General notion of functional space. Topological vector spaces. Normed and pre-Hilbertian spaces, Cauchy-Schwarz inequality, parallelogram identity, polarization identity. Banach and Hilbert spaces. Spaces of sequences. Synthetic introduction to the abstract measure theory and integration theory. Taking limit under the integral sign. Quadratic mean convergence. Spaces of square summable functions. Subspaces of a Hilbert Space; Theorem of projections. Orthogonal decomposition. Orthonormal systems and Hilbertian bases. Generalized Fourier series. Separable spaces. Hilbertian isomorphism; Continuous linear operators. Representation of continuous linear operators. Algebra of continuous operators: unitary operators, projectors, adjoint operators; convergence of series of operators. Fourier series for periodic functions. Trigonometric series. Fourier integral; elementary properties, the Riemann-Lebesgue lemma. Fourier transform in the Schwarz space of fast decreasing test functions. The Hermite basis and its basic properties. The Fourier-Plancherel transform. Tempered distributions as weak limit of square summable functions. Regular and singular distributions. The Dirac’s delta. Distributional derivatives. The Principal Part P1/x distribution. The pointlike charge potential. Further operations: change of variables, product, tensor product. The Fourier transform of tempered distributions; computational rules; explicit examples. Convolution of distributions. Fundamental solutions of a linear differential operator. Fundamental solutions, and the Cauchy problem; Green functions. Fundamental solutions for the diffusion equation, the Schroedinger equation for the free particle, and the wave equation.