Definition 5.21.1 (Vanish at Infinity).label Let $X$ be a topological space, $E$ be a TVS over $K \in \RC$, and $f \in C(X; E)$, then $f$ vanishes at infinity if for every $U \in \cn_{E}^{o}(0)$, $\bracs{f \not\in U}$ is compact.

The set $C_{0}(X; E)$ is the space of all functions that vanish at infinity, equipped with the uniform topology.