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Introduction to Real Analysis

Christopher Heil
Publication Date: 
Number of Pages: 
Graduate Texts in Mathematics
[Reviewed by
Frédéric Morneau-Guérin
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There are many books aimed at guiding novices in the acquisition of basic knowledge in abstract real analysis. For Christopher Heil, the classics on the matter (the books by Folland or Rudin, for instance) stand out from the rest thanks to their structured, effective and systematic treatment of the material.  However, while they are great reference books that every mathematician should keep at hand, they are not well suited to be used as textbooks in an introductory course; ”they should be the second set of books on analysis that you pick up”.
It was with the intention to produce “the analysis text that you read first” and, therefore, break away from the widespread practice of serving, first and foremost, the imperatives of logic and elegance in the presentation of elements when these are difficult to reconcile didactic considerations, that Heil decided to write an Introduction to Real Analysis. This challenge has been met hands down.
This book is intended primarily for students beginning their graduate studies in mathematics but it will also be suitable for well-prepared undergraduates.  The author offers a very airy and well-structured preliminary chapter, in which he reviews the essential prerequisites and takes the opportunity to introduce the notation he will make his own.
Introduction to Real Analysis contains over 30 figures that aid in the understanding of certain results presenting a greater degree of technical difficulty, and well over 400 problems and embedded exercises. Nearly fifty brief clues for the most difficult questions are provided at the end of the book. Furthermore, the preface informs us of the existence of resources to help in solving the exercises (available online) as well as a solutions manual for teachers. Other appeals of the book include the presence of an index of symbols structured by themes, where each symbol is accompanied by a precise reference to the Definition where clarification can be obtained.
Anticipating some frequent questions and misunderstandings, the author devoted an entire chapter to the concept of absolute continuity and the fundamental theorem of calculus. It should also be noted that the chapter devoted to differentiation provides absolute clarity.
It is obvious that the author has taken care to make knowledge acquisition as easy as possible and help students structure their thoughts, including by using mnemonic strategies that help students retain the key ideas of theorems and lemmas. With a clear educational concern, Heil has understood that what does not draw attention has little chance of being thought about, and what is not thought about cannot possibly be learned.​
Frédéric Morneau-Guérin is a professor in the Department of Education at Université TÉLUQ.