Functional Analysis (eBook)

Front Cover 1
Functional Analysis 4
Copyright Page 5
Contents 6
Preface 8
Chapter 1. Preliminaries 12
1. Norms on a Vector Space 13
2. Finite Dimensional Normed Spaces 17
3. Infinite Dimensional Spaces. Hamel and Schauder Bases 22
Chapter 2. Operator Theory 28
1. Compact Linear Operators 28
2. Riesz Theory and Complementary Subspaces 33
3. The Open-Mapping Theorem 38
4. Quotient Spaces of l1 42
5. The Closed Graph Theorem 45
Chapter 3. Linear Functionals 48
1. Special Subspaces of Ia and l1. The Dual Space 48
2. The Hahn–Banach Theorem 55
3. The Banach–Steinhaus Theorem 60
4. The Completion of a Normed Space. Reflexive Banach Spaces 62
Chapter 4. The Weak Topology 66
1. Topology from a Family of Seminorms 66
2. Sets Which Define Seminorms 73
3. Locally Convex Spaces. Kolmogorov's Theorem 76
4. The Hahn–Banach Theorem. Reflexive Banach Spaces 81
Chapter 5. More about Weak Topologies 86
1. Dual Spaces and the Krien–Milman Theorem 86
2. The Eberlein–Smulian Theorem 90
Chapter 6. Applications to Analysis 98
1. Applications to Trigonometric Series 98
2. Miscellaneous Applications 106
Chapter 7. The Theory of Distributions 112
1. Some Function Spaces. Partitions of Unity 112
2. Fréchet Spaces 120
3. The Fourier Transform 125
4. Distributions: Definition and Characterizations 134
5. Distributions: Examples. Properties, and Applications 140
Appendix A: Solutions to Starred Problems in Chapters 1–4 156
Appendix B: Reflexive Banach Spaces 170
References 172
Index 174