In a topological vector space $E$, given $S \subset E$, let $$ S^\perp = \{ \phi \in E^* \mid \forall x \in S,\,\phi(x) = 0 \}. $$ And for $F \subset E^*$, where $E^*$ is the dual space for $E$, let $$ F_\perp = \{ x \in E \mid \forall \phi \in F,\,\phi(x) = 0 \}. $$
In Pedersen's Analysis Now, paragraph 2.3.6, we have the following comment: (with the above notation)
It is immediate that $S \subset (S^\perp)_\perp$, and it follows from 2.3.5 that actually $S = (S^\perp)_\perp$ if $S$ is a closed subspace. By contrast, we cannot expect that $F = (F_\perp)^\perp$ for every norm closed subspace $F$ of $E^*$.
I would like a counter example! :-)
I mean, I'd like a Banach space $E$ and a norm closed subspace $F \subset E^*$ such that $$ F \subsetneq (F_\perp)^\perp. $$