# 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, $C_{0}(X)=C(X)$.

Title vanish at infinity VanishAtInfinity 2013-03-22 17:50:57 2013-03-22 17:50:57 asteroid (17536) asteroid (17536) 6 asteroid (17536) Definition msc 54D45 msc 54C35 zero at infinity vanishes at infinity RegularAtInfinity ApplicationsOfUrysohnsLemmaToLocallyCompactHausdorffSpaces $C_{0}$