MATH 564 Functional Analysis

Metric spaces. Banach and Hilbert Spaces. Linear Operators on Normed Spaces, Bounded and Compact Operators. Spaces of Linear Operators and Convergence. Fundamental Theorems for Normed and Banach Spaces: Hanh-Banach Theorem, Uniform Boundedness Theorem, Open Mapping Theorem, Closed Graph Theorem. Linear Functionals on Hilbert Spaces and Riesz Representation Theorem. Adjoint, Self-adjoint, Unitary and Normal Operators. Spectrum and Resolvent of an Operator. Spectral Properties of Bounded and Compact Operators. Unbounded Operators and Their Basic Properties.