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I have an option to choose between the two books Mathematical Analysis by Tom Apostol and Principles of Mathematical Analysis by Walter Rudin as I was gifted Rudin by a friend and ended up buying the other book as well.I will be indebted if someone told me which one is the tougher one and which one is better for the self-learner (I am in high school and have no access to a professor or anyone).I previously used Calculus Volume I by Tom Apostol and Spivak's Calculus(for the differential calculus bit) and I have no issues about how tough the book is but I would like to choose the book that enables me to understand the subject better without being too compressed or too verbose and guides me better.Thank you.Awaiting your response. :D

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I would try to read Rudin, and if you really don't get something see if you can find it in Apostol. Apostol is much easier to read, but to be honest, at some point in your math you're going to have to read texts that are written like Rudin. They omit details and present the shortest, cleanest proof, not the easiest one. So its good to get used to that now. –  user12014 May 5 '12 at 20:28
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@PZZ There's a reason why physical trainers insist you "warm up" with lighter weights and exercises before beginning a true muscle building workout. A similar principle applies to "mathematical muscle"-you need to warm up before attempting truly strenuous workouts.Remember-mathematics as a structure builds vertically. One should ideally begin at the bottom floor and traverse the stairs at your own pace and direction. –  Mathemagician1234 May 6 '12 at 3:08
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@Mathemagician1234 Physical trainers will also insist you continue to challenge yourself. I have always found that I learn the most when texts continue to present me with new challenges and leave things to think about myself. –  user12014 May 7 '12 at 7:03
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4 Answers

up vote 19 down vote accepted

The best advice I can give you is to do what I did when learning real analysis: Use them both. Apostol has a far better exposition, but his exercises are not really challenging. Rudin is the converse -- superb exercises, but dry and sometimes uninformative exposition. The 2 books really complement each other very well -- especially if you're self-learning.

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Thank you.I intend to do the exercises from Rudin after studying from Apostol. –  Eisen May 5 '12 at 7:27
    
I suggest you do it CONCURRENTLY with studying Apostol,Eisen.In addition,I strongly suggest you look at math.stackexchange.com/questions/2786/companions-to-rudin ,it will give you many other terrific suggestions for additional references. –  Mathemagician1234 May 5 '12 at 17:46
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+1: Well done to expose the false dichotomy. If you have two decent books on the same subject and are not in any great hurry (as no high school student should be as far as learning mathematics is concerned), you will do far better to read both and play them off each other. That's general advice, but it should work especially well in this case. –  Pete L. Clark May 5 '12 at 19:23
    
@Pete Couldn't agree more,Pete-especially in these days when American society is regressing back to the 19th century and those without means may have to rely more and more on self-learning then a formal education. –  Mathemagician1234 May 5 '12 at 19:56
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I would recommend at least using Rudin as a supplement based on my own experience with PoMA and Real and Complex. I did self-study out of PoMA and let me warn you that if you decide to go that route, it will be a very difficult struggle. Rudin presents analysis in the cleanest way possible (the proofs are so slick that they often have more of the flavor algebra than analysis to me honestly) and often omits the intermediate details in his proofs. You should be prepared to sit down with a pencil and paper and carefully verify all the steps in his arguments. I don't want to talk about that though, since you can find that comment on any review of Rudin.

Let me tell you about Rudin problems. You will stare at them for hours--days even--and make absolutely no progress. You will become convinced that the statement is wrong, that the problem is beyond your tool-set, and you may even consider looking up the solution. If you stare at the problems long enough, you will eventually come up with the solution--and realize why he asked the question.

I always find that the hardest part of learning a new field of math is learning what an interesting question looks like. Rudin had exceptional mathematical taste, and that taste shines through both in those often-maligned slick proofs and in his choice of questions. If you take the time to ask why each question was asked, how it fits into the bigger picture, and what in the chapter it connects to, you will learn an incredible amount about the flavor of analysis. Really, if you want to learn how to think like a classical analyst, read Rudin.

As an aside, this may not be the case for you but I find that if a book is too well exposited, it actually detracts from my understanding. Rudin may leave out details, but at least then it is known that you need to fill them in. Doing this forced me to learn a lot of the basic argument techniques in analysis. When using a book that carefully explains all the details, I find that it is a bit too easy to waive my hand at an argument and not spend time really learning it since the argument looks so clear. Admittedly that is possibly because I am, at heart, pretty lazy :)

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I share a similar view that I do not always like being guided along like a kid.But I would like to be cautious at first.Sorry,I cannot accept this one too. –  Eisen May 5 '12 at 7:30
    
I'm sorry,I've never bought this argument. "As an aside, this may not be the case for you but I find that if a book is too well exposited, it actually detracts from my understanding." I'll restate this in a somewhat less polite manner that I heard years ago when visiting MIT:"Only people that are too dumb to see the forest for the trees complain about the Greatness of True Mathematics that is Rudin.Incompetence is always clear in the complaints.Don't worry,Andrew-I'll loan you my calculus book and in a few years,your brain might be ready for Rudin." (continued) –  Mathemagician1234 Mar 27 at 3:26
    
(continued from above) I'm sorry,it's arrogant and even worse,it's disingeniuous. You still have to understand a subject as a beginner and if you have to spend weeks and months struggling with a textbook,what's the point of having one? And that does NOT mean I'm demanding to be spoonfed either-one of my all time favorite textbooks is Herstien's TOPICS IN ALGEBRA. I'd HARDLY call that a spoonfeeding text and it's exercises are legend. The difference is that Herstien gives lots of examples and is very careful.Rudin seems sometimes like he's rushing to finish the book in time for a deadline.:( –  Mathemagician1234 Mar 27 at 3:31
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I emphatically insist that you use Apostol. Rudin is not a bad book, but especially for someone who is looking for a first introduction to higher mathematics it's just too terse, and too unintuitive--also, the problems may be a bit hard. Moreover, Apostol is a fantastic expositor, he will also cover more of the things that someone first seeing analysis should see.

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Apostol certainly covers more material, and Baby Rudin is certainly very concise, but the latter might still be the better choice for someone whose tastes run more towards soft analysis. –  Brian M. Scott May 5 '12 at 6:29
    
@Brian What does "soft" mean in "soft analysis"? –  MJD May 5 '12 at 11:52
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@Mark Terry Tao has a very nice write-up here: terrytao.wordpress.com/2007/05/23/… –  Chris Janjigian May 5 '12 at 16:55
    
@BrianM.Scott When you say Apostol covers more material, do you mean that he's just more explicit in his explanations or he covers all the topics and theorems that Rudin does, plus more? –  AlanH May 2 '13 at 23:06
    
@Alan: Neither, though the latter comes closer. He covers more topics, as I recall, but it's also my recollection that Rudin covers some not covered by Apostol. –  Brian M. Scott May 2 '13 at 23:37
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Perhaps you might want to consider an alternative that you hadn't mentioned. Here is a link to a beautifully presented copy of the lectures given by Fields Medal winner Vaughan Jones for his Real Analysis class. I found them most elegant, self-contained, and very accessible. They are available for free here:

http://sites.google.com/site/math104sp2011/lecture-notes

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+1 for a terrific set of lecture notes by a master. But Apostol is a classic for a reason and I still think it's well worth getting for a student in a serious real analysis course.That being said-these are very nice indeed and free! –  Mathemagician1234 May 21 '12 at 3:27
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