This is the first volume of a planned three-volume set for an advanced undergraduate course on mathematical analysis. This first volume gives a rigorous treatment of real analysis in one variable.
Part one, chapters one and two, is devoted to the foundational part. It starts with an axiomatic treatment of basic set theory up to the axiom of choice, including relations, partial orders, functions and equivalence relations. The second chapter introduces the number systems required for real functions of one real variable, namely the natural numbers, the ring of integers and the field of rational numbers, and the construction, via Dedekind cuts, and order properties of the real field. Complex numbers are introduced in chapter 3.
The second part has eight chapters. Chapters 3 to 6 treat in detail sequences and series of real numbers, the topology of R and limits and continuity of real functions of one real variable. Chapters 7 and 8 are devoted to differentiation and Riemann integration of real-valued functions defined on a real interval. The usual highlights are the fundamental theorem of calculus and some mean value theorems. As is now becoming standard, there is an introduction to Fourier series in chapter 9, including convolutions and the Dirichlet, Fejér and Poisson kernels. The last chapter considers various applications of the theory just developed, introducing some important functions: Beta and gamma functions and Riemann’s zeta function, proving several of their main properties or, in some cases, sketching a proof. An appendix is devoted to a proof to the equivalence of the axiom of choice and Zorn’s lemma.
The competition for this book includes such classics as baby Rudin (3rd edition, McGraw-Hill, 1976), Bartle (and Sherbert)’s The Elements of Real Analysis (4th edition, Wiley, 2011) or Royden (and Fitzpatrick) Real Analysis (4th edition, Prentice Hall, 2010), to mention just a few. Garling is a gifted expositor and the book under review really conveys the beauty of the subject, not an easy task. The book comes with appropriate examples when needed and has plenty of well-chosen exercises as may be expected from a textbook. As the author points out in the introduction, a newcomer may be advised, on a first reading, to skip Part one and take the required properties of the ordered real field as axioms; later on, as the student matures, he/she may go back to a detailed reading of the skipped part. This is good advice.
Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is email@example.com.
Part I. Prologue: The Foundations of Analysis:
1. The axioms of set theory
2. Number systems
Part II. Functions of a Real Variable:
3. Convergent sequences
4. Infinite series
5. The topology of R
9. Introduction to Fourier series
10. Some applications
Appendix: Zorn's lemma and the well-ordering principle