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


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


The converse is clearly true, since the closed unit ball is relatively weak star second countable. 


**Def.** A topological space $X$ is hereditarily Lindelöf if every subspace $Y\subseteq X$ is Lindelöf. 



Relation between the weak star topology and hereditary Lindelöfness
Under CH there exists an example of a non-metrizable compact scattered Hausdorff space $K$ such that the Banach space $X=C(K)$ endowed with the weak topology is hereditarily Lindelof. The non-metrizability of $K$ implies that the Banach space $X=C(K)$ is not separable and then the dual $X^\*$ is not separable as well. This example is due to Kunen and is described in the survey paper of Negrepontis in "Handbook of Set-Theoretic Topology" (1984).


