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Benito comments on The value of learning mathematical proof - Less Wrong
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I have some time this summer to spend learning maths, and I was going to begin studying real analysis with Rudin's "Principles of Mathematical Analysis". I have heard it is the best book if you have the time to study thoroughly, which I do (three almost uninterrupted months, although I plan to learn lots of other maths too). As someone who is mathematically able, but has not done Real Analysis (and will not study it at university) what is your recommendation that I read?
Added: My info comes from the incredivly positive amazon reviews, and the less positive Best Textbooks LW thread.
I wouldn't recommend it for someone's first exposure to analysis. When you first meet a subject, you want to get a sense of how the bits fit together, and what the important concepts and theorems are supposed to "mean" (as opposed to their formal definitions). You learn this by slowly working through examples and thinking about special cases.
Unfortunately, Rudin has very few examples, his proofs are more elegant than enlightening (for the beginner anyway, his proofs are very enlightening if you already know the big picture and are want to know the answers to questions like "Do I really need this strong an assumption for this theorem?"), and develops his theories in a lot more generality than a typical introductory analysis course (Which again, isn't necessarily bad, but you do want to get a feel for how things work in R^n before diving into arbitrary metric spaces).
If you have three months, you might want to spend the first half or so on a more verbose book, and then go over the material again using Rudin. You'd get a deeper understanding, and it might even be faster than just going through Rudin once!
That makes a lot of sense. Thank you.
Added: Much of the praise for baby Rudin suggests that trying to prove each theorem before having seen the proof, is one of the best ways to become a good mathematician. Can you comment on my thought that, after having read a more verbose book, I won't have that same experience? Or is that approach going to work with most real analysis books, so that I could still try to prove everything before it is explained?
Yes, you can try to prove everything before it's explained with pretty much any real analysis book. Just be reasonable about it, if you've gone a few hours without even making partial progress on a theorem, read the proof. A first exposure to analysis doesn't just teach you analysis, it teaches you how to build theories from the bottom up. If you can do that on your first try, great. If you can't (as is a lot more likely), learn how and save the "prove everything on your own" experience for a different subject.
As a current Harvard math grad student I think you should read many different easy books to learn a subject whenever possible, especially if you can find them for free. When you say you are mathematically able it is unclear what level you are at. All of my favorite books for learning involve huge number of exercises, and I recommend you do all of them instead of reading ahead.
For basic real analysis, my favorite book is Rosenlicht's Introduction to Analysis but baby Rudin is pretty good too, and I recommend you flip back and forth between them both.
For learning math in general, I think real analysis is a poor place to start, but that may be personal preference because I have a more algebraic slant. I highly recommend books like Herstein's Abstract Algebra, Mathematical Circles: A Russian Experience, I.M. Gelfand's Trigonometry, and Robert Ash's Abstract Algebra: The Basic Graduate Year, mostly for the wealth of exercises. Some of these are books for small children and I think those are the best sort of books to first learn from.
Thanks; I have pm-ed you for a follow-up.
I agree with SolveIt.