Let $X$ be a Banach space and $Y$ be a separable closed subspace of $X^{\*}$. Is there a separable closed subspace $Z$ of $X$ such that $Y$ is isomorphic to a subspace of $Z^{\*}$? Thank you!



Separable subspaces in dual spaces
Yes. Take a countable dense subset $A$ in $Y$. For each element $a\in A$ take a countable sequence $x\_n$ in a unit ball of $X$ such that $a(x\_n)$ tends to $\|a\|$. Let $Z$ be a closed span of all these sequences.


