Does separability follow from weak-* sequential separability of dual space?

Let $E$ be a Banach space. Suppose that $E'$ is weakly-* sequentially separable, that is, that there exists a countable $D \subset E'$ s.t. every $x' \in E'$ is a limit point of a sequence in $D$. Does it follow that $E$ is separable?

This question arises from a conversation with a friend of mine. He thinks this is true and plans to use it to prove separability of $C(K)$ for a metrizable and compact Hausdorff $K$. I'm not so sure this can work, though. Of course, if $E'$ is norm separable then $E$ is separable, but we are talking about a much weaker topology here.

• What a coincidence, I asked myself the same thing a few days ago (I didn't want to prove separability of $C(K)$ from that, though). – t.b. Jun 5 '11 at 10:34
• Well, not quite the same thing, but still similar though. I admit that I have no idea. – t.b. Jun 5 '11 at 10:47
• @Theo: In that link you proved that the non-separable $\ell^\infty$ has got a weak-$\star$ separable dual. This does not disprove the claim of this question because this dual needs not be sequentially separable, right? – Giuseppe Negro Jun 5 '11 at 11:09
• No, as far as I can see it doesn't tell usanything about your question. I know for sure that the unit ball of the dual of $\ell^\infty$ is not sequentially separable (e.g. because it would be metrizable then, which it isn't), but I suspect this is true for the whole $\ell^{\infty}$. I'm not so used to thinking about separability in a non-metric setting, though. – t.b. Jun 5 '11 at 11:16
• Of course, there are much easier ways to prove that $C(K)$ is separable. – Nate Eldredge Jun 5 '11 at 14:49

Let $JT$ denote the James Tree space, $Q: JT \longrightarrow JT\,^{\prime\prime}$ the canonical embedding, $D_\ast$ a countable norm-dense subset of $JT$, let $D= Q(D_\ast)$ and let $E = JT\,^\prime$. Note that $E$ is nonseparable in the norm topology and that the $w^\ast$-sequential closure of $D$ in $E^{\prime}$ is $E^{\prime}$ since $Q(JT)$ is $w^\ast$-sequentially dense in $JT\,^{\prime\prime}$ (for the last claim, see in particular Corollary 2 of Lindenstrauss and Stegall Examples of separable spaces which do not contain $\ell_1$ and whose duals are non-separable, Studia Math. 54 (1975), p.81--105).
I should mention that until the appearance of the Lindenstrauss-Stegall result cited above, it seems to have been open since the time of Banach whether there could exist a separable Banach space that has nonseparable dual and is $w^\ast$-sequentially dense in its bidual.
• I'd like to point out that $JT$ is a modification of James's famous example of a non-reflexive Banach space that is isometrically isomorphic to its bidual. – t.b. Jun 6 '11 at 7:59