In this course the students continue to study the principles of analysis in functional spaces on a deeper level. They will consider Hahn-Banach theorem and separability, separability, linear topological spaces; weak topologies; spectral analysis; introduction in C*-algebras; applications to integral and differential equations; spectal theorems for self-adjoint operators; Stone-Weierstrass approximation theorem; Lebesgue measures and integral; Fredholm alternative; theory of nonlinear operators; Schauder fixed-point theorem; unbounded operators.