Let $V$ be a vector space with inner product $(\phi,\psi)$ antilinear in the second argument - not necessarily a Hilbert space. Let $\Phi$ be an antilinear functional on $V$.

What are the precise (necessary and sufficient) conditions that must be imposed on $\Phi$ that will guarantee the existence of a sequence or net of vectors $\phi_k\in V$ such that $$\Phi(\psi)=\lim_{k} (\phi_k,\psi) ~~~\forall~ \psi\in V?$$ The answer is given by the Riesz representation theorem when $V$ is a Hilbert space, but I am interested in the general case.

any$\Phi$ (because you can find a $\phi$ that works for any finite set of $\psi$s). On the other hand, if you restrict to sequences, $\Phi$ does have to be bounded if $V$ is complete, but this is not obvious (it follows from the Banach-Steinhaus theorem). $\endgroup$