Stone-Weierstrass theorem

The article on this topic in the Differential Geometry Wiki can be found at: Stone-Weierstrass theorem

Statement

Let $X$ be a compact Hausdorff space and $C(X,\mathbb {R} )$ denote the algebra of continuous functions from $X$ to $\mathbb {R}$. Endow $C(X,\mathbb {R} )$ with the topology of uniform convergence.

Suppose $A$ is a subalgebra of $C(X,\mathbb {R} )$ such that:

• $A$ is unital, in the sense that $A$ contains the constant function $1$
• $A$ separates points, in the sense that if $x\neq y\in X$, then there exists $f\in A$ such that $f(x)\neq f(y)$

Then $A$ is a dense subalgebra of $C(X,\mathbb {R} )$. In particular, if we assume $A$ is closed in $C(X,\mathbb {R} )$, we obtain that $A=C(X,\mathbb {R} )$.

Proof

The result is an application of the Weierstrass approximation theorem, and some clever arguments.