Let $X$ be a separable Banach  space whose dual is not separable. Is the locally convex space $(X,w)$ a hereditary Lindelof space? ($w$ is the weak topology on $X$). What about the converse, is the follwoing implication valid? 

$$ \textrm{herditary Lindelofness}~ \Rightarrow X^*~ \textrm{is separable} $$




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