| Real analysis lectures of Francis Su at Harvey Mudd: http://www.youtube.com/playlist?list=PL0E754696F72137EC These could go with the either the book by W. Rudin or (haven't read) A. Browder. These are not easy for a complete beginner; maybe the lectures can provide some motivation. "Understanding Analysis" by S. Abbott is another rigorous and much easier, but very good, introduction. "Advanced calculus" by Shlomo Sternberg ... a work of profound beauty. This was the book used at Harvard in the 60s for the best freshman students, but it begins in a slow yet deep way with sets, logic, linear algebra, calculus, metric spaces... It's my favourite book on calculus on manifolds. http://www.math.harvard.edu/~shlomo/ Some freely available books by Robert Ash: http://www.math.uiuc.edu/~r-ash/ - in particular, his misleadingly-named "Complex variables" (with W.P. Novinger) is a short, rigorous book on complex analysis. "Naive Set Theory" by Paul Halmos is now available for something like $12. (I have not read the following books.) For number theory, maybe the book by George Andrews? It's very elementary and very cheap, and looks top notch. I'd like to read "The Cauchy-Schwartz master class" (on inequalities) but haven't purchased it yet. There are many books on combinatorics and graphs by the Hungarian school. Probably deserving special attention for discrete math are "Concrete mathematics" by Knuth et al. and "Analysis of algorithms" by Sedgewick and Flajolet (distinct from Sedgewick's "Algorithms"). |