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Real Analysis: Measure Theory, Integration, and Hilbert Spaces (Princeton Lectures in Analysis) (Bk. 3) [Hardcover]

Elias M. Stein , Rami Shakarchi
4.0 out of 5 stars  See all reviews (6 customer reviews)

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Book Description

April 3, 2005 0691113866 978-0691113869

Real Analysis is the third volume in the Princeton Lectures in Analysis, a series of four textbooks that aim to present, in an integrated manner, the core areas of analysis. Here the focus is on the development of measure and integration theory, differentiation and integration, Hilbert spaces, and Hausdorff measure and fractals. This book reflects the objective of the series as a whole: to make plain the organic unity that exists between the various parts of the subject, and to illustrate the wide applicability of ideas of analysis to other fields of mathematics and science.

After setting forth the basic facts of measure theory, Lebesgue integration, and differentiation on Euclidian spaces, the authors move to the elements of Hilbert space, via the L2 theory. They next present basic illustrations of these concepts from Fourier analysis, partial differential equations, and complex analysis. The final part of the book introduces the reader to the fascinating subject of fractional-dimensional sets, including Hausdorff measure, self-replicating sets, space-filling curves, and Besicovitch sets. Each chapter has a series of exercises, from the relatively easy to the more complex, that are tied directly to the text. A substantial number of hints encourage the reader to take on even the more challenging exercises.

As with the other volumes in the series, Real Analysis is accessible to students interested in such diverse disciplines as mathematics, physics, engineering, and finance, at both the undergraduate and graduate levels.

Also available, the first two volumes in the Princeton Lectures in Analysis:

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Real Analysis: Measure Theory, Integration, and Hilbert Spaces (Princeton Lectures in Analysis) (Bk. 3) + Functional Analysis: Introduction to Further Topics in Analysis (Princeton Lectures in Analysis) (Bk. 4) + Complex Analysis (Princeton Lectures in Analysis, No. 2)
Price for all three: $227.68

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Editorial Reviews


"We are all fortunate that a mathematician with the experience and vision of E.M. Stein, together with his energetic young collaborator R. Shakarchi, has given us this series of four books on analysis."--Steven George Krantz, Mathematical Reviews

"This series is a result of a radical rethinking of how to introduce graduate students to analysis. . . . This volume lives up to the high standard set up by the previous ones."--Fernando Q. Gouvêa, MAA Review

"As one would expect from these authors, the exposition is, in general, excellent. The explanations are clear and concise with many well-focused examples as well as an abundance of exercises, covering the full range of difficulty. . . . [I]t certainly must be on the instructor's bookshelf as a first-rate reference book."--William P. Ziemer, SIAM Review

About the Author

Elias M. Stein is Professor of Mathematics at Princeton University. Rami Shakarchi received his Ph.D. in Mathematics from Princeton University in 2002.

Product Details

  • Series: Princeton Lectures in Analysis
  • Hardcover: 392 pages
  • Publisher: Princeton University Press (April 3, 2005)
  • Language: English
  • ISBN-10: 0691113866
  • ISBN-13: 978-0691113869
  • Product Dimensions: 9.5 x 6.4 x 1.3 inches
  • Shipping Weight: 1.6 pounds (View shipping rates and policies)
  • Average Customer Review: 4.0 out of 5 stars  See all reviews (6 customer reviews)
  • Amazon Best Sellers Rank: #462,763 in Books (See Top 100 in Books)

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50 of 52 people found the following review helpful
5.0 out of 5 stars Excellent sourse for graduate analysis July 2, 2005
Format:Hardcover|Verified Purchase
This book is the best book on real analysis I have ever studied. It does a wonderful job in bridging undergraduate level with graduate level analysis. I have not seen any book that makes measure and Lebesgue theory so easy to understand.

The books begins by defining what a "measure" is all about. And the description is so intuitive and geometrical that you would wonder why you weren't taught it this way before. The book then goes into Lebesgue theory and all of it suddenly becomes so easy.

The book has plenty of wonderful examples and a good set of over 30 problems per chapter.

Elias Stein (one of the authors) is a very renowned mathematician, and one need not worry about the accuracy of the proofs in the book--they are "bullet-proof", and at the same time succinct.

If you are struggling with W. Rudin's book on Analysis, this book is a MUST for you.
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47 of 63 people found the following review helpful
2.0 out of 5 stars Suffers from all the flaws of a 1st edition December 17, 2005
This book has a lot of problems. Several sections are poorly written/edited. Several important named theorems are not clearly labeled. Also some of the proofs contain typos or errors. The chapter on differentiation is particularly lacking. The chapter is poorly organized and presented. There is also a glaring TeX error in the chapter.

At Princeton this book is used as part of an undergraduate course, and it shows. This is not the ideal book for a graduate level course in real analysis(though I think it would be very well suited for advanced undergrads). Too much time is spent on Lebesgue measure and integration in the first 2 chapters, and abstract measure theory is not intoduced until chapter 6. Also the Monotone Class theorem is lacking from the chapter on abstract measure theory. Also, the book only touches on functional analysis in the two chapters on Hilbert spaces (where they assume all Hilbert spaces are separable).

On the other hand, the presentations of Lebesgue measure/integration and Hilbert spaces in the book are pretty good. The exercises and problems in teh book (when stated properly) are very good and instructive. Overall this book has a lot of potential to be very good, but seems to be suffering from a lack of revision. Hopefully these issues will be fixed in later editions.
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17 of 23 people found the following review helpful
2.0 out of 5 stars Not as good as the classics December 29, 2009
By Abbey
I just completed a first-semester graduate course in which we used this textbook, and I was very disappointed by this choice. The authors too often gloss over details and omit definitions. Plus there are a few minor mistakes or non-standard definitions (check out the definition of "limit point" on page 3!). It reads much more like a lecture than a textbook, and I found it frustrating not to have a thorough resource to fall back on when my own professor's lecture was unclear. I have always prided myself in my ability to learn from a textbook, as I had no difficulty following Munkres in his "Topology" or Dummit and Foote in their "Abstract Algebra." However, I found this real analysis text to be quite challenging to follow time and time again--even our professor commented on how some proofs were unnecessarily complicated and how certain "trivial" details that had been omitted were not quite so trivial and indeed deserved mention. The only reason I did not give this book one star is that I found the problems to be good.

I am getting ready to purchase a copy of Royden's "Real Analysis" to help me study for my qualifying exam. I wish we had used it all along!
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