I have a follow-up question to On the Riesz representation theorem . Let $V$ be a subspace of a Hilbert space, and let $V^\times$ be the space of all antilinear functionals on $V$, equipped with the weak-* topology.

Let $\Psi$ be a weak-* continuous antilinear functional on $V^\times$. Is there always a vector $\psi\in V$ such that $$\Psi(\phi)=\phi(\psi)~~~\forall~ \phi\in V^\times?$$.