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What are the classical mathematics textbooks of the 20th Century?

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Spivak and Rudin!
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Maybe Steen and Seebach's "Counterexamples in Topology"?
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Arnold's Mathematical Methods of Classical Mechanics was published in the 70s. Definitely a classic in the area of geometric mechanics.
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Kolmogorov and Fomin - analysis

Appel et Goursat - Cours d'Anaylse

Lang - Algebra

Siegel - complex function theory

Dedekind - Was Sind und Was Sollen die Zahlen (1901!)

Atiyah and MacDonald - commutative algebra

Strang - some linear algebra book

Serre - a course in arithmetic

Macclane categories for the working mathematicians
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Hartshorne's AG, unquestionably. Eisenbud's CA in my opinion (not from the 70s but the 20th century nonetheless), but more uncontroversially Zariski and Samuel's. Also Mumford's Red Book. Milnor and Stasheff's book on characteristic classes. Adams's Blue Book on stable and chromatic homotopy, and Ravenel's Green Book on the same (again, the latter was not around in the 70s but is still from the 20th century). Tbh any book that's commonly referred to by its color is probably a classic.

Oh, and EGA. Obviously.

EDIT: If you count physics (and books from the 80s), Shankar's book on quantum should be on the list. Likewise with Feynman's Lectures in Physics series and Taylor's Classical Mechanics. I would also argue for Purcell's EM text, although people seem to have shifted away from it lately in favor of generic "intro physics" bullshit like Giancoli.
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Van der Waerden "Modern Algebra" for sure.

Also Courant/Hilbert "Methods of mathematical physics".

MacLanes "Categories for the working mathematician" is from the 70s.
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Complex Analysis by Lars Ahlfors and Topology from a Differentiable Point of View by John Milnor should both count.
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Paul R. Halmos - *Finite dime sional vector spaces*

Hardy & Wright - *An introduction to the theory of numbers*

Coxeter - *Geometry revisited*

And my absolute favorite:

Polya & Szego - *Problems and theorems in analysis*
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Hermann Weyl- The Concept of a Riemann Surface
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Casella and Berger - Statistical Inference

I know you said math books, and I just listed a book on statistics, but thumbing through the pages you can very well see how this can be considered a math book lol

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