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[parent] vanish at infinity (Definition)

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 $ \Vert f(x)\Vert<\epsilon$ for every $ x \in X-K$, where $ \Vert\cdot\Vert$ 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)$.

Remarks

  • When $ X$ is compact, all functions $ X \longrightarrow \mathbb{C}$ vanish at infinity. Hence, $ C_0(X) = C(X)$.



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See Also: regular at infinity

Other names:  zero at infinity, vanishes at infinity
Also defines:  $C_0$

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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 7 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 629 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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