Stone-Weierstrass Theorem

Theorem
Let $X$ be a compact topological space.

Let $\struct {\map C {X, \R}, \times, \norm \cdot_\infty}$ be the Banach algebra of real-valued continuous functions on $X$.

Let $\AA$ be a subalgebra of $\map C {X, \R}$.

Let $\AA$ be such that it separates points of $X$, that is:
 * for distinct $p, q \in X$, there exists $h_{p q} \in \AA$ such that $\map {h_{p q} } p \ne \map {h_{p q} } q$.

Suppose that $1 \in \AA$.

Then the closure $\overline \AA$ of $\AA$ is equal to $\map C {X, \R}$.

Also see

 * Weierstrass Approximation Theorem, of which the Stone-Weierstrass Theorem is a generalization.