Fundamentals Of Mathematical Analysis | Ilyin, Poznyak

This book is based on the lectures read by authors at Moscow State University for a number of years.

As in Part 1, the authors strived to make presentation systematic and to set off the most important notions and theorems.

Besides the basic curriculum material, this book contains some additional questions that play an important part in various branches of modern mathematics and physics (the theory of measure and Lebesgue integrals, the theory of Hilbert spaces and of self-adjoint linear operators in these spaces, questions of regularization of Fourier series, the theory of differential forms in Euclidean spaces, etc.}. Some of the topics, such as the conditions for termwise differentiation and termwise integration of functional sequences and functional series, the theorem on the change of variables in a multiple integral,

Green’s and Stokes’s formulas, necessary conditions for a bounded functiomto be integrable in the sense of Riemann and in the sense of Lebesgue, are treated more generally and under weaker assumptions than usual.

As in Part 1, we discuss in this book some questions related to computational mathematics, including first of all approximate cal­ culation of multiple integrals in the supplement to Chapter 2 and calculation of the values of functions from the approximate values of Fourier coefficients (A.N. Tichonoff’s regularization method) in the Appendix.

The material of this book, together with that of Part 1 published earlier, constitutes an entire university course in mathematical analysis.