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Real and Complex Analysis (International Series in Pure and Applied Mathematics) [Hardcover]

Walter Rudin
4.7 out of 5 stars  See all reviews (31 customer reviews)

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Book Description

May 1, 1986 0070542341 978-0070542341 3
This is an advanced text for the one- or two-semester course in analysis taught primarily to math, science, computer science, and electrical engineering majors at the junior, senior or graduate level. The basic techniques and theorems of analysis are presented in such a way that the intimate connections between its various branches are strongly emphasized. The traditionally separate subjects of 'real analysis' and 'complex analysis' are thus united in one volume. Some of the basic ideas from functional analysis are also included. This is the only book to take this unique approach. The third edition includes a new chapter on differentiation. Proofs of theorems presented in the book are concise and complete and many challenging exercises appear at the end of each chapter. The book is arranged so that each chapter builds upon the other, giving students a gradual understanding of the subject.

This text is part of the Walter Rudin Student Series in Advanced Mathematics.


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Real and Complex Analysis (International Series in Pure and Applied Mathematics) + Principles of Mathematical Analysis (International Series in Pure and Applied Mathematics) + Topology (2nd Edition)
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Product Details

  • Series: International Series in Pure and Applied Mathematics
  • Hardcover: 483 pages
  • Publisher: McGraw-Hill Science/Engineering/Math; 3 edition (May 1, 1986)
  • Language: English
  • ISBN-10: 0070542341
  • ISBN-13: 978-0070542341
  • Product Dimensions: 9.3 x 6.7 x 1.1 inches
  • Shipping Weight: 1.7 pounds (View shipping rates and policies)
  • Average Customer Review: 4.7 out of 5 stars  See all reviews (31 customer reviews)
  • Amazon Best Sellers Rank: #353,652 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews
115 of 117 people found the following review helpful
5.0 out of 5 stars A Comprehensive Guide to Analysis June 3, 2003
Format:Hardcover
Rudin's Real and Complex Analysis is an excellent book for several reasons. Most importantly, it manages to encompass a whole range of mathematics in one reasonably-sized volume. Furthermore, its problems are not mere extensions of the proofs given in the text or trivial applications of the results- many of the results are alternate proofs to major theorems or different theorems not proved. With that in mind, this book is not appropriate for a course where the instructor wants students to merely understand the theorems well enough to develop applications- the structure of the book is far better suited for a more theoretical course.
For example, the construction of Lebesgue measure is considered one of the most important topics in graduate analysis courses. After this construction, more abstract measures are developed, and then one proves the Riesz Representation Theorem for positive functionals later.
Conversely, Rudin develops a few basic topological tools, such as Urysohn's Theorem and a finite partition of unity, to construct the Radon measure needed in a sweeping proof of Riesz's Theorem. From this, results about regularity follow clearly, and the construction of Lebesgue measure involves little more than a routine check of its invariance properties.
Another example of where Rudin takes a more theoretical approach to provide a more elegant, yet less intuitive proof, is the Lebesgue-Radon-Nikodym theorem. Other books generally introduce signed measures with several examples, and use this result, along with properties of measures to derive the proof. On the other hand, since the first half of the book contains an intermission on Hilbert Space, Rudin uses the completeless of L^2 and the Riesz Representation Theorem for a more sweeping proof.
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24 of 24 people found the following review helpful
5.0 out of 5 stars Persistence Pays. October 8, 2000
By Marete
Format:Hardcover
This Book of Rudin, Like Principles, rewards perhaps above all else, persistence; a virtue that, if we are to believe some professional Mathematicians, is indispensable for the study of Mathematics.
Its true that it is terse and efficient. However, this "short-coming" is to me not a short-coming at all for the simple reason that Rudin makes up for it. How? The problems. Once you get through the proofs, a TON of challenging questions will be waiting at the other end to hammer out of you any illusions about you depth of understanding. In my opinion, this is the greatest strenghth of Rudin's book. STICK with the problems, attack them relentlessly and at the end of it all, you will have learned, a little perhaps, how to think for yourself in Analysis.
As regards the section on Complex Variables, I found it fruitful to read it while supplementing the problems with those of Ahlfors, which is more computational (E.g. Although Rudin discusses complex int., he scarcely provides any problems for this, and the same goes for expansion in Power Series).
Stick with the book, and soon it will be like a classic novel. (At least it is for me)
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29 of 30 people found the following review helpful
By A Customer
Format:Hardcover
The first part of this book is a very solid treatment of introductory graduate-level real analysis, covering measure theory, Banach and Hilbert spaces, and Fourier transforms. The second half, equally strong but often more innovative, is a detailed study of single-variable complex analysis, starting with the most basic properties of analytic functions and culminating with chapters on Hp spaces and holomorphic Fourier transforms. What makes this book unique is Rudin's use of 20th-century real analysis in his exposition of "classical" complex analysis; for example, he uses the Hahn-Banach and Riesz Representation theorems in his proof of Runge's theorem on approximation by rational functions. At times, the relationship circles back; for example, he combines work on zeroes of holomorphic functions with measure theory to prove a generalization of the Weierstrass approximation theorem which gives a simple necessary and sufficient condition for a subset S of the natural numbers to have the property that the span of {t^n:n in S} is dense in the space of continuous functions on the interval. All in all, in addition to being a very good standard textbook, Real and Complex Analysis is at times a fascinating journey through the relationships between the branches of analysis.
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30 of 33 people found the following review helpful
5.0 out of 5 stars A start in math. September 21, 2004
Format:Hardcover
I am a fan of Rudin's books. This one "Real and Complex Analysis" has served as a standard textbook in the first graduate course in analysis at lots of universities in the US, and around the world.
The book is divided in the two main parts, real and complex analysis. But in addition, it contains a good amount of functional and harmonic analysis; and a little operator theory.
I loved it when I was a student, and since then I have taught from it many times. It has stood the test of time over almost three decades, and it is still my favorite. I have to admit that it is not the favorite of everyone I know.
What I like is that it is concise, and that the material is systematically built up in a way that is both effective and exciting.
Some of the exercises are notoriously hard, but I think that is good: It simply means that they serve as work-projects when the students use the book. And this approach probably is more pedagogical as well.
After surviving some of the hard exercises in Rudin's Real and Complex, I think we learn things that stay with us for life; you will be "marked for life!"
Review by Palle Jorgensen, September 2004.
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Most Recent Customer Reviews
5.0 out of 5 stars complex analysis part
I took a graduate level course in complex analysis and Rudin's book was the textbook we used in the course. He made the proofs amazingly simple. I liked it 5 stars
Published 2 months ago by ramin
5.0 out of 5 stars Rudin versus Royden and Fitzpatrick
Continuing on in the Rudin versus Royden debate, to be fair when Rudin gets compared to "Royden" the comparison really should be to this book and not "The Principles of... Read more
Published 6 months ago by Gary Bowers
5.0 out of 5 stars superb framework for basic analysis
This is the main reference I used in my real and complex analysis courses. As an older person, who occasionally teaches math, and for whom math is a hobby, it is not always... Read more
Published 7 months ago by Michael George
5.0 out of 5 stars Graduate Level Real and Complex Analysis
This is the second book in the Rudin series suitable for the first year graduate student who has completed Rudin's first book, "Mathematical Analysis" (Chapters 1-7 and 11)... Read more
Published 7 months ago by Mathematician
5.0 out of 5 stars Daddy Rudin
Walter Rudin is a great expositor. He can one line a proof that would take you three pages, and he could 'one page' a proof that would take your professor three weeks to motivate. Read more
Published 10 months ago by Parrotdick
5.0 out of 5 stars Good photocopy of one of the best introductions to analysis
It's a photocopy, but a very good-quality one: text is very legible, and aligned on the pages.

I never met Rudin, but I grieve for his writing style. Read more
Published 19 months ago by STEVAN R WHITE
5.0 out of 5 stars To paraphrase
"Do not stop reading [this] book. [Professor Rudin] does [things I became to appreciate]."

I hear Professor Rudin writes things down on pieces of paper and collects... Read more
Published 20 months ago by Miriosh
3.0 out of 5 stars Comprehensive and boring
If you need to study for a qualifying exam, get this book. If you want to see why analysis (especially complex analysis) is a beautiful subject, avoid like the plague. Read more
Published on December 2, 2010 by Narada
5.0 out of 5 stars Great Learning Experience
This is truly a well-crafted book. The organization is tight and the book is largely self-sufficient, really only calling upon material covered in his previous book, Principles of... Read more
Published on January 27, 2010 by T. Sznigir
5.0 out of 5 stars One of a Kind
I normally don't review books that already have this many reviews, especially when I agree so much with the reviews that already exist. Read more
Published on March 13, 2008 by Christopher Grant
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