Fundamentals Of Mathematical Analysis Part 2
by V.A. Ilyin; E.G. Poznyak
About the Book
This book is based on lectures delivered by the authors at Moscow State University over several years. As in Part 1, the authors aim to present the material systematically, highlighting the most important notations and theorems. In addition to the core curriculum, the book explores several important questions that play a significant role in various branches of modern mathematics and physics, such as the theory of measure and Lebesgue integrals, Hilbert spaces and self-adjoint linear operators, Fourier series regularisation, and the theory of differential forms in Euclidean spaces.
Some topics, such as the conditions for term-wise differentiation and integration of functional sequences and series, variable changes in multiple integrals, and Green’s and Stokes’ formulas, are treated more generally and under weaker assumptions than usual. The book also addresses computational mathematics, including approximate methods for evaluating multiple integrals and calculating function values from Fourier coefficients.
The material, along with Part 1 (Volume 1), constitutes a comprehensive university course in mathematical analysis. The text allows for some flexibility, as chapters on advanced topics such as Lebesgue integrals, Hilbert spaces, and related supplements can be skipped without affecting comprehension of the main material.
The authors acknowledge their indebtedness to various contributors for valuable advice and criticisms, particularly A. N. Tikhonov, A. G. Svesshnikov, Sh. A. Alimov, L. D. Kudryavtsev, S. A. Lomov, P. S. Modenov, and Ya. M. Zhileikin for their assistance in field theory and approximate methods for evaluating multiple integrals.
Translated from the Russian by Vladimir Shokurov
