there is an

c~ > 0 with

II~'Fti > ~lly'li for every y' c (7'. Take y C c~G#\v(E#). Since v ( E # ) is absolutely convex there is a y' C G' such that

sup

z~v(E#)

I(z, v')l < (y, y')

(Corollary 1.3.1.7). Thus

~ily'll ~ liv'FII = sup I(~, ~'y')l = sup I(vx, v')l < (y, y') Imu,

x a

>ux.

T h e n w is obviously linear and continuous. For x C Ker w, UX

--

WX

=

O~

so t h a t x = 0, i.e. w is injective. Take y E I m u . T h e n

1.4 A p p l i c a t i o n s of B a i r e ' s T h e o r e m

273

y = U X ~

for some x E E . T h e n x-px

E G,

w(x - px) = ux - upx = ux = y ,

i.e. w is surjective. Since I m u is closed ( T h e o r e m 1.2.5.8 b ::v a, P r o p o s i t i o n 1.2.5.2 a => b ) , it is complete. By the Principle of Inverse Operators, w is an isomorphism. Set v" F

> E,

y,

) w-l(qy).

T h e n v is linear and continuous and voq--v.

For x E E , x - px E G ,

u ( x - p x ) -- u x ,

(1E -- V o U)X -- X -- W - I ( q u x )

--- X

--

W--I(?.t(X

--

px))

--- X -- W-I(UX) --

-- X -- (X -- px)

-- px,

so t h a t 1E--VoU--p.

For y E F , (u o v)y

-- u(w-l(qy))

-- qy

and so uov--q.

Proposition

~)

1.4.2.14

T h e r e is a v E s is a c o m p l e m e n t e d

Let E, F

I

be B a n a c h

such that uov subspace of E.

spaces and u E s

F).

= 1F i f f u is s u r j e c t i v e a n d K e r u

274

1. B a n a c h Spaces

b)

T h e r e is a v 6 s

E)

s u c h that v o u = 1E i f f u is i n j e c t i v e a n d Im u

is a c o m p l e m e n t e d s u b s p a c e o f F .

a) Assume first t h a t u o v = 1F for some v 6 s

E ) . Then u is surjective

and v is injective. P u t p:-vou.

Then pop=

vouovou

= vou-p,

i.e. p is a projection in E . For x 6 E ,

(px = o) ~

( w x = o) r

(~

= o),

so t h a t Ker u = Ker p and Ker u is a complemented subspace of E by M u r r a y ' s Theorem (Theorem 1.2.5.8 b ~ a). The reverse implication follows from Proposition 1.4.2.13 and Murray's Theorem. b) Assume first t h a t y o u = 1E for some v 6 s

E ) . Then u is injective.

Put p:--uov.

Then pop=

u ov

ou

ov

=

u ov

=

p,

i.e. p is a projection in F . If y C I m u then there is an x C E with y--ux.

It follows y = ux = uvux - pux C Imp,

so t h a t Im u C I m p .

1.4 Applications of Baire's Theorem

275

The reverse inclusion is trivial, so that Imu = Imp and Im u is a complemented subspace of F (Theorem 1.2.5.8 b =~ a). The reverse implication follows from Proposition 1.4.2.13 and Murray's

I

Theorem. Proposition

1.4.2.15

Let E be a Banach space and p be a projection of E"

onto I m j E . Put F := K e r p and u:E

...... ; ( ~

a)

u is an operator and

b)

Given y' E ( ~

x,

~(jEz) I~

Ilull ~< 1.

there is an x E E such that ~x

= V,

I1~[I _< Ilpll lly'll.

c)

u is a isomorphism iff F is closed in E ~ , .

d)

I f F is closed in E~, and IlpII xn.

and u n : ] K (~)

>IK (~),

Prove the following: a) b)

un C / : ( I K (~)) for every n C IN. lim unx = 0 for every x E IK (~) n----~ oo

c)

sup Ilunll = oc (i.e. the conclusion of the B a n a c h - S t e i n h a u s Theorem does nEIN

not hold). E 1.4.2

Let E , F

be Banach spaces and u : E ~

F a surjective operator.

Show t h a t there is an c~ > 0 such t h a t

II~'y'll _> ~lly'll for every y' E F ' . E 1.4.3

Let E be a vector space and let p < q be complete norms on E .

Show t h a t p and q are equivalent.

1.5 Banach Categories

281

1.5 B a n a c h Categories The set of operators on a Banach space forms a Banach algebra and the whole of Chapter II is devoted to the study of such algebras. Unfortunately this theory cannot be applied to the case of operators between two different Banach spaces. The corresponding general theory is the theory of Banach categories, which is the subject of this section. Note that a Banach algebra is no more than a Banach category with precisely one object. This theory is not developed further in this book and so the reader may choose to omit this paragraph. 1.5.1 D e f i n i t i o n s

Definition 1.5.1.1

( 1 )

(2)(3)

A B~h

~y.t~m is aclass $2

and a map fit defined on $22 such that fit(E, F) is a Banach space for every E, F E $2 and

(E, F) --/: (G, H) ~

A(E, F) n A(G, H) = 0

for every E, F, G, H C F2 . We use the expressions "the Banach system (~Q,fit)" or "the Banach system ,4 over Y2" or, simply, "the Banach system M". The elements of [2 are called the objects o f the B a n a c h s y s t e m and the elements of fit(E, F) (for E, F C $2) are called the m o v p h i s m s o f the B a n a c h system. We put

.A(E) := .A(E, E) for E C $2 and E -~ F :r

E 4

A

F " v:=, x E A ( E , F )

for E, F E s Let A, B, C be Banach systems over the same class F2 . An

(A, B, C)-

multiplication is a map ~ defined on ~23 such that the following holds for every E, F, G c $2 :

~(E, F, G) is a bilinear map

A ( E , F) • U(F, a )

~ C(E, a) ,

such that

Ilyzll _< Ilzll Ilyll

(x, y) ,

~ y~

282

1. Banach Spaces

for all E -~ F Y+ G. A left (resp. right) multiplication on A over B is an (.A, B,.A) (resp. (]3, .A, .A))-multiplication. A unit for such a multiplication is a map 1, defined on $2, such that

1~ E B(E), 1E :/: 0 ==V IIIEII = 1, 1EX = X (resp. XlE = X ) for every E, F E $2 and F--~ E (resp. E-5+ F ). A left and a right multipliA

A

cation on .4 is called compatible if (ax)b = a(xb) for every E, F, G, H E $2 and E-~

F4

A

a-~

H.

An inner multiplication on fit is a left (or right) multiplication on ,4 over ,4 such that (~y)z = ~(yz)

for every E, F, G, H E $2 and E - ~ F--~ G - ~ H . An inner multiplication has a unit if tts left and right multiplications have units. Let ($2,.A), ([2, B) be Banach systems. A left and a right multiplication on ~4 over B are called simultaneously compatible with an inner multiplication on.Aft (xa)y = x(ay) for every E, F, G, H E $2 and E4

A

F4

B

G4

A

H.

A B a n a c h category (unital B a n a c h category) is a Banach system endowed with an inner multiplication (which has a unit). If $2 is a class of Banach spaces, then the map

(E, F ) ,

)s

defined on $22 is a Banach system (Theorem 1.2.1.9). It is a unital Banach category with the usual composition of the maps as multiplication (Corollary 1.2.1.5). We denote it by s or, simply, s

1.5 Banach Categories

Example 1.5.1.2

283

Take p C {0} U [1, cx3] and let q be the conjugate exponent

of p. Let 32 be a class of sets. Then the Banach system

(s, T) ,

~ e~,~(S, T)

on 32 with the multiplication defined in Proposition 1.2.3.5 is a Banach category. I

The claim follows from Proposition 1.2.3.5.

Proposition 1.5.1.3 ( 2 ) Let .A,B,C be Banach systems on the same class 32, ~a an (A, B, C)-multiplication, and take E, F, G E 32. F

~) ~(E, F, G) i~ contin~o~ b) If (XL)tEI is a summable family in A ( E , F) (in B(F, G) ), then

tel

for every x e B(F,G)

t~:_I

tel

tC I

(x E A ( E , F ) ) .

a) follows from Proposition 1.2.9.2 c =:~ a. b) follows from a) and Proposition 1.2.1.16.

I

Proposition 1.5.1.4

( 2 ) Every unital Banach category admits a unique unit for the left multiplication and a unique unit for the right multiplication and they coincide; we call it the unit of the Banach category.

Let (32,A) be a Banach category and 1 (resp. 1') be a unit of the left (resp. right) multiplication of A. Then 1E= 1EI~ = 1~ for every E C 32. Definition 1.5.1.5

I ( 1 ) ( 2 ) ( 3 ) Let (32, A ) b e a u n i t a I B a n a c h

category and take E, F E 32. An element x c A(E, F) is called left invertible (right invertible) if there is an y e A(F, E) such that y X = IE

( x y = IF).

x is called invertible if it is both left and right invertible.

284

1. Banach Spaces

The invertible morphisms of /: are precisely the isomorphisms of Banach spaces. Proposition

1.5.1.6

(~) ) Let A be a unital B a n a c h category, E , F

of A , and x an invertible m o r p h i s m of A ( E , F ) . y E A(F, E)

objects

Then there is a unique

with yx-

1E,

x y = 1F.

Put

X-1 : - - y . X -1 is called the i n v e r s e o f x .

Since x is invertible, there are y, z E A ( F , E ) so that y x -- 1E~

x z ~ 1F.

Then y = y l F = y ( x z ) = ( y x ) z = 1EZ = Z. Corollary

1.5.1.7

m

( 2 ) L~t (~,A) b~ a ~ t a t ,a~ach cat~go~y ~k~

E , F, G C ~ , and E _5+ F -~ G . I f x and y are invertible, then y x is invertible and (yx) -1 - x - l y - 1 .

We have (~-~y-')(yx)

- x-~(y-~)~

(y~)(x-~y -') - y(xx-')y

C o r o l l a r y 1.5.1.8

( 2 )

E , F c Y2, E ~ F , and F - ~

Let ( ~ , A ) E.

- ~-~x = 1~, -~ -

yy-~ -

1~.

m

be a unital B a n a c h category. Take

Then x and y are invertible iff x y and y x

are invertible.

By Corollary 1.5.1.7, if x and y are invertible, then x y and y x are also invertible. Assume now that x y and y x are invertible. Put U :-- xy~

V "-- y x .

Then ~ ( y ~ - i ) = ( ~ y ) ~ - i _ 1~,

so that x is invertible. Hence, y is also invertible.

m

1.5 Banach Categories

Definition 1.5.1.9

(

1)

(

2)

Let ([2, A ) b e a B a n a c h

(E,F),

285

system. The map

~ A(F, E)'

defined on E22 is a Banach system. It is called the dual of A and is denoted by A t. The dual of .At is called the bidual of .4 and is denoted by A" and the dual of M" is called the tridual of A and is denoted by A'".

Definition 1.5.1.10

( 1 )(2) (3) Let ~ be a c l a s s a n d A a Banach category (unital Banach category) over E2. A left A - m o d u l e (unital left A -

module) is a Banach system .4 over $2 together with a left multiplication over A such that (ab)x = a ( b z ) ]or every E, F, G, H c E2 and

E4 F4 G4 H A

A

A

(and such that the unit of A is the unit for the left multiplication). The right A - m o d u l e (unital right A - m o d u l e ) is defined in a similar way.

A is a left and a right A-module (unital left and unital right A-module) in a natural way. Proposition 1.5.1.11

( 1 )

(2)

Let (r

category (unital

Banach category), A a left A-module (unital left A-module), and H a Banach space. Given E, F, G c Y2 , put A H ( E , F) := s

ua : A ( G , E )

>H ,

E), H) ,

x,

>u[ax]

for all E--~ F 2+ G. Then the Banach system .A H with the above multiA

AH

plication is a right A-module (unital right A-module), and, similarly, if we interchange left and right. In particular, A ~ = A' is a right A-module (unital right A-module).

It is easy to see that the maps defined form a multiplication. Take E , F , G , D E I2

286

1. Banach Spaces

and E4

A

F~

A

G--% D. AH

Then (u(ba))[x] = u[(ba)x] = u[b(ax)] = (ub)[ax] = ((ub)a)[x]

for every D--~ E (and A

(ule)[y] = u[lvy] = u[y]

for every D --~ G ), so that A

u(ba) = (ub)a

(ule = u).

m

Definition 1.5.1.12

( 1 ) (~) ) ( ] ) Let ($2,A), (Y2,A)be two Banach categories (unital Banach categories). A (A, A)-module (unital (A, A ) module) is a Banach system over $2 endowed with the structure of a left A module (unital left A-module) and right A-module (unital right A-module) such that the left and the right multiplications are compatible. A A-module (unital A-module) is a (A, A)-module (unital (A, A)-module).

Every Banach category (unital Banach category) A is a A-module (unital A-module). Corollary 1.5.1.13

( 1 ) ( 2 ) Let (~, A), (~, A) be two Banach categories (unital Banach categories) and A a (A, A)-module (unital ( A , A ) module). Then fit' is a (A,A)-module (unital (A,A)-module). If A is a A module (unital A-module), then A' is also a A-module (unital A-module). By Proposition 1.5.1.11, ,4' is a left A-module (unital left A-module) and a right A-module (unital right A-module). Take E, F, G, H C ~Q and E4

A

F-~ G4 A' A

H.

Then (x, (bx')a>- (ax, bx'>- ((ax)b,x'>- (a(xb),x'> = (xb, x'a> = (x,b(x'a)>

for every x C A(H, E). Thus (bx')a = b(x'a),

m

1.5 Banach Categories

Definition 1.5.1.14

287

( 1 ) ( 2 ) ( 3 ) Let A be a B a n a c h category (unital

Banach category). A A-category (unital A - c a t e g o r y ) is a A-module (unital A-module) A

endowed with an inner multiplication (with a unit) such that

each left multiplication on A is compatible with each right multiplication on .4 and that the left and the right multiplication on .A over A are simultaneously compatible with the inner multiplication on yl.

A is a A-category (unital A-category). Every A-category (unital Acategory) is a Banach category (unital Banach category) with respect to the inner multiplication.

288

1. Banach Spaces

1.5.2 Functors Definition 1.5.2.1

(1)

(2)(3)

(~,A),(~,B)be two Banach

Let

systems. A f u n c t o r of M into B is a map f defined on f22 such that f(E,F) c s for every E , F

C E2. The functor f

is called isometric if f ( E , F )

is an

isometry for every E, F C f2. ,4 and B are called isometric if there is an isometric functor of A into B. The map (E, F ) ,

> 1A(E,F)

defined on ~22 is an isometric functor of ,4 into ,4. It is called the identity f u n c t o r of .A. Let ($2,A),

(E2,B), (f2, C) be Banach systems, f a functor of fit into B,

and g a functor of B into C. The map (E, F ) ,

~ g(E, F) o f ( E , F)

defined on ~22 is a functor of .4 into C. It is called the composition of the f u n c t o r s f and g and it is denoted by g o t . Let

(~,M) be a Banach system. Given E, F C ~ , let jEF denote the M(E, F ) . Then the map

evaluation on

(E, F) , ,', jEF defined on f2 2 is a functor of ,4 into its bidual A " . It is called the evaluation f u n c t o r of A . Given E E ~2, put jE Let (~2,.A), ($2, B)

:-- jEE.

be Banach categories (unital Banach categories). A

f u n c t o r of B a n a c h categories (unital B a n a c h categories) of A into B is a functor f of M into B such that f(xy) = f(x)f(y)

(and

f(1E)= 1E)

for every E, F, G E ~2 and

E-5 F 4 A

.4

C.

Let ($2, A) be a Banach category and ,4, B left (right) A-modules. A funct o t of left (right) A-modules of A into B is a functor f of A into B such that

1.5 Banach Categories

f (ax) = a f (x)

289

( f (xa) = f (x)a)

for every E, F, G c $2 and E4

A

F4

A

G.

(E--% F-5+ G ) . A

A

A f u n c t o r of A-modules is a functor of left and right A-modules. Let ($2, A) be a Banach category (unital Banach category) and let .4,13 be two A-categories (unital A-categories). A functor of (unital) A-categories of .4 into 13 is a functor of A-modules of .4 into 13 which is also a functor of Banch categories (unital Banach categories).

Example 1.5.2.2 Banach system

( 2 )

Let $2 be a class of Banach spaces and `4 the

(E,F),

>s

over $2. Given E, F C $2, put

/(E,F):L(E,F)

~A(E,r),

~,

~ ~'

Given E, F, G C ~2 and E--~ F 2+ G , put A

A

V?.t : z

?J, O V

.

Then `4 with this multiplication is a unital Banach category and f is a functor of unital Banach categories of E into `4. `4 is called the transpose unital category of s and f the transposition f u n c t o r of s

The result follows from Theorem 1.3.4.2 and Corollaries 1.3.4.3, 1.3.4.4, and 1.3.4.5. I Proposition 1.5.2.3

( 2 ) Let ($2, A) be a Banach category. Let `4, B be

left (right) A-modules and let f be a functor of left (right) A-modules of .4 into B. Given E , F C F2, define f ' ( E , F ) = f(F, E)'. Then f' is a functor of right (left) A-modules of B' into A ' , called the transpose of f .

290

1. Banach Spaces

Given E, F C .Q,

Take E, F, G E .Q and

E-~ F-~ c. A

B'

Then (x, f ' ( y ' a ) ) - ( f x , y'a) - ( a / x , y') - ( f ( a x ) , y') - (ax, f ' y ' ) = (x, (f'y')a)

for every x C .A(G, E) (Theorem 1.3.4.2 a)). Thus f'(y'a) = ( f ' y ' ) a .

Proposition 1.5.2.4

( 1 ) ( 2 )

I

Let ($2, A ) b e

a Banach category and

,4 a left (right) A-module. Then the evaluation functor on ..4 is a ]unctor of left (right) A-modules into its bidual.

Given E , F E ~2 and x c A ( E , F ) ,

set

x= j A x .

Take E, F, G c f2 and E - ~ F - ~ G. .,4

A

Then (6"-~, x') - (ax, x') = (x, x'a) -- ('~, x' a) = (a~, x')

for every x' C A'(G, E), so a'-~ -

Definition 1.5.2.5

a~.

I

( 1 ) ( 2 ) Let (f2, A ) b e a B a n a c h

a left (right) A-module. Take E , F , G c ~2. Given E-~ F 4 G A'

( E-5,. F S~ G),

A

A

A'

set x x " A(C, E) ---+ IK ,

(~'~ A(a,E)

)~,

a,

) (ax, x') ,

, )(xa, x'>).

category and .4

1.5 Banach Categories

291

If A = A, then it is easy to see that the above composition law coincides with the multiplication introduced in Proposition 1.5.1.11. P r o p o s i t i o n 1.5.2.6 ( 1 ) ( 2 ) Let A be a B a n a c h category a n d " 4 a left (right) A-module. The composition law introduced in Definition 1.5.2.5 is an (A', A, A')-multiplication ((A, .4', A')-multiplication) such that X !

E --+ F _5+ G -% H ~ ( resp. E --% F _2+ G -~ H ~ E4

x !

F-+

(resp. E ~

F-~

G4

( a x ) x ' = a(xx') (x' x)a = x' (xa) ),

H==~ ( x x ' ) a = x(x'a),

G--~ H ~

(ax')x=a(x'x)

),

where a is a morphism of A, x is a morphism of A, and x' is a morphism of "4'. If A is a A-module, then X !

E _5+ F --~ G -+ H ~

(x'a)x = x'(ax),

E -~ r - - ~

(xa)x' = x(ax') ,

G ~

H ~

with the same conventions as above. The first assertion is easy to verify. We have

(b, (ax)x'} = (b(ax),x'} = ((ba)x,x'} = (ba, xx'} = (b,a(xx')} (resp. (b, (x'x)a} = (ab, x'x} - (x(ab),x'} = ( ( x a ) b , x ' ) = (b,x'(xa)}),

(b, (xx')a} = (ab, x x ' } -

((ab)x,x'} = (a(bx),x'} = (bx, x'a} = (b,x(x'a)}

(resp. (b, (ax')x) = (xb, ax') = ((xb)a, x'} =

= (x(ba), x'} = (ba, x'x} = (b, a(x'x)} ) for every H - ~ E , which proves the relations. If A is a A-module, then A

IK,

x'a" : A(G, E)

x,

> (a",xx')

z,

~ ~",~'x>

for E, F, G C .Q and

z~4 F4a A" A' a"x' A ( G, E ) ----+ IK, J

for E, F, G E f2 and

E4A' F~A" a). It is easy to see that if .4 = A, then the above definition coincides with Definition 1.5.2.5. P r o p o s i t i o n 1.5.2.9 ( 1 ) ( 2 ) Let A be aBanach category, .4 a left (right) A-module, and E, F, G, H objects of A . a)

The composition law introduced above is a right (left) multiplication on .4' over A" with the property that E ~-~ F - ~ G _5+ H ~ A"

A

A'

A'

A

A"

1t

x(x'a")= (xx')a"

1.5 Banach Categories

b)

293

If F - ~ G and A'

u "A(G,H)~

A'(F,H),

(u " A ( E , F )

~ A'(E, G) ,

X l

x,

} XX t

". x'x) ,

then

H~ F~u~a "=x'a" Au

(G~ E~u'a"=a"x'). All a) The first assertion is easy to verify. By Proposition 1.5.2.6, (a,x(x'a")l - ( a x , x'a") -- (a", (ax)x' I - (a",a(xx')) = (a, (xx')a")

((a, a" (x'~)) = (a", (~'z)a) - (a", ~' (~a)) -- (xa, a"~') - (a, (a"~')~)') for every H-% E . A

b) We have (x, ulall I ~_ (~tx~a It) - - ( x x l

a 1'} - - ( x ,

xPall I

for every G _5+H A

((x, ~'a") = ( ~ , a") -- (~'~, a") - (~, a"x') for every E-G F ) .

W

A

(1) (2)

Let (~,A)

b e a B a n a c h cat~go~y. 1)e~ne

x"~"-A'(G,E)

)~,

~',

)(y",~'x"),

z" ~ y"- A'(G, E)

~~ ,

~',

~ (~", y"~')

Definition 1.5.2.10

for E , F , G c ~ , EV-~ F ~-~ G. A"

An

-q and ~ are called the right and the left

A r e n s multiplication, respectively (1951). Remark.

q and F- may be different (see Remark to Example 1.5.2.15).

294

1. Banach Spaces

(i)(2)

Proposition 1.5.2.11 Let A be a B a n a c h category, E , F , G , H objects of A, and A a left (right) A-module. Then a pt

E--+ F -~ G Y+ H ::=::vx ' ( b" -q a " ) = (x' b" ) a" A"

(E --+ F .A t

.A'

A pt

-~ G ~-~ H ~ A"

(b" ~- a " ) x ' = b"(a"x')

A"

).

By Proposition 1.5.2.9 a) <x,x'(b" q a")) = (b" q a",xx') = (a", (xx')b") = ( a " , x ( x ' b " ) ) - {x, (x'b")a")

((x, (b" F- a")x') = (b" F- a",x'x) - (b",a"(x'x)) - (b", (a"x')x) = (x,b"(a"x')))\ for every H -5+E.

m

A

Theorem 1.5.2.12

( 1 ) (2)

Let A be aBanachcategory.

a) q and ~ are inner multiplications on A". b) A" together with q (with ~-) is a A-category. We denote this category by A'~ (A~). The evaluation functor of A into A'~ (into A'~ ) is a functor of A categories.

d) If A is a left (right) A-module, then ,4' is a right A'~-module (left A~module). e) A' is a left A'~-module and a right A~-module.

f) If A' is a (A~,A~)-module, then (z" q y") ~ x" -- z" -~ (y" ~ x") for all objects E, F, G, H of A and E~F~GS~H. Apt

Apt

Au

g) If A is a unital Banach category, then A'~, A'~ are unital A-categories. In this case, the evaluation functor of A into these unital categories is a functor of unital A-categories, and A' is a unital left A~-module and a unital right A~-module.

1.5 Banach Categories

h)

295

If j denotes the evaluation functor of A, then x"x -- x" ~ (jEFX) -- x" ~ (jEFX) ,

y x " - - (jGHY) ~- x" = (jGHY) ~ X" for all objects E, F, G, H of A and x I!

E-~ F-+ A

i)

Let E , F , G , H

G -~ H .

A"

A

x/!

be objects of A and take F ~

G. Then the maps

Atl

A"(E,F)

>A " ( E , G ) ,

y",

>x" q y " ,

A"(G,H)

>A " ( E , G ) ,

y",

>y"~-x"

are continuous with respect to the topologies of pointwise convergence. It is easy to see that -~ and ~ are (A", A", A")-multiplications. Let E, F, G, H be objects of A. In order to simplify notation, we adopt the convention that the symbols x, y denote morphisms of A, the symbol x' denotes a morphism of A', and the symbols x", y", z" denote morphisms of A". a) By Corollary 1.5.2.7, A" is a A- module. Let

-+H. By Proposition 1.5.2.11, ((z" ~ y") ~ ~ " , x ' ) = ( x " , ~ ' ( z " ~ y")) - (x", (x'~")~"l -

= (y" ~ ~", ~ ' z " ) - (z" ~ (y" ~ x"), ~'), ((~" ~ y") ~ ~",~') = (z" ~ y",x"~') = (z",y"(x"x')) = (~", (y" ~ ~")x') = (z"~ (y"~ ~"), ~'), x !

for every H - + E . Thus

(z" -~ y") ~ x" -- z" -t (y" ~ x") ,

296

1. Banach Spaces

(z" ~ y") ~ x" -- z" ~ (y" ~- x").

b) Step 1

E -~ F ~ F-5+H ~ "

x(y" ~ x") - (xy") t- x" i x ( y " ~ x") - (~y") ~ ~"

By Proposition 1.5.2.9, (x(y" ~ x"), x') - (y" -~ ~", x'x) - (x", (x' z)~"l -

= (x", ~'(xy")) - ((zy") ~ x", x ' ) ,

(~(y" ~ ~"),~') - (y" ~- x " , x ' z ) = ( ~ " , x " ( x ' x ) ) -

= (y", (x"x')~) = ( x y " , x " ~ ' ) =

( ( x y " ) ~ x",~')

for every H -~ E.

st~p 2 E-~F -~ r - ~ H ~

{ ( ~ " -~ x")~ - y" -~ (x"x) (y"~ x")x - ~"~ (~"~:)

Given H - ~ E ,

((y" ~ x")x, x') - (y" ~ x", xx') = (z", (~x')y") = (:~", :~(x'y"))= (x"x, x'~")= (y"-~ (x"~), x'), ((y" ~ x")x, x') - (y" ~- ~", ~ ' )

= ( y " , ~ " ( : ~ z ' ) ) = (y", ( x " x ) x ' ) -

-

(y"~ (x"~),x')

(Proposition 1.5.2.6). Step 3

E~

-~ G ~ H ~

~ (Sx) (Y"x) ~~ ~x "" -- y"~ y'' ~ (xx") (~") (

Given H - ~ E , ((y"~) ~ ~",~') - ( ~ " , ~ ' ( y " ~ ) ) = (~", (~'y")z) - ( ~ " , z ' y " )

- (y" ~ ( ~ x " ) , ~ ' ) ,

((y"~) ~- ~",~') - (y"~,~"~:') - (y",~(~"~')) - (y", (~x")~') = (y" ~- ( x ~ " ) , x ' )

1.5 Banach Categories

297

(Proposition 1.5.2.6). Step 4 b) Follows from Corollary 1.5.1.13 and the preceding steps. c) By Proposition 1.5.2.4, the evaluation functor j of A is a functor of A-modules. Take E ~ F--~G. Then (j(yx),x')-

(y(jx),x')=

(y, ( j x ) x ' ) -

(jy, (jx)x'} = ( ( j y ) ~ ( j x ) , x ' l ,

( j ( y x ) , x ' I - ( ( j y ) x , x ' } - ( x , x ' ( j y ) } - (jx, x'(jy)) - ((jy) ~ ( j x ) , x ' ) , for every G - ~ E , so that j ( y x ) = (jy) ~ (jx) - (jy) -~ ( j x ) . d) follows from b) and Proposition 1.5.2.11. e) follows from Corollary 1.5.1.13 and d). f) Given H - ~ E ,

((z" ~ ~") ~ ~",~') = (z" ~ y",x"x') - (y", ( ~ " ~ ' ) ~ " ) = = (y",~"(~'z")) = (y"~ ~",~'z") - (z" ~ (y"~ x"),~'). x II

g) Take E ~ F .

Then

( x " ~ (jE1E),X') - (X", (jE1E)X'} - (X", 1 E X ' ) = (X",X'},

((j~l~)

~ ~",~'1 -

( j . l ~ ,x"x') = (1 ~, x"x') = ( X " , x ' l g ) -

(x",X'),

(x" -~ (jE1E), x') = (jE1E , X'X") -- (1 E , X'X") = (X" , 1EX') = (X", X'} , ((jrlF)-t x",x')=

(X",x'(jF1F))-

(X",X'IF) - (X",X')

for every F---~E (Corollary 1.5.2.7, Corollary 1.5.1.13). Thus X" ~- (jE1E) -- X",

(jF 1F) ~ X" = X",

x" -t (jE1E) = x",

(jF1F) -~ X " = X".

Hence A"~ and A"~ are unital A-categories and the evaluation functor of A into these categories is a functor of unital A-categories.

298

1. Banach Spaces

h) We have

(x" F- (jEFX), X') = (X", (jEFX)X') = (X", XX') = (X"X,X') , (x" -d (jEFX),X') -- (jEFX, X'X") -- (X,X'X") = (X"X,X') for every G - ~ E and

( (jaHY) F- X", X') = (jagY, X"X') -- (y, X" X') -- (yx", X'), ((jaHY)-d X",X')= (X",x'(jGHy))= (X",X'y)= (yx",x') ggt

for every H--+F (Corollary 1.5.2.7). i)We have lira (x" --t z",x') -

Ztt---~ytt

for every E ~ F ,

lim (z",x'x") - (y",x'x") = (x" --t y",x')

z"---+ytt

G -~ E , and

lim (z" ~ x", x') = ztt--+ytt lim (z", x"x') = (y", x"x') = (y" F- x", x')

ztt--._~ytt

tt

for every G ~ H ,

Xt

H -+ F .

m

P r o p o s i t i o n 1.5.2.13 Let A be a Banach category and E, F , G , H objects of A. Take F Art ~-~G. Let j denote the evaluation functor of A' and put

u=, : A(F, G)

~, A'(E, G) ,

(resp. u:, "A(F,G)

x,

~. A ' ( F , H ) ,

~ xx' x,

~ x'x)

for every E -~ F (resp. G -~ H ). Then the following are equivalent: At

a)

At

x It

E --+ F =~ x" ~ y" = x" -~ y" AH

(resp. G Art ~-~ g =~ y" F- x" = y" -t x" ). b)

The map A"(E,F) ( resp.A" ( G, H)

~ A"(E, G), >A" ( F, G) ,

y", y" ,

; x" ~ y" > y" -t x")

is continuous with respect to the topologies of pointwise convergence.

1.5 Banach Categories

c)

299

E s p ~ ~".x" = j~(x"~') At

(~p.

II x l l c ~3s H ~ u~, = j~(~'~")

).

A~

d)

E -~ F ~ ux,"x" e jEc(A'(E, G)) At

(~p. a ~AH t

II II ~ ~,~ 9 j~,(A'(P,H))).

a ~ b follows from Theorem 1.5.2.12 i). b ~ a. Let i be the evaluation functor on A. Then iEF(A(E, F)) (resp. icH(A(G, H ) ) ) is dense in A"(E, F) (resp. in A"(G, H ) ) with respect to the topology of pointwise convergence (Corollary 1.3.6.5). By Theorem 1.5.2.12 h), x" F- y" - x" A y"

(resp. y" ~ x" = y" -~ x")

for every y" e iEF(A(E,F)) (resp. y" e iaH(A(G,H)) ). By continuity, ( b ) and Theorem 1.5.2.12 i)), x" k y" = x" q y"

(resp. y" F- x" = y" -~ x")

for every E - ~ F (resp. G - ~ H). AH

A"

a :=~ c. By Theorem 1.5.2.12 h) and Proposition 1.5.2.9 b), (y", ~," ~") = (~", ~x,y' "'~ = (x", ~'y") = (y" ~ x", ~') =

= ( y " ~ x",x') - (y",x"x'} -- (y",jEG(x"x'))

for every G -~ E (resp. AH

(~". ~".~") - (x", ~'~,y) - (x", y"~') - (x" ~ y", x') = - (~"~ y " . ~ ' ) - (y". ~'~")= (y". j~.(x'x")) p!

for every H ~,, F ), so tl X t t

u x,

- jEa(X"X')

It tl (resp. ux, x -- jFu(X'X")).

C =~ d is trivial. d ==> b. By Proposition 1.5.2.9 b), ( ~ " ~ y". ~'} - (x". ~" ~'} - (x" . ~.' y "'~ = ~ . '" ~". ~")

for every E -~ F and G -~ E A"

AI

300

1. Banach Spaces

(resp. (y" -t x", x') - (x", x'y") - (x", u'x,, y"> -A " ( E , G ) ,

(resp. A"(G, H)

y",

> A"(F,G),

> x " ~ y" y",

> y" -~ x")

is continuous with respect to the topologies of pointwise convergence,

m

P r o p o s i t i o n 1.5.2.14 Let (~2, A), (~2, B) be two Banach categories and let f be a functor of Banach categories of A into B. a)

I I E _5+ F 13' v_~G ::v (fFaY )x A

~

I fsa(Y'fEFX)

l I E s13' F 4,4 G :=~ x(fEFY ) = f ~I a ( ( f F a x ) y I ).

b)

1 I E ~ F ~ G ~ (fFay )x" A"

__

13'

till'!

I

E 13' v_~F A-~" G ~ x tJEFY ) EG ~ ~ a ~ .4" .A"

__

J"' E G t Y" l

l/tit

'"" JE, F x "") , It

IEG ((JFa x )Y').

f ~u ( Y " ~ ~'') - ( eJu ~ Y u ~J ~ "( ]~ E" F x " ) f~c(Y" ~ x") - (]i:aY"" ") -~ t.JEF "),

i.e. f" is a functor of Banach categories of .A~ into 13~ (of Jt~ into r2,,~ a) Take G --%E. Then A

(a, (f'ray')x) - (xa, f ' p a Y ' ) - (far(xa),y') - ((fErx)(fc, E a ) , y ' ) = (/~a,

y ' f ~ . ~ > - (a, f ~ ( y ' / ~ ) >

in the first case and (a,x(f~y'))-

(ax, f'EPY')- (fpE(ax), y ' ) -

((faEa)(fpax),y')-

= (faEa, (frax)y'} -- (a, f'~a((fpax)Y')) in the second case. b) Let G - ~ E . By a), A

(X, I''

(JraY ,~)x ,, )

_

( X u,

xfFay') = (x", I'FE((faEx)Y')}

=

1.5 Banach Categories

__

-

301

I1 :EFx , (faEx)y') = (faEX, y ' :"E' : " ) = (x, :'Ea' ~y"'"':Erx"')>

( I'll

in the first case and (x , x" f 'EF y'X, = (x" , (fEFY)X) ' ' ' ' faEX)) = (x" , fGF(Y _- P:lFl a x It .y I faEx) -- (faEX, (:" :..: "")y')

-

-

I /I'll It (x.:'~.(~:,~: )y')>

in the second case. X! c) Let G---+E. By b), A'

(f~.(y" ~ ~") .~') = (y" ~ ~,,. : ~,: ,> - (~/.,~"":~:~'" =

--

Y", ."IF fG

II

II

!

((fEFX )X ) } -

(~11 / I I JEG(Y __-- ( X l l ~ J F E ~ /

-I

Xlt

I :FGYPII

E x a m p l e 1.5.2.15

~i'll

II

11

/!

1

//tl!

!1\

~,:FaY , (f~Fx )X ) - - ~,(JFaY ) ~ (f~Fx"), x'),

), x') =

(y#

I!~)> __ ~ JrEl l F

I!

p

I

~ x , f 3 ~ x ) -- (x", (f~E ')y") = !1 ~ xlPI!jFGylI~)

= ~(]FGYl/rll

II) ~1 ~ ] E F

I1~)~ x l >

.

B

Let ~2 be a class of sets and A the Banach system

(s, T) . ~ :,~(S, T) defined on $22 endowed with the multiplication defined in Proposition 1.2.3.5. Take P, R, S, T e ~2 .

a)

A is a Banach category. For convenience, we settle on the following notations: 1) h , k

denote morphisms of A ,

k' denotes a morphism of

A' and h", k", 6" denote morphisms of A" ; 2) p, r, s, t denote points of P, R, S, T , respectively.

b)

A' may be identified with the A-module

(s, T) ~. > ~',~176 T) , kk" T x R

~, IK,

(t, ~) ,

~~

k(~, t)k'(~, ~) 8

for R - ~ S -~ T , and k'k" T x R - - + lK,

(t, ~) ~

~

k(~, ~)k'(t, ~) 8

fo~ R - S S ~ T .

302

1. Banach Spaces

c) A" may be identified with the map ( S ,T) ,

>E(S, T) ,

where E ( S , T ) is the Banach space introduced in Proposition 1.2.3.7 a) for p - 1 and

cl)

R ~ S -~ T :=v (kk")r

:

f

k(s,-)kT({s}) ,

k(s, .)dkT(s ) - E 8

k II

R -~ S -~ T ~ (k"k)r- ~

~)

k(~, ~)k;', 8

c3) R -+ S -~ T =~ ( [t, r] = k'(t, )dk~r' ~" k'k") f 9 , R -~ S ~-~ T ~

C4)

(kt'kt)[t, r] -- E

k'(s,r)k;'({t}), 8

(h" q k")r - fhTdk'~'(s)

R~ s ~ T~

%)

(h" ~- k")~- E k"({~})h~. 8

d)

A~ is a unital Banach category. For t C T, and A' is a unital left A~-module.

e)

A' is a

(1T)t

iS Dirac measure at t

(AT,A~)-module.

f) P 5 R g S ~4 T ~ (h" ~ k") ~ e" = h" ~ (k" ~ ~")

a) is easy to see. b) The identification follows from Proposition 1.2.3.6 c). Given T --~ R, (h, kk') - (hk, k') - ~ ( h k ) [ ~ ,

~]k'(~, r) -

8~r

= E (Ek(s't)h(t'r))k'(s'r) s,r

- Eh(t,r)(Ek(s,t)k'(s,r)),

t

t,r

/h, k'k) = (kh, k') = ~ ( k h ) [ t ,

s

s]k'(t, ~) -

t~s

E ( E h(t,r)k(r,s))k'(t,~)- ~ h(,, r)(~ k(~,s)k'(~,~)) t~8

r

t,r

s

1.5 Banach Categories

303

c) The identification follows from Proposition 1.2.3.7 b). We have:

c,)

T -+ kR~

: :~ r

i

@k", k')= @", k'k) = ~

S(E ~(',')~'(','))"~"(')-

(k'k)(~, ~)dk"(~)=

r

t

-~ ~,(..,)(I~(,.,),~.,(,))r~t

r,t

r

s

T -+ R ~

(k"k, k')= (k"kk')= ~

(kk')[~,t]dk'~'(t) = 8

8

r

r

8

c3) r 4 e ~

(k, k'k")= (k", kk')= Z / ( k k ' ) [ ~ , ~]ek'r'(~)= 8

-E

s

S(~ ~("r)~'("'))'~"(')t

t,r

c4) T & R ~

(k, k"k') = (k", k'k) = ~ f(k'k)[s, t]dk2'(t) = 8

- ~ S(v- ~(,..)~'(,..)),~'.' (~)- ~ ~,(...)(S~(,..),~,.,(,))__ 8

r

8~r

8~r

t

t,r

s

c5) r -~ ~' R ~

(h" ~ k" , k')= @", k'h") = ~ S @'h")[~,~]ek"(~) = r

r

r

= (h" V k", k') = (h", k"k') = E J(k"k')[s, t]dhT(t ) = 8

8

r

: Efk'(~, )d(E k'r'((~)h:) r

8

304

1. Banach Spaces

d) We have (1r -t k")s -- f f (1T)tdk:'(t) = k~',

(h" ~ lr)~ = f h',:d(lr)~[t'] - h~ k"

h"

for S --+ T --+ R and S -~ T. e)

P -~ R -~ S -~ T ~ : f(E

((k"k')h")[t,p]-

k'(s, r ) k : ( { t } ) ) d h ; ( r ) 8

= E

E

f (k"k')[t,~]dh';(~) (Jk'(s, r)dh;(r))k:({t})-

$

(k'h")[s,p]k;'({t})- (k"(k'h"))[t,p]. 8

f) follows from e) and Theorem 1.5.2.12 f). Remark.

If k T ( { s } ) - 0 for every (r,s) e R x

S incs),then

h" t- k" = 0 for every h". Take S -~ S and for s E S , let hi~' be the Dirac measure at s.

Then h" -t k" -- k" by d). Hence t- and --t may be different. Definition 1.5.2.16 ( 2 ) Let (X?,A) be a Banach category and .4 a A module (A-category). A A-submodule (,4 subcategory) of .4 is a A-module (A-category) 13 such that B(E, F) is a Banach subspace of .4(E, F) for every E, F C ,(2 and such that the multiplications of B are the restrictions of the multiplications of .4. The map defined on f2 2 such that f (E, F) is the inclusion map

u(E, F)

~A(E, F)

for every E, F C ~2 is called the inclusion functor of 13 into .4. P r o p o s i t i o n 1.5.2.17 ( 2 ) Let (O,A) be a Banach category. Let ,,4 be a A-module (A-category) and 13 a A-submodule (A-subcategory) of A . Put

C(E,F)

:=

A(E,F)/U(E,F)

1.5 Banach Categories

305

and let f (E, F) denote the quotient map ,,4(E, F)

. ~ C(E, F)

for E, F E ~ . Define multiplications on C by taking the factorizations. Then C is a A-module (A-category) and f is a functor of A-modules (A-categories). C is called the quotient A-module (quotient A - c a t e g o r y ) o f gl by 13 and is denoted by A/13; f is called the quotient functov of A onto A / B . Let (O,A) be a unital Banach category. If A is a unital A-module, then A/13 is a unital A-module. If A is a unital A-category, then A/13 is a unital A-category and the quotient functor of A

onto A/13 is a functor of unital

A-categories. The proof is a long verification. P r o p o s i t i o n 1.5.2.18

( 2 )

I

Let (a2, A) be a Banach category, .4 a A -

module, and 13 a A-submodule of .4. Given E, F E $2, set B~

:= 13(F,E) ~ .

Then 13~ is a A-submodule of .A' . For E, F E g2 , let f (E, F) denote the canonical isometry (.A'/B~

>B ' ( E , F )

(Proposition 1.3.5.2). Then f is an isometric functor of A-modules of A'/13 ~ into 13'. Take E, F, G, H E ;2 and consider x !

E-~F--+G--~H. A

B~

A

Then {m,m'a) =

(az, z'}

= O,

(y. b.') = ( v b . . ' ) for every

G-~E, 13

so that

H-~F 13

= 0

306

1. Banach Spaces

x'a C B ~

bx' C B ~

Hence B ~ is a A-submodule of A'. Take E, F, G, H E $2, X!

E-~F--+G-2+H, A

.,4'

A

and let q be the quotient functor of ,4' onto .A'/13 ~ . We have

(x, f ((qx')a)) = (x, (qx')a) = (ax, qx') = (ax, f (qx')) = (x, (f (qx'))a) , (y, f (b(qx'))) - (y, b(qx')) = (yb, qx') - (yb, f (qx')) = (y, bf (qx')) for every

G-~E, B

H--~F. 15

m

This proves the last assertion.

Proposition 1.5.2.19 tion functor of .4, and

( 2 ) Let (~,.A) be a Banach system, i the evalua-

(~(A)~

r ) . - (~(.4)(r, E)) o

If M' is a Banach category and i(A) is a submodule of M", then i(A) ~ is an ,t ,,, _submodule. A~'-submodule and an .-'4 We denote by j the evaluation functor of .,4'. Stepl

E ~ F ~-~ G ~ H = = , x ' " x " = O , y " x " ' = O .

We have

(~', x'" x") = (~'", x"x') = o,

(y', y"~'") = (~'", y'y") = o

for every

G-~E, A'

Step 2 E ~

F ~ G ~ i(.4)o ~(.4)o

For G - ~ E ,

H Y-~F. A'

y'" F- x'", y"' -~ x'" c (i(A)~

G) .

1.5 Banach Categories

307

(y'" ~ x'", x") - (y"', x'" x") = O,

(y"' -~ x'", x") = (x'", x"y"') - 0

by Step 1. Step3 E ~ j(A')

F Y-~ G ~-~ H===v i(A)

j(A')

~

y,! ~ x , l y,! -d x "l, z "l ~- y , t z m --t ym are morphisms of

i(A) ~

Take x !

zI

E~F,

G-+H

A'

A'

with XlI!

--

9 JEFX

!

~

Zl!!

~

9 JGH

z

!

.

Then (y'" -~ x'", x") -- (x'", x"y'") -- O,

(y"' ~- x'", x") - (y"', x'" x") - (y"', ( j E r x ' ) x " )

(z'" ~ y"',y") -

-- (y"', x ' x " ) -- O,

(z'",y"'y") - o ,

(z'" -~ y'", y") -- (y'", y"z'"} -- (y"', y " ( j a H Z ' ) ) = (Y'", y"z') -- 0

for every

G~E, HY--~F i(A) i(A) by Step 1 and Corollary 1.5.2.7. Step 4

i ( A ) ~ is an A~'-submodule and an A~'-submodule.

The assertion follows from Steps 2 and 3 together with Proposition 1.3.6.19 d). m

308

1.6

1. Banach Spaces

Nuclear

Maps

Several classes of compact operators on Hilbert spaces are known. These classes have some connection or other to the eP-spaces (p E {0} U [1, oc[). The class of nuclear operators, which are also called trace operators, arises when p = 1. They are the subject of this section. Their strong properties make the theory also applicable to Banach spaces, as shown by Alexander Grothendieck. Since this theory is not pursued further in this book, the reader may skip this section. 1.6.1 G e n e r a l R e s u l t s D e f i n i t i o n 1.6.1.1

( 0 ) (Grothendieck, 1952) Let E , F

be normed spaces.

A map u" E -+ F is called nuclear if there is a family ((x'e, Y~))eE, in E' x F such that

eel

and

eEI

for every x E E . We write s

F) for the set of nuclear maps of E into F

and define

I1~ I ~ - i~f E I1~:11Ily~ll eEI

for u E E , I ( E , F ) , where the infimum is taken over all families ((x:,Ye))eE, in E' • F with the above properties. We set E,'(E) := f ~ I ( E , F ) .

We can replace the indexing set I by IN in the above definition, since

{~ E I I I1~:1111Y~II# 0} is countable. E x a m p l e 1.6.1.2 ( 3 ) L~t T b~ ~ ~t, E .-~0(T) F :-- ~P(T), p E [1, oo], y E F , and take u:E

~F,

x~

~xy.

Then u E f~l(E) iff y E gl(T) and in this case

Ilulll--Ilylll.

(~p. E . - ~ ( T ) ) ,

1.6 Nuclear Maps

309

Assume t h a t y c g l ( T ) . Given t c T , set

x t''E

)IK,

x,

)x(t)y(t).

Then

tCT

tET

E (x' x't}et - E tcT

x(t)y(t)et - ux

tET

for every x C E . Hence u is nuclear and

Assume t h a t u is nuclear. There is a family ((x'~, y~))~e, in E ' x F , such that

i~',ll i y, II < oo LCI

and

for every x E E . Then

LEI

so t h a t

LEI

~EI

for every t E T . Hence

Z ly(t)l ___~

~

tET

~CI

- ~ ~EI

tCT

i~, i~

I(~,, ~:)1 lly, i, =

I(~,, ~:)i _< Z Ily, II IIz:li,

tET

tEI

so t h a t y E gl(T) and

liyill _< ii~Jli 9

I

310

1. Banach Spaces

T h e o r e m 1.6.1.3

s

a)

( 0 ) Let E , F

be normed spaces.

is a vector subspace of f ~ / ( E , F ) and

for every u 9 s (E, F ) . The map

b)

C~(E,F)

>~+,

u,

>lull,

is a norm. Take f_.l(E, F) with this norm. c)

f_,I(E,F) is a dense set o f / ~ I ( E , F ) .

d)

/21(E,F) is complete whenever F is complete.

e)

u 9s

IIr

~ u' 9 s

llullx.

~

Let ((x't, yt))t e, be a family in E' • F

a), b), and c). Take u 9 s with

Ifx:fl Ily~fl < tEI

and such that

LGI

for every x E E . u is linear and

II<x,x:>y~ll = ~ I<x,~:>l Ily~ll ~
kz

(Proposition 1.2.3.8 a) ). Then u

u

u

Ilk lco(Z)ll, = I kll, = I kll

k c Z,'(e~(T),eP(S)),

for every k E fl'p(T,S) and the map

e"p(T,S)

, Z:'(~o(T),eP(S)),

k

~, ~lco(T)

is an isometry.

Step 1

Ca(co(r),

3!k e t~I'p(T, S), o

k I ~o(T) - u, Ilkll _< It-ll,.

Let ((x',, Y,)),c, be a family in co(T)' x gP(S) such that

Y~ IIXII ly, I < o~ and

tCI

for every x c co(T). We identify co(T)' canonically with el(T) (Example 1.2.2.3 e) ) and set

k.TxS

~:,

(t,~),

>(~r

1.6 Nuclear Maps

327

Let q be the conjugate exponent of p. Take t E T . T h e n

sES

sES

tel

sES

sES

_
IK ,

h" gl'p(T,S)

k,

>

~

h(t, s)k(t, s)

(t,s)CT•

belongs to gl'p(T, S)' for every h e

e~,'(T, S), and the map

> el'p(T, S)',

g~'q(T, S)

h,

)h

is an isometry. This proves the corollary. P r o p o s i t i o n 1.6.2.4 ( 1 ) ( 3 ) exponents. Given k C gl,q(s, T), let A

k" gP(T)

II

Let S , T be sets and let p,q be conjugate

~ g~(S),

N

x,

> kx

(Proposition 1.2.3.4 a) ). N

~) If p # ~ ,

th~n k c

L~(e~(T),eI(S)) fo~

g~'q(S,T)

-~/:I(~P(T), ~1(S)),

~y

k e e~,~(Z,T) and th~

map n

k,

)k

is an isometry.

b) If p -

A

cxD, then k E / : I ( g ~ ( T ) , el(S)), N

n

Ilklco(T)lll = IIk]l~ - I kll for every k E gl,l(S, T) and the map

e 1'1(S, T) is an isometry.

> ~1 (C0(T), g1(S)),

k,

0

> k lco(T)

330

1. Banach Spaces

Put E:=

~ eP(T)

t

if if

co(T)

pr p = cx~.

{ 3k E ~"q(S,T), Step 1

u E s

==~

n

klE = ~,

Ilkll _< I1~ I1.

Let (( x'L, YL))~I be a family in E' • gt(S) such t h a t

IIx: I Ily~ll < c~ LEI

and

eEI

for every x e E . Identify E' with

k. SxT

e~(T)

~IK,

(Example 1.2.2.3 d), e)) and set

(~,t),

~(~r

Then

tET

tCT

for every x E IK (T) and s C S. By Proposition 1.2.2.2,

LCI

for every s C S . Hence

k Es

T)

and

Ilkll _< l u l l .

k(s, .) C gq(T)

and

1.6 Nuclear Maps

331

We have n

k~T - k(., t) : ~ T

for every t E T and so N kx

for every x

E

IK (T) .

Since

:

ttX

is dense in E (Proposition 1.1.2.6 c)),

]K (T)

M

klE=~. The uniqueness of k is obvious. {k Step 2

k E ~I'q(S,T)

==~

Cs n

n

IlklEIl~ = Ilkll, = Ilkll.

Given s c S, let x s' : = k ( s , - ) ,

ys:=esS.

Then x 's C gP(T)' and ys c t~1(S) for every s E S,

IIx'~ll Ily~ll = ~ s6S

IIk(~, )llq = I kll
_ y} c E+ = E + ,

yE x ( t ) y ( t ) tET

(Example 1.2.2.3 a)). a) Given x C gq(T), the linear form ~ is positive iff x is positive (Example 1.7.2.8 a)). b) x e eP(T) 7r, whenever x e eq(T). c)

gP(T) ~ = (x' l x' is an order continuous linear form on gP(T)} ,

~P(T) ~ - (x' l x' is an order a-continuous linear form on gP(T)}. d) The map ~ . q ( T ) ----} ~ . P ( T ) 7r ,

x,

) 'x

is an isometry of ordered Banach spaces (Proposition 1.7.1.9, Proposition 1.7.2.7). a) is easy to see. b) Since every element of t~q(T) is a linear combination of positive elements it is sufficient to show that ~ is order continuous for every x C t~q(T). Case 1

p = 1

The assertion follows from the last assertion of Proposition 1.7.2.5 and Example 1.7.2.8 d)). Case 2

p :/: 1

It is obvious that ~ is order continuous if x C IK (T) . Since IK (T) is a dense set of gq(T) (Proposition 1.1.2.6 c)), it follows from Proposition 1.7.2.7 (and Example 1.2.2.3 b)), that ~ is order continuous for x C gq(T). c) follows from the fact that gP(T) is a vector lattice. d)

Case 1

p -~-

By Example 1.2.2.3 d), the map

346

I. Banach Spaces

gq(T) , > e ( T ) ' ,

x,

>

is an isometry of Banach spaces and so the assertion follows from a) and b). Case 2

p = c~

Let x' be a positive order continuous linear form on g ~ ( T ) . We define

x" T

>IK,

t,

>x'(et) .

Then

x(t) - ~ tEA

x'(et) = x(ea) a. b ==v a. Let A be a downward directed countable set of E with infimum 0 and let ~ be its lower section filter. A s s u m e t h a t x ' ( ~ ) d o e s not converge to 0. T h e n there is an c > 0 such t h a t for every x E A there is some y E A such that y<x

and

Ix'(y) l > ~. Thus there is a decreasing sequence (Yn)nE~ in A with infimum 0 such t h a t

I~'(y~)l > for every n E IN. Given n E IN, define

Xn .m Yn -- Yn+l 9

338

1. Banach Spaces

Then

yn ~<m

Xk

k E IN k>n

for every n E IN. By b), =

kEIN

k >_ n

for every n E IN. We deduce the contradiction that 0-n~o~lim I E x'(xk)l = ~-,~limIx'(Yn)l > ~ .

I

kEIN

k >_ n

P r o p o s i t i o n 1.7.2.11 Let E be a a-complete ordered vector space and (x~)ne~ a sequence of order continuous (order a-continuous) linear forms on E such that (x~(x)),e~ converges for every x E E . Then x''E--~IK,

x:

!

; n---~ lim(:X3 Xn(X )

is order continuous (order a-continuous).

Let (X~),EI be an order summable countable family in E+. Given a linear form y' on E , define y" I

) IK,

to

) y'(x~) .

Let (aL)~EI be a bounded family in JR+. By Proposition 1.7.2 10 a =~ b x n E t~l (I) and

for every n E IN. Hence lim E

o~x'~(~)

n-~oc

LEI

--

x' ( E - - - )OZ~XL

"

LEI

It follows that (X~)nEr~ is a weak Cauchy sequence in gl(I). By Schur's Theorem (Theorem 1.3.6.11), x ' E gl(I) and lim IIx" - x'll, - 0.

n--.+oo

1.7 Ordered Banach spaces

Thus

f

349

)

Xt k tEI

~EI

tGI

tEI

By Proposition 1.7.2.10 b => a, x' is order a-continuous. Now suppose t h a t each x 'n (n C IN) is order-continuous and t h a t x' is not order continuous. Then there are a downward directed set A of E with infimum 0 and an c > 0 such t h a t for every x E A, there is a y E A with y < x and

Ix'(y)l > ~. We may construct a decreasing sequence (xn)ne~ in A inductively such that for every n C IN I X t ( X n ) I ~> -C

and k ~ ~

1

~

tx~(x~) I _< - . n

We put X:~-

AXn. nE ]N

Then !

!

Xk(X ) -

lim Xk(X,~ ) - 0

n--~ OQ

for every k C IN and I x ' ( x ) l - lim Ix'(xn)l _> e. n---9.oo

This leads to the contradiction t h a t

< x'(x)l-

lim Ix~(x)l = 0.

I

k--+c~ /

D e f i n i t i o n 1.7.2.12

(

0

)

Let T be a locally compact space. The open set

U of T is called an e x a c t set o f T if it is of the f o r m

u-{~r f o r some x E C(T).

350

1. Banach Spaces

Let if; be the a-algebra on T generated by the exact sets of T . The elements of T, are called Baire sets and the T,-measurable functions on T are called Baire functions. T

is called a Stone space ( a - S t o n e space) if the closure o.f any open

(exact) set of T is open. A hyperstonian space is a Stone space T in which

U

supp#

ttECo(T)~

is dense. By Urysohn's Theorem, every open a - c o m p a c t set of T is exact. The intersection of a finite family of exact sets is exact, as is the union of a countable family of exact sets. If T is metrizable, then every open set of T is exact, so that every Borel function on T is a Baire function. Proposition

1.7.2.13

( 0 )

Let T be a locally compact space. Then the

following are equivalent: a)

T is a Stone (a-Stone) space.

b) If x is a bounded Borel (Baire) function on T , then there is a y E C(T) such that {x ~ y} is meager. c) Every nonempty (countable) family (x~)~e, in C(T)+ has an infimum y in C(T) and

{t E T ly(t ) ~ infx~(t)} tEI

is meager. d)

C(T) is order complete (order a-complete).

e)

Co(T) is order complete (order a-complete).

The function y in b) is unique and

y(T) c ~(T). a =~ b. Let ~R be the set of subsets A of T for which there is a clopen set U of T such that

(A\U) U(U\A) is meager. By a), 9~ contains the open (exact) sets of T . It is easy to see that ~R is a a-algebra. Hence, every Borel (Baire) set of T belongs to 9~. By the definition of 9~, for each A E 9~ there is some y E C(T) for which {CA -~ y} is meager. It follows that b) holds for step functions on T with respect

1.7 Ordered Banach spaces

351

to 9~. Since x is a bounded Borel (Baire) function on T , there is a sequence ( X n ) n ~ of step functions on T with respect to 9t: converging uniformly to x . For every n C IN, there is a Yn C C(T) fbr which {xn ~ Yn} is meager. Since T is a Baire space,

Ilym - y,~llo~ < IIx,,, - xnlloo for m, n C IN. Hence (Yn)nEIN is a Cauchy sequence in C(T). Set y "-- lim yn. n--). oo

Then

{X # y} C U {xn # Yn}. nE]N Hence {x # y} is meager. b ==> c. Define y'T

>r

t,

>infx,(t).

y is a bounded Borel (Baire) function. By b), there is an x E C(T) for which {x # y} is meager. Since T is a Baire space, x is the infimum in C(T) of c :=> d =::> e is easy to see. e ==~ a. Let U be an open (exact) set of T . We are required to prove that U is open. We may assume U to be relatively compact. First assume that U is open and put

.T" "- {x c Co(T)+ Ix ~ ~ } ,

V xeCo(T).

y-

xC~

By Urysohn's Lemma, y is 1 on U and 0 on T \ U . Hence -1

u -

y (]0, ~ [ )

is open. Now let U be an exact set of T and take x E C0(T) such that U-

{x ~ 0}. Let

Y := V ( ~ / ~ (~lxl))

nEIN

e

do(T).

352

1. Banach Spaces

m

Then y is 1 on U and 0 on T \ U . Hence m

-1

u = y (]0, ~ [ ) is open.

m

The last assertion follows from the fact t h a t T is a Baire space. Example 1.7.2.14

( 0 ) Let T be a locally compact space. Take it 9 .h/lb(T)

and let F := Supp it 9 We identify .Mb(T) canonically with Co(T)' a) b)

# 9 Co(T) ~ iff every meager set of T is a # - n u l l set. -6 # 9 Co(T)~ ==vF = F .

c)

If T is Stonian and # 9 Co(T) ~ , then F is open and there is a unique

x 9 Co(T) with = ~.1,1,

I~1 < ~

In this case Ixl = eF. If in addition # is real, then x is also real and ,§

- ~§

,-

- ~-1,1

9

d) T is hyperstonian iff Co(T)co(T). is Hausdorff and in this case for every open nonempty set U of T

there is a ~ 9 Co(T)~_ with S u p p v

compact,

nonempty, and contained in U . a) Take # 9 Co(T) ~ . Let K be a nowhere dense compact set of T . Let 9= {x 9 Co(T) l e g F

1.7.1.4 1.5.1.1

E _5+ F

1.5.1.1

A

ElF

1.2.4.1

f'

1.1.6.24

~n

1.2.6.1 1.1.6.1

~I

/IS f(a, .) f(.,b) f(A) f(x)

NT

f-1

NT

NT NT NT NT

-1

f (B)

NT

-1

f (y)

NT

f ' X--+ Y NT f " X -+ Y . x ~> T(x) F[s,t] NT F[t] NT F 9G

1.2.5.3

{f = g}

NT

{f -r g}

NT

{f > a} gof -~A

im

NT NT, 1.5.2.1 1.7.2.3 1.1.1.1

Im

1.2.4.5

jE

1.3.6.3, 1.5.2.1

jEF IK

1.5.2.1 1.1.1.1

IK[.], IK[., .] N

k

1.2.3.1

1.1.1.1

NT

Symbol Index

373

U

k

1.2.3.1

Ker

1.2.4.5

s

1.2.1.3, 1.5.1.1

/2~

1.5.1.1

/2/

1.2.1.3

/21

1.6.1.1, 1.6.1.3

/2~

1.6.1.13

~P

1.1.2.5

t~P(T) t~~

1.1.2.5 1.1.2.3

t~~

1.1.2.3

t~~

1.1.2.2

g~(T)

1.1.2.2

gP'q(S, T)

1.2.3.2

go'q(S,T)

1.2.3.2 1.1.2.26 NT 1.1.3.3

A/Ib IN INn

1.1.2.1 ~/ Q

1.1.2.1 NT

IR

NT

IR

NT

re

1.1.1.1

Supp f Suppp

NT NT

u'

1.3.4.1

u"

1.3.6.15

U~(t) uT(t) X~ ,.., x -1 (x~)~cI

1.1.1.2 1.1.1.2 NT 1.5.1.6 NT

x'a" 1.5.2.8 xx', x'x 1.5.2.5 (x, x'}, (x', x> x" -t y", x" !- y"

1.2.1.3 1.5.2.10

374

{x I P(x)} NT {x e X IP(x)} NT (.,x')y 1.3.3.3 2Z

NT

z -t- A

1.2.4.1 1.1.1.1

Io~1

1.1.1.1 1.2.4.1

o~A

Ia,/3[, ]a,/3], [a,/3[, [a, ,3] A NT

6st

NT

1.2.2.6 1.2.7.14

5t

~(s, t)

1.2.2.6

]#1 I-I x~

NT NT

tEI

E x(t)

1.1.2.1

tET q

Xn

1.1.6.2

n--p

Y]xL

1.1.6.2

eel

1.3.3.3

~--~(., x'~)yL LEI oo

a'~x,~

1.1.6.22

n--0

~--]Sx~

1.7.2.10

~EI

1

1.2.1.3

1E + • \ (., .)

1.2.1.3, 1.5.1.5 1.2.4.1 NT NT 1.2.1.3

{-I.} NT {. = - } , {. :/: .}, {. > .} - (mod p) NT ~,l1.5.2.10

NT

Symbol Index

V,A

I]" II II-II~ II-I11 II-Iio I1 I1~ V, 3, 3!

375

1.7.2.1 1.1.1.2, 1.2.1.3

1.1.2.5 1.6.1.1 1.1.2.3 1.1.2.2

NT o NT, 1.5.2.1 G 1.2.5.3 [-,-], ]-,-[, [', [, ]-,]

NT

This Page Intentionally Left Blank