Normed and Banach spaces. The Hanh-Banach Theorem and its consequences. Lp space and Hilbert spaces. Orthonormal basis. Linear operators and unbounded linear operators. Compact operators. The Riesz-Fredholm theory. The spectrum of a compact operator. Self adjoint operators. Sobolev spaces. Variational formulations of boundary value problems An introduction to the principles of measure and integration theory, the elements of functional analysis in Banach spaces, and spectral theory in Hilbert spaces, to discuss two (2) classical areas of applications, integral and differential equations. Additional topics will include differential and integral calculus in Banach spaces, fundamental results of distribution theory and Sobolev spaces. 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.