Functional analysis. Entering Hilbert space

✍ Scribed by Hansen, Vagn Lundsgaard

• Publisher: World Scientific Publishing
• Year: 2006
• Tongue: English
• Leaves: 148
• Category: Library

✦ Table of Contents

Contents......Page 10
Preface......Page 8
Preliminary Notions......Page 12
1. Basic Elements of Metric Topology......Page 16
1.1 Metric spaces......Page 16
1.2 The topology of a metric space......Page 20
1.3 Completeness of metric spaces......Page 22
1.4 Normed vector spaces......Page 26
1.5 Bounded linear operators......Page 29
2. New Types of Function Spaces......Page 34
2.1 Completion of metric spaces and normed vector spaces......Page 34
2.2 The Weierstrass Approximation Theorem......Page 39
2.3 Important inequalities for p-norms in spaces of continuous functions......Page 43
2.4 Construction of Lp-spaces......Page 47
2.4.1 The Lp-spaces and some basic inequalities......Page 47
2.4.2 Lebesgue measurable subsets in R......Page 50
2.4.3 Smooth functions with compact support......Page 53
2.4.4 Riemann integrable functions......Page 54
2.5 The sequence spaces lP......Page 56
3. Theory of Hilbert Spaces......Page 60
3.1 Inner product spaces......Page 60
3.2 Hilbert spaces......Page 65
3.3 Basis in a normed vector space and separability......Page 66
3.3.1 Infinite series in normed vector spaces......Page 66
3.3.2 Separability of a normed vector space......Page 67
3.4 Basis in a separable Hilbert space......Page 69
3.5 Orthogonal projection and complement......Page 77
3.6 Weak convergence......Page 82
4. Operators on Hilbert Spaces......Page 86
4.1 The adjoint of a bounded linear operator......Page 86
4.2 Compact operators......Page 93
5. Spectral Theory......Page 100
5.1 The spectrum and the resolvent......Page 100
5.2 Spectral theorem for compact self-adjoint operators......Page 104
Exercises......Page 112
Bibliography......Page 140
List of Symbols......Page 142
Index......Page 144

✦ Subjects

Functional analysis;Hilbert space