In this article we investigate universal approximation theorems.

Stone-Weierstrass Theorem

The Stone’s version of universal approximation theorems consider an algebra of functions.

Suppose $X$ is a compact Hausdorff space and $A$ is a subalgebra of $C(X, \RR)$ which contains a non-zero constant function. Then $A$ is dense in $C(X, \RR)$ if and only if it separates points.