Alternative Undergraduate Analysis Texts

Charles Pugh's Real Mathematical Analysis covers a wide range, starting from real numbers, topology, and basic 1D calculus, and then moving into multivariable calculus, function spaces, and Lebesgue measure/integration, all in a compact 450 pages. The writing is clear and quirky, and there are lots of interesting and hard problems.

Terence Tao has published his notes for undergrad analysis as a book:

http://terrytao.wordpress.com/books/

The original notes can be found on his webpage. I'm not sure exactly what the differences between the notes and the book are.

Nobody has mentioned Folland's "Real Analysis with Applications"?? This was the textbook for my undergraduate real analysis course (measure theory, Banach spaces, Hilbert spaces), and I still go back to it all the time. I am not yet all that experienced (I just finished my third year of graduate school), but overall I have gotten more use out of this book than any other that I own.

It has the most comprehensive swath of applications of analysis of any introductory text I have ever encountered: basic functional analysis, Fourier theory, probability theory, distributions, Hausdorff measures, Haar measure, smooth measures, and more. The early material is covered with all the appropriate detail, while the later material quickly provides the essential definitions and results needed to come to grips with an unfamiliar idea in the literature. Also, the exercises are abundant and uniformly fantastic. My only complaints are that some of the later proofs are hard to read, and there is sadly no discussion of the spectral theorem.