Principles of Mathematical Analysis (International Series in Pure & Applied Mathematics)

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.


Analysis 101   May 13, 2006
Justin Bost (Salisbury, NC)
74 out of 75 found this review helpful

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 convergence of infinite series of numbers use lim sup heavily.

Chapter 4: Limits are finally introduced as the reader remembers them from basic calculus. The only difference is that Rudin works with arbitrary metric spaces, which turns out to be very useful later. Take note of Theorem 4.2. Reformulating the existence of a limit of a function in terms of limits of sequences is a handy theoretical tool that makes a lot of proofs (Rudin's included) much easier to understand. That said, there are no real surprises until Theorem 4.8. You can probably omit the subsection "Discontinuities" with no loss. I say this even though some of the theorems in "Monotonic Functions" use that material in their proofs. Theorem 4.30 in particular (monotone functions on open intervals have at most a countable number of discontinuities) has a much better proof than that Rudin provides. So try and look elsewhere for the proofs of those theorems.

Chapter 5: All the derivative proofs are just like you remember from advanced calculus. The only one that merits special attention is L'Hospital's Rule. Work through it very carefully, it is more subtle than it appears.

Chapter 6: The Riemann-Stieltjes integral can be obtained by only slightly more effort, so Rudin wisely decides to base all of his proofs (through Theorem 6.19) on it. Just be aware that some of the material covered, such as the Fundamental Theorem of Calculus and integration by parts is only discussed for the original Riemann integral. Theorem 6.25 (based on the Cauchy-Schwarz inequality) acquires a special significance in the following chapters, so memorize it!

Chapter 7: By far, this is the most crucial chapter in the book. This is probably the material that you may have had limited or no exposure to in the past. The famous Weierstrass Approximation Theorem (and its generalization by Marshall Stone) is given here. Read this chapter front to back at least four times. Yes, it is that important. Otherwise, the Fourier Theory presented in Chapter 8 will seem like gibberish.

Chapter 8: Expansion of analytic functions via power series is presented here. A brisk, but complete development of the exponential, sine and cosine functions is also featured here. Problem 6 in the exercises at the end of the chapter is worth special consideration. Work it out after you read about the exponential function. The Fourier material is relatively straightforward, although awkward when divorced from measure theory. As Rudin himself notes, the hypothesis that f be Riemann integrable is often unnecessary, so you may want to peek ahead at Chapter 11 while reading many of the proofs, especially Parseval's Theorem. The material on the gamma function is cute, but not really needed.

Chapter 9: The standard treatment of multivariable functions. Rudin's coverage of linear algebra is succinct. Also, the linear algebra has more important uses than merely providing a pathway to "multivariable calculus". The theory of linear operators sketched in Theorems 9.5 to 9.8 will lead you directly to the more abstract theory of Banach spaces. The Banach spaces take a very central role in advanced analysis as can be seen by reading Royden's "Real Analysis". I also recommend supplementary reading for this chapter. A good book to look at is Charles Pugh's "Real Mathematical Analysis" which has an extensive treatment of multivariable functions. Also, you might skim over George Simmon's "Introduction to Topology and Modern Analysis", a great introduction to the abstract theory of operators. This material is only hinted at in Rudin, but comes to its full fruition in Simmons.

Chapter 10: This is Rudin's introduction to differential geometry. I honestly have not given this chapter a thorough reading, but on the surface it looks ok. Most of the deeper theorems from multivariable calculus (excepting the Implicit Function Theorem, discussed in Chapter 9) are treated here, such as the trifecta known as Stoke's, Green's, and the Divergence Theorems, respectively. This chapter is important to anyone going into fields such as partial differential equations.

Chapter 11: This chapter seems to be the one that most people criticize in the book. Rudin gives a perfunctory outline of Lebesgue theory that seems to rob the reader of much needed detail. Indeed, this chapter is a little too lean for my tastes. But in Rudin's defense, he warns the reader at the outset that "proofs are only sketched in some cases, and some of the easier propositions are stated without proof". Hence, I recommend just giving this chapter a light read, then go to another book (such as Royden) for the real proofs. As expected, Rudin discusses some of the seminal results of Lebesgue theory, including, but not limited to: the Monotone Convergence Theorem, the Dominated Convergence Theorem, and the correspondence of the Lebesgue integral with the ordinary Riemann integral (whenever the latter exists). The Riesz-Fischer Theorem from Fourier analysis appears here. Lebesgue's Dominated Convergence Theorem (Theorem 11.32) is worth a careful reading. Afterwards, look at Exercise 12 in Chapter 7 for a simpler version of the DCT using the familiar Riemann integral. The proof is not that difficult.

It goes without saying that the exercises are extremely important and should all be attempted. Unless you are brilliant, odds are that at least a few will elude you. Nonetheless, many important results and counterexamples are listed in the exercises, so you will benefit from working them. Be warned though that Rudin will intermingle easy with very difficult problems.

An obvious problem is the outrageous price. Unfortunately, this book is essential reading, so you'll just have to cough up the dough or look for it cheap elsewhere. It is a good book to learn from and a fantastic reference. I don't know if I would call it the "best analysis book ever". But its current edition was released 30 years ago, so that says something about its popularity.

P.S. Once you've finished Rudin, the book by Pugh referenced above is a good read to "pull it all together". There are some well-thought out problems that will both challenge and inspire you to learn more at the same time.

Highly recommended.


Getting started in math.   September 21, 2004
Palle E T Jorgensen (Iowa City, Iowa United States)
29 out of 32 found this review helpful

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.



If you are serious about doing math...   May 3, 2004
Giovanni Dimatteo (Padova, Italy)
30 out of 31 found this review helpful

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.