A strict systematized statement of the foundations of functional analysis and subtle questions of the theory of functions of a real variable is given. The basis was a course of functional analysis, read by A.N. Kolmogorov for several years at the Faculty of Mechanics and Mathematics, Moscow State University M.V. Lomonosov. Academician A.N. Kolmogorov developed a program for a new discipline (called "Analysis III"), incorporating elements of set theory, metric and normalized spaces, Lebesgue measure and integral theory, and linear operators in Banach and Hilbert spaces.