This module covers basic functional analysis and selected applications. It is intended for graduate students in mathematics. Major topics: (1) Norms and seminorms, Banach and Fréchet spaces, Hahn-Banach and separation theorems, Uniform Boundedness Principle, Open Mapping and Closed Graph Theorems. (2) Dual spaces, uniformly convex and reflexive spaces, Radon-Nikodým Theorem and the dual of Lp, Banach-Alaoglu’s Theorem, Mazur’s Theorem, adjoint operators. (3) Compact operators, compactness of adjoint, spectral theory and Fredholm alternative for compact operators, application to differential equations. (4) Hilbert space and operators on Hilbert space, Lax-Milgram Theorem, Fourier series, spectral theorem for compact self-adjoint operators, application to differential equations.