Proposition 10.12.2 ([III.3.3, SW99]).label Let $E, F$ be TVSs over $K \in \RC$, $\sigma \subset 2^{E}$ be an ideal, and $A \subset B_{\sigma}(E; F)$, then the following are equivalent:

1. (1)

   $A \subset B_{\sigma}(E; F)$ is bounded with respect to the $\sigma$-uniform topology.

2. (2)

   For each $V \in \cn_{F}(0)$, $\bigcap_{T \in A}T^{-1}(V)$ absorbs every $S \in \sigma$.

3. (3)

   For every $S \in \sigma$, $\bigcup_{T \in A}T(A)$ is bounded in $F$.