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# 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.

Type of Math Object:

Theorem

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Reference

## Mathematics Subject Classification

46E15*no label found*

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