Stone-Weierstrass theorem

The article on this topic in the Topology 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 $x\ne y\in X$, then there exists $f\in A$ such that $f(x)\ne f(y)$

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 $A=C(X,\mathbb{R})$.

Applications

The Stone-Weierstrass theorem is used to prove that certain restricted spaces of functions, are dense in the space of all continuous functions, for some pairs of spaces.

For a complete list of applications, refer:

Category:Applications of Stone-Weierstrass theorem

Proof

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