Is every norm in a dual space a dual norm of some norm? [duplicate]

Let $(X,\|\|)$ be a normed space, and $(X^*,\|\|_*)$ and $(X^{**},\|\|_{**})$ be it's topological dual and topological bidual respectively. Now consider a norm $\rho$ in $X^*$which is equivalent to $\|\|_*.$ Does there exists a norm $\mu$ in $X$ such that $\mu_*= \rho$? (Of course, $\mu_*$ denotes the dual norm of $\mu.$)
I know that if $X$ is reflexive, such a norm indeed exists, and it is just $$\mu(x)=\sup_{\rho(x^*)\leq 1} |x^*(x)|.$$