myArmoury.com: Real Analysis (3rd Edition)

Real Analysis (3rd Edition)

by Halsey Royden

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Edition: Paperback

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Other items of interest:
• Abstract Algebra
• Principles of Mathematical Analysis, Third Edition
• Counterexamples in Analysis (Dover Books on Mathematics)
• Real and Complex Analysis (International Series in Pure and Applied Mathematics)
• Topology (2nd Edition)

Customer Reviews:

Classic text, but a poor reference.
Review date: 2009-02-08

Halsey Royden's Real Analysis has become the de facto standard for teaching a graduate course on real analysis and integration. It has, however, become a bit dated. First off, the method of developing the Lebesgue integral before general measure theory is out of style. It is now generally accepted that learning the (relatively easy concept of) general measure theory first, and then the Lebesgue measure as an example, is a superior pedagogical approach.

That said, Royden is very good at explaining things in more detail; it is both a complement and a criticism that the book manages to cover a good deal less than Folland's text in almost twice the length. Complementary in the sense that the book motivates the material and gives explanations without leaving the reader to any important developments, critical in that the book is more or less useless as a reference. What's more is the fact that in the book's 444 pages it only manages to cover about half of what Gerald Folland goes through in his shorter book; this makes the ridiculous price of the book even less justifiable.

All of this said, I would still recommend this book for study. It explains well and would be a good read for self study. As for the criticisms that label the book as either "too difficult" or "too dense," disregard them. Those who make these claims are probably just not very good with Analysis. For a book that is truly awful, see M.A. Armstrong's Basic Topology.

Really Great for Certain Topics
Review date: 2007-08-18

Great for the bookshelf but really pretty hard - you need a good proof course and a year suffering through baby Rudin - and believe me, you will suffer - but will be a better person (and mathematician) for it.

Maybe good as a supplement, or a first time looking at the material
Review date: 2007-02-09

There are three books that are usually used for a first graduate course in analysis, including measure theory, namely Rudin's Real and Complex Analysis, Folland's Real Analysis, and Royden's book. Of the three, I would say Royden's book is the easiest, both in terms of the exposition, material, and exercises. Of the three, Royden is the only one to fully develop the Lebesgue measure and the associated integral before developing a more general theory of measure and integration. Furthermore, he does not develop Hilbert and Banach space theory, the very basics of functional analysis, to anywhere near the extent that Folland and Rudin do.

There is some debate as to whether it is better to start with the Lebesgue integral, and then talk about abstract integration, or the other way around. Personally, I found the development of the Lebesgue integral a bit tedious; the whole thing works a bit better when you first talk about abstract integration, which really isn't a terribly difficult concept, prove the basic integration theorems, then show how to construct an outer measure, and suddenly, the Lebesgue measure and integral just falls into place. I'm not sure anything is lost in the process.

The biggest shortcoming in this book would have to be the exercises: for the most part, they are not very difficult, particularly when you compare them to say, Rudin's text. For the most part, the exercises are fairly trivial, and if they are difficult, or require a bit of creativity, Royden often gives you lots and lots of hand-holding, sometimes even in the form of sketching out the proof for you. In spite of the relatively low difficulty level, most of the exercises are fairly instructive, in so far as they highlight, elucidate, and expand upon the material.

For the most part, this book is not bad. It makes a good supplement to a book like Rudin or Folland, as it is less abstract, and does a better job motivating the material. The exercises here can work well if you want some extra practice that won't take up too much time. If you're a student of econ, or physics, or you just feel like learning graduate-level real analysis, then this book is probably adequate (although I should qualify that statement by saying that I know nothing of econ and little of physics). But if you are a serious student of mathematics, particularly the pure variety, this is really not the book you should be using. It is just too easy.

Classic text on measure & integration theory
Review date: 2006-08-22

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.

Not very instructive to newcomers
Review date: 2006-05-10

I'm currently earning an MS in Mathematics, and am currently completing the last few weeks of my Real Analysis course using this book as its text. On the whole, I don't like this book.

I'm sure to people whom analysis comes naturally this is a fine book, and they can learn a lot from it. Also, I believe that a good professor could deliver Royden's text so that any student would have a good time learning it.

However I'm not a natural analyst, and in the absence of a good teacher I'm forced to rely on the text alone. I have had a terrible time, specifically:
-Incessant omissions in homework problems leave me wondering under what conditions I'm proving the theorem.
-the book references some fairly obscure ideas without explanation (which analyists may be familiar with, but I have no idea about, e.g. "a standard diagonal argument")
-almost no time is spent discussing common methods for proving certain species of analytical proofs
-some of the notation is outdated (according to at least one professor) and could be cleaned up
-I think one too many important proofs were "left to the reader"

On the upside:
-On the whole, the proofs are interesting and diverse
-the problems I did understand taught me much
-Royden's repetition is a good learning device and makes it easier to find some information

Product Details/Specifications:

Authors:
Halsey Royden

Recording label: Prentice Hall
Publishers:
Prentice Hall

Manufacturer: Prentice Hall
EAN: 9780024041517
Binding: Paperback
Dewey decimal number: 515.8
ISBN: 0024041513
Number of items: 1
Number of pages: 444
Publication date: 1988-02-12