Lemma 4.1.5 (Seminorm Properties).label Let $E$ be a TVS over $K\in \curl{\R,\com}$ and $\rho:E\to[0,\infty)$ be a seminorm on $E$, then the following are equivalent:
- (1)
$\rho$ is uniformly continuous
- (2)
$\rho$ is continuous
- (3)
$\rho$ is continuous at $0$
- (4)
$\curl{x\in E:\rho(x)<1}\in \cali{N}_{E}(0)$
Proof. It suffices to prove $(4)\implies (1)$. Let $x,y\in E$ and $r>0$. If $x-y\in \curl{x\in E:\rho(x)<r}=r\curl{x\in E:\rho(x)<1}\in \cali{N}_{E}(0)$ then $\rho(x-y)<r$$\square$