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 $C_0(X)$

Remarks

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