Theoretical Books on Mathematics

[Kalvino]

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.

 

[MidgetDwarf]

I wouldn't say Simmons is a rigorous book. Yes, it is a good book, however, it is very hand wavy.

 

[HallsofIvy]

I am surprised that you do not have Walter Rudin's real analysis book listed.

 

[MidgetDwarf]

introduction to ordinary differential equations by coddington , i would consider theoretical at the elementary level. Everything is proved, starts with complex numbers 1st page!

 

[bacte2013]

How about Tom Apostol's Mathematical Analysis and Paul Halmos' Finite Dimensional Vector Space"?