Stone-Weierstrass theorem for locally compact spaces
StoneWeierstrassTheoremForLocallyCompactSpaces
2009-01-01 21:54:29
2009-01-02 14:24:57
Definition
Stone-Weierstrass theorem
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
The following results generalize the Stone-Weierstrass theorem (and its \PMlinkname{complex version}{StoneWeierstrassTheoremComplexVersion}) for locally compact spaces. The cost of this generalization is that one no longer deals with all continuous functions, but only those that vanish at infinity.
\subsection*{Real version}
{\bf Theorem -} Let $ X$ be a locally compact space and $C_0(X, \mathbb{R})$ the algebra of continuous functions $X \to \mathbb{R}$ that \PMlinkname{vanish at infinity}{ VanishAtInfinity}, endowed with the sup norm $\Vert \cdot \Vert _{\infty}$. Let $\mathcal{A}$ be a subalgebra of $C_0(X; \mathbb{R})$ for which the following conditions hold:
\begin{enumerate}
\item $\forall x, y \in X, x \ne y, \exists f \in \mathcal{A} : f(x) \neq f(y)\;$, i.e. $ \mathcal{A}$ separates points.
\item For each $x \in X$ there exists $f \in \mathcal{A}$ such that $f(x) \neq 0$.
\end{enumerate}
Then $\mathcal{A}$ is dense in $C_0(X; \mathbb{R})$.
\subsection*{Complex version}
{\bf Theorem -} Let $X$ be a locally compact space and $C_0(X)$ the algebra of continuous functions $X \to \mathbb{C}$ that vanish at infinity, endowed with the sup norm $\Vert \cdot \Vert _{\infty}$. Let $\mathcal{A}$ be a subalgebra of $C_0(X)$ for which the following conditions hold:
\begin{enumerate}
\item $\forall x, y \in X, x \ne y, \exists f \in \mathcal{A} : f(x) \neq f(y)\;$, i.e. $\mathcal{A}$ separates points.
\item For each $x \in X$ there exists $f \in \mathcal{A}$ such that $f(x) \neq 0$.
\item If $f \in \mathcal{A}$ then $\overline{f} \in \mathcal{A}\;$, i.e. $ \mathcal{A}$ is a self-adjoint subalgebra of $ C(X)$.
\end{enumerate}
Then $\mathcal{A}$ is dense in $C_0(X)$.