Preface v
Chapter 0. Introduction 1
x0.1. Linear partial di erential equations 1
Chapter 1. A rst look at Banach and Hilbert spaces 5
x1.1. Warm up: Metric and topological spaces 5
x1.2. The Banach space of continuous functions 14
x1.3. The geometry of Hilbert spaces 19
x1.4. Completeness 24
x1.5. Bounded operators 25
Chapter 2. Hilbert spaces 29
x2.1. Orthonormal bases 29
x2.2. The projection theorem and the Riesz lemma 34
x2.3. Operators de ned via forms 35
x2.4. Orthogonal sums and tensor products 37
Chapter 3. Compact operators 41
x3.1. Compact operators 41
x3.2. The spectral theorem for compact symmetric operators 43
x3.3. Applications to Sturm{Liouville operators 46
x3.4. Fredholm theory for compact operators 49
Chapter 4. Almost everything about Lebesgue integration 55
x4.1. Borel measures in a nut shell 55
x4.2. Measurable functions 64
x4.3. Integration | Sum me up Henri 66
x4.4. Product measures 70
Chapter 5. The Lebesgue spaces Lp 75
x5.1. Functions almost everywhere 75
x5.2. Jensen Holder Minkowski 77
x5.3. Nothing missing in Lp 80
x5.4. Integral operators 83
Chapter 6. The main theorems about Banach spaces 87
x6.1. The Baire theorem and its consequences 87
x6.2. The Hahn{Banach theorem and its consequences 91
x6.3. Weak convergence 97
Chapter 7. The dual of Lp 103
x7.1. Decomposition of measures 103
x7.2. Complex measures 106
x7.3. The dual of Lp, p < 1 109
x7.4. The dual of L1 and the Riesz representation theorem 110
Chapter 8. Bounded linear operators 115
x8.1. Banach algebras 115
x8.2. The C algebra of operators and the spectral theorem 120
x8.3. The Stone{Weierstra theorem 124
Appendix A. Zorn's lemma 127
Bibliography 129
Glossary of notation 131
Index 133
