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 Calculus: 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.