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.



On the Riesz representation theorem
Such a net exists for any $\Phi$ (in fact, there is a canonical such net). First, note that if $F\subseteq V$ is a finite-dimensional subspace, then there is a unique $\phi\_F\in F$ such that $\Phi(\psi)=(\phi\_F,\psi)$ for all $\psi\in F$. The collection of such $F$ form a directed set under inclusion, and the net $(\phi\_F)$ will have the desired property.


