Plan for the course 
Hilbert spaces Basic theory. Projections, Orthonormal basis, Linear functionals (Riesz representation). 
Banach spaces Uniform boundedness principle, Open mapping theorem, HahnBanach extension theorem, Dual space, Weak topologies, BanachAlaoglu theorem. 
Compact operators Integral kernels. Fredholm alternative. Spectral decomposition for compact symmetric operators on Hilbert space. 
Spectral theorems 1. Bounded symmetric operators on Hilbert space. 2. Bounded normal operators on Hilbert space. (If we have time)

Various applications We shall try to include as many concrete applications or examples as time permits. Tentatively some of them could be  Duals of various spaces, Fourier series, Fixed point theorems, Integral operators of importance in mathematical physics, Weakly compact subsets of L^{p}, invariant measures on compact groups, Moment problems, MüntzSzász theorem etc.
