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Introduction to Real Analysis

Robert G. Bartle and Donald R. Sherbert
John Wiley
Publication Date: 
Number of Pages: 
[Reviewed by
P. N. Ruane
, on

Joint review of:

  • An Introduction to Real Analysis, by Robert G. Bartle and Donald R. Sherbert
  • Real Analysis, by Gabriel Klambauer

Despite almost identical titles, these are two very different books; readers would need to be very familiar with the contents of the first if they were to have any chance of making progress with the second. However, the books almost coincide regarding their expected readerships, which are loosely described as straddling the undergraduate/postgraduate divide. Moreover, each of the books is claimed to be of relevance to the needs of applied scientists etc. Yet another common feature is that they have now been around for many years, meaning that some MAA members may possibly have a working knowledge of either, or both, publications.

The Introduction to Real Analysis seems to be a third edition (or later) and is said to be accessible to those students who have studied calculus. But what type of student and which sort of calculus? In my view, if this is to be a first encounter with analysis, then students should be (in the UK) good honours candidates or (in the USA) strong maths majors. For most students, however, this treatment would be too challenging as an introduction to real analysis. Although, as suggested by the authors, initial motivation could be provided by means of a course on calculus, wherein analytical ideas are gradually introduced in "practical" contexts (as in [1]) 

For many reasons, I am very impressed with this book by Bartle and Sherbert. It represents a serious incursion into the heart of real analysis, and lays strong foundations for subsequent study of more advanced topics. It is well structured across eleven chapters and, despite its level of difficulty, the material is presented with clarity and precision. But another notable strength concerns the wealth of worked examples, together with the exercise sets provided at the end of each section. Each set contains a good variety of problems, ranging from the routine to the more open-ended sort, and there are hints and partial solutions for many of these. 

The following examples may give an indication as to the level of treatment given in this book: 

  1. The concept of limit of a function is defined in terms of cluster points, quickly followed by an equivalent definition based upon convergent sequences. At the same time, students would be getting to grips with very tricky ε–δ techniques and other concepts, such as that of "bounded neighbourhoods."
  2. As with many of the chapters, that on continuity is conceptually very dense and examines this idea in relation to Cauchy sequences, Bernstein's approximation theorem and gauges (not to mention discussion of uniform continuity). 
  3. Differentiation is explained in terms of functional notation and then in terms of the Carathéodory derivative. Discussion of Leibniz notation is avoided and, in this respect, the treatment would be out of step with typical prior student experience. Obviously, tutors would have to consider whether this approach suits their purpose in terms of subsequent courses on differential equations or differential forms etc. 

All this, by the way, is excellent mathematics, and the book could well provide a sound basis for a course for second or third year maths specialists. Commencing with two chapters on set-theoretic foundations, and a serious discussion of the real number system, it takes the reader on a journey through the exacting terrain of classical real analysis, terminating at the threshold of Lebesgue integration and point-set topology. Nonetheless, those who decide to base a course upon this book may have to select an abbreviated pathway through all this material, for there seems too much here for a one semester course. 

Klambauer's book was first published by Elsevier in 1973, and is now another of those Dover reissues, which so fittingly rescue quality publications from the realms of obscurity. Stated as meeting "the needs of beginning graduate students in mathematics" and providing a review for those who have to face comprehensive examinations, it is suggested that only modest pre-requisite knowledge is necessary. In addition, the presentation is described as being "concise", whilst the approach is said to "proceed from concrete situations to more abstract theory", claims which must come under a little scrutiny, I think. 

The first chapter, Lebesgue Measure on R1, begins some way ahead of where the previous book left off, and the very first sentence gives an entirely unmotivated formal definition of sequential covering classes. This is followed by an ever-deepening account of the theoretical underpinnings of Lebesgue measure, together with nineteen important theorems, all in the space of twelve pages. Throughout the book, there is a notable shortage of worked examples, hardly any historical commentary and a dearth of additional expository discourse. For example, having no specialised knowledge in this area myself, the book made me curious as to the background of P. J. Daniell, whose ideas are central to the two penultimate chapters. With no bibliography, nor any list of references provided by Klambauer, I turned to my old friend Google, who provided me a little history and explained, amongst other things, that Daniell's ideas (circa 1917) reduced the dependence of integration upon the development of a workable measure theory.

In summary, this is one of those books that, in days of yore, would be classed as a "Treatise", because it is really a formal account of much of the relevant the matter on integration theory for real-valued functions, and would therefore serve the admirable purpose of being a handbook for postgraduate students specialising in this field. The first three chapters cover Lebesgue measurable functions, Lebesgue integrals and Lebesgue function spaces (all on R1). Chapter 4 discusses differentiation and absolute continuity followed by chapters on abstract measure, outer and product measure. There is a brief change of direction at this point, with chapter 7 devoted to a condensed introduction (or revision) of topological and metric spaces. Then there are those two chapters on the methods of P. J. Daniell and the Stone-Daniell integral.

As for the author's belief that the narrative proceeds from the concrete to the abstract, I can see little evidence of such a process in this book . For although the book exudes a high degree of mathematical integrity, the presentation is so condensed that all but the most able postgraduate students would find it off-putting. Therefore, despite the provision of a large number of exercises at the end of each chapter, I recommend it, at most, for supplementary use to a taught postgraduate course on integration theory, but certainly not as a self-tuition manual.   


[1]  Calculus: Single and Multivariable (4th Edition), by Hughes-Hallett, Gleason, McCallum et al. Wiley (2005).

Peter Ruane is retired from university teaching and now dabbles in as many creative diversions as possible.



The Real Numbers.

Sequences and Series.


Continuous Functions.


The Riemann Integral.

Sequences of Functions.

Infinite Series.

The Generalized Riemann Integral.

A Glimpse into Topology.



Photo Credits.

Hints for Selected Exercises.