Principles of Mathematical Analysis [Hardcover]

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.

This book is hyped up a lot by intimidating professors (and competitive students), but does not deliver the goods. Many people feel that Rudin is concise and effective. But to me, Rudin is terse and weak.

It is not hard to discover why his book is in fact so ineffective. The reason is that he is trying to cover too much ground in too few pages.

The core of this goal, is probably a sick conspiracy: to achieve the impossibe --- to be the most bought math book in history (required text for every math curriculum), yet at the same time cover all the difficult topics that 99% of Math majors will never master without graduate studies.

This all reaches a peak in his neglectful treatment of multivariate functions. It would be a shame if a student really had to learn Multivariate analysis from this book. (However, Rudin is good to keep handy if you are doing problems from Spivak's book.)

The end result, is that this book is extremely demanding for even the eager student, who is seeing it for the first time. Nobody I know, in result, has benefited much from this book.

One final criticism. For those, like myself, who haven't worked all the problems in this book, Rudin is a pretty terrible reference. I once had the misfortune of trying to reference his proof of L'Hospitals. In conclusion, I found it easier to reprove L'Hospital myself than to read his cryptic use of the real axioms.

Now with so many criticisms, I must explain why I have given 4 stars.

There comes a point in time, for any respectable math student, that he must develop the ability to solve difficult, abstract problems with little explanation of how and why.

In this regards, Rudin's book could be an extremely valuable resource. He has left a trail (THE PROBLEMS!!) which goes through many crucial ideas in Mathematics. Few books, at the undergraduate level, have such a vast amount of problems - aimed at the budding math student. In this respect, Rudin should get no less than 5 stars.

But I stand at 4. Regretfully, Mathematics departments everywhere have forced the Rudin pedagogy on everyone. I believe the student should make this choice (i.e. which books to study in detail).

And since it was forced on me, I have a voice in this matter: This book should not be on the undegraduate curriculum. And in fact, I don't like his style, I don't like this book, and I'll do problems elsewhere, thanks.

-TM

p.s. If you happening to be struggling through the book at this time, here is some advice: Keep your freshman Cal book handy. Don't become a victim, and don't go through this course not knowing how to prove the limit laws, the definition of a derivative, Mean value theorem, derivative laws the proof of the fundamental theorem of calculus, and theorems involving integrals of continuous functions, convergence divergence tests, power series representations, partial derivatives. Note that all of these topics are indeed in a freshman cal course. (Well, this is what popped into my head, not a formal and complete list..)

It is here where calculus actually can become very useful. For example you can define the logarithm, exponential function - and this leads to a definition of a real exponent without using inf / sup 's as Rudin does in a Chapter 1 problem.