Definition 10.11.4 (Space of Bounded Linear Maps).label Let $E, F$ be TVSs over $K \in \RC$, $\mathfrak{S}\subset 2^{E}$ be an upward-directed system, and $k \in \nat$. The space $B_{\mathfrak{S}}^{k}(E; F)$ is the set of all $k$-linear maps $T: E^{k} \to F$ with $T(S^{k}) \in B(F)$ for all $S \in \mathfrak{S}$, equipped with the $\bracsn{S^k| S \in \mathfrak{S}}$-uniform topology.

Let $\fB \subset 2^{E}$ be the collection of all bounded subsets of $E$, then $B_{\mathfrak{S}}(E; F) = B(E; F)$ is the space of bounded linear maps from $E$ to $F$.