An Introduction to Functional Analysis

Cover......Page 1
Title Page......Page 2
Copyright Page......Page 3
Dedication......Page 4
Preface\0......Page 6
Contents\0......Page 10
I. Topological Vector Spaces (TVS)\0......Page 16
1. Definition and Basic Properties\0......Page 18
2. Quasi-normed and Normed Linear Spaces (NLS)\0......Page 28
3. Metrizable TVS\0......Page 46
4. Bounded Sets in a TVS\0......Page 52
5. Linear Operators and Linear Functionals\0......Page 58
6. Quotient Spaces\0......Page 74
7. Finite Dimensional TVS\0......Page 80
II. The Three Basic Principles\0......Page 86
8. The Hahn-Banach Theorem\0......Page 88
8.1 Applications of the Hahn-Banach Theorem in NLS\0......Page 92
8.2 Banach Limits\0......Page 101
8.3 The Moment Problem\0......Page 104
9. The Uniform Boundedness Principle (UBP)\0......Page 106
9.1 Bilinear Maps\0......Page 120
9.2 The Nikodym Boundedness Theorem\0......Page 127
9.3 Fourier Series\0......Page 134
9.4 Vector-Valued Analytic Functions\0......Page 138
9.5 Summability\0......Page 140
10. The Open Mapping and Closed Graph Theorems\0......Page 152
10.1 Schauder Basis\0......Page 161
III. Locally Convex TVS (LCS)\0......Page 170
11. Convex Sets\0......Page 172
12. Separation of Convex Sets\0......Page 180
13. Locally Convex TVS\0......Page 184
13.1 Normability\0......Page 197
13.2 Krein-Milman Theorem\0......Page 199
14. Duality and Weak Topologies\0......Page 204
15. The Bipolar and Banach-Alaoglu Theorems\0......Page 214
16. Duality in NLS\0......Page 224
17. Polar Topologies\0......Page 246
18. The Mackey-Arens Theorem\0......Page 252
19. The Strong Topology and the Bidual\0......Page 258
20. Quasi-barrelled Spaces and the Topology j3*(E, E\')\0......Page 266
20.1 Perfect Sequence Spaces\0......Page 271
21. Bornological Spaces\0......Page 276
22. Inductive Limits\0......Page 282
IV. Linear Operators\0......Page 292
23. Topologies on Spaces of Linear Operators\0......Page 294
 \0......Page 0
24. Barrelled Spaces\0......Page 310
25. The UBP and Equicontinuity\0......Page 320
26. The Transpose of a Linear Operator\0......Page 324
 26.1 Banach\'s Closed Range Theorems\0......Page 340
 26.2 The Closed Graph and Open Mapping Theorems for  LCS\0......Page 345
 26.3 Vector Integration\0......Page 349
 26.4 The Space of Schwartz Distributions\0......Page 362
 26.5 The Lax-Milgram Theorem\0......Page 381
 26.6 Distributions with Compact Support\0......Page 386
 26.7 A Classical Theorem of Borel\0......Page 394
27. Projections\0......Page 396
28. Compact Operators\0......Page 402
 28.1 Continuity Properties of Compact Operators\0......Page 412
 28.2 Fredholm Alternative\0......Page 416
 28.3 Factoring Compact Operators\0......Page 420
 28.4 Projecting the Bounded Operators onto the Compact  Operators\0......Page 431
29. Weakly Compact Operators\0......Page 438
30. Absolutely Summing Operators\0......Page 446
V. Spectral Theory\0......Page 456
31. The Spectrum of an Operator\0......Page 458
32. Subdivisions of the Spectrum and Examples\0......Page 466
33. The Spectrum of a Compact Operator\0......Page 472
 33.1 Invariant Subspaces and Lomonosov\'s Theorem\0......Page 476
34. Adjoints in filbert space\0......Page 480
35. Symmetric, Hermitian and Normal Operators\0......Page 484
36. The Spectral Theorem for Compact Symmetric Operators\0......Page 500
36.1 Hilbert-Schmidt Operators\0......Page 509
37. Symmetric Operators with Compact Resolvent\0......Page 516
38. Orthogonal Projections and the Spectral Theorem for Compact Symmetric Operators\0......Page 524
39. Sesquilinear Functionals\0......Page 530
40. The Gelfand Map for Hermitian Operators\0......Page 534
41. The Spectral Theorem for Hermitian Operators\0......Page 538
42. Banach Algebras\0......Page 548
43. Commutative Banach Algebras\0......Page 560
44. Banach Algebras with Involutions\0......Page 572
45. The Spectral Theorem for Normal Operators\0......Page 582
Notation\0......Page 588
Appendix: Hilbert Space\0......Page 590
References\0......Page 606
Index\0......Page 612