Complete metric spaces. Fixed point theorem. Theorem of embedded balls. Baire theorem. Compactness: equivalent definitions. Arzela-Ascoli theorem. Normed linear spaces and applications: Banach spaces. Spaces of continuous linear applications. Dual space. Cauchy-Schwarz inequality. Parallelogram law. Orthogonalization. Hilbert spaces. Fourier series. Bessel's inequality. Parseval equality. Riesz-Fisher theorem. Hahn-Banach theorem. Minkowski function. Separation theorems. General form of linear functionals: functional in spaces of sequences; in Hilbert spaces. Riesz theorem. Weak convergence in a normed space and its dual. Banach-Steinhaus theorem. Compactness in the dual space. Topological vector spaces: Theorem of Kolmogorov. Weak topologies. Banach theorem of the inverse. Symmetric compact applications: Hilbert's Theorem. Fredholm theory.