Let $V$ be a vector space with inner product $(\phi,\psi)$ antilinear in the second argument. Let $\psi$ be an antilinear functional on $V$.

What are the precise (necessary and sufficient) conditions that must be imposed on $\psi$ that will guarantee the existence of a sequence or net of vectors $\psi_k\in V$ such that $$\psi(\phi)=\lim_{k\to\infty} (\psi_k,\phi)?$$