Kevin Shenoy

The only real self-study Real Analysis book!! This is THE BEST Real Analysis Book. I completed the first book today(and will begin with Analysis-II in some days) Literally no other book is so profound as this one. This book is just not about learning Real Analysis, but also about being a part of the process of making mathematics rigorous in your mind, thats the apt description I can think about! Suggestion for students who want to study Real Analysis properly: After being proficient in Calculus and (well 70 or above retainment in understanding the concepts etc.) go for a foundations book , like a very basic book on fundamentals of mathematics. I went for Naive Set Theory(Paul R. Halmos)(I didnt finish the whole book, but half of it.) Then I had started with 'A Foundation Course in Mathematics(By Ajit,Kumaresan, and Bhaba), and finished this whole book, then I went for Real Analysis ( and the best one in this is Analysis by Tenrence Tao, and this is not because I solely tried from this book, NO! I tried out Analysis by Rudin, Apostol, Hardy(course on pure mathematics), the abbot book, basic course in real Analysis (by s kumerasan), and even some more; none of these explained the concepts in a way in which a student can link the concepts linearly (most of the time, either the proofs are very difficult to grasp, or without a clear cut geometrical representation of the intuition behind the proof , one cant understand it, in case of Tao's book 95-98 percent of the time no such hurdles were faced and the way in which Prof. Tao explains things and gives questions on these concepts, let's one absolutely have a command onn that particular concept( but all of this is only true if you are really paying attention to all the things given in the book) Once you are done with both the books of Tao, go for Mathematical Analysis by Apostol and Principles of Mathematical Analysis by Rudin, only after you are done with a introductory to intermediate level book(the best one being of Tao) and then with Rudin and Apostol, can you really state that you 'know' Real Analysis. On a side note( if you can do real Analysis along with your course (in clg) its best, I didnt have such a case, because I started to do Real Analysis properly after entering M.Sc. Math, but nevertheless in case of the Tao Books self-study is possible (if you are good/some what good at calculus).