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286 of 294 people found the following review helpful
5.0 out of 5 stars
Analysis 101,
By Justin Bost (Salisbury, NC)  See all my reviews
This review is from: Principles of Mathematical Analysis (International Series in Pure and Applied Mathematics) (Hardcover)
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 RiemannStieltjes 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 CauchySchwarz 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 RieszFischer 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 wellthought out problems that will both challenge and inspire you to learn more at the same time. Highly recommended.
515 of 561 people found the following review helpful
5.0 out of 5 stars
Book should be called "Tada! You're a mathematican!",
This review is from: Principles of Mathematical Analysis (International Series in Pure and Applied Mathematics) (Hardcover)
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 the thing, but now? Why spit on your luck? And if you'r a student and find the book too hard? Try harder. That's the point. If you did not crave intellectual work why are you sitting in an analysis course? Dig in. It will make you a better person. Trust me. Or you could just change your major back to engineering. It's more money and the books always have lots of nice pictures. In conclusion: Thank you Dr. Rudin for your wonderfull book on analysis. You made a man of me. Rock
124 of 134 people found the following review helpful
5.0 out of 5 stars
Remembered with reverence,
By
This review is from: Principles of Mathematical Analysis (International Series in Pure and Applied Mathematics) (Hardcover)
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 twosemester 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.
47 of 50 people found the following review helpful
5.0 out of 5 stars
If you are serious about doing math...,
This review is from: Principles of Mathematical Analysis (International Series in Pure and Applied Mathematics) (Hardcover)
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 getyourhandsdirty 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 getyourhandsdirty 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.1Ch.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.
24 of 24 people found the following review helpful
5.0 out of 5 stars
These Five Stars Need an Explanation,
By Margaret A. Murray (Blacksburg, VA USA)  See all my reviews
This review is from: Principles of Mathematical Analysis (International Series in Pure and Applied Mathematics) (Hardcover)
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 19992000 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?
19 of 19 people found the following review helpful
5.0 out of 5 stars
Essential for any serious Math Undergraduate,
By longhorn24 "longhorn24" (New York, NY)  See all my reviews
This review is from: Principles of Mathematical Analysis (International Series in Pure and Applied Mathematics) (Hardcover)
Rudin's Principles of Mathematical Analysis (often called Baby Rudin, in contrast to his wellknown graduate analysis text) is essential reading for any mathematics undergraduate student for three reasons. First, while the key proofs are generally found in most undergraduate texts, Rudin's signature "theoremproof" terseness and the inclusive nature of the text (Rudin does not cheat by using the horrendous "the details are left to the reader" qualifier) shows students, possibly for the first time in their study of mathematics, that a proof can be both concise and complete.
Also, after a quick introduction of the real and complex number fields, Rudin quickly presents enough topological concepts to truly understand analysis on the real line, and more generally, finitedimensional Euclidean space. This presentation is beneficial because shows how many of the properties of functions in these familiar spaces are actually topological, and it lays a foundation to build a more abstract understanding of analysis on metric spaces (or even more general spaces) in later courses. In the exercises, for example, Rudin asks the readers to generalize many topological concepts to metric spaces. While the second chapter includes most of the topological properties needed to really understand the rest of the course, the third chapter is an excellent introduction to sequences and sums. Rudin's topological introduction allows him to present convergence in terms of Cauchy sequences on metric spaces, a useful strategy that makes for a better understanding of uniform continuity and convergence later. He then quickly but completely (no pun intended) proves the basic results like the root and ratio tests and gives a few examples that the reader is expected to generalize in later problems. Rudin's chapters on integration and differentiation are like most other texts, with the exception that Rudin generalizes the Riemann integral to the RiemannSteijtles case, which is useful in both applications and as a demonstration of a more general form of integration. While this is not a building block to the Lebesgue theory, it is useful because it shows that the Riemann integral is merely an introduction to integration that works in only "nice" cases. The seventh chapter, after which a reader is ideally prepared for later study of analysis, is on uniform continuity and convergence, two properties which play an important role in function theory and which require a thorough understanding of topology on the real line and Cauchy sequences. It even has a proof of Weierstrass' approximation theorem. This chapter is loaded with excellent problems and, with the first six chapters, forms the basic onesemester undergraduate analysis course at most schools. The eigth chapter introduces familiar functions in a new light, such as the gamma and trigonometric functions. While not essential reading for an undergraduate, it allows the reader to apply the more abstract concepts in the earlier chapters to known functions. The ninth and tenth chapters, on integration on forms, is not a great introduction to the material because it is too terse and dense with simplistic notation that makes the proofs harder to grasp than they really are. Then again, this material often constitues (with a few other concepts) an entire semester at some schools. While Spivak's Calculus on Manifolds has a better proof of the Inverse and Implicit Function Theorems, it is very expensive ($40 for a short book that also skimps on notation and examples). I would recommend getting a copy of Munkres' outofprint Analysis on Manifolds to understand this material. This book complements the first seven chapters of Rudin well and together, with maybe another topic (such as Rudin's last chapter on the basics of Lebesgue integration) makes for a nice yearlong course in undergraduate analysis. Finally, the last chapter, although brief, should be read throughly and I would recommend that all readers try the problems. It introduces Lebesgue integration painlessly, while throwing in a few basic concepts that the reader will generalize in a later course. For example, Rudin introduces regularity (which can be assumed for Lebesgue measure, since R^n is one of the "nicest" topological spaces), a concept that will appear over and over again for all measures. Also, he introduces L^2 space, which is essential to understand the basics of Hilbert theory and Parseval's equation and rudiments of Fourier analysis. After studying the first seven chapters of Rudin, along with the last chapter (using the ninth and tenth chapters as a reference along with a more descriptive introduction to calculus on forms), the reader is prepared to tackle graduate analysis. Luckily, much of the material needed for an advanced course is covered in the exercises (for example, convexity and the Holder and Minkowski inequalities). I strongly recommend, first, a study of pointset topology using Munkres' introductory text. Since the reader has already learned some of the basics working with Rudin's book (and some key results doing the exercises), this won't be too difficult and will make almost every future course in analysis much easier. I give Rudin's book, despite some weaknessess in the material on differential forms, five stars because the first seven chapters cover more than enough for the money paid (even if you buy it new and there are thousands of used copies floating around) the last chapter is an added bonus. Also, the problems are so challenging that a course in pointset topology or complex analysis will be much easier. More importantly, the thorough nature of the book allows for selfstudy.
19 of 19 people found the following review helpful
5.0 out of 5 stars
Unbeleivably enlightening after a torrid workout,
By
This review is from: Principles of Mathematical Analysis (International Series in Pure and Applied Mathematics) (Hardcover)
"Little Rudin" has this reputation of being a particularly difficult book to read, especially if its your first time learning analysis. And true, Rudin is a particularly difficult taskmaster, especially in his exercises. However, learning through Rudin provides unbelievable insight into many aspects of real analysis, that you wont find anywhere else(and its even more helpful if you use Rudin as your first analysis text, rather than let your mind be corrupted by other lesser analysis texts). After struggling through this book(and I mean doing the exercises too), youll know what I mean when you pick up any of the "analysis for babies" books. For example, standard analysis texts tend to talk about the the fact that continuous functions on closed intervals are uniformly continuous for pages and pages. Rudin however simply talks about continuous maps on compacts subsets of metric spaces in general, rather that going through just R. Not only does this approach help you understand continuous functions on R better, but in fact prepares you well for continous maps in general metric spaces. My favrouite section in the book is the RiemannStieltjes Integration, a chapter which again provides an unbelievable clear discussion of integration on the reals. And remember, if you get stuck on any particular section, dont simply skip it. Working through and trying to make sense of it provides an impeccable understanding of the concept that you cant hope to get anywhere else. And lastly, if this has convinced you to pick up a copy of Rudin to slave through, GOOD LUCK!
16 of 16 people found the following review helpful
5.0 out of 5 stars
the best,
By A Customer
This review is from: Principles of Mathematical Analysis (International Series in Pure and Applied Mathematics) (Hardcover)
Rudin's elementary analysis text is simply the best. This is mostly attributable to its very precise and compact language. (I wish all proofs were written like this.) The book is easy to read and largely understandable. It makes good first time reading as well as a handy reference.
(By first time reading, I don't mean that the text is recommended for people without formal exposure to proofsRudin's text simply assumes that one is already familiar with the rules and concepts of basic logic. On the other hand, if one has already had some exposure to proofs in a nonanalysis context (i.e. number thoery), then Rudin's book can serve as a very good intro text to analysis.) Many other analysis texts are simply too verbose, taking paragraphs and paragraphs to describe concepts that can be easily/better said in just a few words. As a result they can't be read in a feasible amount of time. I also dislike texts that are not written in the definitionaxiompropositionproof mannerI find that to be a lack of mathematical discipline and consistency. And I doubt that one can gain much mathematical maturity by reading texts of this sort. If one wants to seek mathematical motivation behind real analysis or a starter's book that introduces basic logic, something that the Rudin text clearly lacks, I *highly* recommend "The Lanaguage of Mathematics" by Keith Devlin. This is a popular math book that is suitable for all levels, and it successfully communicates the basic ideas and motivation behind many branches of math. It also covers most of the important historical development of classical mathematics with many engaging stories. Another book that I can really recommend as a more understandable substitute for Rudin's text is "Elementary Mathematical Analysis" by Colin W. Clark. This book is the most understandable intro analysis text I've seen and contains sufficient coverage of logic and mathematical motivation to be a standalone text. However, it's no longer in print but should be available at most school's libraries.
33 of 37 people found the following review helpful
3.0 out of 5 stars
Poor introduction, Outstanding reference,
By A Customer
This review is from: Principles of Mathematical Analysis (International Series in Pure and Applied Mathematics) (Hardcover)
I was terrified by this text my freshman year in college. Unfortunately, this book was the only required book for the class. The main difficulty is that the book resembles a magnificent outline of the material more than a text. The shortest, most elegant proof of anything is invariably choosen and there is little motivation given for the material. Thus, I found this book to be difficult to use to learn how to do mathematics. On the other side, if you know the basic ideas of analysis, then this book is a remarkable, clear, and elegant place to review and extend your knowledge. I therefore would HIGHLY recommend it as a companian text for an analysis course or as a reference. My rating therefore is an average: as an introduction to analysis by itself, it rates one star; as a supplement to another text, as a review text, or as a reference, it clearly rates five stars.
23 of 25 people found the following review helpful
5.0 out of 5 stars
A book I cannot praise enough.,
By Kyle R. Hofmann (San Diego, CA, USA)  See all my reviews
This review is from: Principles of Mathematical Analysis (International Series in Pure and Applied Mathematics) (Hardcover)
My first introduction to this book was in high school. I do not think that I have ever read a more fascinating book. Its clear explanations and exciting results engrossed me for hours at a time; I read and read and read; then I thought and thought and thought; and I repeated this for days at a time. Despite others complaints about Rudin's style, I found and still find this book wonderful. It may not be for everyone: it is, as one reviewer noted, in a Bourbakian style, and consequently does not always present much motivation. Any reader who finds this a problem should skip ahead a few pages to see where Rudin is going. Another aspect of this style is the very terse "outline" feel that another reviewer noted: Rudin has short proofs, no pictures, and fewer examples than many other analysis texts. This was not a problem for me; the proofs are very elegant and were an excellent introduction to the art of writing proofs. No difficult visualization is required; I was able to visualize all the pictures I needed in my head. The definitions are stated clearly enough that the student can construct his own examples, and the included examples are very good and clarify the most difficult portions of the text. The exercises, too, are very good. They range from trivial to mystifying, but all expand the reader's knowledge of the material. In summary, I think anyone who learns analysis should be exposed to this book. Reading it can be very difficult, and it is not unreasonable to spend hours on a page, but those hours will have been well spent.

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Principles of Mathematical Analysis (International Series in Pure and Applied Mathematics) by Walter Rudin (Hardcover  January 1, 1976)
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