I hope this is in the right section, otherwise please move it. This might be a long post and I hope someone will give it a shot. It is about books but not really a comparison between them. I will give some background first, so you might skip to the end (- Books * -) if you are only interested in the title question. With graduate I will mean masters level and above although I am generally used to bachelor and master being undergraduate and graduate meaning PhD students. - Background - I am a masters student in engineering. Focus is generally on computation and deriving formulas (theory) but no really what mathematicians call proofs which is something I am not familiar with the slightest. Some of what we do and use in our courses includes: tensors, FEM, CFD, control theory to make the list very short. So I do know some (some might call it a lot) of applied mathematics. I would like to be able to study Functional Analysis, higher level PDE (well the analysis part of it) and some mathematical aspects on classical mechanics to name a few subjects. These are generally courses given by the mathematical institution belonging to their masters program. As such they are mathematically rigorous, at least more then I'm used to. A typical text book can be theorem, proof, trivial example, theorem, proof and so on. The exercises are mostly show and prove this and that. The prequisitions listed for these courses are Calculus and Linear Algebra which is something I'm quite accustomed to (the computational part of them at least) and passed with good grades. But I believe it's a little misleading. - Books Studied - The book used in our Calculus courses was: Calculus: A Complete Course, Robert A. Adams Whole book used with the Linear Algebra book and other lecture notes over three courses. Proofs of trigonometric stuff, induction and that sort was covered in those courses but it was three years ago and it was not in the "theorem, proof" styling. The book in Linear Algebra courses: Linear Algebra and its Applications, David C. Lay Whole book used over two courses with the Calculus book. Lecturers notes on Transforms (Fourier, Laplace, with applications to ODE, PDE). Lecture notes on tensors. ODE, PDE have been introduced a lot trough mechanic courses and sometimes had some rigour (solutions belong to this and that space with set notations, not really proofs and really brief). Statistics (and probability) course (starting with basics and ending with basic linear regressions). I feel that I'm not really prepared to take on, say, Functional Analysis as the professor is basically rambling up axioms, theorems and proofs on the black board. The book used is: Introduction to Hilbert Spaces with Applications, L. Debnath, P. Mikusinski. - Books to Buy - What I want to learn is mathematical rigour, logic, notation (set notation mostly, I don't know if a book on Set Theory is really what I want though) and just gain mathematical maturity. I guess a good starting point would be to have exercises on material I know but that asks to prove this and that. Although I would like to learn some new area along the rigour way to do mathematics. Right now I just don't see that a proof really proves anything and I'm even less likely to prove something my self. This has lead me to the following books. I want a book from both areas. I cannot stress this enough, much logical reasoning and all the standard notations for proofs are something I want in the book and of course figures to illustrate the theory are welcome. Calculus books by Spivak or Apostol (both volumes). Maybe Courant, but what or how many volumes? I'm leaning towards Apostol as it covers multivariable calculus although I don't care much for the Linear Algebra part (see why below). I have read most posts on PF about these two books and they both seam great. What I cant decide on is if there is any point in buying these. Might it be better to buy something like: Analysis, Steven R. Lay. I can't say I'm really into "Real Analysis" although I might want to take it on some day but for now I want to gain maturity and learn all about proofs and the used notations (sets, subsets, and all those notations you could think of). From what I can see (brief look in the library) neither Apostol nor Spivak use for example set notations in their proofs but that might belong more to Linear Algebra. Linear Algebra books by Friedberg or Hoffman and Kunze. I'm leaning towards Friedberg as I believe it's enough (The Hoffman and Kunze might be to pure for what I'm seeking). I dont know if it will cover any new material (not needed) but from what I have heard and if it is anything like Hoffman and Kunze (which I have glanced at) it is rigorous. - End Notes - So, was my Calculus And Linear Algebra books pure **** and should I refresh my knowledge with more rigorous books or move to some introduction to real analysis? Do you guys have any ideas on other areas to study, books to self-study, before taking on a Functional Analysis course with the mentioned book above or would it be enough to gain some maturity from some (in that case which) of the mentioned books on the buy list. I was thinking of reading trough: Book of Proof, at least the Fundamentals which should be enough on logic and sets. Well budget is not infinite so one (all volumes) in each area mentioned (or the Real Analysis mentioned). I want books that have at least solutions for half of the exercises. Not books for which answers are impossible to find. - No Need to Read - I don't want, nor do I have the intention, to do Calculus my whole life but as an engineer I feel that this is so much more useful then say number theory. And for PhD Studies later on it might be good to have maturity as I guess control theory articles can use their fair share of math. More Probability and Statistics is something I plan on taking later when I have developed some maturity.