Quotient normed spaces; reflexivity; weak topologies; compactness in spaces of continuous functions: the Ascoli-Arzelà theorem; elements of operator theory: spectrum, adjoint operator, compact operators, the spectral theorem for compact selfadjoint operators, unbounded operators, closed operators; elements of spectral theory in Banach algebras; Gelfand theory for commutative Banach algebras; introduction to C*-algebras; the Gelfand representation theorem for commutative C*-algebras.