This book is the second edition of the third part of lecture notes corresponding to the material of the third semester lectures on mathematical analysis for students in mathematics (for the first part see [Zbl 1280.26003], and for the second part [Zbl 1280.26004]). These lectures were delivered by the author at the mechanical-mathematical faculty of the M. V. Lomonosov Moscow State University. This edition is published in the framework of the programme “Lectures of NOTs (Scientific-educational Center)” realized by the V. A. Steklov Mathematical Institute of RAS. The high educational standard of the mechanical-mathematical faculty of MSU is well-known. Therefore, it is possible to deliver lectures at the deep rigors. In the present book the material of the third semester is presented mainly on the base of traditional point of view. The author supposes that students have to understand first the classical ideas and results, then entering for themselves into deep modern generalizations of these results. The only exclusion is the material related to orthonormal systems in Hilbert space. This chapter contains few necessary results from functional analysis which are in style of classical mathematical analysis. They serve to deliver lectures on Fourier series on the proper level. The book consists of five chapters: “Number series”, “Functional sequences and series”, “Parameter-dependent integrals”, “Orthonormal systems in Hilbert space”, “Fourier series on the trigonometric system”. The main material is illustrated by a large collection of examples, and the principal ideas are discussed in their historical retrospective. At the end of some chapters few exercises for self-education are presented.

Reviewer:

Sergei V. Rogosin (Minsk)