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An Introduction to Modern Analysis

Vicente Montesinos, Peter Zizler and Václav Zizler
Publication Date: 
Number of Pages: 
[Reviewed by
Tushar Das
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The word modern in the title of an analysis text throws this reviewer back to two classics, both now available online. The first, from over a century ago, is Whittaker and Watson’s A Course of Modern Analysis (1st ed, 1902); the second, from about half a century ago, is Dieudonné’s Foundations of Modern Analysis (1st ed, 1960). Even a brief perusal of these excellent textbooks will shed some light on what the term modern has come to mean in describing a course in analysis.

Whittaker and Watson’s text, subtitled as “an introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions”, was primarily a quick course in the function theory of a single complex variable that went on to spend the majority of its 600-plus pages on the study of (what are now called) special functions.

Dieudonné’s book, on the other hand, is written in an altogether different style — terse, abstract and très rigoureux. Not unexpected of a founding member of Bourbaki. He starts with primers on naive set theory, real numbers and metric spaces, moves on to vector spaces with norms and inner products before spending a chapter proving the Stone-Weierstrass and Arzelà-Ascoli theorems along with some applications. The next chapter on (several variable) differential calculus is written in the setting of continuous multilinear functions on Banach spaces and the clarity that followed from this coordinate-free rendition was taken up in later textbooks by writers like Henri Cartan and Serge Lang. The last trio of chapters included one on analytic functions, one on existence theorems for ordinary differential equations, and a wonderful denouement on elementary spectral theory that draws from all the preceding 300 or so pages.

The book under review (which we abbreviate as MZZ), published as a 21st century introduction to modern analysis, is closer in spirit to Dieudonné’s text, but there are substantial differences. There is no classical nor modern complex analysis, no calculus/analysis of several real variables, and no development of general measure theory. The core of the text is an exposition of single real-variable calculus, and it ends with a beautiful exposition of basic functional analysis. This last chapter ends with rather sophisticated material aimed at whetting the appetite of budding Banach space theorists. Both advanced undergraduates and graduate students will enjoy the proofs in this text, and would do well to try their hand at the vast collection of problems (the large majority of which come with hints) that are collected at the end of the book. Instructors and students will no doubt relish this treasure trove of problems — the exercises and hints account for a quarter of the book’s 863 pages!

The authors begin with an introduction to the real numbers (representation, cardinality, topology), move on to sequences and series, Lebesgue measure on \(\mathbb{R}\), real-valued functions, and sequences and series of of them, metric spaces, and integration theory (the Riemann and Lebesgue integrals). These topics consist of the first 7 chapters that cover over 50% of the book. Chapters 8 through 10 are each brief forays into convex functions, Fourier series and descriptive statistics.

We then arrive at the final Chapter 11 that sets this book apart from other standard texts at this level, the “Excursion to Functional Analysis”. The writing here is distinguished by its clear proofs and strong emphasis on geometric intuition. Infinite dimensional scenarios are natural generalizations to consider when one moves beyond the study of finite dimensional vector spaces. In our opinion this is basic material that every serious student of mathematics should have some exposure to before they leave college. This encounter may be viewed as a first date, where the elegance of coordinate-free mathematics could easily enchant the beginning student. Those who were mystified or intimidated or turned off by infinite dimensional spaces in the past should read this chapter from beginning to end.

The writing is uneven at parts, which is unsurprising given a book of this length. For example, it was somewhat incongruous to find the Cantor middle-third set appear as the fixed point of a contraction mapping on the space of all nonempty compact subsets of a complete metric space in Section 7.1.6 titled “Some Applications of the Riemann Integral and the Fixed Point Theory to the Theory of Ordinary Differential and Integral Equations”. Such a result would fit more naturally at the end of Section 6.11.1 (The Banach Contraction Principle). Furthermore, there is no reference to who proved such a theorem, which is of quite recent vintage compared to the surrounding material, nor a nod to a standard textbook that would contain such material, e.g. Falconer’s Fractal Geometry.

This brings me to my second point, viz. that the book could be peppered with a number of further directions for the curious student to delve deeper. For instance there is no mention of Hausdorff measure and dimension, or general measure theory. Similarly, though there is a very brief chapter on the basic notions of descriptive statistics there is no mention of the measure-theoretic underpinnings of modern probability theory, or any of its burgeoning tributaries (e.g., ergodic theory) and the plethora of interactions with harmonic and functional analysis. As an editor, I would have suggested splitting the book in two volumes and including an exposition of analysis in \(\mathbb{R}^n\) before the final chapter on functional analysis. I probably would have liked more spectral theory than the brief treatment in Section 11.5.

Functional analysis in general, and Banach space theory in particular, are subjects close to the authors’ hearts. They “try to illustrate to some extent how ‘abstract’ functional analysis emerges from the waters of real analysis as a lighthouse to orientate and overlook the whole sea.” Their deep engagement is further evidenced by the fact that two of the authors (Montesinos and V. Zizler) are co-authors of two substantial textbooks on functional analysis and Banach space theory. At the very least, the book under review (MZZ) serves the necessary preliminaries to read the authors’ more advanced texts:

  1. M. Fabian, P. Habala, P. Hájek, V. Montesinos Santalucia, J. Pelant, and V. Zizler, Functional Analysis and Infinite-Dimensional Geometry, CMS Books in Mathematics, Springer 2001.
  2. M. Fabian, P. Habala, P. Hájek, V. Montesinos, and V. Zizler, Banach Space Theory. The Basis for Linear and Nonlinear Analysis, CMS Books in Mathematics, Springer 2011.

When all has been said and done, the authors must be congratulated on writing a useful textbook that includes plenty of bonuses for both students and instructors.

Tushar Das is an Assistant Professor of Mathematics at the University of Wisconsin–La Crosse.