- 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: 6.8 x 0.8 x 9.5 inches
- Shipping Weight: 1.7 pounds (View shipping rates and policies)
- Average Customer Review: 36 customer reviews
- Amazon Best Sellers Rank: #600,881 in Books (See Top 100 in Books)
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Real and Complex Analysis (Higher Mathematics Series) 3rd Edition
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This is an excellent book that combines real and complex analysis into one course. A good thing about using this book is that one can complete a course in both subjects in one year affording them room in their graduate corriculum to study an additional mathematical area. Furthermore, it is good to see the two topics combined into one course showing applicability of Real Analysis in areas of Complex Analysis, such as Fourier Transforms. Also, topics in Functional Analysis are provided later in the book.
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.
The first chapter and the chapter on Hilbert spaces are favorites of mine. It makes a good reference, but one should be sure to study other authors as well. The organization of the book is innovative and for that it is stimulating.
It's good, but frustrating at times, when one is drawn in by an elegantly stated theorem only to be inexplicably let down by a proof which does the job, but leaves out the motivating ideas. Rudin has a talent for making difficult things clear, but one should exercise caution, because he also has a talent for making simple things appear difficult.