Proposition 9.3.3. Let $E$ be a bornologic space, $F$ be a locally convex space, and $T \in \hom(E; F)$, then the following are equivalent:
$T$ is continuous.
$T$ is bounded.
Proof. (1) $\Rightarrow$ (2): By Proposition 8.5.3.
(2) $\Rightarrow$ (1): Let $\rho: F \to [0, \infty)$ be a continuous seminorm, then $\rho \circ T$ is a seminorm on $E$ that is bounded on bounded sets. Since $E$ is bornologic, $\rho \circ T$ is continuous. Therefore $T$ is continuous by Proposition 9.2.1.$\square$