TITLE: On the Riesz representation theorem
QUESTION [5 upvotes]: 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.

REPLY [9 votes]: 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.