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Advanced real analysis. Along with a companion volume Basic real analysis. (English) Zbl 1095.26002

Cornerstones. Basel: Birkhäuser (ISBN 0-8176-4382-6/hbk; 0-8176-4442-3/ebook; 0-8176-4407-5/set). xxii, 465 p. (2005).
This book is a companion volume to Basic Real Analysis (2005; Zbl 1095.26001). It systematically develops concepts and tools presented there, however this volume is devoted to more serious and up-to-date problems in functional analysis and its applications in harmonic analysis, theory of ordinary and partial differential equations, probability theory. Advanced Real Analysis contains Preface, Dependence among chapters, Guide for the readers, List of figures, Notation and terminology, 9 Chapters, Hints for solutions of problems, Index of notation, and Index. The volume contains more than 200 problems and a separate section gives hints or complete solutions to them. As Basic Real Analysis the book seems to be completely unified, carefully reasoned, rich in concepts, methods and results, and useful for all mathematicians dealing with modern analysis and its applications.
Chapter I Introduction to boundary-value problems is concerned with two problems: separation of variables in boundary value problems for partial differential operators and “operator” version of the Sturm-Liouville theory for ordinary differential equations of the second order. Chapter II Compact self-adjoint operators presents the classical spectral theorem for compact self-adjoint operators in Hilbert spaces, Hilbert-Schmidt theorem, and elementary theory of operators of trace class. Chapter III Topics in Euclidean Fourier analysis is devoted to Harmonic Analysis on \({\mathbb R}^N\); here the reader can find a delicate account of the theory of Sobolev spaces, basic results about harmonic functions and \(H^p\)-spaces, the Calderón-Zygmund theorem about linear singular operators and its applications to the multidimensional Riesz transform, Beltrami equation, multiple Fourier series, and multidimensional singular operators of the trace class. Chapter IV Topics in functional analysis deals with basic results usually united under the name Functional Analysis. The basic part of the chapter is written in the framework of topological vector spaces. After standard definitions and results here are described the spaces \(C^\infty(U)\), \(C^\infty_{\text{com}}(U)\) and distributions, weak and weak-star topologies, the Banach-Alaoglu theorem about the weak compactness of balls in conjugate spaces, the Stone representation theorem, convex sets and locally convex spaces, the Krein-Milman theorem about extreme points, the Schauder-Tikhonov fixed-point principle, the Gelfand transform for commutative \(C^*\)-algebras, and the spectral theorem for bounded self-adjoint operators. Chapter V Distributions is devoted to more delicate properties of distributions connected with topology on their space, elementary operations and the operation of convolution for distributions, the Fourier transform of distributions, fundamental solution for the Laplacian.
Chapter VI Compact and locally compact groups investigates ways where groups play an essential role; after some sketch about general and locally compact topological groups \(G\), the chapter presents an existence and uniqueness theorem of invariant Haar measures, describes relations between left and right Haar measures and unimodular groups, studies conditions under which one can define a regular Borel measure on a quotient group; the second part of the chapter deals with the spaces \(L^p(G)\), representations of compact groups, and Fourier analysis on compact groups including the Peter-Weil theorem. Chapter VII Aspects of partial differential equations provides an introduction to partial differential equations; the chapter is concerned with the following topics: local solvability in the constant coefficient case, maximum principle in the elliptic second-order case, parametrix for elliptic equations with constant coefficients, basic results about pseudodifferential operators needed to obtain parametrix for linear elliptic equations with smooth coefficients. Chapter VIII Analysis on manifolds, in general, is devoted to the explanations how pseudodifferential operators are defined and work on smooth manifolds; here are the basic sections of the chapter: differential calculus on smooth manifolds, vector fields and integral curves, identification spaces, vector bundles, distributions and differential operators on manifolds, pseudodifferential operators on manifolds. Chapter IX Foundations of probability presents elements of probability theory as a system of models, based on measure theory, of some real-world phenomena; the chapter gives basic notions of probability theory, in particular, independent random variables, Kolmogorov extension theorem (in particular, the existence of infinite sets of independent random variables with specified distributions), and strong law of large numbers.
The volume ends with Hints for solutions of problems, Selected references, Index of notation, and Index. The volume of Advanced Real Analysis requires of the reader more serious knowledge in such parts of analysis as measure theory, Hilbert and Banach spaces, however, it is also useful both as a course text and for selfstudy. One can see, that its contents presents many areas of pure mathematics, as well as applied mathematics, including statistics, mathematical physics, and differential equations. I think that this volume turns to be useful for any mathematical library and can be recommended to students and lecturers.

MSC:

26-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to real functions
46-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis
47-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operator theory
42-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to harmonic analysis on Euclidean spaces
43-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to abstract harmonic analysis
22-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to topological groups
35-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations
60-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory

Citations:

Zbl 1095.26001
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