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Foundations Theoretical Books on Mathematics

  1. Aug 1, 2015 #1
    What are some rigorous theoretical books on mathematics for each branch of it? I have devised a fantastic list of my own and would like to hear your sentiments too.

    Elementary Algebra:

    Gelfand's Algebra
    Gelfand's Functions & Graphs
    Burnside's Theory of Equations
    Euler's Analysis of the Infinite
    Bellman's Introduction to Inequalities
    Umbarger's Logarithms

    Elementary Geometry:

    Kiselev's Geometry
    Lang's Geometry
    Gelfand's Trigonometry
    Gelfand's Method of Coordinates
    Gutenmacher's Lines & Curves

    Overview: Serge Lang's Basic Mathematics


    Spivak's Calculus
    Apostol's Calculus
    Courant's Introduction to Calculus & Analysis
    Simmons' Calculus with Analytic Geometry
    Hubbard's Vector Calculus

    Linear Algebra:

    Lang's Introduction to Linear Algebra
    Axler's Linear Algebra Done Right
    Friedberg's Linear Algebra
    Hoffman-Kunze's Linear Algebra
    Roman's Advanced Linear Algebra

    Real Analysis:

    Binmore's Mathematical Analysis
    Pugh's Real Mathematical Analysis
    Folland's Real Analysis
    McDonald's A Course in Real Analysis

    You may make additions to my list or add more branches like Topology, Complex Analysis and Differential Geometry if you like, but remember; the books should focus on the "Why?" rather than the "How?" or in other words; should be highly theoretical. Books like Stewart's Calculus don't classify as being theoretical.
  2. jcsd
  3. Aug 1, 2015 #2
    I wouldn't say Simmons is a rigorous book. Yes, it is a good book, however, it is very hand wavy.
  4. Aug 1, 2015 #3


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    I am surprised that you do not have Walter Rudin's real analysis book listed.
  5. Aug 1, 2015 #4
    introduction to ordinary differential equations by coddington , i would consider theoretical at the elementary level. Everything is proved, starts with complex numbers 1st page!
    Last edited: Aug 1, 2015
  6. Aug 1, 2015 #5
    How about Tom Apostol's Mathematical Analysis and Paul Halmos' Finite Dimensional Vector Space"?
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