Definition 10.11.9 (Bounded Convergence Topology).label Let $E, F$ be TVSs over $K \in \RC$, $\fB \subset 2^{E}$ be the collection of bounded subsets of $E$, then the $\fB$-uniform topology on $L(E; F)$ is the topology of bounded convergence.

The space $L_{b}(E; F)$ denotes $L(E; F)$ equipped with the topology of bounded convergence.