Contents
Preface xi
Chapter 1. Introduction 1
§1.1. Topological spaces 1
§1.2. Measure and integration 8
§1.3. Exercises 21
Chapter 2. Normed spaces and operators 25
§2.1. Banach spaces 26
§2.2. Linear operators 39
§2.3. Hilbert spaces 45
§2.4. Convolutions and summability kernels 52
§2.5. The Riesz-Thorin interpolation theorem 59
§2.6. Applications to linear differential equations 63
§2.7. Exercises 69
Chapter 3. Fr´ echet spaces and Banach theorems 75
§3.1. Fr´ echet spaces 76
§3.2. Banach theorems 82
§3.3. Exercises 88
Chapter 4. Duality 93
§4.1. The dual of a Hilbert space 93
§4.2. Applications of the Riesz representation theorem 98
§4.3. The Hahn-Banach theorem 106
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