I am currently compiling a comprehensive set of documents for personal use that will act as reference material (similar to programming documentation) for all the major fields of Mathematics.

Most notably:

  • Statistics
  • Combinatorics
  • Calculus (Differential/Integral/Multivariable/Vector/Tensor)
  • Linear algebra

Some others:

  • group theory
  • set theory
  • topology

Since I believe it would be useful to have at hand such reference material for future projects and works. (Rather than dig through mountains of papers to confirm esoteric/minute properties)

However, many of the textbooks I am currently are using are long, thick, and rather than a report-like structure, is much more verbose in unnecessary content and lacks a comprehensive table/list of desired properties of varying theorems/transforms/etc.

So I was wondering if there are any recommendations of texts for me to use as a foundation for my own set of documents.

Preferably short due to the number of topics I wish to cover in available time. All difficulty level welcome from introductory to advanced (preferably somewhere in the middle).

I am a first year graduate student who indeed just wants to freshen up his "linear regression and inferentials"

  • $\begingroup$ Some universities give that kind of list, but first you should specify the expected level (middle is a bit vague), and in what area you will specialize, as it could change the answer. Are you a student? There is no perfect mathematical library suitable to everybody, especially if you want to keep it small. As is, I fear the question is too broad. For instance, statistics has so many subfields there could be an entire list devoted to them. $\endgroup$ – Jean-Claude Arbaut Mar 28 '16 at 11:59
  • $\begingroup$ @Jean-ClaudeArbaut by middle - I mean covering all of the contents of an introductory course but for someone who has already done it. More as a review. Also like the post says (all levels are welcome) $\endgroup$ – AlanSTACK Mar 28 '16 at 12:13
  • $\begingroup$ My problem is: I could recommend Brockwell & Davis' "Time Series: Theory and Methods" or Agresti's "Categorical Data Analysis", but if you are not going to specialize in statistics, I doubt it would be of any use to you. And all "introductory" courses are likely to not go beyond inferential statistics and linear regression, which will be far from enough if you are really into statistics. Not easy to answer such a question. Do you just want a list to prepare, say, your master's degree? But even it that case, there is some specialization to expect. $\endgroup$ – Jean-Claude Arbaut Mar 28 '16 at 12:19
  • $\begingroup$ @Jean-ClaudeArbaut I am a first year graduate student who indeed just wants to freshen up his "linear regression and inferentials" $\endgroup$ – AlanSTACK Mar 28 '16 at 12:43
  • $\begingroup$ Ah, you should have started your question with this! :) $\endgroup$ – Jean-Claude Arbaut Mar 28 '16 at 12:45

I give the Amazon links not to push you to buy (they are often too expensive, and if you can get them at your university library, go ahead!), but to have a look at tables of contents and reviews.

Statistics: Schervish, "Theory of Statistics". Probability would probably(!) be useful: Borovkov, "Probability Theory".

Forget calculus and learn analysis, once and for all :)

Real analysis: J. Yeh "Real Analysis : Theory of Measure and Integration" (covers Lebesgue's integral and a good deal of functional analysis).

Also in analysis: the two Rudin books, "Principles of Mathematical Analysis and "Real and Complex Analysis"

Groups and linear algebra: either Serge Lang "Algebra" or Dummit & Foote "Abstract Algebra". They may be too abstract (you will find Galois theory, modules, etc. in them), so if you are only looking after matrix theory, I could suggest Axler "Linear Algebra Done Right" (I don't own this one though, but I believe it's quite appreciated). If you need numerical linear algebra: Golub & Van Loan "Matrix Computations", Demmel "Applied Numerical Linear Algebra" or Trefethen & Bau "Numerical Linear Algebra"

Topology: Munkres, "Topology", but many texts are worth a look, even if they are not as known. I like Newman "Elements of the Topology of Plane Sets of Points" for instance, and Steen & Seebach "Counterexamples in Topology" is a jewel, to really understand the limits of many theorems.

You didn't mention complex analysis, numerical analysis, differential geometry, and many others, so I'll let you update the question if you feel the need.

  • $\begingroup$ Is calculus that bad compared to analysis? $\endgroup$ – AlanSTACK Mar 28 '16 at 14:59
  • 1
    $\begingroup$ @Alan No, it's a tongue-in-cheek comment: calculus is often said to be a simplified and less rigorous version of analysis. $\endgroup$ – Jean-Claude Arbaut Mar 28 '16 at 15:04

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