This course covers topics in Real Analysis and Functional Analysis and their applications. It starts with a review of the theory of metric spaces, the Lp spaces, and the approximation of real functions. It proceeds to the theory of Hilbert spaces, Banach spaces and the main theorems of functional analysis, linear operators in Banach and Hilbert spaces, the spectral theory of compact, self-adjoint operators and its application to the theory of boundary value problems and linear elliptic partial differential equation. It concludes with approximation methods in Banach spaces.