Proposition 9.6.6. Let $E$ be a locally convex space over $K \in \RC$, then
For any subspace $M \subset E$, $x \in M \setminus E$, and continuous seminorm $\rho: E \to [0, \infty)$, there exists $\phi \in E^{*}$ such that
$\phi \le \rho$.
$\dpb{x, \phi}{E}= \inf_{y \in M}\rho(x + y)$.
For any $x \in E$ and continuous seminorm $\rho: E \to [0, \infty)$, there exists $\phi \in E^{*}$ with $\phi \le \rho$ and $\dpb{x, \phi}{E}= \rho(x)$.
If $E$ is Hausdorff, then for any $x, y \in E$, there exists $\phi \in E^{*}$ with $\dpb{x, \phi}{E}\ne \dpb{y, \phi}{E}$.
Proof. (1): Let $\rho_{M}: E \to [0, \infty)$ be the quotient of $\rho$ by $M$, then $\rho_{M} \le \rho$ is a continuous seminorm on $E$ by Definition 9.3.1. Let $\phi_{0}: Kx \to K$ be defined by $\lambda x \mapsto \lambda \rho_{M}(x)$. By the Hahn-Banach theorem, there exists $\phi \in \hom{E; K}$ such that $\dpb{x, \phi}{E}= \rho_{M}(x)$ and $\phi \le \rho_{M} \le \rho$.
(2): By (1) applied to $M = \bracs{0}$.
(3): By (2) applied to $x - y$.$\square$