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.

Recommended Books