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Real Analysis: Modern Techniques and Their Applications, by Gerald B. Folland
Ebook Real Analysis: Modern Techniques and Their Applications, by Gerald B. Folland
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An in-depth look at real analysis and its applications-now expanded and revised.
This new edition of the widely used analysis book continues to cover real analysis in greater detail and at a more advanced level than most books on the subject. Encompassing several subjects that underlie much of modern analysis, the book focuses on measure and integration theory, point set topology, and the basics of functional analysis. It illustrates the use of the general theories and introduces readers to other branches of analysis such as Fourier analysis, distribution theory, and probability theory.
This edition is bolstered in content as well as in scope-extending its usefulness to students outside of pure analysis as well as those interested in dynamical systems. The numerous exercises, extensive bibliography, and review chapter on sets and metric spaces make Real Analysis: Modern Techniques and Their Applications, Second Edition invaluable for students in graduate-level analysis courses. New features include:
* Revised material on the n-dimensional Lebesgue integral.
* An improved proof of Tychonoff's theorem.
* Expanded material on Fourier analysis.
* A newly written chapter devoted to distributions and differential equations.
* Updated material on Hausdorff dimension and fractal dimension.
- Sales Rank: #59220 in Books
- Brand: Folland, Gerald B.
- Published on: 2007-05-01
- Original language: English
- Number of items: 1
- Dimensions: 9.50" h x 1.10" w x 6.45" l, 1.60 pounds
- Binding: Hardcover
- 416 pages
From the Back Cover
An in-depth look at real analysis and its applications-now expanded and revised.
This new edition of the widely used analysis book continues to cover real analysis in greater detail and at a more advanced level than most books on the subject. Encompassing several subjects that underlie much of modern analysis, the book focuses on measure and integration theory, point set topology, and the basics of functional analysis. It illustrates the use of the general theories and introduces readers to other branches of analysis such as Fourier analysis, distribution theory, and probability theory.
This edition is bolstered in content as well as in scope-extending its usefulness to students outside of pure analysis as well as those interested in dynamical systems. The numerous exercises, extensive bibliography, and review chapter on sets and metric spaces make Real Analysis: Modern Techniques and Their Applications, Second Edition invaluable for students in graduate-level analysis courses. New features include:
* Revised material on the n-dimensional Lebesgue integral.
* An improved proof of Tychonoff's theorem.
* Expanded material on Fourier analysis.
* A newly written chapter devoted to distributions and differential equations.
* Updated material on Hausdorff dimension and fractal dimension.
About the Author
GERALD B. FOLLAND is Professor of Mathematics at the University of Washington in Seattle. He has written extensively on mathematical analysis, including Fourier analysis, harmonic analysis, and differential equations.
Most helpful customer reviews
24 of 26 people found the following review helpful.
I get it now
By Kinga
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.
40 of 47 people found the following review helpful.
Could have been great
By Henry Rivers
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.
9 of 9 people found the following review helpful.
solid graduate-level textbook
By Tom B.
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.
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