Functional Analysis - MA 223 (Fall 2010)

Manjunath Krishnapur

Tuesday, Thursday (5:00-6:30 PM, in LH-1)

Homeworks Due onComments
Homework 1 Aug 13th. Correction Problem 1, τ is an isometry only if α=1; Problem 5(1), the definition should be changed to ||x||=inf{r>0: s-1x for all s>r} or else V={(x,y): x2+y2=1} is a counterexample
Homework 2 Sep 11th.
Homework 3 Sep 14th.
Homework 4 Oct 5th.
Homework 5 Oct 19th.

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.

  1. Linear Algebra - up to spectral theorem for symmetric matrices.
  2. Point set topology - a first course. For most purposes metric spaces suffice.
  3. Measure theory and Real Analysis - Lebesgue integration and (preferably) Lp spaces.