Proposition 8.2.3. Let $E$ be a TVS over $K \in \RC$, and $\rho: E \to [0, \infty)$ be a pseudonorm, then the following are equivalent:
$\rho \in UC(E; [0, \infty))$.
$\rho \in C(E; [0, \infty))$.
$\rho$ is continuous at $0$.
The topology on $E$ contains the topology induced by $\rho$.
Proof. $(4) \Rightarrow (1)$: By Definition 8.2.2, for each $r > 0$, $\rho^{-1}([0, r)) \in \cn_{E}(0)$. Thus for any $x, y \in E$, if $x - y \in \rho^{-1}([0, r))$, then $\abs{\rho(x) - \rho(y)}\le r$. Therefore $\rho \in UC(E; [0, \infty))$.$\square$