Let $X$ be a Banach space, and $X^*$ its dual. Which subspaces of $X^*$ are duals of subspaces of $X$?
My idea is the following: we know that if $Y\subset X$ is a subspace, $Y^\perp$ its annihilator is closed in $X^*$ and $Y^*\cong X^*/Y^\perp$. So I would say that $Z\subset X^*$ is a dual if and only if it is of the form $X^*/N$, $N=Y^\perp$ for some $Y$ subset of $X$, so if and only if $N=(N_\perp)^\perp$ (the lower $\perp$ is the pre-annihilator, i.e. the set of elements of $X$ for which $f(x)=0$ for all $f$ in the set of functionals we are considering), so if and only if $N$ is weakly*-closed. The latter comes from $(N_\perp)^\perp=\overline N^{w^*}$.
Is this ok? And is there any further characterisation of such $Z$? And moreover, under which generality can I extend such ideas (topological vector spaces, etc.)?
Thank you in advance.