Lemma 9.1.7. Let $E$ be a TVS over $K \in \RC$ and $[\cdot]: E \times E \to [0, \infty)$ be a seminorm on $E$, then the following are equivalent:
$[\cdot]$ is uniformly continuous.
$[\cdot]$ is continuous.
$[\cdot]$ is continuous at $0$.
Proof. $(3) \Rightarrow (1)$: Let $\eps > 0$, then there exists $V \in \cn(0)$ such that $[x] < \eps$ for all $x \in V$. In which case, for any $x, y \in E$ with $x - y \in V$, $\abs{[x] - [y]}\le [x - y] < \eps$. By Proposition 8.1.6, $[\cdot]$ is uniformly continuous.$\square$