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Basic real analysis. Along with a companion volume Advanced real analysis. (English) Zbl 1095.26001

Cornerstones. Basel: Birkhäuser (ISBN 0-8176-3250-6/hbk; 0-8176-4441-5/ebook; 0-8176-4407-5/set). xxi, 653 p. (2005).
Basic Real Analysis and Advanced Real Analysis (2005; Zbl 1095.26002) are two companion volumes of a comprehensive treatment unifying numerous branches of classical and functional analysis that can, with more or less assurance, be named as real analysis. Any textbook or manuscript in Analysis presents the author’s point of view on the subject and its contents essentially depends on aims and targets the author is standing against and also depends on the author’s style and the level of generality. The framework of these books is to present basic concepts and tools used in real analysis, to describe basic connections between real analysis and other parts of mathematics; to write all this material in a clear and transparent form, giving new topics and theories from particular to general, with examples that illustrate this material; to present the list of problems allowing the reader himself to estimate the level of the assimilation of ideas and constructions new for him.
Basic Real Analysis contains Preface, Dependence among chapters, Guide for the readers, List of figures, Acknowledgments, Standard notation, 12 Chapters, Appendix, Hints for solutions of problems, Selected references, Index of notation, and Index. Chapters I–IV are devoted to some fundamentals of real variables and functions, usually omitted in standard courses of Calculus, Chapters V–XII deal with Lebesgue theory in Euclidean and abstract spaces, Fourier series and Fourier transforms for the Lebesgue integral, and also the basics of Hilbert and Banach spaces related to the Lebesgue integral. The volume contains more than 300 problems and a separate section gives hints or complete solutions to them. I can not say that the account is systematical or exhaustive (the author himself mentions only two omissions: Stokes’ theorem and differential forms, and the use of Complex Analysis; I, as a reviewer, could add many others), however, in general, the book seems to be completely unified, carefully reasoned, rich in concepts, methods and results, and indubitably useful as for students in Real Analysis so also for teachers in this field.
Some information about the contents of the book. Chapter I Theory of calculus in one real variable deals with the following delicate problems in Analysis: reals, sequences, continuity, interchange of limits and uniform convergence, Riemann integral and Taylor’s theorem with integral remainder, power series and special functions, summability, Weierstrass approximation theorem and Fourier series; as one can see, the chapter is concerned with many subjects; naturally, this chapter is the most mosaic. Chapter II Metric spaces presents topological theory of metric spaces; here one can find basic definitions of the theory, numerous examples of metric spaces, and a sufficiently exhaustive account of their properties as compactness, completeness, connectedness, Baire category theorem and theorem about completion; Arzelà-Ascoli and Stone-Weierstrass theorems for \(C(S)\) for compact metric \(S\) are also presented. Chapter III Theory of calculus in several real variables states the following facts: operator norms and exponential of a matrix, differentiation and partial derivatives, partitions of unity, inverse and implicit function theorems, Riemann integrals and Fubini’s theorem for it, change of variables in the Riemann integral. Chapter IV Theory of ordinary differential equations and systems contains elements of the theory: existence and uniqueness theorems (by Picard-Lindelöf), the simplest results about dependence on initial condition and parameters, the basic facts about linear equations and systems including serious solutions of the second-order linear equations.
Chapter V Lebesgue measure and abstract measure theory presents basic notions and results, Lebesgue’s integration theory including Fubini’s theorem and definitions and elementary properties of \(L^1\), \(L^2\), and \(L^\infty\). Chapter VI Measure theory for Euclidean space deals with more special properties such as: Lebesgue and Borel measures on \({\mathbb R}^n\), comparison of Riemann and Lebesgue integrals, change of variables for Lebesgue integrals, Sard’s theorem, Hardy-Littlewood maximal theorem, Riesz-Fischer theorem; Stieltjes measure on the line and Dirichlet-Jordan theorem are also considered. The short Chapter VII Differentiation of Lebesgue integrals on the line deals with a standard set of Lebesgue theorems about the differentiations including Lebesgue decomposition of functions with bounded variation. Chapter VIII Fourier transform in Euclidean space is devoted to the classical theorems about Fourier transform in \(L^1({\mathbb R}^n)\) and \(L^2({\mathbb R}^n)\), Schwartz space \(S({\mathbb R}^n)\), and also Poisson summation and integral formulas; as well basic results about Hilbert transform in \(L^2({\mathbb R}^n)\) are presented. Chapter IX \(L^p\) spaces deals with the theory of the basic results in analysis whose formulations are essentially connected with the concept of \(L^p\) spaces; here are convolution theorems, Jordan and Hahn decompositions of measures and Radon-Nikodým theorem, Riesz theorems about linear functionals, and even Marcinkiewicz interpolation theorem.
The last three chapters are the most abstract ones in the volume. Chapter X Topological spaces deals with the general theory of topological spaces and presents basic definitions and properties, numerous examples, fundamental constructions, and most important results in this field: Tikhonov product theorem, Uryson’s lemma, some theorems about metrization, Ascoli-Arzelà and Stone-Weierstrass theorems for \(C(S)\) for a compact topological \(S\). The small Chapter XI Integration on locally compact spaces is devoted to the analysis of relations between abstract integration theory and integration in topological spaces; here one can find Riesz representation theorems, theory of regular Borel measures (in particular, on \({\mathbb R}^n\)). The last Chapter XII Hilbert and Banach spaces gives three basic principles of analysis in the framework of Hilbert and Banach spaces: Hahn-Banach, Banach-Steinhaus theorem and Banach interior mapping and closed graph theorems.
Appendix at the end of the volume contains the following sections: sets and functions, mean value theorem and some consequences, inverse function theorem in one variable, complex numbers (although the author writes that the complex analysis in the volume is omitted, he uses complex numbers in many places), classical Schwarz inequality, equivalence relations, linear transformations, matrices and determinants, factorization and roots of polynomials, partial orderings and Zorn’s lemma, cardinality. The volume ends with Hints for solutions of problems, Selected references, Index of notation, and Index.
Summing, I can repeat that this volume is useful and interesting for all who deal with analysis and its applications. I suppose, any mathematical library must have a copy of this book and can recommend it to students and lecturers.

MSC:

26-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to real functions
28-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to measure and integration
33-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to special functions
34-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordinary differential equations
42-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to harmonic analysis on Euclidean spaces
44-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to integral transforms
46-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis
47-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operator theory
54-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to general topology

Citations:

Zbl 1095.26002
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