Show that every locally convex topological vector space is a seminormed space

Question

Show that every locally convex topological vector space is a seminormed space

Answer 1

Consider a weak topology for a normed space $X$, the weak topology is generated by $p_{f}(\cdot)=\left|\left<\cdot,f\right>\right|$ for a continuous linear functional $f\in X'$, so the weak topology is a locally convex space.

And it is a standard theorem saying that, for that weak topology to be induced by a metric, a necessary and sufficient condition is that $X$ being finite dimensional.