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I am currently majoring in pure mathematics, and would like to purchase textbooks that would not only assist with content covered in the courses, but will serve as a great reference throughout my studies. Furthermore, if you believe that I will only ever use the textbook for the semester in which I am studying the subject, then please do not recommend it. Here are my second year mathematics courses:

  • Several Variable Calculus
  • Linear Algebra
  • Theory of Statistics
  • Mathematical Computing
  • Fundamental Analysis and Abstract Algebra
  • Theory and Application of Differential Equations
  • Complex Analysis
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    $\begingroup$ De Groot and Schervish is not bad for theory of statistics. $\endgroup$ – Michael Hardy Oct 3 '14 at 2:18
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    $\begingroup$ See math.stackexchange.com/questions/674257/…, especially math.stackexchange.com/a/674273/589. $\endgroup$ – lhf Oct 3 '14 at 2:29
  • $\begingroup$ Cheers, that's a ton of fantastic information for me to look through! @MichaelHardy: Cheers, after a quick search it appears to be a solid textbook. However, I don't know how useful it will be to me as a pure major. $\endgroup$ – eloiprime Oct 3 '14 at 2:37
  • $\begingroup$ @eloiPrime : If you want to learn its material, you'll find it useful. $\endgroup$ – Michael Hardy Oct 3 '14 at 16:34
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Here are some books which I think can be classified as 'reference books.' Some are less rigorous than the material you'll learn in advanced pure math courses, but will always be helpful to you. I say this as an opinion, as they have been helpful to me throughout my university career.

I didn't cover all of the areas you listed but hopefully this helps.

Abstract Algebra - Dummit and Foote, Abstract Algebra. This book is humungous and contains pretty much everything you'll need in undergraduate algebra.

Calculus - For single variable stuff I say Spivak's Calculus. For multivariable, I still look back at James Stewart's Multivariable Calculus. Although this isn't a pure math book, it contains all of the main theorems, and more, you'll learn in 2nd year. It also contains tons of examples. When you learn differential geometry you'll go back and look at these theorems in a different light. A more rigorous multivariable calculus book that is worth storing on your shelf is Spivak's Calculus on Manifolds.

Real Analysis - I think J. 
Marsden
 and
 M.
Hoffman,
 Elementary
 Classical
 Analysis is a great reference.

Linear Algebra - Friedberg, Insel, Spence, Linear Algebra. I still use all the time.

Complex Analysis - L.
Alhlfors,
 Complex 
Analysis contains all the fundamentals.

Topology - Munkres, Topology. I think everyone who has studied topology knows this book. A definite reference book to have.

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    $\begingroup$ One of the questions in my calculus problem set was taken from Spivak's Calculus! It was a doozy of an integral. Cheers for taking the time to give such a detailed answer! $\endgroup$ – eloiprime Oct 3 '14 at 2:41
  • $\begingroup$ I agree with most of these, but I think Stewart's books, or at least the ones I've seen, are less than top-notch. $\endgroup$ – Robin Goodfellow Oct 3 '14 at 3:17
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You don't need to buy them(having a tablet would help), you know what to do. Fridberg and Insell's Linear Algebra and Linear Algebra by Hoffman and Kunze are both golden linear algebra books. Worth to have a look at is 'baby Rudin'(complex analysis), An intro to Analysis by Wade(formal mutivariable calculus).

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