vanish at infinity

Let $X$ be a locally compact space. A function $f:X\\longrightarrow\\mathbb{C}$ is said to vanish at infinity if, for every $\\epsilon>0$ , there is a compact set $K\\subseteq X$ such that $|f(x)|<\\epsilon$ for every $x\\in X-K$ , where $\\|\\cdot\\|$ denotes the standard norm (http://planetmath.org/Norm2) on $\\mathbb{C}$ .

If $X$ is non-compact, let $X\\cup\\{\\infty\\}$ be the one-point compactification of $X$ . The above definition can be rephrased as: The extension of $f$ to $X\\cup\\{\\infty\\}$ satisfying $f(\\infty)=0$ is continuous at the point $\\infty$ .

The set of continuous functions $X\\longrightarrow\\mathbb{C}$ that vanish at infinity is an algebra over the complex field and is usually denoted by $C_{0}(X)$ .

0.0.1 Remarks

•
When $X$ is compact, all functions $X\\longrightarrow\\mathbb{C}$ vanish at infinity. Hence, $C_{0}(X)=C(X)$ .

单词	VanishAtInfinity
释义	vanish at infinity Let $X$ be a locally compact space. A function $f:X\\longrightarrow\\mathbb{C}$ is said to vanish at infinity if, for every $\\epsilon>0$ , there is a compact set $K\\subseteq X$ such that $\|f(x)\|<\\epsilon$ for every $x\\in X-K$ , where $\\\|\\cdot\\\|$ denotes the standard norm (http://planetmath.org/Norm2) on $\\mathbb{C}$ . If $X$ is non-compact, let $X\\cup\\{\\infty\\}$ be the one-point compactification of $X$ . The above definition can be rephrased as: The extension of $f$ to $X\\cup\\{\\infty\\}$ satisfying $f(\\infty)=0$ is continuous at the point $\\infty$ . The set of continuous functions $X\\longrightarrow\\mathbb{C}$ that vanish at infinity is an algebra over the complex field and is usually denoted by $C_{0}(X)$ . 0.0.1 Remarks • When $X$ is compact, all functions $X\\longrightarrow\\mathbb{C}$ vanish at infinity. Hence, $C_{0}(X)=C(X)$ .
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