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Real and Complex Analysis (Higher Mathematics Series) 3rd Edition
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This text is part of the Walter Rudin Student Series in Advanced Mathematics.
- ISBN-100070542341
- ISBN-13978-0070542341
- Edition3rd
- PublisherMcGraw Hill
- Publication dateMay 1, 1986
- LanguageEnglish
- Dimensions6.6 x 0.8 x 9.5 inches
- Print length483 pages
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- Publisher : McGraw Hill; 3rd edition (May 1, 1986)
- Language : English
- Hardcover : 483 pages
- ISBN-10 : 0070542341
- ISBN-13 : 978-0070542341
- Item Weight : 1.58 pounds
- Dimensions : 6.6 x 0.8 x 9.5 inches
- Best Sellers Rank: #614,624 in Books (See Top 100 in Books)
- #249 in Mathematical Analysis (Books)
- #1,637 in Mathematics (Books)
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Also, Rudin does not discuss some of the more advanced or interdisciplinary topics such as distribution theory (Sobolev spaces, weak derivatives, etc.) or applications of measure theory to the probability theory, both explored in the book by Folland. Last but not least, it's worth noting that contrary to the common practice, Folland includes many end-of-chapter notes where he outlines some important historical aspects of the development of the topics, and also gives a few references for further study. For example, in the notes section at the end of the chapter on Lebesgue integration, he mentions --and briefly outlines-- the basics of the theory of "gauge integration" (aka Henstock-Kurzweil theory) which serves to construct a more powerful integral than that of the Lebesgue's. As another instance, having already defined and used "nets" within the chapter on topology, in the end-notes Folland also introduces "filters" and "ultrafilters". These are all machineries which have been developed to play the role of the metric space sequences in general (locally Hausdorff) topological spaces, but for some historical reasons, ultrafilters have nowadays taken a backseat to the nets (the older general topology books usually prove the Tychonoff theorem using ultrafilters). All said, I can recommend taking up Royden as your very first approach to measure theory, then based on how well you think you have learned the first course, move on to either Rudin or Folland for a more advanced treatment. Please note that the other books I have mentioned above do not discuss complex analysis, a subject which is also masterfully presented in Rudin. There are however a few other equally well-written complex analysis books to pick from, for example John B. Conway's classic from the Springer-Verlag graduate text series, or L.V. Ahlfors's wonderful monograph, to name just a couple.
Like much of Rudin's other writings, this book is written from an advanced perspective. This often leaves newcomers to analysis and first year graduate students scratching their heads in confusion, and it's partly the reason why it's rarely used in qualifying exam sequences when compared to Folland's text. However, if you plan to use analysis in your research, then Rudin will give you a perspective that is difficult to find in other places.
In particular, his treatment of real analysis in the first 9 chapters is heavily influence by functional analysis. This is most striking in chapter 2, where he proves the Reisz Representation Theorem which shows an equivalence between positive measures and positive linear functionals (hiding within this generality is a construction of the Lebesgue Measure). Another example of this is his treatment of Fourier Series/orthogonal decompositions of L^2 in chapter 4.
The same trend continues in the complex analysis section of the text. Here, the presentation of the material is heavily influence by H^p space theory. For instance, he gives a treatment of harmonic extension via convolving with the Poisson measure, something you'll find in chapter 1 of any H^p space book like Koosis. He even has a chapter on H^p space theory. His treatment of the basics of complex analysis uses real and functional analysis freely. If I recall correctly, his proof of Runge's Theorem uses the Hahn-Banach Theorem.
So this book has an almost orthogonal treatment of complex analysis to the more traditional, geometry-based, texts like Alhfors, Gamelin, or Krantz's. I think the geometry-based approach is best for a first-semester course in complex analysis, but the student interested in analysis should probably work through chapters 10 onward in their own free time at some point before research gets too heavy.
As is usual for Rudin, the writing is sharp and clean. On a first read, it may seem too crisp, but then the reader grows in understanding, and then the writing appears perfect. The exercise sets are great, as is usual for Rudin. Bashing your head against these problems will definitely improve your thinking.
In short, Rudin's Principles of Mathematical Analysis made me want to be an analyst, but his Real and Complex Analysis taught me how.
Unfortunately I lost all my belongings and have had to replace my math books. And McGraw-Hill seems to have lost its integrity. As you can see, the photos here are not the same as the photos on Amazon (which are of the high-quality printing). The most obvious giveaway is the printed red “McGraw-Hill” on the spine - on Amazon this part is gold and embossed. But the entire cover is a cheaply laminated low-quality printing. The paper quality is also much lower - not dirt-cheap but it feels more like printer paper than “Rudin paper.” I am a book dork and have fond memories of the way this book felt in my hands. It’s always ridiculous to spend $100 on a relatively small book, but the craftsmanship used to justify the price. Very disappointing to see that this book is now part of the standard grad student textbook racket.
I have not noticed any QC problems like misprints or missing pages, and everything is quite readable. The paper is cheap but seems fairly robust to tears. So the book is not a terrible con job, but Amazon should really update the picture - and drop the price!
Until then you should try to find it used, with the nicer cover.
Reviewed in the United States 🇺🇸 on April 28, 2021
Unfortunately I lost all my belongings and have had to replace my math books. And McGraw-Hill seems to have lost its integrity. As you can see, the photos here are not the same as the photos on Amazon (which are of the high-quality printing). The most obvious giveaway is the printed red “McGraw-Hill” on the spine - on Amazon this part is gold and embossed. But the entire cover is a cheaply laminated low-quality printing. The paper quality is also much lower - not dirt-cheap but it feels more like printer paper than “Rudin paper.” I am a book dork and have fond memories of the way this book felt in my hands. It’s always ridiculous to spend $100 on a relatively small book, but the craftsmanship used to justify the price. Very disappointing to see that this book is now part of the standard grad student textbook racket.
I have not noticed any QC problems like misprints or missing pages, and everything is quite readable. The paper is cheap but seems fairly robust to tears. So the book is not a terrible con job, but Amazon should really update the picture - and drop the price!
Until then you should try to find it used, with the nicer cover.















