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Real Analysis

American Mathematical Society
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Frank Morgan has produced a polished set of lecture notes that, alas, would not serve as a good analysis/advanced calculus textbook for most students. The book consists of 135 pages of text (including exercises) divided into 30 chapters. Each chapter is a math bite, truly offering a “simple and sophisticated point of view,” as the book’s AMS website claims. The style is informal, the proofs are “efficient” as claimed on the back cover, and concepts are well motivated.

In addition to elegant treatments of real numbers and limits, the topology of the real line, derivatives, the Riemann integral, infinite series, and metric spaces, Morgan’s text contains some unexpected topics: the Cantor set and fractals, Lebesque integration (two pages, no proofs), Fourier series and applications (“Strings and Springs”), volumes of n-balls and the gamma function, and equicontinuity (Ascoli’s Theorem). There is no discussion of multivariable calculus.

A fundamental weakness of the book is the general lack of worked out examples and detailed explanations that most students need. On average, there are fewer than ten exercises at the end of each (short) chapter. Any instructor using this text would have to supplement in a big way.

In contrast, a book such as Introductory Analysis: The Theory of Calculus (Second Edition) by J. A. Fridy (Academic Press, 2000) is still a relatively compact text (with more than twice the number of pages) that treats both single and multivariable calculus and addresses the needs of a typical student in a more appropriate way. For example, Fridy’s text treats uniform continuity in roughly six pages with five examples and twelve exercises, in contrast to Morgan’s one page, no examples, and eight exercises. For the Fundamental Theorem of Calculus, the score is three pages, one example, and eight exercises for Friday versus two pages, no examples, and four exercises for Morgan. These statistics are typical and suggest there simply isn’t enough detailed exposition of standard material in Morgan’s text to make it a viable book for adoption in a typical analysis/advanced calculus course.

The summarizing analogy that occurs to me is that of a nouvelle cuisine meal — prime ingredients well prepared, an elegant presentation, but not necessarily enough food to satisfy. This is a well-written book that anyone teaching introductory analysis should have on his or her shelf, but it is not a book that many of us would choose to adopt for class use.

Henry Ricardo ( is Professor of Mathematics at Medgar Evers College of The City University of New York and Secretary of the Metropolitan NY Section of the MAA. His book, A Modern Introduction to Differential Equations, was published by Houghton Mifflin in January, 2002; and he is currently writing a linear algebra text.

Date Received: 
Thursday, August 11, 2005
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Frank Morgan
Publication Date: 
Henry Ricardo

Part I: Real numbers and limits

  • Numbers and logic
  • Infinity
  • Sequences
  • Functions and limits

Part II: Topology

  • Open and closed sets
  • Continuous functions
  • Composition of functions
  • Subsequences
  • Compactness
  • Existence of maximum
  • Uniform continuity
  • Connected sets and the intermediate value theorem
  • The Cantor set and fractals

Part III: Calculus

  • The derivative and the mean value theorem
  • The Riemann integral
  • The fundamental theorem of calculus
  • Sequences of functions
  • The Lebesgue theory
  • Infinite series $\sum a_n$
  • Absolute convergence
  • Power series
  • Fourier series
  • Strings and springs
  • Convergence of Fourier series
  • The exponential function
  • Volumes of $n$-balls and the gamma function

Part IV: Metric spaces

  • Metric spaces
  • Analysis on metric spaces
  • Compactness in metric spaces
  • Ascoli's theorem
  • Partial solutions to exercises
  • Greek letters
  • Index


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Wednesday, September 26, 2007


akirak's picture

I like this elegant and attractive book, though occasionally the author moves a little too fast. In equation 3.8(2) on page 18, he takes the limit inside the function ex without even a comment. He takes it for granted that the reader knows and understands the space of real numbers as decimal expansions, and uses this without further comment in proving results such as: compact sets contain their maximums.

In spite of these comments, I believe this book has the potential to be a fine course text in the hands of a motivated instructor. The key ideas are presented in succinct clear style. Indeed, there are 37 chapters in 180 pages. Topics of special interest to the instructor or students can be fleshed out. From Chapter 22 on, many interesting nontrivial applications are discussed, each quite briefly. Again, instructors would probably want to flesh out some of these chapters.