Math 713 Spring 2012 Lecture Notes on Functional Analysis

1. Topological Vector Spaces 1 1.1. The Krein-Milman theorem 7 2. Banach Algebras 11 2.1. Commutative Banach algebras 14 2.2. ∗–Algebras (over complexes) 17 2.3. Problems on Banach algebras 20 3. The Spectral Theorem 21 3.1. Problems on the Spectral Theorem (Multiplication Operator Form) 26 3.2. Integration with respect to a Projection Valued Measure 27 3.3. The Functional Calculus 34 4. Unbounded Operators 37 4.1. Closed, symmetric and self-adjoint operators 37 4.2. Differential operators 41 4.3. The Laplacian over R 43 4.4. The Laplacian in one dimension 45 4.5. A non-self-adjoint version of the Laplacian 47 4.6. von Neumann’s criteria for self-adjointness 49 4.7. The spectral theorem for unbounded self-adjoint operators 51 4.8. Quadratic forms and the Friedrichs extension theorem 54 5. Compact Operators 59 5.1. Riesz Theory of Compact Operators 61 6. Semigroups of Operators 66