I'm an english major with a vivid interest in mathematics,I've read and enjoyed What Is Mathematics? by Courant and Robbins (does this count as some background?),and I've decided to begin a serious study of mathematics, I've looked up many books and judged them by the reviews on ams ,maa and zbmath,and narrowed it down to the following:

Calculus/Analysis: Mathematical Analysis I,II By Zorich(what are the exact prerequisites for this one?) ,Problems in Mathematical Analysis by Boris Demidovich .

Complex analysis: Visual complex analysis by Needham.

Abstract/linear algebra: A Course in Algebra by E. B. Vinberg,Algebra by Michael Artin.

Geometry: Modern Geometric Structures And Fields by Novikov and taimanov.

Topology:O.Ya. Viro, O.A. Ivanov, Kharlamov and Netsvetaev, Elementary Topology: Textbook in Problems.

Differential Equations: the books by V. I. Arnold.

Methods in physics:Mathematical Methods of Classical Mechanics by V. I. Arnold ,A Course in Mathematics for Students of Physics by Bamberg and Sternberg.

Do you have any comments on the choice of books ? any better suggestions? any suggestions for other subjects ?

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    $\begingroup$ First go through one book from beginning to end. As much as you can. Then it might create a new interest. Focus on that book. And repeat this cycle. $\endgroup$ – Nicolas Bourbaki Aug 11 '14 at 22:29
  • $\begingroup$ You might have difficulty with some of the definitions and notation. Much of this is covered in a typical calculus class. Perhaps having a basic calculus text next to you as you read the Calculus/Analysis text you have chosen might be a good idea. You can find them at a used book store pretty easily. $\endgroup$ – Paul Sundheim Aug 11 '14 at 22:32
  • $\begingroup$ A good way to start is Proofs and fundamentals, by Ethan D. Bloch. $\endgroup$ – Miguelgondu Aug 11 '14 at 23:04
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    $\begingroup$ Number theory can be a very fun way to get started in math. You might want to check out some of the popular introductory number theory books. $\endgroup$ – littleO Aug 11 '14 at 23:23
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    $\begingroup$ You might be rushing things a bit. If you haven't done mathematics for a while, it might be a good idea to first review high school algebra and trigonometry -- really master it if you haven't already. Then try first-year algebra and calculus (for beginners). My guess is you are probably not ready for an intensive proof-oriented course in analysis or abstract algebra. You really have to master the computational aspects of mathematics as a base to draw on before you start to do complicated proofs. $\endgroup$ – Dan Christensen Aug 12 '14 at 4:09

A good book for a first look at analysis is Bartle's real analysis. A very good introductory linear algebra text is Linear Algebra Done Right, by Axler. Another good linear algebra text is finite dimensional vector spaces by Halmos. For basic topology, nothing beats Munkres. Another good abstract algebra text is Basic Algebra 1 by Jacobson; another is Hungerford. You should also learn some number theory. For number theory, I'd say Davenport's Elementry number theory or Strayers.

  • $\begingroup$ Thanks! do they contain some physical applications? $\endgroup$ – madmorty Aug 11 '14 at 23:33
  • $\begingroup$ Not really. They might have some, but it would be minor. Linear algebra books usually talk about some applications. Abstract algebra books do sometimes as well. But, you can't really look too much at applications in an introduction. You'll only be able to look at the most elementary applications $\endgroup$ – RougeSegwayUser Aug 11 '14 at 23:35
  • $\begingroup$ Do you any comment on zorich's book on analysis? $\endgroup$ – madmorty Aug 11 '14 at 23:48
  • $\begingroup$ No I don't, sorry. I have never even heard of that book. Honestly, just look at a few different analysis texts online (there are ways to find them), and select one that you find easy to read. $\endgroup$ – RougeSegwayUser Aug 12 '14 at 21:07

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