Proposition 8.11.7. Let $E, F$ be TVSs over $K \in \RC$ and $\net{T}\subset L(E; F)$ and $T \in L_{s}(E; F)$. If
There exists a dense subset $S \subset E$ such that $T_{\alpha} x \to Tx$ strongly for all $x \in S$.
$\bracs{T_\alpha|\alpha \in A}$ is uniformly equicontinuous.
then $T_{\alpha} \to T$ in $L_{s}(E; F)$.
Proof. Let $x \in E$, $U \in \cn_{F}(Tx)$, and $V \in \cn_{F}(Tx)$ be balanced such that $V + V + V \subset U$. By (b), there exists a balanced neighbourhood $W \in \cn_{E}(0)$ such that $T(W) \cup \bigcup_{\alpha \in A}T_{\alpha}(W) \subset V$. By (a), there exists $y \in S \cap (x + W)$ and $\alpha_{0} \in A$ such that for all $\alpha \ge \alpha_{0}$, $T_{\alpha} y - Ty \in V$. In which case, for any $\alpha \ge \alpha_{0}$,
\[T_{\alpha} x - Tx = \underbrace{T_\alpha x - T_\alpha y}_{\in V}+ \underbrace{T_\alpha y - Ty}_{\in V}+ \underbrace{Ty - Tx}_{\in V}\in U\]
$\square$