Real Analysis

ISBN: 0024041513
ISBN 13: 9780024041517
By: Halsey Lawrence Royden Jr.

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About this book

Real Analysis, Fourth Edition, covers the basic material that every reader should know in the classical theory of functions of a real variable, measure and integration theory, and some of the more important and elementary topics in general topology and normed linear space theory. This text assumes a general background in mathematics and familiarity with the fundamental concepts of analysis. Classical theory of functions, including the classical Banach spaces; General topology and the theory of general Banach spaces; Abstract treatment of measure and integration. For all readers interested in real analysis.

Reader's Thoughts


I learned how to construct the Reals from the Integers.


Didn't care for the way this book was organized. The first half was a bunch of theorems on the real line, and the last half was all almost identical theorems on arbitrary topological spaces. I've started Rudin's and so far feel like I'm getting more out of that.


possible text for math 525a


Royden's text begins with a careful development of Lesbesgue measure and integration, with a discussion of differentiation and the L^p spaces. The book also provides a good introduction to metric spaces, including a discussion of Banach and Hilbert spaces. Also included is a very insightful and user friendly introduction to topological spaces. The later portions of the text present a well written development of abstract measure theory including signed measures, the Radon-Nikodym Theorem, Fubini and Tonelli Theorems, and the Riesz Lemma. Overall, the book is an indispensible tool for serious mathematics students. It is a very readable introduction to ideas central to mathematics as well as an invaluable reference.


If I can, I would give it one and a half. I read only seven chapters of the book.The merits are that it is a slow introduction to Lebesgue measure and integration. On the other hand, a lot of non-trivial theorems are left to the reader, and the author proves only very simple theorems. It feels like a math problem book, without solutions. Folland's book is much better than this one.


I only read chapters 2-6.

Imran Qureshi

this proof is the main text of the book. was left confused and baffled a bit more than most math books


Wonder bar.

Peter Frazier

This, together with Rudin's "Real Analysis", is one of the standard texts on the subject. I personally like Royden a little more -- it has a slightly more conversational tone (but not overly so), and it covers more functional analysis than Rudin does.


One of the must-haves for a beginner in learning analysis; concise and not deep.


good measure theory book, title is misleading, but this is a classic text. a great book to learn from

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