Stone–Weierstrass theorem
Weierstrass showed polynomials approximate any continuous function on an interval. Stone asked the natural follow-up: what is it about polynomials that makes this work, and could other families of functions do the same? The Stone–Weierstrass theorem distills the answer into a few structural conditions, vastly generalizing the original.
Setting: let X be a compact space (for example a closed bounded subset of R^n) and C(X) the continuous real-valued functions on it. Suppose A is a subalgebra of C(X) — a collection closed under addition, multiplication, and scalar multiples. If A contains the constant functions and separates points (for any two distinct points there is a function in A taking different values on them), then A is dense in C(X) under the sup norm. Every continuous function is a uniform limit of members of A.
Polynomials on [a, b] satisfy these hypotheses (they form an algebra, contain constants, and the single function x already separates points), so the classical theorem is the special case. The two conditions are genuinely necessary: a family that cannot tell two points apart, or that is not closed under multiplication, can miss continuous functions. A complex-valued version needs the extra requirement that A be closed under complex conjugation.
On the circle, the trigonometric polynomials (finite combinations of sin nx and cos nx) form a conjugation-closed algebra that separates points and contains constants. Stone–Weierstrass then yields that every continuous periodic function is a uniform limit of trigonometric polynomials.
A trigonometric version, recovered as a special case.