Analysis I

Introduction to Calculus and Analysis, by Richard Courant and Fritz John. This book clearly explained many of the motivating examples of the concepts in Calculus, which was important to me as a beginning student. Too many times I read math textbooks that simply present a list of definitions, theorems, corollaries, and whatnot, failing to guide readers from one point to another. I agree that for advanced level of math, there're probably no good motivations or 'storyline' that weave these constructs together (they're purely symbolic); however, for a layman level, I feel it's crucial to at least make the readers understand why and how these concepts come to be, so that they can grasp at least a little bit of the 'logic' of math. Similarly, Analysis I by Terence Tao has a 'Why do analysis' that has made me actually enjoy analysis.

Does math count as "technical?" I was originally a Physics major and lately I've been on a kick of filling in the mathematics that was used in my Physics classes but that I feel like was never really gone into in much depth. My current reading list is: - "Analysis I" (and II) by Terence Tao (I finished the first volume and am now on the second, but I consider them really one book) - "Understanding Analysis" by Stephen Abbott - "Topology Through Inquiry" by Starbird and Su - "Introduction To Topology And Modern Analysis" by George F. Simmons The Terence Tao books are amazing so far. Extremely readable introduction to Real Analysis. Abbott also came highly recommended and from reading the first couple chapters I can see why as it also seems to be very readable. I don't know if it would be a better introduction than Tao, but it covers mostly the same material and I think having two different perspectives will really help solidify things for me. Once I finish those, I'll see whether I want to go deeper into Topology or move to Complex Analysis or Differential Geometry.…

I've tried reading Rudin a couple times, but it's a bit of a slog. There are easier analysis texts. Abbott's Understanding Analysis is good, though a bit basic. I'm currently reading Terence Tao's Analysis I, which is very good if you're in the right frame of mind for it. The first 150 pages are spent building the real numbers from scratch, starting with set theory and the Peano axioms. You successively construct the naturals, the integers, the rationals, and then the reals (as equivalence classes of Cauchy sequences of rationals). It's fun to see how the sausage is made, but I can also admit that when I was just starting out in math I might have found this book unbearably tedious.