# Relation between the weak star topology and hereditary Lindelöfness

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.

• could you sketch why $X^\ast$ separable (strongly?) implies that $X$ is weakly hereditarily Lindelöf? – Henno Brandsma Jun 5 '18 at 13:00
• @HennoBrandsma: Combination of two points get the result: (1) Let $Y$ be separable Banach space then, the closed unit ball of $Y^*$ is weak-star second countable. (2) The relative weak star topology of $X^{**}$ on $X$ is just the weak topology. – Ali Bagheri Jun 5 '18 at 14:50

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).
• Is $C(K)$ hereditarily Lindelöf in the weak topology? – Henno Brandsma Jun 6 '18 at 18:23
• interesting that $C_p(K)$ and $C(K)$ (in the traditional weak topology) are the same when $K$ is scattered compact. I really have to look at Zemadeni again.. I looked at $C(X)$ for another one of Kunen's example: his compact $L$-space. Nice area, IMHO. – Henno Brandsma Jun 10 '18 at 22:36
• @HennoBrandsma It seems that you are right: the weak topology and the topology of pointswiese convergence are different on $C(K)$ even for infinite scattered $K$. What is true that they coincide on bounded subsets of $C(K)$. But this is sufficient for our purposes. – Taras Banakh Jun 11 '18 at 6:58