> [!definition] > > Let $X$ be a [[Topological Space|topological space]] and $f \in C(X)$ be a real/complex-valued [[Space of Continuous Functions|continuous function]]. $f$ **vanishes at infinity** if for all $\varepsilon > 0$, $\bracs{x: |f(x)| \ge \varepsilon}$ is [[Compactness|compact]]. > [!definition] > > Let $X$ be a topological space, then > $ > C_0 = \bracs{f \in C(X): f \text{ vanishes at infinity}} > $ > is the collection of all functions that vanish at infinity. > [!theorem] > > All functions that vanish at infinity are [[Space of Bounded Continuous Functions|bounded]].