Foundations of analysis
About one year ago, the author published his excellent textbook ``Complex variables'' [Providence, RI: American Mathematical Society (AMS) (2011; Zbl 1234.30002)]. Like that profound introduction to the classical theory of functions of one complex variable, the book under review evolved from the author's notes for a basic undergraduate course at the University of Utah. As it is pointed out in the preface, this course on foundations of real analysis is geared toward students who have completed three semesters of calculus and one semester of linear algebra, by which the prerequisites for the present book are already outlined. Also, as the author particularly emphasizes, the book reflects the two main goals of this specific course, namely to help students develop the necessary mathematical maturity and sophistication needed for senior or graduate level mathematics courses, on the one hand, and to present a first rigorous treatment of calculus from scratch on the other. Although the topics covered in the current analysis primer are quite standard, their treatment is certainly rather special and didactically refined. In fact, the rich material is presented in a fashion that combines lucidity, rigor, conciseness, straightforward explanations, and instructive examples in a masterly manner, thereby displaying an exceptionally high degree of didactical effectivity and user-friendliness. Actually, this text has been used by the author and his colleagues for several years at the University of Utah. It has undergone many improvements, additions, and corrections in the course of these years, and appears now in a well-tested, highly polished design.NEWLINENEWLINE As for the precise contents, the book comprises eleven chapters, each of which is subdivided in several sections.NEWLINENEWLINE Chapter 1 gives an outline of the construction of the field of real numbers. Starting with some basics on sets and functions, the natural numbers (via Peano's axioms), integers and rational numbers, Dedekind cuts in the field of rational numbers, and finally the complete, Archimedean, ordered field of real numbers are introduced.NEWLINENEWLINE Chapter 2 treats sequences and limits of sequences, including the usual limit theorems, Cauchy sequences, and the Bolzano-Weierstrass theorem. Chapter 3 turns to a rigorous study of the concept of continuity for real-valued functions. Properties of continuous functions, uniform continuity, and uniform convergence are the main topics in this context. Chapter 4 provides rigorous proofs of the standard theorems from calculus, based on limits of functions and the precise notion of derivative. The definite Riemann integral is introduced in Chapter 5. The existence and the main properties of the integral are proved in detail, together with the two form of the fundamental theorem of calculus and applications to establishing the natural logarithm, the exponential function, and the notion of improper integrals. Chapter 6 is devoted to infinite series, including tests for their convergence, absolute and conditional convergence, power series, and Taylor expansions.NEWLINENEWLINE The second half of the text deals with real-valued functions of several variables and vector analysis. Chapter 7 analyzes the Euclidean topology of the \(n\)-dimensional Euclidean space \(\mathbb R^n\), with focus on convergent sequences of vectors, open and closed sets, compact sets, and connected sets. Continuous functions of several variables and sequences of functions are studied in Chapter 8, in particular under topological aspects and with a view toward the concept of uniform convergence. This chapter also contains some relevant linear algebra. Chapter 9 provides a rigorous development of differentiation in several variables, including the total differential of a vector-valued function on \(\mathbb R^n\) via affine approximation, Taylor's formula in several variables, the inverse function theorem, and the implicit function theorem. Chapter 10 discusses Riemann integration over Jordan domains in \(\mathbb R^n\), with the highlights being Fubini's theorem and the ``general change of variables formula'' for integrals over the smooth image of a compact Jordan region in \(\mathbb R^n\). For these more complicated theorems, the author gives rigorous and detailed proofs as well as some instructive applications.NEWLINENEWLINE The final chapter, Chapter 11, is devoted to basic topics from vector calculus. Using the modern approach via differential forms, the author studies integration over curves and surfaces in Euclidean space, which culminates in the respective classical integral theorems of Green, Gauss, and Stokes. In an optional section at the end of this chapter, integrals over \(p\)-chains and \(p\)-cycles in \(\mathbb R^n\) are briefly introduced, and the general form of Stokes's theorem is stated as an illustration. Furthermore, the classical forms of the integral theorems in low dimension are derived from their modern forms as well.NEWLINENEWLINE Finally, there is an appendix on infinite sets at the end of the book. The author discusses here, with full proofs of the respective results, cardinalities, countable sets, uncountable sets, and the equivalences of the axiom of choice, Zorn's lemma, and the well-ordering theorem.NEWLINENEWLINE Each section of the text concludes with a large list of related exercises, which differ widely in level of abstraction and degree of difficulty. Also, each section contains a wealth of examples and worked problems illustrating the material of the respective part of the text. Moreover, these examples are also designed to teach students how to tackle the exercises for that section. A short bibliography and a carefully compiled index can be found at the end of the book.NEWLINENEWLINE Apart from the various features of this excellent introduction to real analysis, as they were already mentioned in the beginning of this review, another particular advantage deserves mention. Namely, especially in the second part of the book, the author has followed the philosophy of giving more abstract outlooks based on the concrete concepts developed in the text, thereby motivating students for further, more advanced study of analysis, topology, differential geometry, and functional analysis.NEWLINENEWLINE All together, the book under review must be seen as a highly welcome and useful enrichment of the existing vast textbook literature in the field of basic real analysis.
scientific article; zbMATH DE number 6064534