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.

$\forall x,y\in X,x\neq y,\exists f\in\mathcal{A}:f(x)\neq f(y)$
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
Canonical name	StoneWeierstrassTheorem
Date of creation	2013-03-22 12:42:06
Last modified on	2013-03-22 12:42:06
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	9
Author	rspuzio (6075)
Entry type	Theorem
Classification	msc 46E15