View source for Stone-Weierstrass theorem

A broad generalization of the classical [[Weierstrass theorem|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 $C_0$.



====Comments====
The expository article [[#References|[a4]]] is recommended in particular.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.W. Cheney,   "Introduction to approximation theory" , Chelsea, reprint  (1982)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Schönhage,   "Approximationstheorie" , de Gruyter  (1971)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M.H. Stone,   "The generalized Weierstrass approximation theorem"  ''Math. Mag.'' , '''21'''  (1948)  pp. 167–183; 237–254</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M.H. Stone,   "A generalized Weierstrass approximation theorem"  R.C. Buck (ed.) , ''Studies in Modern Analysis'' , '''1''' , Math. Assoc. Amer.  (1962)  pp. 30–87</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M.H. Stone,   "Applications of the theory of Boolean rings to general topology"  ''Trans. Amer. Math. Soc.'' , '''41'''  (1937)  pp. 375–481</TD></TR></table>