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

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  StoneWeierstrass theorem 

Canonical name  StoneWeierstrassTheorem 
Date of creation  20130322 12:42:06 
Last modified on  20130322 12:42:06 
Owner  rspuzio (6075) 
Last modified by  rspuzio (6075) 
Numerical id  9 
Author  rspuzio (6075) 
Entry type  Theorem 
Classification  msc 46E15 