Category:Stone-Weierstrass Theorem

This category contains pages concerning Stone-Weierstrass Theorem:

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 $\norm \cdot_\infty$.

This page or section has statements made on it that ought to be extracted and proved in a Theorem page. In particular: A definition page for Banach algebra $\map C {X, \R}$ and its verification? You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by creating any appropriate Theorem pages that may be needed. To discuss this page in more detail, feel free to use the talk page.

This article, or a section of it, needs explaining. In particular: By the definition of Banach algebra, $\map C {X, \R}$ should be a commutative ring. Can it be confirmed that this is indeed the case? You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

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 $\map C {X, \R}$.