# Stone-Weierstrass theorem

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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 compactum $X$ with the topology of uniform convergence, i.e. the topology generated by the norm

$$\|f\| = \max_{x\in X}\left|f(x)\right|, \quad f\in C(X),$$

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 $C_0$.