Basic topological concepts, Metric spaces, Normed linear spaces, Banach spaces, Bounded linear functionals and dual spaces, Hahn-Banach theorem. Bounded linear operators, open-mapping theorem, closed graph theorem. The Banach- Steinhaus theorem. Hilbert spaces, Riesz representation theorem, orthogonal complements, bounded operators on a Hilbert space up to (and including) the spectral theorem for compact, self-adjoint operators.