Weierstrass’ Approximation Theorem shows any continuous real-valued function on a closed bounded interval can be arbitrarily and uniformly approximated by polynomials; that is, there is a sequence of polynomials which uniformly converges to the function.

The theorem was considerably generalized by Stone to the following: let X be a compact Hausdorff space and let C(X) denote the algebra of continuous real-valued functions on X with the uniform topology; then any subalgebra A of C(X) is dense in C(X) if for any distinct points x,y in X there exists f in A such that f(x) ≠ f(y).