Stone-Weierstrass Theorem

Theorem

Compact Space

Let $T = \struct {X, \tau}$ be a compact topological space.

Let $\map C {X, \R}$ be the set of real-valued continuous functions on $T$.

Let $\times$ be the pointwise multiplication on $\map C {X, \R}$.

Let $\struct {\map C {X, \R}, \times}$ be the Banach algebra with respect to $\norm \cdot_\infty$.

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Let $\AA$ be a unital subalgebra of $\map C {X, \R}$.

Suppose that $\AA$ separates points of $X$, that is:

for distinct $p, q \in X$, there exists $h_{p q} \in \AA$ such that $\map {h_{p q} } p \ne \map {h_{p q} } q$.

Then the closure $\overline \AA$ of $\AA$ is equal to $\map C {X, \R}$.

Locally Compact Hausdorff Space

Let $X$ be a locally compact Hausdorff space.

Let $\struct {\map {\CC_0} {X, \R}, \norm {\, \cdot \,} }$ be the Banach algebra of real-valued continuous functions vanishing at infinity on $X$.

Let $\AA$ be a subalgebra of $\map {\CC_0} {X, \R}$ such that:

$(1) \quad$ for each $x, y \in X$ with $x \ne y$ there exists $h_{x y} \in \AA$ such that $\map {h_{x y} } x \ne \map {h_{x y} } y$

$(2) \quad$ for each $x \in X$ there exists $f_x \in \AA$ such that $\map {f_x} x \ne 0$.

Then $\AA$ is everywhere dense in $\struct {\map {\CC_0} {X, \R}, \norm {\, \cdot \,} }$.

Source of Name

This entry was named for Marshall Harvey Stone and Karl Theodor Wilhelm Weierstrass.