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The third edition of this well known text continues to provide a solid foundation in mathematical analysis for undergraduate and first-year graduate students. The text begins with a discussion of the real number system as a complete ordered field. (Dedekind's construction is now treated in an appendix to Chapter I.) The topological background needed for the development of convergence, continuity, differentiation and integration is provided in Chapter 2. There is a new section on the gamma function, and many new and interesting exercises are included.
This text is part of the Walter Rudin Student Series in Advanced Mathematics.
- Sales Rank: #80547 in Books
- Published on: 1976-01-01
- Original language: English
- Number of items: 1
- Dimensions: 9.30" h x .70" w x 6.10" l, 1.20 pounds
- Binding: Hardcover
- 342 pages
Most helpful customer reviews
139 of 150 people found the following review helpful.
Remembered with reverence
By Paul J. Papanek
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.
1 of 1 people found the following review helpful.
Like drinking math out of a fire hose
By Y. Wang
'Baby Rudin' is an introductory text in analysis for the serious student of mathematics. This was the text used for the first semester of Harvard's freshman real analysis course (Math 25a, modestly titled "Honors Multivariable Calculus and Linear Algebra") back in 2004. A majority of student in the course had been exposed to proof writing, through high school olympiads, upper division university mathematics, or both. Although the professor (Tom Coates, now of Imperial College London) was an excellent lecturer and was very helpful and patient, students lacking this sort of background ("mathematical maturity") had substantial difficulty keeping up with the course. Those of us who remained in the course after the first few weeks found ourselves spending 15 to 20 hours a week working problems assigned from Rudin (which range from routine verifications to problems barely within the grasp of the most talented student in the class) and additional problems supplemented by the instructor.
As many previous reviewers have already noted, "Principles of Mathematical Analysis" (PMA) is simply the best out there in terms of clear, concise mathematical exposition of one-variable advanced calculus (Chapters 1-8). While the entire book is succinct and fantastically organized, chapter 2 on metric space topology deserves particular mention. This concisely written chapter (23 pages) is strategically placed to serve as a cornerstone for the rest of the book. Assuming only rudimentary familiarity with set and functions, Rudin includes only what is needed: just a taste of topology to enable the subsequent discussions of continuity and convergence in a general and abstract setting. Several interesting ideas not included in the text of chapter 2 (like the Baire category theorem) are included in the exercises. Though the abstraction is challenging for the beginner, the reward is substantial, enabling celebrated theorems (e.g., Arzelà-Ascoli, Stone-Weierstrass) to be proved cleanly and quickly later on. The usual topics (limits, continuity, differentiation, and Riemann(-Stieltjes) integration) are covered in the subsequent chapters 3-7, followed by a discussion on power series at the beginning of chapter 8. A discussion of the rigorous definitions of special functions (exp, log, sin, cos, Gamma), the fruits of one's labors, nicely rounds out a comprehensive one-semester course in real analysis. (Back then, our course covered chapters 1-7, and a few sections of chapter 8. It ended up being the course with the biggest workload for me that semester, by far!)
Rudin covers analysis on several variables (incl. differential forms and the generalized Stokes Theorem) and Lebesgue measure and integration in the three remaining chapters. These chapters have been the subject of criticism, mostly due to the bare bones coverage of these intricate topics. In my opinion, they really are too lean to be pedagogically useful introductions. However, they are useful as "cliff notes" after already learning them. Rudin's outlines of these topics do not provide the reader with all of their formalisms and machinery or their many implications, so one is well-advised to find other resources for an introduction to these topics. For multivariable calculus, Spivak’s "Calculus on Manifolds" is flawed but still the best out there, though it should be supplemented by Halmos’s "Finite-Dimensional Vector Spaces" or Axler’s "Linear Algebra Done Right" to provide the requisite abstract linear and multilinear algebra background. Bartle’s "Elements of Integration and Lebesgue Measure" provides a clear though not particularly thorough introduction to Lebesgue theory.
With the exception of a short introductory paragraph at the beginning of each chapter, the book consists almost entirely of a series of statements labeled Definition, Theorem, or Corollary (in that order) with only an occasional Example or Remark squeezed in. (The Examples and Remarks are extremely important -- they generally address subtleties or common points of confusion for the beginner.) This format may be shocking or frustrating for some. However, the beauty of the mathematics is allowed to speak for itself through Rudin's austere exposition. Flipping through my tattered copy of the book from 2004, I still marvel at the elegance and polish of the proofs -- completely rigorous, yet without a wasted symbol or syllable. Precisely for these reasons, however, students will find themselves reading and re-reading a proof to try to figure out how Rudin managed to construct it and/or where the idea behind it came from. To get the most out of PMA, the diligent student should have pencil and paper in hand to try to write their own less elegant but plainly motivated proofs, preferably before even looking at Rudin's proof. Also, Rudin only occasionally remarks upon why the hypotheses of a theorem cannot be weakened. Thus, to fully understand a theorem, the diligent student should frequently think about constructing concrete examples or counterexamples.
A professor teaching from this text should ideally motivate the definitions (and even some of the theorems) presented in PMA during lectures. (It's not always clear why he proves something; Rudin does not distinguish between main theorems and lemmas.) However, if you are teaching yourself, a less formal exposition with more examples and explanations with occasional diagrams to facilitate intuition will probably be useful. Pugh’s "Real Mathematical Analysis," comes to mind. (It covers more topics than PMA, but adopts a conversational tone throughout. In addition, the exceptionally interesting and difficult exercises in Pugh’s text are an added bonus!)
A final remark about the style: if PMA sometimes reads like a collection of lecture notes, it's probably because it started out that way. It originated when Rudin, a freshly minted Ph.D., started teaching real analysis as a lecturer at MIT in 1950. Back then, there were very few analysis textbooks in English. Texts like Hardy's venerable "A Course of Pure Mathematics" and Titchmarsh's advanced text "The Theory of Functions" present a classical view of the subject without reference to topological notions. There were no suitable modern treatments of real analysis at the time, so he decided to write one for his course. The first edition of PMA was published only three years later in 1953. To varying extents, PMA inspired the treatment of modern ideas in almost all subsequent English language textbooks in analysis.
0 of 0 people found the following review helpful.
Rudin, oh Rudin
By Stephen R Garth
What can I say its a classic, though better as a second analysis book.
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