Lemma 11.1.9.label Let $E$ be a TVS over $K \in \RC$ and $[\cdot]: E \to [0, \infty)$ be a seminorm on $E$, then the following are equivalent:
- (1)
$[\cdot]$ is uniformly continuous.
- (2)
$[\cdot]$ is continuous.
- (3)
$[\cdot]$ is continuous at $0$.
- (4)
$\bracs{x \in E| [x] < 1}\in \cn_{E}(0)$.
- (5)
$\bracs{x \in E| [x] \le 1}\in \cn_{E}(0)$.
Proof. $(5) \Rightarrow (1)$: Let $x, y \in E$ and $r > 0$. If
\[x - y \in \bracs{x \in E|[x] \le r}= r\bracs{x \in E|[x] \le 1}\in \cn_{E}(0)\]
then $[x - y] \le r$.$\square$
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