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vanish at infinity
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(Definition)
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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 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 $C_0(X)$
- When $X$ is compact, all functions $X \longrightarrow \mathbb{C}$ vanish at infinity. Hence, $C_0(X) = C(X)$
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Cross-references: compact, field, complex, algebra, continuous functions, point, continuous at, extension, one-point compactification, compact set, function, locally compact
There are 10 references to this entry.
This is version 2 of vanish at infinity, born on 2008-02-22, modified 2008-02-22.
Object id is 10321, canonical name is VanishAtInfinity.
Accessed 2090 times total.
Classification:
| AMS MSC: | 54C35 (General topology :: Maps and general types of spaces defined by maps :: Function spaces) | | | 54D45 (General topology :: Fairly general properties :: Local compactness, $\sigma$-compactness) |
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Pending Errata and Addenda
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