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.
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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
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .
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
. . . . . . . . . . . . . . . . . . . .
159 159
1.3.1
The Banach Theorem . . . . . . . . . . . . . . . . . . . .
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
1.3.3
The Hahn-Banach Theorem
1.3.4
. . . . . . . . . . . . . . .
171
. . . . . . . . . . . . . . . .
180
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 . . . . . . . . . . . . . . . . . . . . . . .
240
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
Theorem
. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .
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
1.6
1.7
228
Banach Categories
264 280
. . . . . . . . . . . . . . . . . . . . . . . . .
281
1.5.1
Definitions . . . . . . . . . . . . . . . . . . . . . . . . . .
281
1.5.2
Functors
. . . . . . . . . . . . . . . . . . . . . . . . . . .
288
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
308
Nuclear Maps 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
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
. . . . . . . . . . . . . . . . . . . . . . .
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
Examples
. . . . . . . . . . . . . . . . . .
Function and the Neumann
Series
. . .
of R i e s z a n d G e l f a n d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . .
2.4
Involutive Banach Algebras
32
69
.....
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3
3
. . . . . . . . . . . . . . . . . . . . .
69 82 114 125 153 161 174 197
. . . . . . . . . . . . . . . . . . . .
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 The Strong Topology . . . . . . . . . . . . . . . . . . . . 6.3.2 Bidual of a C*-algebra . . . . . . . . . . . . . . . . . . . 6.3.3 Extension of the Functional Calculus . . . . . . . . . . . 6.3.4 Von Neumann- Algebras . . . . . . . . . . . . . . . . . . 6.3.5 The Commutants . . . . . . . . . . . . . . . . . . . . . . 6.3.6 Irreducible Representations . . . . . . . . . . . . . . . . 6.3.7 Commutative von Neumann Algebras . . . . . . . . . . . 6.3.8 Representations of W*-Algebras . . . . . . . . . . . . . . 6.3.9 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
-~ 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 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
c < 5
n=p+l Suppose that /5' E Uro_l~l(a). ~ Then ]/31 < r0, and so n-1
n-1
E
n--1
E
_ E
m=0
m=0
r 0 7"0
-- ?2r0
m=0
1
n=0
( ~ 2mo~ )-m-1 xo II
\m=O
x~ < n=p+l \m=O
~ n=p+l m=O
oo
n=p+l (Corollary 1.1.6.13, Corollary 1.1.6.10). There is a 5' > 0 with ~ m o~n-m-1
X n --
n=0 \m=O
?.,,ctn - l x n
n=0
0. There is an m C IN such that
lluxm- UXnll
m . Then
IIXm
Xnll < I llU(Xm O~
xn)ll
l ll~
?-tXnll
OL
OLC OL
for every n C IN with n > m . Hence (xn)ne~ is a Cauchy sequence and so a convergent sequence in E . It follows that y-
lim uxn = u( lim xn) e u ( E ) . n---+ OO
n--+OC~
Thus u(E) is closed. d) follows from b) and c).
m
72
1. Banach Spaces
Proposition
1.2.1.19 ( 0 )
For each x' = (x't)tE1
Let (Ee)ee, be a finite family of normed spaces.
YI E~, defi~2e
E
eEI
9-" 11 E~ ---, ~,
(x~)~,. , Z ~:(x~).
eel
tel 1
~) x~'E (11,~,E,)' a~d IIx~ll--II~'l : ( ~ II~:ll~)~
for every x' C [I E~. eEI
b)
The map ,.,.,
II,:
X I t,
,
eel
) Xt
eCI
is an isometry. a) Given x -
(xe)ee, E 1-I Ee, e6I
I~(x)l- I ~--~~:(x~) I 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 9 IK 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 9 c IK ~ such that ( , ) i s for every k C INn,
fulfilled. Then
1.2 Operators
E
j=l i=1
j=l
i=1
83
aij~ik
i=1
i=1
j=l i=l
It follows
k=l
k=l i=1 m
n
i=l
j=l
j=l i=1
)
d) We have that
m E I~kll 2 -- ~ k=l m k=l
~k2~kl
m E ~kl~k2 k=l m E
k=l
-
2
m
_~/~2 (~'~k=lIOLkll2-t- k=l ~ IOZk2'2) )~2r" (~k=l IOZkl]2)(~-~k=llXk212) E IOQIOLj2 -- OQ2OLjl i,j=l
-- E i,j=l
ailOgj2 -- Cti2Ctjl
)(
k--1
-~kl C~k2
ailO~j2 -- O~i20Ljl
)
=
m --i,3.~=1 (OZill 20Lj212-I- 1~i2121OLjll2) -- ~ ('~il~i20LjlOL-j2-~OLilOli2OLjlOlj2)i,j--1 m -~k l Ctk2 : 2 (k~ 1 IOZkl,2/ (k_~l ,OLk212) -- 2 k--1
The assertion now follows from c). E x a m p l e 1.2.2.8 ( 0 ) u E/:(IK 2) and let
m
Take IK 2 endowed with the Euclidean norm. Take
5
be the matrix associated to u.
84
1. Banach Spaces
a) I1~12 =
= 1- ('c~12 + 1/312+ 1'712+ 15[2+ i('oz'2 + 1/312+ 13'12+ ,5i2)2 - 4,c~5 -/3'7i2 ) 2 b) o~,~ ~r ==~
-
sup{]c~ +
~1, I ~ - ~1}.
a) follows from Example 1.2.2.7 d). b) Take ~o,~b E IR such that
and put 0 .= 2(r
~).
Then I~ + 91 ~ =
rI~1~ ~
+ I~1~'r
~ = fl I~l • I~l~ ~
- I~l ~ + I~l ~ + 21~119J co~
so that 0 sup{Ic~ +/~12, Ic~ -/~12 } - I~1~ + I~l ~ + 21c~11911cos ~1. On the other side, ~_
9=1= _ i 1~12~=~o _ j~l~[~_
= I~14 + 1 9 1 4 -
121~ = +
21~1=1 = -
l l~j ~ _ 191=~,ol~ =
21~1~191~coso,
4 Io~2 - ~212=
o
1.2 Operators
= 4 (Iozl 4 + 2lo~l=l~l 2 + I/3l 4 -
85
la} 4 - I ~ l 4 + 2lc~l~-I~l 2 cos o) =
= 8[al2[/312(1 + cosO) -- 16Ic~[~[/3[2 cos 2
0
By a),
1(
- ~
-
21~1 ~ + 2 ~ + 4 1 ~ 1 1 ~ 1 1 c o s ~ 1
0 I~1 ~ + I~1 ~ + 21~11fll I cos ~1 - s u p ( l ~
0)
-
+ ill2,1~ _ ill2}.
c) follows from b). Remark.
Example
b) will be generalized in Example 7.1.4.10 h). 1.2.2.9
Take p C [1, cx~] U {0}. Given x e gP, define
xr'IN
a)
m
0 x(n-1)
>I[4,
n,
>
xe 9IN -----+ IK,
n,
> x ( n + 1).
/fn=l ifn~l,
x r , xg C gP and
for every x c gP. Define
b)
u~ 9 gP
gP,
x~
xr
(the right shift on t?),
ui " gP
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 ) ,
e'(T)
F
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)
~
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 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
d~.) k(., t) 9 co(S) e)
(~esp. c(S) );
k(e'(T)) c co(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~
I9 9
e(T)# )
95
n
= Ilkll
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 T
k~ = k(x - ~ 4 ) + ~ k ~ = k(~ - ~ 4 ) + ~ ~
k(., t) c ~(z).
tET
g) We set
h-R•
~,
N
(r,t), "(~k~T)(~).
Then 2
E
2
t E T rE R
(r,t)EIRxT
N T
tET
~
12--
tET
tET
sES
(Proposition 1.2.1.4 a)), so that h e tF'2(R, T) and
We have T
for all (r, t) C R x T , so that
2
98
1. Banach Spaces
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
IN:,
(s,t),
,~ (ueS)(t).
Then
k(s, . ) = uef e e~(T) and
whenever s 9 S . Thus k 9 t~~'~(S, T ) . Given s 9 S , u S kG - k ( s , ' ) - ue s , U
andso k=u
on IK (s) .By b), k = u ,since IK (s) is dense in ~1 (S) (Proposition
1.1.2.6 c)). dl =~ d2 follows from esS 9 fp(S) and the fact that U S
k(~, . ) - k~ e co(T) (e c(T)) for every s E S.
u
d2 =~ dl. Given x e ]K (s), k x 9 co(T)
(e c(T)) and dl) follows from b)
and Proposition 1.1.2.6 c) (and Example 1.1.2.3). Remark.
I
If p ~= 1, then the map u
~q"(S,T)
>s
k.
>k
need not be surjective. Indeed, if S - T and 1 < p < r , then the embedding u
gP(S) c K ( S ) is not of the form k.
1.2 Operators
Definition
1.2.3.9
1 )
(
103
A family (A~)~e, of sets is called disjoint if A~ N A x = 0
for any distinct t, A E I . Proposition
1.2.3.10
(
1 ) Let S,T
be sets and k a function S x T --+ IK
such that k(s, .) E/~I(T) for every s 6 S . Then the following are equivalent: M
a)
kx e g~176 whenever x e g ~ ( T ) .
b)
kx e e~(S) So~ ~r
c)
ke s 6
N
d)
n
T
~cx~
9 e Co(T).
(S) for every countable subset B of T .
There is a sequence (C~)n~IN in IR+\{0} with the property that for every disjoint sequence ( B n ) n ~
in q3s(T ) there is a sequence (an)ne~ in IR N
such that a,, > c,~ for every n 6 IN and that kx C f ~ 1 7 6 the map given by 0
x'T
> 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
(s,t),
;sup{rek(s,t),0}
k2:SxT
>IR,
(s,t),
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
c(S)) and has norm
a =~ b. Given t E T , e T and e T belong to c(T) and
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 lim inf (keB)(Sn) --(keB)(Sn+l ) [ > O. Given n c IN, put
Ca "= { k ( ~ , ") # 0}
B
and
c:=Uc,. ncIN
Then C is countable and n T
n T
(keB)(sn)-(kec)(Sn)
for every n E IN. Thus 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
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
s
teB,~\Bn-1
teT\B,~
for every n C IN. Take n C IN and a s s u m e t h a t the sequences have been cons t r u c t e d up to n -
1. Since lim k(., t) -- a(t) a
for every t E B n - 1 , there is a finite subset A of S with
Ik(~, t)
s -
a(t)l
tET
~
k(s2~
tcT\B2n
I~(s2~,,~l - ~
Ik/s2n,,t-o~,/I - ~
tCB2n-2
tEB2n\B2n-1
-~ tET\B2n
I~(s2n,~/I-
Ik~2n 1,'~-a/'/I -
tEB2n-2
~ tcT\B2n-1
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)
M
x c e~(T)
k(.,t) e Co(S) for every t e T , and given c > O, there is a finite subset 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
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
( 0 ) If E, F are finite-dimensional normed spaces, then every bijective linear map E ~ F is an isomorphism, m
C o r o l l a r y 1.2.4.9
C o r o l l a r y 1.2.4.10
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 ( 0 ) Let E , F
C o r o l l a r y 1.2.4.11
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 - ~.
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
127
be the quotient map. By Proposition 1.2.5.2
-+ E / G
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,
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.
I
Remark. Lindenstrauss (1967) proved that every infinite-dimensional complemented 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
(
) 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
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
I ~
v',
The set
has the desired properties. Now we prove the last assertion. We may assume that A ~- 0. We put
c~ := 1 + sup Ix'(x)l e ]R+. xEA
Then, for each x c A,
Ii.'(m)l_< 1, C~ so that ~x' C V , i.e. x' E a V .
1.2 Operators
L e m m a 1.2.6.3
135
( 0 ) Let x' be a linear form on the vector space E , and
(Xtt)eCI a finite, nonempty family of linear forms on E with
N Ker x: C Ker x'. tEK
Then x' is a linear combination of the x: (5 9 I ) .
We may assume that the family (x~)~i is linearly independent. We prove the lemma by complete induction on the cardinality of I . Take A 9 I and put J := I\{A}. By the hypothesis of the induction, there is an
x~ ( n Kerx:)\Kerx~ ~EJ
(if J = q), replace the intersection by E). Put y' "=X'--
Xt
(X)
,
X'~ (X) x~ "
Take
rCJ
Then
(
)
, x~(y) x~ Y-x~(x) X =0, !
x~! (x) x C
Ker x~'
o - x' (y) - ~'~ (y) x'(x) = y' (y)
~'~ (x)
N Ker x'~ C Ker y'. tEJ
By the hypothesis of the induction, y' is a linear combination of the x'~ (~ C J ) . Hence x' is a linear combination of the xt~ (5 C I). I L e m m a 1.2.6.4
( 0 ) Let A be a finite subset of the vector space E . Let
F be a vector space of linear forms on E and x" a linear form on F which is bounded on
136
1. Banach Spaces
{x' C F' l x c A ==a Ix'(x)l < 1}. Given x E E , define
Y" F
~ IK,
z'~
", z ' ( z ) .
Then x" is a linear combination of (X)xEA" In particular, there is an x E E with ~ -
x".
We may assume that A is not empty. Then N Ker2 C K e r x " , xcA and the assertion follows from Lemma 1.2.6.3. /
C o r o l l a r y 1.2.6.5
[
I
\
0
)
Let E
be a n o r m e d space, F
a vector subspace
of E ' , and x" a continuous linear f o r m on F E . Then there is an x C E such that
(x, .)IF -- x". By Proposition 1.2.6.2, there is a finite subset A of E such that x" is bounded on {x' E E ' l x
C A ==~ [z'(x)] _< 1}.
The assertion now follows from Lemma 1.2.6.4. P r o p o s i t i o n 1.2.6.6
I
( 0 ) I f E is a n o r m c d ,space, then the map
Ek
~+,
.~" *ll~'l
is lower semicontinuous and E '# is a closed set of E 'E .
Given x C E , the map
E~
~+,
~',
~x'(x)l
is continuous. The assertion now follows from Proposition 1.2.1.4 b). Remark.
E '# is even a compact set of E ~ . (See Theorem 1.2.8.1).
I
1.2 Operators
Proposition
1.2.6.7
137
If E is a normed space then the map
is continuous. Take (uo, vo) C s
# x s
A a finite subset of E , and c > 0. T h e r e
are neighbourhoods b/, 12 of uo and vo in s
and s
respectively,
such t h a t
II~o~- ~o~o~ I < ~ , for all u E b/, v C ]2, and x E A . It follows
IJ~v~- ~o~o~ll _< IJ~w- ~vo~li + I1~o~- ~o~o~ll < c
c
c
< l l ~ - ~o~ll + ~ < ~ + ~ = for all 'u E / g , v C 12, and x C A. Hence the above m a p is continuous,
i
138
1. Banach Spaces
1.2.7 C o n v e x Sets D e f i n i t i o n 1.2.7.1
( 0 )
The subset A of the vector space E is called convex (resp. absolutely convex) if a A + 13A c A
for every a, 3 E IR+ (resp. a, 3 E IK) with
+
+)3
It is easy to see that a subset A of a vector space is absolutely convex iff it is convex and aACA
for every a E IK with [a[ _< 1. P r o p o s i t i o n 1.2.7.2
( 0 ) 1] E is a normed space, then
{xCE[[[x[[ IK,
rl
>
if r - - S
(1 + Ix(s)l]
if r = t
0
if r = T \ { s , t } ,
0
z "T
> IK,
r'
~,
if r = s
( 1 + Ix(s)l Ix(t)l]~
x(t)
if r = t
x(r) T h e n y, z 9 g l ( T ) #
if r = T \ { s , t } .
and
Ix(~)l Ix(t)l Ix(~)I + Ix(t)l y + Ix(~)l + I~(t)I z = x. This c o n t r a d i c t s the h y p o t h e s i s t h a t x is an e x t r e m e p o i n t of t~l(T) # . Hence ix(t)l = 1 for s o m e t 9 T .
i
T h e above result a d m i t s a generalization. Example
1.2.7.13
Let # be a Radon measure on the Hausdorff space T .
Then x 9 L I ( # ) # is an extreme point of L I ( p ) # iff Ix(t)p({t})l - 1
for some
t 9 T.
Ix(t)#({t})l-
for s o m e
t 9 T.
First a s s u m e t h a t 1
Take y , z E L I ( # ) # a n d a e ] 0 , 1[ w i t h a y + (1 - a ) z - x . Then
a y ( t ) + (1 - a ) z ( t ) - x(t) , so t h a t
144
1. Banach Spaces
y(t)
- z(t) = ~(t)
Since x , y , and z all vanish #-a.e. on T \ ( t ) ,
y=z-x. Thus x is an extreme point of LI(#) # . Now assume that x is an extreme point of LI(#) #
Assume further t h a t
Ix(t)p({t})l # 1 for every t C T . By Proposition 1.2.7.11,
Ilxll
- 1. Hence there are two disjoint
compact sets K, L of T with oe
9- / ~ I~1dl,I # 0,
9 - f~ I*1 d #1 # O.
Define T y . _ c~ + / 3 x e T + xer\(KuL) ' o~
Z
"--
T
+ ~ x e T + XeT\(KuL).
Then y, z C L 1(#)# a::d c~ + / 3 y + a + /3z = x . This contradicts the hypothesis t h a t x is an e x t r e m e point of L 1(#)#. Hence
Ix(t)p({t})l- 1 for some t C T . Example
1.2.7.14
m
Let T be a Hausdorff space, .h4b(T) the Banach space
of bounded Radon measures on T (Example 1.1.2.26), and take p C .hdb(T) # Then # is an extreme point of .A4b(T) # iff there is a t C T such that Suppp-
{t},
Ip({t})l-
1.
M " - {~' C .h4b(T) ] ~, is positive and ~ ( T ) -
is a face of .A/Ib(T) # and
1}
1.2 Operators
145
{at It E T} is the set of extreme points of All, where at denotes the Dirac measure on T at the point t E T (i.e. at(A) =eA(t) for every Borel set A of T ). Assume t h a t
# is an extreme point .hdb(T) # . Assume further that the
support of # contains at least two points. T h e n there are disjoint Borel sets A and B of T such that A tO B : S u p p p ,
IpI(A)
--/: o,
IpI(B) :/: o.
Define 1 #A - - i P l ( A ) e A - # ,
1
#s -
I~I(B)eB
.#.
Then #A, PB E A//(T) # ,
I~I(A) + I#I(B) _< 1,
# =
I#I(A)~A + I#I(B)~B,
contradicting the hypothesis that p is an extreme point. Hence there is a t E T with Suppp={t},
I>({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
,~(~'J)~ A l~ + ; >
B 9=
~}.
Then p
(i,j)EB
n=O
-
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 9 IN
~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(,
xe" ~
n, n,
Show that"
a) x e ~p(2z) ~ Xr, Xe e ~P(~), I x~llp --Ilxell~ --Ilxll~. Co(2Z)) =:> xr,xe C c(~)
b) x c c(2Z)
(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+.
y'(y) 0. Then
z'(z))]
1.3 The Hahn-Banach Theorem
161
~ o ( , ( 1 +~)-z'( 1 y)+ z,~(~y)) =~p(y+x)-p(~x+y), 1 ,
z0(-~
ly))
lim inf z~ < z'(x) < lim sup xn. n---+ o o
n--+oo
Moreover, XI
Ur
--
Xt
X!
for every such x ' , where Ur (resp. u~ ) denotes the right (resp. left) shift on
Define p g~ 9
>IR,
x,
>limsup n--+ oo
--
T~
.~n ~_~ X
m
m--1
converges
X m
n m--1
1 _
nCIN
,
162
1. Banach Spaces
y'
F
9
>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 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
tel
tel
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) 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),
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 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,
2) x' o u t = x' o u = x' for every t c T , 3) x'(x) e
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) 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 1
1,
(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
-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 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 II~'ll = 1,
x'(a)
= 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
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
U) I ~
~t t
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)
v E /:(G,H)
u E s
Let E , F , G , H
be normed spaces. Take
and let ((y~,zt))te, be a finite family in F ' •
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)
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
sES
tET
t6T
(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
The assertion follows now from Example 1.2.2.7 d).
m
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
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
(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
(Corollary 1.3.3.5). But then y' E (F) ~ = which is a contradiction. Hence
= (~176176 (Proposition 1.3.5.4),
=
Proposition 1.3.5.8 s
( 0 )
Let E , F
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 I
(Corollary 1.3.3.8 a), Proposition 1.3.5.8). Remark.
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. d) follows from c) and Proposition 1.3.5.4. Proposition
1.3.5.13
( 0 )
Let T
I
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
.T'. Hence u,#1
1
I1~11 v, I1~- vii
u
.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
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 ::
x (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
=
I P ~ I ( T \ K ) _
q'y'
is an isometry. Let j : F ~ ~ E ' be the inclusion map. Then
q'=jou so t h a t
jl (Corollary 1.3.4.5). Now Ker j ' = F ~176 is an isometry (Theorem 1.3.5.12 and the factorization v of j' t h r o u g h E " / F ~176 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 ~
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
1
(Corollary 1.2.1.5, Theorem 1.3.4.2 b)). c) By a), 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).
b)
(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 W
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)
-
= 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)
(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) Proposition
(Im w").
m
1.3.6.25
( 7 )
Let E , F be n o r m e d spaces and u C s
(Im jE')
(Im u'") = Im(jE, o u') = Im (u'" o jF')-
Then
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'
y' l
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 )
I
be a Banach space and (u~),e, a finite
Let E
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,
-x'EImv=Keru
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
Let T be a compact space and let C(T)T denote the set L e m m a 1.3.7.14 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
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
a) Let j 9F ~ E
241
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
IK ,
x"" 2t4b(Z)
#J ) f eB d#
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
P r o p o s i t i o n 1.3.8.13 Every bounded sequence in a reflexive Banach space has a weakly convergent subsequence. 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.
I
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
Then
(:),o m--0
Oonm
/o
Oo, n
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
m=0
n=p+l
o0
p
Ilf(~) - ~
(~ - :3o)~Ymll ~ I f(~) -- ~
n=0
(~ - ~0)~x~ll+
n=O
P
+ll Z
P
(" - "0) nz` - Z
(" - Z~
+
m=0
n--O
oo E
E
E
m=p+ 1
Since c is arbitrary, oo
f (a) - ~
(a - ao)mYm .
m=O
Since a and /3o are arbitrary, f is analytic. P r o p o s i t i o n 1.3.10.4
( 0 ) Let E , F be Banach spaces. Take u E s
Let U be an open set of IK and f " U - + E an analytic function. Then u o f is analytic. oo
Take a0 E U. There is a power series ~
tnxn in E and an r > 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. I T h e o r e m 1.3.10.6
( 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
for n E ~
and x ~ E E ~.Hence
IIx~ll_ ~ for every n E IN (Corollary 1.3.3.8 b) ). Thus lim sup lixnii! 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) 9 Hence 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/
1 '
-1
C(S,)
fv(e it)
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
dt =
t 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
nu(E#)-
u(nE#)-
n E IN
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 I is an interior point of u(U) and u(U) is thus open. 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" (Proposition 1.3.6.16), so that
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 ' ) ,
j
j
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
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 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~
1.
there is an x E E such that ~x
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.
The claim follows from Proposition 1.2.3.5.
I
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 and g a functor of B into C. The map (E, F ) ,
a functor of fit into B,
~ 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 evaluation on M(E, F ) . Then the map (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
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
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 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.
c ~ H ~ u~,
= j~(~'~")
).
A~
d)
E -~ F ~ ux,"x" e jEc(A'(E, G)) At
(~p. a ~AH t
~ ~,~ 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)
> A"(F,G),
y",
> 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)
E _5+ F v_~G ::v (fFaY )x A
fsa(Y'fEFX)
E s13' F 4,4 G :=~ x(fEFY ) = f ~ a ( ( f F a x ) y ).
b)
1 I E ~ F ~ G ~ (fFay )x"
E 13' v_~F A-~" G ~ x tJEFY )
__
J"' E G t Y"
'"" JE, F x "") ,
IEG ((JFa x )Y').
f ~ ( Y " ~ ~'') - ( J ~ Y J ~ (" ]~ E" F x " ) f~c(Y" ~ x") - (]i:aY"" ") -~ t.JEF "),
EG ~ ~ a ~ .4" .A"
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
( I'll
:EFx , (faEx)y') = (faEX, y ' :"E' : " ) = (x, :'Ea' ~y"'"':Erx"')>
in the first case and (x , x" f 'EF y'X = (x" , (fEFY)X) ' ' ' ' faEX)) = (x" , fGF(Y
in the second case. X! c) Let G---+E. By b), A'
--
Y", fG
((fEFX )X ) } -
(~11
Xlt
E x a m p l e 1.5.2.15
/!
~,:FaY , (f~Fx )X ) - - ~,(JFaY ) ~ (f~Fx"), x'),
), x') =
(y#
I!
p
I
~ x , f 3 ~ x ) -- (x", (f~E ')y") =
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~
i
@k", k')= @", k'k) = ~
(k'k)(~, ~)dk"(~)=
r
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
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
= (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
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) =
= 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
This proves the last assertion.
Proposition 1.5.2.19 tion functor of .4, and
m
( 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 --
9 JEFX
~
~
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 9 s
~ u' 9 s 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 ~
IK ,
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
0 and x0 9 A. There is a v 9 ~
II~- vii
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,
for m, n C IN. Hence (Yn)nEIN is a Cauchy sequence in C(T). Set y "-- lim yn. n--). oo
Then
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
{x c Co(T)+
xeCo(T). 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
352
1. Banach Spaces
m
Then y is 1 on U and 0 on T \ U . Hence m
-1
u = y (]0, ~ [ ) is open. The last assertion follows from the fact t h a t T is a Baire space. Example 1.7.2.14
m
( 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 1.2.3.2
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 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
x(t)
Xn
1.1.6.2
n--p
Y]xL
1.1.6.2
eel
~--~(., x'~)yL 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~
I1 I1~
375
1.7.2.1 1.1.1.2, 1.2.1.3
1.1.2.5 1.6.1.1 1.1.2.2
3, 3! NT o NT, 1.5.2.1 G 1.2.5.3 [-,-], ]-,-[, [', [, ]-,]
NT
