Functional analysis

Title page......Page 1
ISBN 978-93-86286-09-3......Page 3
Preface......Page 4
Contents......Page 10
Chapter I Preliminaries......Page 12
1 Relations on a Set......Page 12
2 Linear Spaces and Linear Maps......Page 16
3 Metric Spaces and Continuous Functions......Page 30
4 Lebesgue Measure and Integration on R......Page 53
Chapter II Fundamentals of Normed Spaces......Page 73
5 Normed Spaces......Page 73
6 Continuity of Linear Maps......Page 94
7 Hahn-Banach Theorems......Page 115
8 Banach Spaces......Page 135
Chapter III Bounded Linear Maps on Banach Spaces......Page 149
9 Uniform Boundedness Principle......Page 149
10 Closed Graph and Open Mapping Theorems......Page 177
11 Bounded Inverse Theorem......Page 193
12 Spectrum of a Bounded Operator......Page 203
Chapter IV Spaces of Bounded Linear Functionals......Page 227
13 Duals and Transposes......Page 227
14 Duals of LP([a, b]) and C([a, b])......Page 246
15 Weak and Weak* Convergence......Page 271
16 Reflexivity......Page 291
Chapter V Compact Operators on Normed Spaces......Page 313
17 Compact Linear Maps......Page 313
18 Spectrum of a Compact Operator......Page 328
19 Predholm Alternative......Page 344
20 Approximate Solutions......Page 357
Chapter VI Geometry of Hilbert Spaces......Page 378
21 Inner Product Spaces......Page 378
22 Orthonormal Sets......Page 392
23 Approximation and Optimization......Page 413
24 Projection and Riesz Representation Theorems......Page 431
Chapter VII Bounded Operators on Hilbert Spaces......Page 452
25 Bounded Operators and Adjoints......Page 453
26 Normal, Unitary and Self-Adjoint Operators......Page 471
27 Spectrum and Numerical Range......Page 494
28 Compact Self-Adjoint Operators......Page 515
Appendix A Fixed Points......Page 539
Appendix B Extreme Points......Page 552
Appendix C Sturm-Liouville Problems......Page 564
Appendix D Unbounded Operators and Quantum Mechanics......Page 582
Bibliography......Page 602
List of Symbols......Page 608
Index......Page 612