[nextos]

I certainly agree. I'm trying to re-learn advanced calculus and analysis from a rigorous standpoint, as I think it is crucial for developing deep knowledge in probability theory, among other things.

Slightly tangential but, while there are many lovely books for linear algebra (like Halmos, Axler or Hoffman & Kunze), as a newcomer I don't find analysis literature so exciting. The standard, Rudin, is really synthetic Bourbaki-style. I like short and precise books, but I found it really removes most intuition. Any good books you happen to like? Perhaps Pugh or Zorich?

[stiff]

Among the many texts almost the only one I really liked and learnt from is Courant's "Differential and Integral calculus" and the newer edition "Introduction to Calculus and Analysis". It doesn't do the typical modern division of topics and instead treats single-variable calculus, multi-variable calculus and real analysis in its two volumes in a single long sequence, but the writing style is very pleasant and the exposition very intuitive, and it includes a lot of physics applications. Hardy "A course of pure mathematics" is great too, but it is much more, well, pure, but it stays relatively intuitive and the clarity with which he writes is unparalleled, many things I first really understood from this book. Those are old texts though and notation and details of exposition differ here and there from modern standards.

I have the book by Pugh, but that one is pure^2, even as far as analysis texts go, the problems are difficult and there are no solutions, so I think it would work only for people very in love with absolutely pure mathematics and most likely only in an academic setting, while I am interested in applications and self-studying. From modern texts, given your interests, I would look at "Understanding analysis" by Abbott and "Measure, Integral and Probability" by Capinski, both pleasant to read and together providing a not too steep path toward measure-theoretic probability.

[nextos]

Thanks for taking the time to write such a thorough reply. I find Abbott a bit imprecise sometimes, but Courant is a fantastic book. Could you also mention to some of your favorite math references, in particular those that deal with probability theory and statistics?

[ekm2]

I found it easier to handle Analysis by following the sequence:Spivak ->Apostol->Rudin.

[nextos]

I think I am ready to tackle Rudin, but I don't really like it. I'm not alone there. Arnol'd is said to have called it Bourbakian propaganda. That's a bit extreme, but I certainly dislike books that strive to remove intuition:

http://pauli.uni-muenster.de/~munsteg/arnold.html

[wyclif]

Interested in a range of book recommendations for Calculus on up.