Functional analysis (lecture notes)

Preface......Page 6 0.1. Linear partial differential equations......Page 8 1.1. Warm up: Metric and topological spaces......Page 12 1.2. The Banach space of continuous functions......Page 20 1.3. The geometry of Hilbert spaces......Page 24 1.4. Completeness......Page 29 1.5. Bounded operators......Page 30 2.1. Orthonormal bases......Page 32 2.2. The projection theorem and the Riesz lemma......Page 37 2.3. Orthogonal sums and tensor products......Page 38 2.4. Compact operators......Page 40 2.5. The spectral theorem for compact symmetric operators......Page 42 2.6. Applications to Sturm-Liouville operators......Page 44 3.1. Borel measures in a nut shell......Page 48 3.2. Measurable functions......Page 57 3.3. Integration --- Sum me up Henri......Page 58 3.4. Product measures......Page 63 4.1. Functions almost everywhere......Page 66 4.2. Jensen Hölder Minkowski......Page 68 4.3. Nothing missing in Lp......Page 71 5.1. The Baire theorem and its consequences......Page 74 5.2. The Hahn-Banach theorem and its consequences......Page 78 5.3. Weak convergence......Page 84 6.1. Decomposition of measures......Page 90 6.2. Complex measures......Page 93 6.3. The dual of Lp, p