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 $$\Phi(\psi)=\lim_{k\to\infty} (\phi_k,\psi)?$$ The answer is given by the Riesz representation theorem when $V$ is a Hilbert space, but I am interested in the general case.

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.

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 $$\psi(\phi)=\lim_{k\to\infty} (\psi_k,\phi)?$$

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 $$\Phi(\psi)=\lim_{k\to\infty} (\phi_k,\psi)?$$ The answer is given by the Riesz representation theorem when $V$ is a Hilbert space, but I am interested in the general case.