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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).

I can choose between 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 Apostol 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 have no issues about how tough the book is, but I would like the book that enables me to understand the subject better without being too compressed or too verbose and guides me better.

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    $\begingroup$ 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. $\endgroup$
    – user12014
    May 5, 2012 at 20:28
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    $\begingroup$ @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. $\endgroup$
    – Mathemagician1234
    May 6, 2012 at 3:08
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    $\begingroup$ @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. $\endgroup$
    – user12014
    May 7, 2012 at 7:03

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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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  • $\begingroup$ Thank you.I intend to do the exercises from Rudin after studying from Apostol. $\endgroup$
    – Eisen
    May 5, 2012 at 7:27
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    $\begingroup$ 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. $\endgroup$
    – Mathemagician1234
    May 5, 2012 at 17:46
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    $\begingroup$ +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. $\endgroup$
    – Pete L. Clark
    May 5, 2012 at 19:23
  • $\begingroup$ @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. $\endgroup$
    – Mathemagician1234
    May 5, 2012 at 19:56
  • $\begingroup$ @Mathemagician1234 Sorry to contact you here, I didn't know where else. I know you have a great deal of experience with books, so I was wondering if I could ask you for a recommendation? I have to learn semi-direct products and nilpotent groups. My class uses Dummit and Foote, but I really hate that book (for one, they seem to make things unnecessarily abstract). Would you happen to know of a good book which goes over any of the following topics in comparable depth to Dummit and Foote? 1) Semidirect products 2) Nilpotent groups 3) Automorphisms (useful for (1)). Thank you so much! $\endgroup$
    – Ovi
    Dec 12, 2019 at 1:19
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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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  • $\begingroup$ 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. $\endgroup$
    – Eisen
    May 5, 2012 at 7:30
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    $\begingroup$ 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) $\endgroup$
    – Mathemagician1234
    Mar 27, 2014 at 3:26
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    $\begingroup$ (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.:( $\endgroup$
    – Mathemagician1234
    Mar 27, 2014 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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  • $\begingroup$ 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. $\endgroup$
    – Brian M. Scott
    May 5, 2012 at 6:29
  • $\begingroup$ @Brian What does "soft" mean in "soft analysis"? $\endgroup$
    – MJD
    May 5, 2012 at 11:52
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    $\begingroup$ @Mark Terry Tao has a very nice write-up here: terrytao.wordpress.com/2007/05/23/… $\endgroup$
    – Chris Janjigian
    May 5, 2012 at 16:55
  • $\begingroup$ @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? $\endgroup$
    – TheRealFakeNews
    May 2, 2013 at 23:06
  • $\begingroup$ @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. $\endgroup$
    – Brian M. Scott
    May 2, 2013 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:

https://sites.google.com/site/math104sp2011/lecture-notes then at bottom is a link to pdf file, just tried it and works.

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    $\begingroup$ +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! $\endgroup$
    – Mathemagician1234
    May 21, 2012 at 3:27
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    $\begingroup$ The link is not working anymore. :) $\endgroup$
    – Hosein Rahnama
    Sep 9, 2017 at 9:29
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    $\begingroup$ @H.R. I just added an edit and it works. $\endgroup$
    – user12802
    Sep 9, 2017 at 13:26
  • $\begingroup$ According to the footnote on Page 1 of his lecture notes, it seems that he is using a special book. Do you know what that book is? $\endgroup$
    – Hosein Rahnama
    Sep 9, 2017 at 13:40
  • $\begingroup$ @H.R. I can"t remember it, but other than assigned HW probs, I never looked at it. The notes are really his own treatment of the material, which makes them so special. Just hang in there past the first few obligatory stage-setting pages, and you won't be disappointed. $\endgroup$
    – user12802
    Sep 9, 2017 at 16:02
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You can try "Curso de análise" vol1 and vol2 by Elon Lages Lima too

In this series of books ideas under a topological analysis approach and to some extent intuitive occur . It is harder to read than Apostol but less than Rudin, the problems are not easy so working on them will help to absorb the ideas of analysis.

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I can answer from England. In 1959 I read Tom as a Balliol undergrad. It was his 1st edition which started with a brilliant exposition of the Riemann-Stieltjes Integral. His 2nd edition was mundane. Noone to date has picked up that he used Finer Partitions rather than a Mesh size. I read Rudin. Both Real and Complex.I felt proofs were not fully rigorous. By today's standards. But Math is evolving.

I could recommend Zorich in the translation. Both volumes.See Springer as publisher. He teaches at Moscow State University.

A seminal work is still Ahlfors Complex Analysis. He won a Fields Medal for it.

I had caught up on Banach spaces with a publication from The Canadian Mathematical Society. Jeffrey G Thomas.

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