I'm starting to read Baby Rudin (Principles of mathematical analysis) now and I wonder whether you know of any companions to it. Another supplementary book would do too. I tried Silvia's notes, but I found them a bit too "logical" so to say. Are they good? What else do you recommend?
[Math] Companions to Rudin
analysisbook-recommendationreference-requestsoft-question
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First of all: you shouldn't give up on problems after 30 minutes. Take a break, try a different problem, maybe wait a few days and try again -- you'll gain a lot more from the problem if you struggle and solve it yourself. Having access to solutions can be helpful, but you don't want to find yourself relying on them. (There's a phrase that gets thrown around a lot: "If you can't solve a problem then there's an easier problem you can't solve; find it").
Baby/Blue Rudin is a great book for an introduction to the basics of analysis (beyond one-variable "advanced calculus"). After that I'd suggest looking at the 'Lectures in Analysis' series written by Elias Stein and Rami Shakarchi (Stein was actually Terrence Tao's advisor). These books cover introductory Fourier analysis, complex analysis, measure theory, and functional analysis. Along the way the authors expose you to all kinds of in-depth and enlightening applications (including PDEs, analytic number theory, additive combinatorics, and probability). Of all the analysis textbooks I've looked at, I feel like I've gained the most from the time I've spent with Stein and Shakarchi's series -- these books will expose you to the "bigger picture" that many classical texts ignore (though the "classics" are still worth looking at).
I've skimmed through parts of Terrence Tao's notes on analysis, and these seem like a good option as well (though I looked at his graduate-level notes, I don't know if this is what you're referring to). I've always gotten a lot out of the expository stuff written by Tao, so you probably can't go wrong with the notes regardless. If you feel like you need more exercises, don't be afraid to use multiple books! Carrying around a pile of books can get annoying, but it's always helpful to see how different authors approach the same subject.
I can recommend you two books. They are old (1990 & 1974), but I believe they are (SO SO) awesome.
FIRST:
Problems in Real Analysis - A Workbook with Solutions
Charalambos D. Aliprantis, Owen Burkinshaw
Academic Press
1990
SECOND:
Exercises in Real and Complex Analysis with Solutions
Walter Rudin
1974
Also, here a book with (DJVU) format:
Best Answer
1) Introduction to real analysis by Bartle and Sherbert
2) Methods of Real Analysis by R.R. Goldberg
3) Mathematical Analysis by Tom Apostol
4) Real and Abstract Analysis by Karl Stromberg.
5) A radical approach to real analysis by David M Bressoud by MAA.
The first book is a very good book for a beginner. The next two are classics. (4) is also very good in case you want to read something advanced. The last one keeps entertaining you with some interesting examples as well as some interesting history of Real Analysis.
Happy Reading!!