Proposition 4.2.2.label Let $E,F$ be a locally convex spaces and $T\in hom(E;F)$ then the following are equivalent:

1. (1)

   $T$ is uniformly continuous

2. (2)

   $T$ is continuous.

3. (3)

   $T$ is continuous at $0$.

4. (4)

   For every continuous seminorm $\braks{\cd}_{F}$ on $F$, there exists a continuous seminorm $\braks{\cd}_{E}$ on $E$ such that $\braks{Tx}_{F}\leq \braks{x }_{E }$ for all $x\in E$.