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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: #373940 in Books
- 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
4 of 4 people found the following review helpful.
My favorite analysis textbook
By Keenan J. Kidwell
I first encountered this textbook when I used it to study for the analysis prelim at UT the summer after my first year of graduate school. I had been studying Rudin's Real and Complex Analysis during the semester, and while Rudin is elegant and beautiful, this book is far more comprehensive and, most importantly to me, systematic. From the very beginning, Folland develops the theory of Borel measures on the real line in generality (whereas Rudin's "construction" of Lebesgue measure based on the Riemann integral is, in my opinion, incomprehensible). The treatment of general measure theory in this book is worth the purchase price on it's own, but Folland includes lots of material going well beyond the standard course on measure theory and integration. The most valuable chapters non-measure-theoretic chapters for me are the ones covering point set topology and the rudiments of functional analysis (Hilbert and Banach spaces, and a little bit about general topological vector spaces, but not much).
Graduate school is now behind me, but Folland's book continues to be my favorite reference for analysis and even point set topology. Admittedly I have not looked for analysis books in many years, but that's because Folland has met all my needs (in the interest of full disclosure I should say I'm a number theorist, not an analyst, but for what I do a good knowledge of analysis is essential).
If you're interested in the book to supplement a course in measure theory and abstract real analysis, or for self-study, I would say that the only background knowledge is the material in a standard undergraduate introduction to real analysis (perhaps sticking to metric space topology and treating Riemann integration rigorously). More important for reading the book is a capacity to understand and appreciate abstraction, and a reasonable level of mathematical maturity. I took an undergraduate course in general topology (using Munkres) which covered the material in Folland (most of it anyway) over the course of a semester, and I think this did more to prepare me for Folland (and advanced mathematics in general) than any other course I took. If you are comfortable reading and writing proofs (general topology being an excellent subject with which to cultivate these skills), then I think you will enjoy and learn a great deal from Folland's book.
2 of 2 people found the following review helpful.
Comprehensive and challenging
By Kindle Customer
While trying to review and strengthen my foundations in measure theory and integration, I found this book.
It is tough reading to be sure, but it is easier to work through than Papa Rudin, and feels more "up to date" than Rudin.
I like Folland because he does not waste undue time developing the basics of measures. He sprints right through those (Chapters 1,2, and 3) and into Point-Set Topology in Chapter 4. I really enjoyed that chapter, since he goes into the applications to analysis (mostly locally compact Hausdorff spaces) pretty thoroughly. To compare with Royden (3rd edition), you have to wait until 2/3 of the way through to get to this point. By Chapter 5, you are ready for Functional Analysis, which is what I am teaching myself now. (I chickened out of Functional Analysis in grad school, and this has bothered me ever since!)
I will have to edit this review once I have gone through the rest of the book. Unlike 3rd edition Royden (I haven't seen the 4th edition, so I can't comment on that), by the time you finish Folland it appears that you will be ready for research in various branches of math that depend heavily on analysis.
Therefore, I would have to highly recommend this book. Just consult a more "basic" reference for your first look at measure theory. Jump into this book for a broader view of things.
4 of 4 people found the following review helpful.
The Best! book on the subject, hands down.
By Garbanzo Bill
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, Donald Cohn's "Measure Theory" makes a great supplementary text, along with the Folland book. If not Folland, then try Cohn.
EDIT: 12/04/2015
There are many other great books on "Real Variables." Let's face it: A one-year course in Real Variables, using this text, may cover only half of the book, leaving out much of the supplementary material on Harmonic Analysis/PDE's; and if a course does have time for some of the supplementary material, it might be preferable to go in a different direction (depending on the audience). For instance, the supplementary topics in Cohn's book include Polish spaces (and some of the beginnings of descriptive set theory), as well as the Bochner integral (none of which is included in Folland's text [for the exception of an exercise on vector-valued integration]). If you look at Halmos' Measure Theory or at the 3rd Edition of Royden, you'll find some very important and technical aspects of mappings between measure spaces which aren't in Folland's text. In Volume I of Dunford & Schwartz, you'll find the theory for finitely additive set functions, in general, as well as discussions of function spaces and spaces of measures (as Banach spaces).
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