|Plan for the course |
|Hilbert spaces Basic theory. Projections, Orthonormal basis, Linear functionals (Riesz representation). |
|Banach spaces Uniform boundedness principle, Open mapping theorem, Hahn-Banach extension theorem, Dual space, Weak topologies, Banach-Alaoglu 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 Lp, invariant measures on compact groups, Moment problems, Müntz-Szász theorem etc.