Rating: 
Summary: A start in math.
Review: 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.
Rating:
Summary: For a second read in Complex Analysis
Review: I'm a Electronic Enginner and I like to read mathematics just for fun. This book is for a graduate student but if you want to read it try to use other book with more examples ( in complex analysis ). To work with proof use "How to read and do proofs" of Daniel Solow wish is a good book, too. With these books you won't need any teacher to get a good understanding of it.
Rating: 
Summary: Generally poor, but there aren't good alternatives
Review: Many consider this book the "Bible" for graduate level analysis. Unfortunately, the dialect used is somewhat obstruse. It is a bit less terse than "little Rudin" (Introduction to Analysis), but is a bit spotty in comprehensiveness. The problem is that the alternatives have problems. Simon Hall's book doesn't give as many proofs and doesn't cover the subject matter as much (dealing more with functional analysis). Royden is probably better, but I have not read it much, and I think it is less advanced from what I hear. Folland didn't excite me either. Maybe it is just a difficult subject to write a good text on, but I bet someone will come along and write a better one to take over, like Gallian replaced Herstein in algebra.
Rating:
Summary: A Comprehensive Guide to Analysis
Review: 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. In the real analysis section, Rudin covers advanced topics generally not covered in a first course on measure theory. The chapters on differentiation and Fourier analysis are key examples of this. Rudin uses maximal functions to develop the Lebesgue Point theorem and results from complex analysis, and provides an incredibly thorough proof of the change-of-variables theorem. The ninth chapter, on Fourier transforms, relies heavily on convolutions, which are developed as a product of Fubini's theorem. This, in turn, is used to prove Plancherel's theorem and the uniqueness of Fourier transforms as a character homomorphism. The tenth chapter, on basic complex analysis, essentially covers an entire undergraduate course on the subject, with added results based on a solid knowledge of topology on the plane. Once a solid foundation on the topic is laid, Rudin can develop more advanced topics from Harmonic analysis using general results from real analysis like the Hahn-Banach theorem and the Lebesgue Point theorem (for Poisson integrals). Most of the basic results from the power series perspective are covered in the text, but while the geometric view is examined, it is still done in a very analytic, formula-based way that does not allow the reader to gain too much intuition. Nonetheless, all the basic results are covered, and Rudin uses these to develop the main theorems, such as the Mittag-Leffler and Weierstrass theorems on meromorphic functions, and the Monodromy Theorem and a modular function used to prove Picard's Little Theorem. As an introductory text, even for advanced students, Rudin should probably be accompanied by more descriptive texts to develop better intuition. In fact, I would recommend Folland's Real Analysis and Ahlfors' Complex Analysis for self-study, because the problems are easier and one can learn better through those. With a good instructor, though, Rudin's text is concise and elegant enough to be both useful and enjoyable. It is also a good test to see how well one REALLY knows the subject.
Rating:
Summary: A Comprehensive Guide to Analysis
Review: 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. In the real analysis section, Rudin covers advanced topics generally not covered in a first course on measure theory. The chapters on differentiation and Fourier analysis are key examples of this. Rudin uses maximal functions to develop the Lebesgue Point theorem and results from complex analysis, and provides an incredibly thorough proof of the change-of-variables theorem. The ninth chapter, on Fourier transforms, relies heavily on convolutions, which are developed as a product of Fubini's theorem. This, in turn, is used to prove Plancherel's theorem and the uniqueness of Fourier transforms as a character homomorphism. The tenth chapter, on basic complex analysis, essentially covers an entire undergraduate course on the subject, with added results based on a solid knowledge of topology on the plane. Once a solid foundation on the topic is laid, Rudin can develop more advanced topics from Harmonic analysis using general results from real analysis like the Hahn-Banach theorem and the Lebesgue Point theorem (for Poisson integrals). Most of the basic results from the power series perspective are covered in the text, but while the geometric view is examined, it is still done in a very analytic, formula-based way that does not allow the reader to gain too much intuition. Nonetheless, all the basic results are covered, and Rudin uses these to develop the main theorems, such as the Mittag-Leffler and Weierstrass theorems on meromorphic functions, and the Monodromy Theorem and a modular function used to prove Picard's Little Theorem. As an introductory text, even for advanced students, Rudin should probably be accompanied by more descriptive texts to develop better intuition. In fact, I would recommend Folland's Real Analysis and Ahlfors' Complex Analysis for self-study, because the problems are easier and one can learn better through those. With a good instructor, though, Rudin's text is concise and elegant enough to be both useful and enjoyable. It is also a good test to see how well one REALLY knows the subject.
Rating:
Summary: Excellent, often intriguing treatment of the subject
Review: 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.
Rating:
Summary: The Holy Bible of Real Analysis
Review: This book has no doubt about it. It is a HOLY BIBLE of Real Analysis Course. Trully speaking, if you know this book means you know mathematics. The book is written in very nice form, the excercises is extremely hard and challenging. Rudin put a lot of efford to make this book became clear in analysis. There is no joke stuff in his book, everything you step to another chapter, it must be related one and another. This book is designed truly for only who wants to become MASTER PIECE IN mathematics course, no doubt about it. Another Suggested reading H L Royden,this book is pretty similar to Rudin. No doubt about it also. Both of them truly are designed for MASTER PIECE.
Rating:
Summary: The "Bible"
Review: This book is no frills, but the pay off is insight into not only a beautiful subject but the mind of one of the masters in the field. This book is worth every penny and should be read by anyone who is serious about Analysis. A big word of caution to prospective buyers... You must have considerable mathematical maturity inorder to benefit from this book. Rudin's book is not a fly by the seat of your pants, I just want to be published book. This is the real deal and if I could I would give this book 8 stars.
Rating:
Summary: TRULY A MASTERPIECE
Review: 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)
Rating:
Summary: Best (math) book ever written
Review: This text is a model of mathematical style. The usual Rudin stuff: concise and elegant proofs, great chanllenging exercises and that undefinable sense of quality -mathematical taste- pervading all the book. 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.
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