An Introduction to Banach Space Theory [1 ed.] 0387984313, 9780387984315

Table of contents :
Contents......Page 8
Preface......Page 10
A Few Notes on the General Approach......Page 12
Synopsis......Page 14
Dependences......Page 19
Acknowledgments......Page 20
1.1 Preliminaries......Page 21
1.2 Norms......Page 28
1.3 First Properties of Normed Spaces......Page 37
1.4 Linear Operators Between Normed Spaces......Page 44
1.5 Baire Category......Page 55
1.6 Three Fundamental Theorems......Page 61
1. 7 Quotient Spaces......Page 69
1.8 Direct Sums......Page 79
1.9 The Hahn-Banach Extension Theorems......Page 90
1.10 Dual Spaces......Page 104
1.11 The Second Dual and Reflexivity......Page 117
1.12 Separability......Page 129
1.13 Characterizations of Reflexivity......Page 135
2 The Weak and Weak Topologies......Page 157
2.1 Topology and Nets......Page 158
2.2 Vector Topologies......Page 181
2.3 Metrizable Vector Topologies......Page 205
2.4 Topologies Induced by Families of Functions......Page 223
2.5 The Weak Topology......Page 231
2.6 The Weak* Topology......Page 243
2. 7 The Bounded Weak* Topology......Page 255
2.8 Weak Compactness......Page 265
*2.9 James's Weak Compactness Theorem......Page 276
2.10 Extreme Points......Page 284
*2.11 Support Points and Subreflexivity......Page 290
3.1 Adjoint Operators......Page 303
3.2 Projections and Complemented Subspaces......Page 315
3.3 Banach Algebras and Spectra......Page 325
3.4 Compact Operators......Page 339
3.5 Weakly Compact Operators......Page 359
4 Schauder Bases......Page 369
4.1 First Properties of Schauder Bases......Page 370
4.2 Unconditional Bases......Page 388
4.3 Equivalent Bases......Page 406
4.4 Bases and Duality......Page 419
*4.5 James's Space J......Page 431
5 Rotundity and Smoothness......Page 445
5.1 Rotundity......Page 446
5.2 Uniform Rotundity......Page 461
5.3 Generalizations of Uniform Rotundity......Page 479
5.4 Smoothness......Page 499
5.5 Uniform Smoothness......Page 513
5.6 Generalizations of Uniform Smoothness......Page 524
A Prerequisites......Page 537
B Metric Spaces......Page 541
C The Spaces lp and l......Page 549
D Ultranets......Page 561
References......Page 567
List of Symbols......Page 585
Index......Page 589