Analysis (Q5919153)

This textbook offers a concise, nevertheless comprehensive introduction to the principles of real analysis as it is taught in the first three semesters at German universities. Certainly, there is a vast quantity of analysis primers within the international textbook literature, which raises the question of what could make the present new text different at all. In fact, the distinguishing feature of the author's one-volume introduction to analysis is its unique precision and terseness of presentation, which leads to a high degree of efficiency and lucidity likewise. In the preface the author points out his guiding didactic principles through the following three short statements: 1. All that can be said -- can be said tersely. 2. The best motivation for a mathematical fact is a clear and simple proof. 3. As for a good text, not only the mentioned things are important, but also the withheld ones. Indeed, these principles characterize the author's approach very aptly, regardless of their general acceptance by experts in mathematical education. Anyway, the book under review provides an excellent introduction to the main topics of undergraduate real analysis, leading the reader from the set-theoretic and topological foundations to the theorem of Stokes on differentiable manifolds, and that on just about 400 pages. As for the precise contents, the text comprises eighteen chapters and two short appendices. Chapter 1 is of preliminary nature and develops the basic facts from set theory, including a little propositional logic, sets and mappings, products of sets, relations, and the principle of mathematical induction. The following seven chapters form Part I which bears the title ``Differential and integral calculus''. This part covers the standard material on one-dimensional real analysis as follows. Chapter 2 gives an axiomatic introduction to the field of real numbers, culminating in the concept of Dedekind completeness (instead of Cauchy completeness, which is then proved in Appendix B at the end of the book). Chapter 3 discusses sequences and series of real numbers, together with their convergence properties and criteria. Chapter 4 treats continuous functions of one variable, the main theorems they satisfy, and the fundamental elementary functions such as the log-function, the exp-function, and the trigonometric functions. Chapter 5 turns to differential calculus in one variable, whereas Chapter 6 deals with the Riemann integral, including improper integrals. Chapter 7 is devoted to sequences of functions, thereby focussing on the concept of uniform convergence, power series, Taylor series, and Fourier series. The abstract material is illustrated by many concrete examples and applications. Chapter 8 provides the necessary material from general topology. In this context metric spaces, complete metric spaces, continuous mappings between metric spaces, the notions of connectedness and compactness, the Arzelà-Ascoli theorem, and a first introduction to normed real vector spaces are briefly explained. In the sequel, Part II of the book (containing Chapters 9--12) gives introductions to multivariable real analysis and the allied advanced general topology, respectively. Chapter 9 is dedicated to differential calculus in \(\mathbb{R}^n\), with the local inverse function theorem and the implicit function theorem as highlights. Integration in \(\mathbb{R}^n\) is discussed in the subsequent Chapter 10, and that at first in its simplest form via iterated one-dimensional integration. This requires the restriction to functions with compact support, but already this elementary approach turns out to suffice for the study of continuous vector fields on spheres. Chapter 11 gives a brief, very first introduction to ordinary differential equations, while Chapter 12 complements the discussion of Chapter 8 by treating more abstract topological spaces. This chapter contains, among other topics, central theorems on compact spaces (à la Urysohn and Tychonov), the Stone-Weierstrass theorem, some basic facts on Hilbert spaces and general Fourier series, Baire's theorem and Tietze's extension theorem. The following Part III is formed by the subsequent four chapters and comes with the heading ``Measure and integration theory''. Chapter 13 develops the basic measure theory, with a special view toward the Lebesgue measure. Chapter 14 gives an introduction to the Lebesgue integration theory, including the Riesz representation theorem and Hahn's decomposition theorem for complex-valued measures. The study of \(L^p\)-spaces with respect to a given measure is conducted in Chapter 15, with the highlight being the theorem of Lebesgue-Radon-Nikodým and its consequences. Chapter 16 treats product measures and product integrals, with Fubini's theorem as the central result in this context. Part IV contains the remaining two chapters of the book and is entitled ``Integration on manifolds''. Chapter 17 introduces differentiable manifolds in \(\mathbb{R}^n\) and differential forms on them, which prepares the reader for the concluding Chapter 18, where the theorem of Stokes for smooth, oriented submanifolds of \(\mathbb{R}^n\) is proved, along with Poincaré's lemma, a brief introduction to holomorphic functions, and Brower's fixed point theorem as a prominent application of the theorem of Stokes. Appendix A discusses the existence and uniqueness of \(\mathbb{R}\) via Dedekind cuts and Dedekind-complete fields, and Appendix B provides the comparison between Dedekind-completeness and Cauchy completeness of the field \(\mathbb{R}\), as already mentioned before. Each chapter ends with a series of related exercises, and many instructive examples illustrate the versatile material throughout the text. Overall, the book stands out by its high degrees of clarity, conciseness, rigor, and elegance of presentation, which makes it a perfect source and companion for students and instructors likewise, be it for basic courses or for self-study.