Read the original document by opening this link in a new tab.
Table of Contents
Contents
Preface iv
Chapter 1. Banach and Hilbert spaces 1
1.1. Linear spaces and linear operators 1
1.2. Normed spaces 7
1.3. Banach spaces 15
1.4. Inner product spaces 19
1.5. Hilbert spaces 25
1.6. Fourier series 28
Chapter 2. Bounded linear operators 39
2.1. Bounded linear functionals 39
2.2. Representation theorems for linear functionals 42
2.3. Hahn-Banach theorem 47
2.4. Bounded linear operators 56
Chapter 3. Main principles of functional analysis 69
3.1. Open mapping theorem 69
3.2. Closed graph theorem 74
3.3. Principle of uniform boundedness 76
3.4. Compact sets in Banach spaces 83
3.5. Weak topology 87
3.6. Weaktopology. Banach-Alauglu's theorem 91
Chapter 4. Compact operators. Elements of spectral theory 94
4.1. Compact operators 94
4.2. Fredholm theory 98
4.3. Spectrum of a bounded linear operator 101
4.4. Properties of spectrum. Spectrum of compact operators. 103
Chapter 5. Self-adjoint operators on Hilbert space 110
5.1. Spectrum of self-adjoint operators 110
5.2. Spectral theorem for compact self-adjoint operators 113
5.3. Positive operators. Continuous functional calculus 116
5.4. Borel functional calculus. Spectral theorem for self-adjoint operators 122
Bibliography 126
Index 127
Summary
These notes are for a one-semester graduate course in Functional Analysis, which is based on measure theory. The course covers the theory of Banach and Hilbert spaces, bounded linear operators, main principles of functional analysis, compact operators, spectral theory, and self-adjoint operators on Hilbert space. The material is compiled from various textbooks and includes topics such as the Hahn-Banach theorem, open mapping theorem, and Fourier series. The author is grateful to the students for their corrections and suggestions.