Principles of Mathematical Analysis

I liked parts of it, but it's just not for me. I'm used to others, and it ended up just fine. I think this is better than Abbott and Ross for sure. If you like it, more strength/power for you. But hey... That is my take.

Principles of Mathematical Analysis by Walter Rudin can rightly be called "the Bible of classical analysis". I have seen it cited in more books than I can count. And after a full year of working through the book in graduate school, I can see why. As many other reviewers here have pointed out, this book requires more than a little of that magical quality called "mathematical maturity". Simply defined, "mathematical maturity" is the ability to read between the lines and fill in the gaps in a given mathematical text. While Rudin certainly provides an encyclopedic account of basic analysis in metric spaces, he does leave some gaps (many are intentional) in his proofs. So be alert when you read this book, and if anything in his super short, slick proofs is not 100% clear, be prepared to fill in the details yourself. Also, remember that Rudin's way of presenting proofs is not always the most instructive when first learning the material. There is an implicit challenge to the reader to see if he or she can provide a more expository proof. Although I can say that when the classical proof suffices, Rudin usually does not deviate from it. Some of the highlights/weaknesses of the book are the following: Chapter 1: The material in this chapter is of course standard. However, Rudin supplements the chapter with an appendix on the construction of the real field from the field of rationals via the notion of Dedekind cuts. After reading many, many analysis books, I can tell you that it is difficult to find an explicit construction of the reals in books on an elementary level. Thus, while certainly not required to appreciate the rest of the text, I do recommend at least a casual perusal of the appendix just to see that "it can be done". Chapter 2: Rudin may seem to go a little overboard in his presentation on basic topology, but trust me, it will *all* be used later. So do not gloss over anything in this chapter. In particular, note how the notion of compactness is not defined a priori by any metric space ideas. However, in metric spaces, compactness does imply certain useful properties. One that is used again and again is the equivalence of compactness and sequential compactness in metric spaces. Thus, after moving on to Chapter 3 and beyond, I advise you to look back at Chapter 2 often. Chapter 3: One notable feature is that Rudin does not attempt to discuss limits per se before discussing numerical sequences and series. This may make you a little uncomfortable at first, but it turns out that this approach works best. Again, everything in this chapter is essential to the rest of the book. My only gripe with this chapter is the material on "upper and lower limits", better known as lim sup and lim inf. I feel that he should have expanded the discussion in this section a little more. In particular, his Theorem 3.19 should have had a proof supplied in the text. One of the reasons I feel this way is because the Root and Ratio tests for c

OK... Deep breaths everybody... It is not possible to overstate how good this book is. I tried to give it uncountably many stars but they only have five. Five is an insult. I'm sorry Dr. Rudin... This book is a good reference but let me tell you what its really good for. You have taken all the lower division courses. You have taken that "transition to proof writing" class in number theory, or linear algebra, or logic, or discrete math, or whatever they do at your institution of higher learning. You can tell a contrapositive from a proof by contradiction. You can explain to your grandma why there are more real numbers than rationals. Now its time to get serious. Get this book. Start at page one. Read until you come to the word Theorem. Do not read the proof. Prove it yourself. Or at least try. If you get stuck read a line or two until you see what to do. Thrust, repeat. If you make it through the first six or seven chaptors like this then there shall be no power in the verse that can stop you. Enjoy graduate school. You half way there. Now some people complain about this book being too hard. Don't listen to them. They are just trying to pull you down and keep you from your true destiny. They are the same people who try to sell you TV's and lobodemies. "The material is not motivated." Not motivated? Judas just stick a dagger in my heart. This material needs no motivation. Just do it. Faith will come. He's teaching you analysis. Not selling you a used car. By the time you are ready to read this book you should not need motivation from the author as to why you need to know analysis. You should just feel a burning in you chest that can only be quenched by arguments involving an arbitrary sequence {x_n} that converges to x in X. Finally, some people complain about the level of abstraction, which let me just say is not that high. If you want to see abstraction grab a copy of Spanier's 'Algebraic Topology' and stare at it for about an hour. Then open 'Baby Rudin' up again. I promise you the feeling you get when you sit in a hottub for like twenty minutes and then jump back in the pool. Invigorating. No but really. Anyone who passes you an analysis book that does not say the words metric space, and have the chaptor on topology before the chaptor on limits is doing you no favors. You need to know what compactness is when you get out of an analysis course. And it's lunacy to start talking about differentiation without it. It's possible, sure, but it's a waste of time and energy. To say a continuous function is one where the inverse image of open sets is open is way cooler than that epsilon delta stuff. Then you prove the epsilon delta thing as a theorem. Hows that for motivation? Anyway, if this review comes off a combative that's because it is. It's unethical to use another text for an undergraduate real analysis class. It insults and short changes the students. Sure it was OK before Rudin wrote t

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

I stumbled onto this discussion by accident, and then remembered that Rudin's book had been my Analysis text very many years ago, in a two-semester upper division course, for undergrad math majors. Personally, I've long since left behind the formal pursuit of math, but keep a fond appreciation for those years of study.I recall that at the beginning of my Analysis course I hated Rudin's book, and then after a few weeks found that I was beginning to tolerate it, even appreciate it. By the end of the course, under the tutelage of my wily professor, I came to regard the book and its author with near veneration. I still remember being forced to work through the problem sets, grumbling at the beginning, and then achieving that sense of exhilaration one feels when a dimly understood idea suddenly becomes blazingly clear, and another tantalizing idea is close behind. Perhaps such experiences, which are both intellectual and emotional, are part of the "maturity" that seasoned mathematicians try to cultivate in their students. In any case, I'm convinced that Rudin's book, at least in the hands of a skillful teacher, can help bring a dutiful student to mathematical maturity.After all this reminiscing, I'm going to dig out a copy, and see if I can recapture some of those memorable moments of discovery.

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?