Let $X$ be an F-space (space with topology given by complete, translation-invariant metric). Let $E\subset X^*$, such that $$\rho(f):=\sup_{\varphi\in E} |\varphi(f)|<\infty.$$
Show that $\rho$ is a continuous seminorm.
The fact that $\rho$ is a seminorm is trtivial, but I don't have any idea how to prove the continuity. We have a metric $d$ from $X$. If there would be formula:$$d(x,y)=\rho(x-y)$$
we'd get contiunuity form: $$|\rho(f)-\rho(g)|=||\rho(f)|-|\rho(g)||\le |\rho(f-g)|<\delta,$$ so for $\varepsilon=\delta$ we'd have continuity. But we don't use anywhere the fact how the seminorm is defined (by supremum), so I don't think that this lead to solution (I've just written my observations).
Any hint to this problem?