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Stone–Weierstrass theorem

A subalgebra of continuous functions on a compact space that separates points and contains constants is dense in the full algebra.

Stone–Weierstrass theorem

Stone–Weierstrass theorem: Let $K$ be a compact Hausdorff topological space and let $C(K,\mathbb{R})$ denote the real-valued continuous functions on $K$ . Let $A\subseteq C(K,\mathbb{R})$ be a subalgebra that contains the constant functions and separates points of $K$ . Then $A$ is dense in $C(K,\mathbb{R})$ with respect to the supremum norm : for every $f\in C(K,\mathbb{R})$ and every $\varepsilon>0$ there exists $g\in A$ such that

\sup_{x\in K}|f(x)-g(x)|<\varepsilon.

This theorem explains many uniform approximation results as density statements in the space of continuous functions ; for example, the Weierstrass approximation theorem arises from a suitable choice of $K$ and $A$ .