Let $X$ be a  Banach space. Is the following implication valid? 

$$  (X,w) \textrm{ is herditary Lindelof}~ \Rightarrow X^*~ \textrm{is separable} $$




Def. A topological space $X$ is hereditary Lindelof if every subspace $Y\subseteq X$ is Lindelof.