Rating: 
Summary: A masterpiece
Review: I absolutely agree with Professor Jorgensen.
I loved it when I was a student of physics, and I still love it because I tend to consider it as my personal standard in Classical Mathematical Analysis (and not only): sort of a "pacemaker" which sets the qualitative level to go back to just every time one is a little confused about what to do and where to go ;)
Rating:
Summary: Very Good for the Sufficiently Prepared Reader
Review: I am a beginning graduate student who learned real analysis over a decade ago from a different book, and decided to read Rudin and work through many of the exercises by way of review. In this context, I am happy to say that the book has been an absolute pleasure to read: it is a wonderfully efficient, clean presentation of the fundamentals with many good (sometimes quite tough) exercises to solidify understanding and introduce relevant related material.Rudin's book is not a good place to start if one does not already have some mathematical maturity, in particular, the ability to understand and produce proofs (e.g., induction, contradiction, contrapositive, epsilon-delta arguments). These skills can of course be developed concurrently if the reader is taking a university course from a lecturer who can flesh out the material, but the book itself assumes this background. Also, although Rudin develops analysis from first principles, there is not much in the way of motivation for concepts such as continuity, the derivative, and the integral; the reader needs to have seen this material before, perhaps in an introductory calculus course, or again to have access to a lecturer who will provide this motivation. Assuming these prerequisites are met, the diligent and persistent reader is in for a real treat. Most of the single-variable topics (i.e., chapters 1 through 8) that Rudin covers are presented in the most economical, efficient, and downright beautiful exposition that I have seen. There are a few topics that could be improved. First, Rudin's definition of connected sets is not quite standard, and as a result his proofs here are a bit more convoluted than one finds elsewhere. Second, his proof of L'Hopital's rule handles all cases simultaneously, and ends up being more obscure than if he had treated them separately. Third, and this is not uncommon among real analysis books, the treatment of power series would be more insightful in the context of complex variables. Fourth, the proofs of the convergence of the important "special sequences" (in Theorem 3.20(d), for those with a copy at hand) are quite slick and quite pretty, but would greatly benefit from the inclusion of a few intermediate steps. Fifth, the chapter on sequences and series includes many, but not all, of the standard convergence tests; it would be nice for the sake of reference to have them all here. That these are essentially ALL of the complaints I have concerning chapters 1 through 8 will, I hope, suggest just how highly I regard the rest of the exposition. There is not much else that I would change. In particular, I think very highly of the chapters on sequences and series and the Riemann-Stieljes integral. Beyond chapter 8, I think the book falls apart somewhat. The treatment of multivariate calculus seems very rushed, and the lack of motivation really becomes a problem in the chapter on integration of differential forms. This material is treated efficiently, but in considerably more detail and, I think, much more comprehensibly, in Browder's "Mathematical Analysis" (Springer, 1996). Chapter 11 on Lebesgue integration is also too rushed, and also somewhat old fashioned, developing the theory on sigma-rings instead of sigma-algebras. Browder's coverage of this material is more modern (if also a bit tersely), and he uses the Lebesgue integral for his integration on forms section, which I much prefer. Other options: the Lebesgue integral is given a very nice treatment in Bartle's "The Elements of Integration and Lebesgue Measure", a tragically overpriced Wiley Classic, or one can simply defer the Lebesgue theory until one is ready to read an introductory graduate-level text, such as Folland, (big) Rudin, Royden, or Wheeden and Zygmund.
Rating:
Summary: Getting started in math.
Review: I am a fan of Rudin's books. This one "Principles of Matheamtical Analysis" has served as a standard textbook in the first serious undergraduate course in analysis at lots of universities in the US, and around the world.
The book is divided in the three main parts, foundations, convergence, and integration. But in addition, it contains a good amount of Fourier series, approximation theory, and a little harmonic analysis.
The foundational part begins with a beautiful and axiomatic approach to the real number system: It is based on Dedekind's cuts. It is exceptionally well presented, and is beautifully illustrated with examples and with exercises.
I loved the book 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 for this level. I have to admit that the book is not the favorite of everyone I know. And there are a lot of books out there now that cover more or less the same. Still, to me, Rudin's book is the best!
What I especially like about it is that it is concise, and that the material is systematically built up in a way that is both effective and exciting.
The exercises and just at the right level. They can be assigned in class, or students can work on them alone. I think that is good, and the exercises serve well as little work-projects. This approach to the subject is probably is more pedagogical as well.
I think students will learn things that stay with them for life.
Review by Palle Jorgensen, September 2004.
Rating:
Summary: this is besk
Review: I am a student(math major)from China .Now I'm reading this book,I learned a lot of from this book.If you are REALLY intrested in math,I can give YOU a reading list(this is also my textbook):
1.Introduction to calculus and analysis
by Richard Courant, Fritz John 2.Principles of Mathematical Analysis
by Walter Rudin
Rating:
Summary: Yet Another Popular (YAP) Math Text
Review: I am editing my original review, so it appears I'm giving 5 stars. But now I would give it 3 stars.
I think this book is extremely annoying. It is not at all how I would present a course in undergraduate analysis.
Real Analyis the way I would do it:
Real Numbers (an ordered field): have the least upper bound property. If you haven't learned that by the time of your senior Analysis course, you are behind, and will not catch up. Therefore, this section should be in the appendix or in the bibliography.
Real Numbers are a topological space: Add +/- infinity and you get a compact metric space. If you are a senior, you should be ready for this much abstraction, including: the open balls are a basis for R^n. (Perhaps schools will follow my advice and merge point set and analysis into the same course!)
Limits happen in Metric Spaces: Continuous functions are great for working with, and we can differentiate things in R^n. You should know linear algebra before starting this Analysis course.
Integration: Spivak's chapter 3 shows the correct way to introduce integration to a Math student.
Sequences of Functions: Recall, function spaces are topological and metric too. Abstraction leads to clarity!
additional: Should add manifolds in addition to the inverse function theorem.
Baby Rudin is a good reference, since all the undergraduate Analysis results are in there, but still... The book is just boring to tears to read. Case in point: L'Hosptals rule. Man that proof is a terse one if I have ever seen one, and it clearly could be done better.
Rating:
Summary: Great intro to the subject
Review: I used this book in my intro to analysis class during my freshman year. The book is very well organized -- from the construction of whole number, notion of cardinality, metric space, all the way towards 'raw calculus'. I don't see the point some people had made that this is too highbrow for an intro text -- the book is written essentially for the math-inclined students so it is not as 'example-ridden' and 'instinctive' as other intro texts are. The examples are few but essential and instuctive, and most concepts can be visualized easily by the intended audience of this book (last time I checked, there is no illustration in the whole book). Some of the exercises can be quite challenging too. I'll definitely recommend this as a text for an intro to analysis class.
Rating:
Summary: These Five Stars Need an Explanation
Review: I write this review from the perspective of a mathematician who first encountered this book as an undergraduate in the 1970s and who has most recently had the enjoyable experience of teaching from it during the 1999-2000 academic year. "Baby Rudin" is like no other "elementary" text I have ever encountered. I agree with the other reviewers who criticize the book for its lack of pictures, its lack of historical motivation, its lack of "soul." Yet, in the hands of a professor who is prepared to present the pictures, the motivation, and the "soul" that the text itself lacks, this book can form the basis of a deep, rich introduction to the glorious world of real analysis. Every time I return to this book I discover new and wonderful things in it. For example, in his treatment of the limits of elementary sequences (that are "normally" treated using the log and the exponential function), Rudin uses the binomial theorem with a deftness and facility that contemporary students rarely encounter. Although Rudin's text presents minimal historical background, it is at the same time more faithful to the historical development of the subject than any other text I can think of. That the book is small and easy to carry around is no disadvantage. Who says that a calculus book has to be the size of the Manhattan phone directory to be valuable?
Rating:
Summary: Working through this book is the wisest choice...
Review: Some say you shouldn't use this as your first analysis text book. However, if the reader is properly familiar with the construction of proofs (say from seriously studying for math contests), it is more than ok to use it as the first book, even for independent study: That's what I did.
Although at first it did take up a LOT of time to understand the topics (this was my first exposure to higher mathematics), I could work through it and solve the exercises at the end of each chapter I went through.
One of the nicest things about this book is the way that it allows you to actually think, the ideas you get trying to prove its theorems are completely different from the ones given by the book, so what you think is complemented very nicely. If this is used as a first book, it will also make a lot of questions arise that you will either answer yourself (because the book got you in love with analysis) or push you to work on the following chapter.
I am thankful for this book, the INSIGHT I got of analysis from working through such dense material was impressive and some other books in analysis just feel different. The beauty of this book when used as a first text is how it exposes you to beautiful math in such a complete way that you can really decide if you want to be a mathematician or not.
I am particularly fond of the problem sets at the end of Ch. 2 (Point Set topology) and the Chapter on Special functions (7, I think). The beauty of what I learned in this book made me realize how much I wat to become a mathematician. It is perfect. Math is perfect :-)
Rating:
Summary: If you are serious about doing math...
Review: then I suggest you use this book for your introduction to analysis. I divide up my critique into the following sections: Content:
The author of this book expects you to be comfortable with mappings, set theory, linear algebra, etc. I would recommend that you use either Munkres' book on topology, or (if you can't afford that) the Dover book, Introduction to Topology by Bert Mendelson (you should read all of Ch. 3 BEFORE starting Rudin if you want to pick up on which things could be even more general than they are in Rudin - refer to earlier chapters if you don't recognize something). I suggest also looking at continuity in one of the topology books I mentioned. Also, look up the following things and at least know what they are before getting past Ch. 4, so you have some supplemental language to use: Banach space, boundary, basis for a topology, functional.
Like I said, this book is for serious people, and it requires strong focus for you to pick up on all the subtle arguments made through his examples. I do not agree with some people who say this book is bad for an introduction, in fact I think it is the best because Rudin REFUSES to be tied down to single variable concepts which could be explained just as easily in the context of more general spaces. If you are one of those kids who think's you're great at math because you do well in competitions, steer clear; your place is playing with series, inequlities, and magic tricks. If you are a get-your-hands-dirty kind of mathematician, then you should never let this book leave your side. Readability:
I think that it may be a different style than most people are used to, but once you get past that I think I would call the readability nearly perfect. He strips away most general useless commentary (for example, in Gallians poor algebra book, "In high school, students study polynomials with integer coefficients, rational coefficients, and perhaps even complex coefficients"). In Rudin, you get no nonsense -- only math. The real trick to getting in his swing of things is to MAKE SURE YOU COMPLETE HIS PROOFS. They are extremely slick and often are polished in such a way that it's like his little secret. If you can't do one on your own, just ask the prof in office hours or put it aside for later. The proofs are not presented in this way as to imply that you should just accept them, he wants you to dig in and justify the intermediate steps for yourself, so do it and you'll be good by Ch. 3, I promise. Exercises:
Many exercises in this book are often found as theorems in other books. What's so unique about this book is that very few problems are solved by simple definition pushing, especially as you go further into the book. That's why I call this the get-your-hands-dirty book, because you'll be forced to, and believe me you'll recognize changes in the way you think if you do this diligently. So, do as many exercises as you can, esp in Ch. 2 and Ch. 4, they will help you the most in this book. What's great about the problems is that they challange you to make REAL connections between ideas and create your own equivalent ways of thinking about the subject. I often have to conjecture and prove several lemmas to avoid wimping out and using "clearly" in my proofs. Suggestions:
If you really really love math and know in your heart that you need to get better to be happy in life, you should cover Ch.1-Ch.6 before Juior year of college and finish it before grad school. I also suggest using this book as a stepping stone to more advanced books -- see Halmos' Measure Theory and know it before grad school. Finally, DO NOT BE AFRAID! You really have to commit to this book before getting into it, do not be afraid. My best advice to any mathematician is to know your weaknesses, BUT to respond promptly to them.
Rating:
Summary: If you are serious about doing math...
Review: then I suggest you use this book for your introduction to analysis. I divide up my critique into the following sections: Content:
The author of this book expects you to be comfortable with mappings, set theory, linear algebra, etc. I would recommend that you use either Munkres' book on topology, or (if you can't afford that) the Dover book, Introduction to Topology by Bert Mendelson (you should read all of Ch. 3 BEFORE starting Rudin if you want to pick up on which things could be even more general than they are in Rudin - refer to earlier chapters if you don't recognize something). I suggest also looking at continuity in one of the topology books I mentioned. Also, look up the following things and at least know what they are before getting past Ch. 4, so you have some supplemental language to use: Banach space, boundary, basis for a topology, functional.
Like I said, this book is for serious people, and it requires strong focus for you to pick up on all the subtle arguments made through his examples. I do not agree with some people who say this book is bad for an introduction, in fact I think it is the best because Rudin REFUSES to be tied down to single variable concepts which could be explained just as easily in the context of more general spaces. If you are one of those kids who think's you're great at math because you do well in competitions, steer clear; your place is playing with series, inequlities, and magic tricks. If you are a get-your-hands-dirty kind of mathematician, then you should never let this book leave your side. Readability:
I think that it may be a different style than most people are used to, but once you get past that I think I would call the readability nearly perfect. He strips away most general useless commentary (for example, in Gallians poor algebra book, "In high school, students study polynomials with integer coefficients, rational coefficients, and perhaps even complex coefficients"). In Rudin, you get no nonsense -- only math. The real trick to getting in his swing of things is to MAKE SURE YOU COMPLETE HIS PROOFS. They are extremely slick and often are polished in such a way that it's like his little secret. If you can't do one on your own, just ask the prof in office hours or put it aside for later. The proofs are not presented in this way as to imply that you should just accept them, he wants you to dig in and justify the intermediate steps for yourself, so do it and you'll be good by Ch. 3, I promise. Exercises:
Many exercises in this book are often found as theorems in other books. What's so unique about this book is that very few problems are solved by simple definition pushing, especially as you go further into the book. That's why I call this the get-your-hands-dirty book, because you'll be forced to, and believe me you'll recognize changes in the way you think if you do this diligently. So, do as many exercises as you can, esp in Ch. 2 and Ch. 4, they will help you the most in this book. What's great about the problems is that they challange you to make REAL connections between ideas and create your own equivalent ways of thinking about the subject. I often have to conjecture and prove several lemmas to avoid wimping out and using "clearly" in my proofs. Suggestions:
If you really really love math and know in your heart that you need to get better to be happy in life, you should cover Ch.1-Ch.6 before Juior year of college and finish it before grad school. I also suggest using this book as a stepping stone to more advanced books -- see Halmos' Measure Theory and know it before grad school. Finally, DO NOT BE AFRAID! You really have to commit to this book before getting into it, do not be afraid. My best advice to any mathematician is to know your weaknesses, BUT to respond promptly to them.
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