# Self-studying through an undergraduate math course. Need Tao-like textbooks!

I'm a physics undergraduate student who always enjoyed math, and briefly studied it at a university but for various reasons (laziness, youth) gave up and changed 'majors'. But I always wanted to go through an undergraduate math course in my own time, unconstrained by class, etc. Now that I've passed all my exams I was thinking of doing something over the summer.

I had a look at Terry Tao's free lecture notes from an analysis course he taught and I was absolutely shocked at how good they are. I love the verbosity and how he motivates every bit of information. From what I read, he wrote an Analysis textbook which I intend to get.

My question is, are there any other similar (in the sense of their exposition) textbooks for subjects such as Topology, Algebra (Linear and Abstract - from my brief studies I've come to believe that I'm an absolute algebra antitalent, but I'm hoping it's because I didn't have anything else than fairly dry lecture notes to study from, and let's be honest, I didn't study very much) and of course more advanced Analysis, Probability and Statistics?

I too am a physics undergrad(in progress) specializing in electronics. And I am also somewhat inclined towards mathematics. Here are my suggestions for self studying:

1. Abstract Algebra : Charles Pinter offers an amazing textbook which is also brilliantly capturing. You will soon start reading up other topics in AA. Then you can check out Artin's book and then I.N.Herstein's classic, Topics in Algebra.
2. Linear Algebra - Gilbert Strang's long standing edifice has become the standard textbook for Linalg in many universities. But Strang's book is not rigorous and scarcely contains any proofs. For a rigorous approach (but with a lot of motivation) look for Poole's "Linear Algebra". You could consider Paul Halmos' Finite Dimensional Vector Spaces as a bit of an advancement, although its a classic.
3. Topology: James Munkres is considered to be a Bible kind of book by many students. However, I started reading topology from the book by the Hungarian mathematician, Ákos Császár.
4. Analysis: Hmm. Walter Rudin's classic on Real Analysis is another standard text in universities, however I found many other books for supplement. Kolmogorov's book(is out of print, you can get a pdf copy)is interesting. You could take up Pure Mathematics by G.H.Hardy: the book is available on Gutenberg. Also, Tom Apostol's books are unmissable.
5. Probability and Statistics: W. Feller's probability texts, two volumes are my constant reference books on Prob. And Stat. They are very well written and require a little bit of patience.
6. Number Theory: Hardy and Wright, Ramanujan's lost notebooks, G.Andrews, D.M.Burton,Waclaw Sierpinski and many many more classics. On a general basis, George Polya's books: How to solve it?, Mathematical Reasoning and induction, etc. have been my choice and they still are on my bookshelf.

EDIT: Found this link, contains some good books that I have forgotten to mention, Books that every student "needs" to go through

• I personally don't think Rudin's book is good in either explanation or being systematical. – Eric Nov 14 '17 at 14:13

Here is the link for all of the classes he's taught. He's made lecture notes for some of them.

https://www.math.ucla.edu/~tao/

Here is his blog.

https://terrytao.wordpress.com/

Have a look throgh this list (and others) at IMPA: https://impa.br/publicacoes/colecao-matematica-universitaria/ A lot of very good and accessible books, often better than everything I can find in english. Mathematical Portuguese is fairly accessible. Especially the books (all of them) by Elon Lages Lima, but try them all!