The course consists of a synthetic but logically coherent introduction to the elements of Functional Analysis on which the mathematical formalism of quantum mechanics is based. The traditional elementary theory of Hilbert spaces will be accompanied by an "effective" introduction to the theory of temperate distributions. Without renouncing logical coherence, the use of abstract aspects of the theory of topological vector spaces will be minimized, while the emphasis will be placed on the most frequently used elementary distributional techniques, such as the Fourier transform, and Green functions.