Real Analysis: A Constructive Approach

Mark Bridger
Publisher:
AMS
Publication Date:
2019
Number of Pages:
302
Format:
Hardcover
Series:
Pure and Applied Undergraduate Texts
Price:
82.00
ISBN:
978-1470451448
[Reviewed by
Bill Satzer
, on
09/22/2019
]
[Note:  This is a reprint edition.  See our review of the first edition of this book.]

Here is another new textbook for undergraduate analysis, but it’s far from a traditional approach. This one is dedicated to being entirely constructive. There is no “there exists an x” without a specific procedure to construct or compute that x. Most of the readers of this review probably learned analysis, as I did, in a traditional way. When we see constructivist approaches, we tend to think that they involve a lot of work to get basically the same thing, or even a bit less. This book might possibly convince you of their value.

Interval arithmetic is key to the author’s approach and - among other things - provides the basis for defining real numbers. Some readers may be familiar with the idea of interval arithmetic from numerical analysis where it is a useful method for bounding rounding errors and measurement errors in computations. In this book, it is an analytical tool.

The author approaches the subject with careful consideration. He understands that the undergraduate analysis course is often a student’s introduction to precise reasoning and careful assembly of proofs. He also knows that many analysis students don’t go on to graduate study in mathematics but want or need a rigorous foundation of the basic calculus they’ve learned to support work in the sciences. He believes that his approach based on interval analysis offers them another tool for thinking about the accuracy and uncertainty of measurements.

The construction of real numbers is paramount. The author begins with rational numbers, rational intervals $[r, s]$ (where $r$ and $s$ are rational), and families of such intervals. A real number is defined as a fine and consistent family of rational intervals. (A family of intervals is fine if for each positive rational $\epsilon$ there is an interval in the family of length less than $\epsilon$. A family is consistent if every pair of intervals in the family intersects.) Before we see this definition there is a good deal of background work on the details of interval arithmetic. After the definition, there is more to do to develop the arithmetic of real numbers. Completeness of the real numbers follows after developing the notion of limiting families of intervals, and the limit as the unique real number contained in each interval.

Another aspect of the constructivist approach is the law of $\epsilon$-trichotomy: given any tolerance $\epsilon > 0$ and any two real numbers $x$ and $y$, then $x < y$, $y < x$, or $x$ and $y$ are within $\epsilon$ of each other. Using this and a bisection-like algorithm, the author proves an inverse function theorem for functions that satisfy upper and lower bounds on the usual difference quotient. The inverse function theorem is used in turn to define nth roots, as well as general exponential and logarithm functions.

By this point, a reader of this review can see how different this approach looks from the more traditional introduction to analysis. It continues to be distinctly different throughout. The author emphasizes uniform continuity over pointwise continuity and uniform differentiability (a derivative function on an interval) over pointwise differentiability. Once again, the author is consistent in using a concept that is constructively definable, whereas the alternative is not.

One advantage of this treatment of analysis is that it makes proofs of the existence of the Riemann integral and the fundamental theorem of calculus very nearly transparent. Most of the work has already been done.

The text also includes treatments of limits, sequences, and series that looks not terribly different from the usual approach. Sequences and series of functions are treated in a later chapter that includes Taylor series and power series. A final chapter briefly touches on complex numbers and Fourier series.

This is a thoroughly rigorous development of the theoretical basis of calculus. Several aspects of it are quite elegant. The author believes that nothing of importance for undergraduates is left out. How true this is depends on one’s sense of “nothing of importance”. Most of the usual anomalies of standard analysis such as the Cantor set and nowhere differentiable functions never appear. Even non-continuous functions are relegated to an appendix. I wonder how students who learn analysis this way and go on to graduate study in mathematics would deal with their first graduate analysis class.

Bill Satzer ([email protected]), now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications ranging from speech recognition and network modeling to optical films. He did his PhD work in dynamical systems and celestial mechanics.