- Series: Higher Mathematics Series
- Hardcover: 483 pages
- Publisher: McGraw-Hill Education; 3 edition (May 1, 1986)
- Language: English
- ISBN-10: 0070542341
- ISBN-13: 978-0070542341
- Product Dimensions: 6.6 x 1.1 x 9.5 inches
- Shipping Weight: 1.7 pounds (View shipping rates and policies)
- Average Customer Review: 4.7 out of 5 stars See all reviews (38 customer reviews)
- Amazon Best Sellers Rank: #338,904 in Books (See Top 100 in Books)
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Real and Complex Analysis (Higher Mathematics Series) 3rd Edition
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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.Read more ›
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)
The book covers the standard material on 'real variable' (measure theory') in a masterful and compact way; then it goes through the standard complex analysis to a level deeper than usual and showing in a very original way its intertwining with real variable. The final third of the book is devoted to more specialized topics.
Just a warning: the construction of Lebesgue measure is based on Riesz representation theorem, whose lengthy proof is imposed to the reader in chapter 2. It is really tough, and makes this chapter much harder to read than the rest of the book.
If you want to learn REAL mathematics, this is the book for you, you'll learn not only the subject matter, but a great style as well.
Most Recent Customer Reviews
This is the best book I know that rigorously works out the theory of holomorphic functions in the language of analysis. Read morePublished 3 months ago by Jordan Bell
I like it, but the papers was not thick enough. Nothing wrong with the product, but it's papers are very thin that I can see the page behind each.Published 6 months ago by Khazam
One problem: The cover of my received book is another than the one pictured on your website and has a golden yellow colour. Read morePublished 11 months ago by Rolf M. Mjelde
I find this book more clear and to the point than Royden's in introducing integration and measure theory to beginners with a modest background in mathematical analysis. Read morePublished 13 months ago by Nintendo
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 starsPublished on April 27, 2014 by ramin
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 morePublished on December 22, 2013 by Gary Bowers