Top Ten Favorite Theorems

When I was an undergraduate at Hastings College, the English department released a list of the 25 essential novels that everyone should read. One of the Mathematics instructors there decided that if the English department could insist that every well-educated person should have read these 25 books, then he could post a list of the essential mathematical theorems that every well educated person should know. This is not that list. This list consists partially of essential theorems that everyone should know, but also contains theorems that I have found personally useful, or that I find particularly pleasing in some way. In fact, one theorem is included merely because I like the name.

Fundamental Theorem of Algebra Fundamental Theorem of Calculus Banach-Steinhaus Theorem Arzela-Ascoli Theorem Cayley-Hamilton Theorem Kharitinov's Theorem Residue Theorem Stokes' Theorem Heine-Borel Theorem Bolzano-Weierstrass Theorem