Richard Beals' Analysis: An Introduction is a serious textbook for serious students. Intended for advanced undergraduates, this book demands as much personal maturity from the reader as it does mathematical sophistication.
The distinguishing feature of this book is its breadth. It is typical for an introductory analysis text to treat certain fundamental topics with great care, making only passing references (if that) to more sophisticated applications. Beals takes a rather different approach. He clearly views this book as being an introduction to the entire area of analysis, rather than an exposition of a predetermined set of topics. Less than half of the book is dedicated to material which (in the reviewer's experience) would generally appear in a standard introductory course. The remainder deals with more advanced topics, as well as a variety of applications. The last third of the book, in fact, is devoted exclusively to Fourier series and differential equations.
While, technically speaking, this book could be used for a first course in analysis, the title is perhaps something of a misnomer. The important introductory concepts are all discussed, precisely and completely, but often as a stepping-stone to more sophisticated results. Take, for example, the chapter that deals with continuity. Beals spends less than six pages (including exercises) discussing the general properties of continuous functions; after that, he shifts his attention to the spaces C([a,b]) and the Weierstrass Approximation Theorem. While one could argue that six pages are sufficient to his purposes, this transition might seem a bit precipitate to someone encountering these concepts for the first time.
Beals' writing style is characterized by a certain austere elegance. The author has an admirable command of the English language, and he appears unaffected by the excessive informality that has afflicted so many undergraduate textbooks. Apart from a few casual remarks in the introduction, there is virtually no "padding" anywhere in the text. The lemmas, propositions, theorems, and corollaries come in rapid succession, with very little commentary in between. Beals clearly expects a level of discipline from his readers that is comparable to his own.
Analysis: An Introduction is most appropriate for an undergraduate who has already grappled with the main ideas from real analysis, and who is looking for a succinct, well-written treatise that connects these concepts to some of their most powerful applications. Beals' book has the potential to serve this audience very well indeed.
Christopher Hammond is Assistant Professor of Mathematics at Connecticut College.