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Real Analysis, 3rd Edition
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This is the classic introductory graduate text. Heart of the book is measure theory and Lebesque integration.
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Product details
Hardcover: 444 pages
Publisher: Pearson; 3rd edition (February 12, 1988)
Language: English
ISBN-10: 0024041513
ISBN-13: 978-0024041517
Product Dimensions:
5.9 x 1.1 x 9.2 inches
Shipping Weight: 2.1 pounds
Average Customer Review:
3.8 out of 5 stars
24 customer reviews
Amazon Best Sellers Rank:
#97,716 in Books (See Top 100 in Books)
I am a researcher and a scientist trained in engineering and computer science. As I progressed in my career, I felt the need of deeper meaning of everyday math I encountered.Somehow, I became aware of Hilbert spaces. To understand the concept, I started reading about them. Quickly, I realized the concept of Analysis. I took a stab at the topic following Elementary Real Analysis by Thomson and Bruckner. It was a very good introduction. I enjoyed reading it. It provided me the foundation, I was looking for.Then I struggled through five or six different graduate level texts. Eventually, I found Royden’s Real Analysis. I have only read the first two chapters, and everything suddenly started making sense and gel together.I am awestruck, how Royden has figured out a way to organize all the concepts in some kind of natural sequence, and the amount of details on each concept. With that said, I have used internet to flush through the details on per need basis.Simply, I have become a fan of Royden’s text. Real analysis helps me in everyday scientific tasks as well. I have a new vantage point.
This book, as far as I consider it, is a classic. Although it has some errata and stuff, it is one of the best books ever written in Real Analysis. I am happy to managed get this specific edition since the author didn't manage to publish its fourth edition, while it was Fitz Patrick the one who edited that edition. I think that books are more valuable when the authors are the only ones who edit them, otherwise I don't think they're worth reading. At least that's my opinion.Then, about the book, it takes to know some modern algebra, set theory and basic analysis to study from this fine book. I haven't seen a book that specially teaches Daniell's integral and this one does. The book is rich on references where one could further study material related to real analysis to the higher level. I think this book will be very useful to serious mathematicians and to those self-studying guys who love the rigour.
I had this as my reference for my first graduate analysis course. It is absolutely excellent and I vastly prefer it to Rudin. Covers some basics of metric spaces, then some fairly nice amounts of topology, measure theory, topological groups, etc. Very pleased :)
Excellent
I appreciate the approach Royden chooses in beginning with Lebesgue measure first then addressing general measure at the end of the book. For those of you frustrated by this approach, you can jump right to chapters 11 and 12 for a general treatment first. While Royden doesn't prove every theorem, enough are proved to give you an idea of how to approach the problem sets. If you take your time and follow each theorem's proof (and of course make your own notes after you understand each proof), you'll get what's going on. I thought some of the exercises were very tough given the fairly straightforward theorems Royden proves in the chapters (I didn't particularly care for the Cantor set problems, amongst others). It is interesting to contrast the slow pace of chapters 1-4 (even with chapters 1&2 essentially skimmed) with the quicker treatment in chapters 11&12. Towards the end, I very much appreciated how chapters 3&4 complemented chapters 11&12 (and vice versa). Very nicely done.
Many people criticize this book as unclear and unnecessarily abstract, but I think these comments are more appropriately directed at the subject than at this book and its particular presentation. I find this classic to be one of the best books on measure theory and Lebesgue integration, a difficult and very abstract topic. Royden provides strong motivatation for the material, and he helps the reader to develop good intuition. I find the proofs and equations exceptionally easy to follow; they are concise but they do not omit as many details as some authors (i.e. Rudin). Royden makes excellent use of notation, choosing to use it when it clarifies and no more--leaving explanations in words when they are clearer. The index and table of notation are excellent and contribute to this book's usefulness as a reference.The construction of Lebesgue measure and development of Lebesgue integration is very clear. Exercises are integrated into the text and are rather straightforward and not particularly difficult. It is necessary to work the problems, however, to get a full understanding of the material. There are not many exercises but they often contain crucial concepts and results.This book contains a lot of background material that most readers will either know already or find in other books, but often the material is presented with an eye towards measure and integration theory. The first two chapters are concise review of set theory and the structure of the real line, but they emphasize different sorts of points from what one would encounter in a basic advanced calculus book. Similarly, the material on abstract spaces leads naturally into the abstract development of measure and integration theory.This book would be an excellent textbook for a course, and I think it would be suitable for self-study as well. Reading and understanding this book, and working most of the problems is not an unreachable goal as it is with many books at this level. This book does require a strong background, however. Due to the difficult nature of the material I think it would be unwise to try to learn this stuff without a strong background in analysis or advanced calculus. A student finding all this book too difficult, or wanting a slower approach, might want to examine the book "An Introduction to Measure and Integration" by Inder K. Rana, but be warned: read my review of that book before getting it.
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