Analysis I

This book is the first part of a three-volume introduction to analysis, based on courses taught by the authors at various universities. The book is self-contained and is prepared mainly for students and instructors beginning courses in analysis. Together with theoretical material, the book also contains problems, exercises and other supplementary material allowing it to be used for self-study. The main feature of the book is its modern and clear presentation. The reader will find classical notions mixed with advanced ones (e.g. the discussion of Banach spaces and algebras). The book consists of five chapters, the first containing necessary foundation material - basics of logic; sets, functions, relations, and operations; natural numbers and countability; groups, rings, fields and polynomials; rational, real and complex numbers; vector and affine spaces and algebras. Chapter 2 deals with limits (and sums of series) of real numbers. The notion of continuity of functions is treated in chapter 3, which also introduces basic elementary functions. Differentiability, the mean value theorem and the Taylor theorem are described in the fourth chapter. The last chapter contains a discussion of sequences of functions (in particular uniform convergence, continuity and differentiability for sequences of functions, analytic functions and polynomial approximations). The material is presented in a fresh form and it is pleasant to read. Abstract concepts are presented using specific applications, which helps to build a solid understanding of modern mathematical analysis.

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