Category:Stone-Weierstrass Theorem
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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 $\map C {X, \R}$.
Pages in category "Stone-Weierstrass Theorem"
The following 3 pages are in this category, out of 3 total.