The manual contains brief theoretical information about the main modules of functional analysis: the theory of compressive operators, the theory of Fourier series in Hilbert space and the theory of linear operators. The focus is on applying the theory to well-known computational methods: solving equations of different types by simple iterations, approximating functions with different orthogonal bases, minimizing functionality by the Ritz method, solving linear operator equations of differential and integral type by approximate methods, in particular the Galerkin method.