Plenty of books at the graduate level - Spivak's Calculus and Rudin's Introduction to Functional Analysis should get you most of the way through mahmud's list of things to learn
After all these, you should be able to pick and choose for yourself. Enjoy!
I hear what you are saying. By "grad level" I only meant that you need to be very comfortable with doing proofs to make any progress. Clumsy phrasing on my part.
I am completely self taught in mathematics (and computer science).I have no real idea what is undergrad level and what is grad level in a real university. It took me a good long while before I tackled Spivak :-D. Apologies for any confusion caused!
Oh, it wasn't really saying anything; I thought you knew something I didn't. I got Spivak hearing that it would be good, and it is, I really like his writing style, but its going to take me a long while before I can tackle Spivak.
How did you get to the point you could tackle Calculus? I still don't know how to convince myself if my answers are correct or if I'm making assumptions that I shouldn't.
EDIT for Reply: Thanks for the link to Strang. I think I'll begin reading it this weekend. :)
"How did you get to the point you could tackle Calculus? "
After working through Strang's Calculus and learning how to tackle proof questions (strangely enough I got confident enough to attempt proof questions after working through a good chunk of "Concrete Mathematics" (Knuth)) I then found what was impenetrable before in Spivak looked attemptable, if not easy. It helps that the end result of most proof questions are known in advance and the problem is finding/writing down the proof itself.
Might be some confusion here between Spivak's "Calculus on Manifolds" and his book just called "Calculus." The former is a more advanced book that might be used for part of a beginning graduate course. I say there might be confusion mainly because _I_ got quite confused when someone mentioned "Calculus" and I assumed they were talking about "Calculus on Manifolds."
I know you asked the question to Mahmud, but here is what I think.
To start with, Undergrad texts (the best I've found) to prepare you for Machine Learning.
Calculus (Strang) available free http://ocw.mit.edu/ans7870/resources/Strang/strangtext.htm
Linear Algebra(Strang) not free.
Probability( by Bertsekas a new edition just came out. expensive but VERY good).
Information Theory (also useful in ML) David McKay. freely available at http://www.inference.phy.cam.ac.uk/mackay/itprnn/book.html
Convex Optimizaton by Boyd. (free http://www.stanford.edu/~boyd/cvxbook/). Videos available on the Stanford site. http://see.stanford.edu/see/courses.aspx
Plenty of books at the graduate level - Spivak's Calculus and Rudin's Introduction to Functional Analysis should get you most of the way through mahmud's list of things to learn
After all these, you should be able to pick and choose for yourself. Enjoy!