Proposition 9.2.1. Let $E, F$ be locally convex spaces and $T \in \hom(E; F)$, then the following are equivalent:
$T$ is uniformly continuous.
$T$ is continuous.
$T$ is continuous at $0$.
For every continuous seminorm $[\cdot]_{F}$ on $F$, there exists a continuous seminorm $[\cdot]_{E}$ on $E$ such that $[Tx]_{F} \le [x]_{E}$ for all $x \in E$.
Proof. $(1) \Leftrightarrow (2) \Leftrightarrow (3)$: By Definition 8.5.1.
$(2) \Rightarrow (4)$: $x \mapsto [Tx]_{F}$ is a continuous seminorm on $E$.
$(4) \Rightarrow (3)$: Let $U \in \cn_{F}(0)$ be convex, circled, and radial, then its gauge $[\cdot]_{U}$ is a continuous seminorm on $F$ by Definition 9.1.10. Thus there exists a continuous seminorm $[\cdot]_{E}$ such that $[Tx]_{U} \le [x]_{E}$. In which case, $V = \bracs{x \in E| [x]_E < 1}\in \cn_{E}(0)$ with $T(V) \subset U$. Therefore $T$ is continuous at $0$, and continuous by Definition 8.5.1.$\square$