# Stone-Weierstrass theorem

Let $X$ be a compact space and let $C^{0}(X,\mathbb{R})$ be the algebra of continuous real functions defined over $X$. Let $\mathcal{A}$ be a subalgebra of $C^{0}(X,\mathbb{R})$ for which the following conditions hold:

1. 1.

$\forall x,y\in X,x\neq y,\exists f\in\mathcal{A}:f(x)\neq f(y)$

2. 2.

$1\in\mathcal{A}$

Then $\mathcal{A}$ is dense in $C^{0}(X,\mathbb{R})$.

This theorem is a generalization of the classical Weierstrass approximation theorem to general spaces.

Title Stone-Weierstrass theorem StoneWeierstrassTheorem 2013-03-22 12:42:06 2013-03-22 12:42:06 rspuzio (6075) rspuzio (6075) 9 rspuzio (6075) Theorem msc 46E15