The first, and perhaps most telling, comment on this book is simply, “I really like it.” For someone who has been around for a good while, that is not a common sentiment for a text on a topic that has been subject to so many different treatments.
So what caused this unexpected response to an undergraduate text in real analysis? The title is not misleading when it describes the text as “a radical approach,” but that alone would not make it successful. The treatment reminded me of my very first experience with this subject, when the professor would frequently warn us about the dangers of relying on intuition while we were learning the rigors of analysis. Essentially, the argument was that intuition would too often prove to be faulty.
By following a fundamentally historical development of the subject, Bressoud shows exactly where the pioneers fell short of the now accepted standards of theoretical justification. They accepted the intuitively obvious even when, on occasions, the so-called obvious was not true. This is achieved through a careful exposition, including historical background where appropriate, but without the book becoming a dry treatise on the history of real analysis—it is anything but!
The examples are thematic, and they illustrate the points while advancing the reader’s understanding, much as the songs in a good musical advance the story rather than interrupt it. They illuminate the points where additional care and rigor are needed. The text then proceeds to answer the problem. The approach is therefore a journey of discovery, in which the pitfalls of incomplete intuition or understanding are exposed in order to help the reader appreciate why the excruciating care of rigorous analysis is needed.
The mere title of the first chapter, “Crisis in Mathematics: Fourier’s Series,” is intriguing to any student embarking on this course. Beginning with a simple example of heat flow in a (long) rectangular plate, the issue of resolving the conflict between the initial and boundary conditions in the trigonometric series solution is examined. The difficulties with the solution and the issues of term-by-term integration and differentiation are motivated by this example. In this first chapter, students and instructors are challenged to extend their understanding of concepts and examples through Web resources, and through Mathematica or Maple scripts, which are both clearly identified throughout the text and exercises.
The Web resources consist of additional portable document format (PDF) files, and Mathematica or Maple scripts that can be downloaded and used to explore further. Additional Mathematica and Maple designations are found in the plentiful exercises. The designation tells the student that Mathematica and Maple code is available to aid understanding appropriate to the particular question. By removing this code from the text and making it readily available on the Web, the book is kept to a reasonable size, and remains much more readable.
Prompted by the example of Fourier’s series, chapter 2 starts the careful thematic development with a study of infinite series. Beginning with Archimedes, it continues with geometric series and the discovery of the radius of convergence for such a series. Explorations into the calculation of &pgr; lead to logarithmic and harmonic series, and then Taylor series. Throughout the chapter, missteps are taken to illustrate the problems, and then solutions to the problems are provided to overcome the oversights. These deal with rearrangements of series, rates of convergence, accuracy of approximations, and, ultimately, the remainder term for Taylor series expansion. In all cases, historical references are included both to illustrate the painful process of even the great mathematicians reaching for full understanding, and to provide human interest and appropriate attribution.
A similar approach characterizes the subsequent chapters: “Differentiability and Continuity,” “Convergence of Infinite Series,” “Understanding Infinite Series,” and “Return to Fourier Series.”
Application of the ideas is never far from the surface, and is a refreshing change for a rigorous treatment of real analysis, in my experience. For example, the final two sections of chapter 3 cover consequences of continuity and the mean value theorem. Chapter 4 concludes with convergence of Fourier series. In chapter 5, “Understanding Infinite Series,” differentiation and integration of power series and a trigonometric series are revisited, and a good motivation for the need for uniform convergence is provided.
Ultimately, the book covers all of the standard material for a first course in real analysis, but in a thoroughly entertaining way that should lead the student to a deeper appreciation, as well as understanding, of the material. Reading this book makes me want to get back to teaching an analysis course, so I can try it for myself! As I said from the outset, I really like this book. I recommend it to anyone who is looking for ways to liven up this material for their students.