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Preface
7
Contents
9
Hilbert Space Preliminaries
11
1.1 Normed Linear Spaces
12
1.2 Orthogonality
20
1.3 Hilbert Space Geometry
22
1.4 Linear Functionals
25
1.5 Orthonormal Bases
29
1.6 Exercises
33
Operator Theory Basics
40
2.1 Bounded Linear Operators
40
2.2 Adjoints of Hilbert Space Operators
43
2.3 Adjoints of Banach Space Operators
50
2.4 Exercises
52
The Big Three
57
3.1 The HahnÒBanach Theorem
58
3.2 Principle of Uniform Boundedness
63
3.3 Open Mapping and Closed Graph Theorems
69
3.4 Quotient Spaces
74
3.5 Banach and the Scottish Caf ï e
75
3.6 Exercises
76
Compact Operators
85
4.1 Finite-Dimensional Spaces
85
4.2 Compact Operators
88
4.3 A Preliminary Spectral Theorem
95
4.4 The Invariant Subspace Problem
102
4.5 Introduction to the Spectrum
104
4.6 The Fredholm Alternative
107
4.7 Exercises
109
Banach and C*- Algebras
114
5.1 First Examples
115
5.2 Results on Spectra
117
5.3 Ideals and Homomorphisms
127
5.4 Commutative Banach Algebras
131
5.5 Weak Topologies
134
5.6 The Gelfand Transform
139
5.7 The Continuous Functional Calculus
147
5.8 Fredholm Operators
150
5.9 Exercises
153
The Spectral Theorem
163
6.1 Normal Operators Are Multiplication Operators
163
6.2 Spectral Measures
171
6.3 Exercises
189
Real Analysis Topics
193
A.1 Measures
193
A.2 Integration
196
A.3 Lp Spaces
202
A.4 The StoneÒWeierstrass Theorem
203
A.5 Positive Linear Functionals on C( X)
204
References
206
Index
208
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