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I am Quasar. A quick two-liner about my background - Computer science undergraduate, with a standard curriulum(single & multi-variable calculus, linear algebra, elementary probability and statistics) worked as programmer for five years before switching fields and now a risk analyst at an investment bank.

I like to self-learn. I enjoy solving probability problems(from Joe Blitzstein's book, William Feller is an awesome read!), or reading about elementary analysis; it makes me tick. I have about 3-4 hours of study time in the evenings.

I am enrolling myself in for a two-year graduate level course(MSc. in mathematics) through distance mode at a local university here. The course material gets delivered to your home, and I would be required to self-learn.

I know, there are no shortcuts. I am dilligent, energetic and determined to take serious efforts, learn as much as possible, get to a point, where I can write mathematical proofs myself, learn new areas like measure theory, functional analysis, solve ODEs and PDEs and gain mathematical maturity.

Here's the course curriculum:

First year

  • 110 Modern Algebra
  • 120 Real Analysis
  • 130 Differential geometry and differential equations
  • 140 Analytical mechanics and Tensor analysis

Second year

  • 210 Complex Analysis
  • 220 Set Topology and Functional Analysis
  • 230 Graph Theory
  • 240 Mathematical Statistics

I know that, topics like real analysis are introduced to students of pure mathematics at an undergraduate level. I would like to therefore, specifically ask, given that I am not a pure math undergrad, what are some of the best books I can buy to self-learn these subject areas? Also, how do I know, that I've mastered a topic for sure?

It would be real nice, if you could suggest two books on each topic - one which provides great intuition and another which is more rigorous.

Thanks a tonne!

Quasar

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    $\begingroup$ Topology by Munkres is an excellent book (particularly the first half is pretty relevant to your course) $\endgroup$ – Bernard W May 16 '17 at 4:08
  • $\begingroup$ @Bernard8, what about real analysis? I have a copy of Bartle's book. $\endgroup$ – Quasar May 16 '17 at 4:26
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    $\begingroup$ I'm curious which institution you're studying with (at?). You've listed a very wide range of topics to study at a graduate level. I've set up mathim.com/quasarchat if you'd rather chat there. $\endgroup$ – mephistolotl May 16 '17 at 4:28
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    $\begingroup$ Dummit and Foote's Abstract Algebra, 3e is pretty much the industry standard when it comes to modern algebra. If you have no background to real analysis, Goldberg's Methods of Real Analysis, 2e can ease you into the subject. However, the textbook is out of print and solutions are somewhat difficult to come by unless you search on Google for what might seem like hours. Rudin is the industry standard in a lot of regards, but it's a very terse book. $\endgroup$ – Decaf-Math May 16 '17 at 4:29
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    $\begingroup$ I think this reading list is good for people starting graduate school in math: math.columbia.edu/programs-math/graduate-program/… Basically, read Artin in algebra, Rudin in analysis and Ahlfors in complex analysis. After that perhaps start with Lang's Real and Functional Analysis. That will teach you enough topology. If these books are too difficult for you, then ask again for books that cover similar material but are a bit easier (e.g. Apostol in analysis). $\endgroup$ – user49640 May 19 '17 at 18:12

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