The concept of the dual space is well known and in some cases computably up to an isometric isomorphism, i.e. $c_0^* \cong \ell_1$ or $\ell_p \cong \ell_q$.
What if we are given a space $X$ and want to find an space $E$ such that $E^* = X$ (or $\cong$)? What constrains do we have to put on $X$ to find such a space $E$?
Could this be a way to show that given $X$, it must be complete, since the dual space of a normed space is complete?