Stone-Weierstrass theorem

A broad generalization of the classical Weierstrass theorem on the approximation of functions, due to M.H. Stone (1937). Let $C (X)$ be the ring of continuous functions on a compact $X$ with the topology of uniform convergence, i.e. the topology generated by the norm

∥ f ∥ = max_{x \in X} | f (x) |, f \in C (X),

and let $C_{0} \subseteq C (X)$ be a subring containing all constants and separating the points of $X$ , i.e. for any two different points $x_{1}, x_{2} \in X$ there exists a function $f \in C_{0}$ for which $f (x_{1}) \neq f (x_{2})$ . Then $[C_{0}] = C (X)$ , i.e. every continuous function on $X$ is the limit of a uniformly converging sequence of functions in $C_{0}$ .