Basic Functional Analysis: Banach spaces, Hilbert space, Hahn Banach theorem, open mapping theorem, closed graph theorem, uniform boundedness principle, Alaoglu's theorem, Frechet spaces.

Fourier Analysis: Fourier series and transforms, Fourier inversion and Plancherel formula, estimates and convergence results, more topological vector spaces, Schwartz space, distributions.

Spectral theory: spectral theorem for bounded self-adjoint operators, specializations to compact operators and/or extensions to unbounded operators, as time permits.