Lecture Notes

This manuscript provides a brief introduction to Functional Analysis. It covers basic Hilbert and Banach space theory including Lebesgue spaces and their duals (no knowledge about Lebesgue integration is assumed).

MSC: 46-01, 46E30
Keywords: Functional Analysis.

The text is available as DVI (809K), or pdf (941K) version. Any comments and bug reports are welcome!

Preface
1. Introduction
1. Linear partial differential equations
2. A first look at Banach and Hilbert spaces
1. Warm up: Metric and topological spaces
2. The Banach space of continuous functions
3. The geometry of Hilbert spaces
4. Completeness
5. Bounded operators
6. Sums and quotients of Banach spaces
3. Hilbert spaces
1. Hilbert spaces
2. The projection theorem and the Riesz lemma
3. Operators defined via forms
4. Orthogonal sums and tensor products
4. Compact operators
1. Compact operators
2. The spectral theorem for compact symmetric operators
3. Applications to Sturm-Liouville operators
4. More on compact operators
5. Fredholm theory for compact operators
5. Almost everything about Lebesgue integration
1. Borel measures in a nut shell
2. Measurable functions
3. Integration - Sum me up Henri
4. Product measures
6. The Lebesgue spaces L^p
1. Functions almost everywhere
2. Jensen ≤ Hölder ≤ Minkowski
3. Nothing missing in L^p
4. Integral operators
7. The main theorems about Banach spaces
1. The Baire theorem and its consequences
2. The Hahn-Banach theorem and its consequences
3. Weak convergence
8. The dual of L^p
1. Decomposition of measures
2. The dual of L^p, p<∞
3. The dual of L^∞and the Riesz representation theorem
9. Bounded linear operators
1. Banach algebras
2. The C^* algebra of operators and the spectral theorem
3. The Stone-Weierstraß theorem
10. Appendix: Zorn's lemma

Bibliography
Glossary of notations
Index