Does anyone have an recommendations for an Analysis textbook for beginners? I'm looking for a textbook that's like Calculus Third Edition by Smith and Minton in that it's hardcover, has good illustrations and is easy to understand, but I don't want anything that skimps on the proofs. Thanks
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I highly recommend Spivak's Calculus with the solutions manual. – Clarinetist Dec 26 '14 at 04:53
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For calculus with proofs, Tom Apostol's Calculus Vol. 1 is quite good. For something that's more difficult that you would use in a real analysis course at a university, Rudin's Principles of Mathematical Analysis is a standard textbook, although the density of that book might not exactly suit self-study. – A.S Dec 26 '14 at 04:54
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1See also: Good book for self study of a First Course in Real Analysis – Dec 26 '14 at 05:14
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Best book hands down is Advanced Calculus by Fitzpatrick. – user60887 Dec 26 '14 at 05:21
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For self-study one could do a lot worse than to begin with Edward D. Gaughan’s Introduction to Analysis before moving on to something more advanced: roughly speaking, it goes back and does the basic theory of the topics typically covered in freshman calculus of one variable. It was designed to be transitional between calculus and graduate-level real analysis, and I remember it as being both rigorous and quite gentle; we used it in a transitional course between calculus and a much harder and more general undergraduate real analysis course.
I liked to use the second edition of Robert G. Bartle’s Elements of Real Analysis when I taught the more advanced real analysis course; it’s especially good on the topological aspects of analysis in $\Bbb R^n$, and on the whole it’s very clearly written.
Brian M. Scott
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+1 for Bartle's Elements of Real Analysis. It was the text for my first analysis course, and in hindsight I'm glad the professor chose to use it instead of the more customary Rudin. Bartle's book is more user-friendly, covers more topics, and has much better exercises, in my opinion. Unfortunately, my copy is in several pieces now and apparently the only version in print is a $170 paperback edition :-( – Dec 26 '14 at 06:41
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Understanding Analysis by Stephen Abbott.
Elements of Real Analysis by Charles Denlinger
Bhaskar Vashishth
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Real Mathematical Analysis by Charles C. Pugh
Elementary Analysis: The Theory of Calculus by Kenneth A. Ross
Math.StackExchange
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In my (elementary) real analysis class we used a book by Stephen Lay called Analysis With an Introduction to Proof. The reviews are generally positive and I thought it was a great book. My only gripe is that there are no solutions to even/odd numbered exercises except for one or two hints for each section. Also, it's incredibly expensive considering it is about the size of an iPad. In terms of content, it was excellent. There were many examples presented and the proofs were straightforward and easy to comprehend. The text started by covering logic and proof techniques, and even provided some "fill in the blank" proof questions that really eased you into it. I would recommend it for a first text in analysis.
MathMajor
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Since you have not accepted an answer - and they are all good - I will suggest my favorite.
These are free notes (they almost read verbatim) of a RA course given by Vaughan Jones (Fields Medal winner - aka Nobel for math). They are complete and offer his own treatment, which is indubitably excellent: