Customer Reviews

Real Analysis: Modern Techniques and Their Applications (Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts)

5 star:		(7)
4 star:		(5)
3 star:		(1)
2 star:		(1)
1 star:		(1)

 

 

I speak as a graduate student in applied math. I really like this book but was bothered by its flaws. Nevertheless, with a good instructor, this text can make for a good learning experience.

Positives: The book is well organized. It builds in a reasonable way so that I could focus on the material in the book and develop my understanding as I went. The book is reasonably well contained. Outside of a reasonable level of basics (a BA or BS in math) the proofs and most of the problems use material developed earlier in the text. I found the book very interesting -- I especially liked the topics presented in the last few chapters.

Negatives: Lots of typos - the author's errata sheet is woefully incomplete. Too few expamples. Too condensed - sometimes to the point of incomprehensibility or even error. The contents of a whole course may be condensed in to a single chapter or even a single section.

Things to be aware of: You should be comfortable with advanced calculus, topology, set theory, and algebra (linear and modern). It also helps to have had some basic real analysis. I highly recommend that you've seen Fourier transforms, Dirac deltas (distributions), and continuous probability. You aren't going to learn these here - you're going to see how measure theory is applied to them.

What I like most about the book is its concise but broad coverage of the fundamentals of real and functional analysis. Although I am not a mathematician, my main interest is solving various engineering problems using numerical methods. A solid background in real and functional analysis would be necessary for deeper understanding of various numerical methods. I wish I had known this book the first time I felt the need to study basic modern analysis. This books has exactly the material I wanted to learn.

In addition to the theorems and proofs, the author tells you why some theorems are important and how they can be used (of course also in a concise way). I found this type of "hints" are extremely helpful. The historical notes at the end of each chapter are also interesting to read.

Folland's book is extremely terse, but very elegant. It covers topics in great generality. I haven't read the whole book,but the parts I read were excellent. Its a damned hard book, but very precise and conveys a great deal of information. Its certainly not easy reading, it makes the reader (at least it made me) think quite a bit since the proofs are very terse. I think that the amount learned is certainly worth the effort. A classic and one that I would definitely recommend!

Strictly from a student's perspective, this is a good textbook in real analysis. The way the material is presented is logical, whatever that means, and consistent. The author doesn't assume a student knows much and you can never go wrong with this assumption. However, this book has TOO MANY typos, so if you've never seen this stuff before, it's not easy to follow. Don't trust this book. Hopefully, next edition will fix this problem. This is a good book and it can become a classic, but, first, they have to correct all the typos. For those, who want a really good book in Analysis which has no typos, I recommend Rudin, "Real and Complex Analysis".

My knowledge of this book originates from a course in measure and integration theory a couple of years ago. Back then I honestly didn't like the book very much. It is very terse, sometimes tending towards incomprehensible. The exercises were great though - hard. Exercises should be hard - that's the best way to learn mathematics.

Anyway - since then i have come to like the book more and more. Whenever I'm wondering of a definition or theorem I often find it in this book presented in a precise and rigorous manner. No redundant or unnecessary information is given - just exactly what I'm looking for. The book is perfect if you have a basic knowledge of a subject and want to fresh your memory or expand your mathematical horizon.

BUT as first time reading the book is almost useless. This, in particular, is the case for chapters 5,6,7,...,12. These chapters are close to incomprehensible if you haven't got a basic knowledge of the subjects allready.

This is the second time I've re-reviewed this book.

First off, I am not a mathematician. I was trained as an engineer, and have recently started studying more advanced mathematics to apply it to my research. The only undergrad math course I'd taken before using this book was the standard analysis course. I initially used this book for a first graduate course in real analysis. Even with a professor, going through the book was incredibly difficult, and I had to resort to another book (Wheeden and Zygmund) as well as extensive notes provided by the professor. This experience made me loathe the book.

A few months after the course, having gained more exposure in this area, I returned to the book, and was surprised to find that I had finally started to understand why the author had organized it the way he had. Now, 6 months and another grad course in analysis later (operator theory), I think the book is worth its weight in gold.

First off, let's outline the cons. At first sight, the book takes brevity to the brink of lunacy. A (very) respectable first graduate course in analysis is covered in the first 100 pages. Dense doesn't even begin to cover it. Major results are relegated to the exercises, whole topics are compressed into a section (sometimes two or three are crammed into one), and even the proofs are presented with the barest minimum of explanation. The whole book is about 370 pages, and has enough material for about 4-5 courses. The exercises range from doable to extremely difficult. You also have chapters on everything from point set topology to harmonic analysis (abstract and otherwise) to probability to functional analysis. Heck, even fractals and manifolds pop up by the end.

The truth, however, is that all of these cons are actually pros in disguise. I know most engineers secretly think that the word 'elegant' used to describe mathematics textbooks is basically code for 'stupendously bad exposition', but the simple truth is this: analysis is hard. Sometimes brutally so, and the more you beat your brains against it, the better. If you're looking for a quick and easy explanation of the Lebesgue integral, this really isn't your book. Especially if you're not used to thinking as abstractly as is required here (I certainly wasn't).

I freely admit that I wouldn't recommend this as a first textbook to my worst enemies (my worst enemies are other engineers). But for mathematicians, the brevity might actually be useful. And once you know the basic material (the first three chapters), the book becomes an invaluable resource.

This book is in my opinion for those undergraduate student that wish to make a step towards a graduate level. Some of the complaints by previous reviewers are not justified. I agree that it requires hard work if you are used to an undergraduate level, but the book is actually very pedagogical and not a bit too terse. You cant demand that a book fills in every gap for you, because mathematics is about understanding things yourself. So I dont understand why someone complained about typos. There are hardly any present, and typos can actually be good, because you have to stay alert and learn to trust your own reasoning. It contains insightful remarks that give you the motivation for the study. The proofs are elegant and perfectly rigorous and the theorems are general. The exercises are ok... not too easy not too difficult.
I havent read any more advanced books, but this is by far the best I have read so far, and after finishing chapter 4 I feel ready for "graduate" texts, which before felt insurmountable.

This books covers a lot of ground, and its main strength is that it draws connections between areas of analysis not normally presented together in standard college or graduate level courses. Every chapter either begins from first principles or builds on previous chapters, making the book logically self-contained. Each chapter is broken into several sections, each with its own set of exercises. Some exercises are fairly straightforward, and others require more work and thought, though some hints are given. I found that most of the exercises were pretty well integrated to the rest of the text, and I would recommend that anyone using this book attempt to do most of them to really learn the material.

In the first three chapters, Folland presents the rudiments of measure theory, integration, and signed/complex measures. All the standard theorems are here, e.g. the Monotone Convergence Theorem, Fatou's Lemma, the Dominated Convergence Theorem, the Radon-Nikodym Theorem, the Lebesgue Differentiation Theorem, etc. The proofs are fine, and fairly intuitive explanations are offered for the material throughout.

In each of these first three chapters, general, abstract material is presented first, after which it is specialized and further developed to the case of Euclidan space. This is in contrast to the approach used by some other authors, such as Royden and Stein/Shakarchi (which are also excellent books), who develop Lebesgue measure on Euclidean space in detail, and then repeat what is essentially the same construction for abstract measure spaces. However, Folland's approach, though not as redundant, requires, I think, greater mathematical sophistication than these other authors'.

Chapters 4 and 5 constitute a rapid introduction to, respectively, point set topology and functional analysis. The purpose of these chapters is not to develop these topics in great depth (for instance, there is no discussion of the spectral theorem in the functional analysis chapter), but rather to give some general language and theory that will be used throughout the remainder of the book.

This brings us to Chapter 6 on L^p spaces, which again covers fairly standard material (Holder and Minkowski inequalities, completeness of L^p, Riesz Representation Theorem for the Dual of L^p, etc.), as well as what I would consider a more advanced topic, interpolation of L^p spaces, namely the Riesz-Thorin and Marcinkiewicz interpolation theorems. Though I might recommend other texts for this, such as the classic Stein/Weiss Fourier Analysis on Euclidean Spaces (which is essential reading anyway for anyone learning analysis), I think Folland's treatment is well-integrated to the rest of the chapter and gives good insight into this material.

Chapter 7 introduces the theory of Radon measures, focusing on the dual space to the space of linear functionals on a locally compact Hausdorff space. The basic notions here make heavy use of the generalities presented in Chapters 4 and 5. The main theorem is the Riesz Representation Theorem for the Dual of C_0(X), which Folland carefully motivates and proves.

Chapters 8 and 9 introduce Fourier analysis and distribution theory, respectively. These chapters give a clean, though necessarily incomplete introduction to these subjects. These areas are vast, and though Folland provides an excellent treatment of the topics he chooses to cover, invariably much has been left out. Again, the alternative reference that comes to mind for Fourier analysis is Stein/Weiss; for distributions, I think Rudin's Functional Analysis, and his Real and Complex Analysis, do a pretty good job as a basic reference. What I like about Folland's treatment, however, is that it is well-integrated to the other chapters in the book.

Chapter 10 introduces probability theory, and again, though it doesn't cover very many topics in this area and feels a bit rushed, it serves to illustrate the interrelations between probability and the other parts of analysis covered in the book. For instance, the Central Limit Theorem is first stated and proved in the language of Fourier analysis, which is then translated into the language of probability, highlighting a fundamental connection between these subjects which many standard treatments do not make clear.

Chapter 11, the last chapter, is a brief introduction to a hodge podge of further topics, including topological groups and Hausdorff measure. The treatment here is cursory, though Folland still manages to give insight into these topics.

As I noted earlier, Folland's style of exposition requires a good deal of sophistication from the reader; I would not recommend this book for learning real analysis for the first time, but only after having looked at some other, more introductory-level texts, such as Royden, Stein/Shakarchi, or Rudin's Principles of Mathematical Analysis. The proofs, though quite good, can sometimes be a little terse; readers must work out some details on their own. I did find one slight error in the chapter on Fourier analysis (of course, it is entirely possible the error is my own and not Folland's!), but it was not central to the text and hardly cause for complaint. I appreciate the Notes section at the end of each chapter, which contain broad comments on the material, delving into the history of each topic and giving numerous outside references which I have found very useful to follow up on.

As I said before, Folland's main strength is his ability to weave many different threads of analysis into one coherent picture. This book is extremely well thought out and carefully planned. I would strongly recommend reading this to any advanced undergraduate or beginning graduate student who wants a deeper appreciation and understanding of the essential topics in real analysis.

This is a great analysis book as others have discussed.

However, the publisher has done a really shoddy job
on the new edition. The print looks like a cheap photocopy
of an original manuscript. Or even a copy of a copy.
It is outrageous that they can charge so much for such
poor quality work. If I were Folland, I'd be really mad.
Also, the new cover is weak- bring back the old
brown cover!

5+ stars for the text, 1 star for the book itself.

The Folland book is, by far, my favorite book on the subject. The exposition is extremely clean and concise (perhaps "dense" would be a better word). The text requires a little bit of work on the reader's part (some small gaps to fill), aside from the exercises (which is a good thing, in my opinion, for a text at the graduate level). It contains a very hefty amount of mathematics, as it functions as an introduction to Measure Theory/Integration, Topology, Functional Analysis, and Fourier Analysis.

Comparing with other books:

Rudin's "Real and Complex Analysis" isn't quite as comprehensive, regarding real variable theory. Also, the exercises in Rudin aren't quite as gentle.
The Royden, Wheeden/Zygmund, Stein/Shakarchi, and Kolmogorov/Fomin books are far less substantial, as texts and references.

I recommend the Folland book, though the Rudin book is good to have. Also, Cohn's "Measure Theory" makes a great supplementary text, along with the Folland book. If not Folland, then try Cohn.