Customer Reviews

Introduction to Classical Real Analysis (Wadsworth & Brooks/Cole Mathematics Series)

5 star:		(4)
4 star:		(0)
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1 star:		(1)

 

 

This is the best book ever written on introductory classical real analysis. Better than other well regarded "classics", but sadly out of print (shame on all math instructors!). As the title implies, there is no abtract measure or integration theory, nor any functional analysis, but many theorems are stated in the context of general metric or even topological spaces. All the usual topics (for this level) are covered: Sequences and Series, Limits and Continuity, Differentiation, Elementary Functions and Integration. Lebesgue's measure is introduced in Chapter 2 and used in every chapter afterwards. The last chapter is the real treat: a wonderful introduction to Trigonometric Series. In the words of the author, this chapter is "a dessert that rewards the reader's hard labor expended in learning the fundamental principles of analysis".

Contrary to what another reviewer states, the book discusses R^n explicitily in the last 50 pages of the chapter on Integration (topics include integration on R^n, iteration of integrals, differential calculus in higher dimensions and transformation of integrals in R^n). And of course, R^n is also included implicitly in any theorem that's stated in terms of metric/topological spaces.

Probably the only shortcoming that anyone could find in this book is one that was also mentioned in another review: the lack of figures. Personally I like it that way, but that is just a matter of preferences, and in any case the author had a very good reason for not including any graphs/figures in his book: He is blind.

I can't end this review without including some excerpts from the author's preface (since Amazon doesn't have it):

"The subject is ... 'real analysis' in the sense that none of the Cauchy theory of analytic functions is discussed. Complex number, however, do appear throughout. Infinite series and products are discussed in the setting of complex numbers. The elementary functions are defined as functions of a complex variable. I do depart from the classical theme in Chapter 3, where limits and continuity are presented in the contexts of abstract topological and metric spaces."

"I have scrupulously avoided any presumption at all that the reader has any knowledge of mathematical concepts until they are formally presented here...for example, the number pi is not mentioned until is has been precisely defined in Chapter 5."

"One significant way in which this book differs from other texts at this level is that the integral we first mention is the Lebesgue integral on the real line."

"I sincerely hope that the exercise sets will prove to be a particularly attractive feature of this book. I spent at least three times as much effort in preparing them as I did on the main text itself...A great many of the exercises are projects of many parts which, when completed in the order given, lead the student by easy stages to important and interesting results."

The proofs are detailed, and there are plenty of important theorems that are worked-out. Don't most of us pay less attention if a theorem is an exercise or worse skip them altogeter? (Edit: I am not putting down its exercise problems or even the need of exercise problems in general. One cannot just read about other people's proofs. One has to practise generating his own proofs to check his understanding. Having said that some books don't have enough work-out proofs or discussions to get a feel for the topic. Sometimes I want the milder pleasure of comprehension. Also most people do not work out ALL of the exercise problems. Maybe the author should have also written that he expects every problem to be worked out. Some authors state important theorems in the main text and leave the proofs as exercises. And making the font of the exercises to be smaller than the main text is not a good practice. This book's exercises include important results, and one should read and do as many of them as possible.)

So far as I can tell, this book does not discuss R^n (Edit: I meant I did not see Stokes theorem, wedge product, Jacobian, which baby Rudin has - vector calculus stuff, while it does have multiple integrals.), but it does introduce Lebegue integral in a concrete manner. (thank you!)
If you read about Cantor sets from Rudin only, you are missing out. Read about the Cantor sets in this book!
My only complain is not enough graphs and illustrations, but this book is better than the baby Rudin (Principles of Analysis).
(Edit: I also found the undefined term vector lattice annoying.
The best INTRODUCTORY analysis book is by Gaskill - Foundation of Analysis. It has rigorous proofs, supplemented, not replaced, by diagrams for better understanding where relevant, and follow-up discussions on what happened on the proofs. While Gaskill does not replace this book or Rudin, it is helpful to read before baby Rudin. Also, elementary does not equal comprehensibility. Fitzpatrick's book is elementary/introductory, but the book and its figures are no more illuminating to a beginner than Rudin. I don't remember any graph from Rudin either.)

I am one of Karl's PhD students.

It is a sad, almost criminal, fact that this wonderful book is out of print. It provides a rigorous and accurate presentation of elementary real analysis using the Lebesgue integral. No, there are not diagrams, but this book is for those who want proofs, not just warm fuzzy hand-waving. The standard theory is stated accurately and proven in detail in the main part of the book, but the true strength of this book is in the exercises. There, step-by-step, an incredible range of good examples and mind-bending theorems are presented. Do you want an example of a differentiable function with bounded derivative that is monotone on no interval? It's there. Do you want a proof that *most* differentiable functions are like this (in a well-defined sense)? It's there.

The primary failure of this book, though, is its index. Karl knew exactly how he wanted to describe each result, but almost nobody else knows his system. It would be a real service to the mathematical community to re-write the index and get this book into the Dover series or at least into print again.

The first sentence in the Preface reads as follows: "This volume has evolved from lectures that I have given at the University of Oregon and at Kansas State University during the past twenty years." Twenty years plus a serious professor gives rise to a masterpiece which is worth being read by any mathematician who is minimally interested in Real Analysis. In order to properly place the mathematical level, the general content of the book is not an example of what can be understood by university first-year students, but rather it provide the reader with a solid background of non-trivial and fundamental results once he or she is acquainted with a typical first course on Mathematical Analysis. The book is profound, accurate, and contains a splendid collection of exercises (again from the Preface: "I spent at least three times as much effort in preparing them as I did on the main text itself"). Prof. Karl R. Stromberg is co-author (with his teacher Edwin Hewitt) of another very great book entitled "Real and Abstract Analysis". Thanks, professor Stromberg; this is one of the best books on Mathematics I have ever read.

I was shown this book recently by a tutee asking WTF its proof of the Heine-Borel theorem is about.

I couldn't help...

There appears to be but a single diagram in all of the order of 550 pages of this book, and that as a concessionary afterthought following an excruciating number theoretic proof of the countability of the rationals; to point out that the rationals can also be counted by the familar diagonal counting process.

Presenting the Lebesgue integral before the Riemann integral indeed makes sense since, as the author points out in his forward, it's no more difficult than the Riemann, but not if the object of the exercise is apparently to make either as hard (the author would call it 'uncompromising') as possible. For example the student quickly learns that the Lebesgue integral forms a vector lattice space whatever that is, and too bad if she doesn't know because there's nothing in the book, leafing through it quite carefully and checking the index, which explains what such a thing, if it indeed really exists, actually <is>.

Well, it's very comprehensive and seriously weighty for sure. But I suggest you buy this book off the shelf after a browse to check it's really your kind of maths book, and funny old you if it is: you must be extraordinarily 90% brilliant and what's a mind like yours doing in a book like this one has to enquire ... have a go at Russell' and Whitehead's 'Principia' for example, now there's uncompromising...

Me, I think the book is quite simply dreadful, worth no more than a single star (fair, for the single diagram).

I genuinely pity Dr. Stromberg's students.

William Boyd
www.vanyka.co.uk