Proposition 9.7.7. Let $E$ be a locally convex space and $\rho: E \to [0, \infty)$ be a continuous seminorm, then $\rho: E_{w} \to [0, \infty)$ is lower semicontinuous and Borel measurable with respect to the weak topology.
Proof. Let $x \in E$, then there exists $\phi_{x} \in E^{*}$ such that $\dpn{x, \phi_x}{E}= \rho(x)$ and $|\phi_{x}| \le \rho$. Thus
\[\rho(x) = \sup_{y \in E}\dpn{x, \phi_y}{E}\]
is lower semicontinuous and Borel measurable by Proposition 4.22.2.$\square$