Let $ V $ be a vector space over a field $ k$. Give $ k $ the discrete topology and give $ V^\ast $ the coarsest topology for which the maps $ j_v: V^\ast \to k$ for all $ v \in V $, defined by $ j_v(u^\ast) = u^\ast(v) $, are continuous. How do I show that the closed subspaces of $ V^\ast $ are precisely the annihilators $ W^\bot $ of subspaces $ W $ of $ V$?
I am able to show that $ W^\bot $ are closed subspaces of $ V^\ast $, since $ W^\bot = \bigcap_{w \in W} j_w^{-1} (0) $, but I am not able to show the converse.
If $F \subseteq V^\ast$ is a closed subspace, define
$F^\perp = \{v \in V: \forall f \in F: f(v)=0\}=\bigcap_{f \in F} f^{-1}[\{0\}]$
and note that is a linear subspace of $V$, as an intersection of subspaces.
By definition $F \subseteq (F^\perp)^\perp$ and use weak$^\ast$ closedness of $F$ to see the reverse inclusion.