Understanding Analysis

Out of this list, the books I am familiar with, are great (Hilbert-Courant, Spivak, Korner's books). At the same time, even with extensive mathematical training, I haven't read them from start to finish. I wouldn't even like to say "read". For someone who's not used to mathematical reading, some of these books require careful study. That means generating examples to understand results (theorems), trying your own conjectures, proving things yourself etc. Over time, one becomes familiar with most/all the material in a book but the knowledge might have been acquired through various books (and courses) over time. Also, mathematics is a massive field. The first question would be what kinds of mathematics would you like to get better at. There are great books in analysis. If you are starting out with a solid calculus knowledge, try Abbott's Understanding Analysis [1] or Duren's Invitation to Classical Analysis [2]. For asymptotic methods in PDEs, try Bender and Orszag [3] which is a wonderful book. But again, this might not be your cup of tea at all and there are more abstract or formal books like Rudin's.…

Supposing you have at least high school mathematics under the belt: 1. Book of Proof by Hammack [1] 2. Tom Apostol Calculus Vol 1 (stop at Linear Algebra) 3. Linear Algebra by Insel, Spence & Friedberg 4. Understanding Analysis by Stephen Abbott 5. Tom Apostol Calculus Vol 2 These are books with excellent exposition, and solution manuals are available when needed. You are absolutely required to do all the exercises. To learn mathematics is to do mathematics. Two other excellent books are Spivak - Calculus, the problems in this book are very good, but much more difficult than Apostol. For more advanced analysis there's Walter Rudin's Principles of Mathematical Analysis (baby Rudin), which on its own is too difficult as a first exposure to analysis, but there's a cool professor that recorded an entire semester of baby Rudin lecture videos [2]. This is as close as it gets without being enrolled in a university. [1] https://www.people.vcu.edu/~rhammack/BookOfProof/ [2] https://www.youtube.com/watch?v=ab41LEw9oiI&list=PLun8-Z_lTk... (https://www.youtube.com/watch?v=ab41LEw9oiI&list=PLun8-Z_lTkC6qJF1sVh3_Hx7aL6FPd0IN) (the first video is a bit blurry, it gets better)

I couldn't disagree more. I studied Analysis in the Uni, and even in that environment Rudin is pretty bad. For a total newcomer that book will leave you completely helpless. Also, solutions are a must have, without them you are almsot totally lost. In their absence, it is OK to ask on StackExchange or #math on EFnet. First let's start with a few books to prep you for college-level maths: * https://www.amazon.com/How-Study-as-Mathematics-Major-ebook/... (https://www.amazon.com/How-Study-as-Mathematics-Major-ebook/dp/B00B9CIORY) * https://www.amazon.com/How-Read-Proofs-Introduction-Mathemat... (https://www.amazon.com/How-Read-Proofs-Introduction-Mathematical-ebook-dp-B00I8XF6TQ/dp/B00I8XF6TQ) ; or * https://www.amazon.com/Numbers-Proofs-Modular-Mathematics-Al... (https://www.amazon.com/Numbers-Proofs-Modular-Mathematics-Allenby/dp/0340676531) ; or * https://www.amazon.com/How-Prove-Structured-Daniel-Velleman-... (https://www.amazon.com/How-Prove-Structured-Daniel-Velleman-dp-1108439535/dp/1108439535) (I believe you can find solutions to the 2nd edition online) For Single-Variable 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 like Pugh too. All three are sitting on my bookshelf :). Coming from very little formal math training, and trying to self-study analysis, I would bounce from Rudin if I got stuck, go to Abbott for a confidence boost, and try out some Pugh for a different viewpoint. Pugh might have been just right in terms of rigor and accessibility.

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.

Echo the above. Pairing little Rudin with a more user-friendly book like Understanding Analysis by Abbott is a great way to learn the material if you get stuck.