Stone-Weierstrass theorem

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

1.
$\\forall x,y\\in X,x\eq y,\\exists f\\in\\mathcal{A}:f(x)\eq f(y)$
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.

单词	StoneWeierstrassTheorem
释义	Stone-Weierstrass theorem Let $X$ be a compact space and let $C^{0}(X,\\mathbb{R})$ be the algebra of continuous real functionsdefined over $X$ . Let $\\mathcal{A}$ be a subalgebra of $C^{0}(X,\\mathbb{R})$ for which the followingconditions hold: 1. $\\forall x,y\\in X,x\eq y,\\exists f\\in\\mathcal{A}:f(x)\eq f(y)$ 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.
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