vanish at infinity

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

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

The set of continuous functions $X⟶ℂ$ 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 $X⟶ℂ$ vanish at infinity. Hence, $C_0(X)=C(X)$.