If $V$ is a vector space over some field, we have a canonical map $\phi_V:V\to V^{**}$ from $V$ to its double dual. The map $\phi_V$ depends naturally on $V$, and its scalar multiples are the only natural transformations form the identity functor on the category of vector spaces to the double dual functor.
The map $\phi_V$ is injective for all $V$, and surjective exactly for those $V$ that are finite dimensional. This implies that my question is only interesting for infinite dimensional vector spaces.
Can one characterize the image of $\phi_V:V\to V^{**}$?
The image is certainly not an arbitrary subspace (whatever that might mean…) For example, the image is a closed subspace for all Banach norms on $V^{**}$ that come from a Banach norm in $V$.