Functional Analysis I

Banach and Hilbert spaces, bounded linear operators; Hahn Banach, Baire Category, Uniform boundedness and Banach Steinhaus Theorem, open mapping/closed graph theorem; convexity; dual spaces; weak and weak* topologies; Banach-Alaoglu; reflexive spaces; Uniformly Convex Spaces; Application to L^p Spaces; Compact operators, Spectral theory of self-adjoint compact operators. Sobolev spaces.