Q:

Is the ball of a reflexive space weakly metrizable?

We know from Banach-Alaoglu that if $E^*$ is separable then the unit ball is metrizable with the weak topology.
Now is this true for reflexive spaces? I.e., if a Banach $E$ (eq. $E^*$) is reflexive, is its ball weakly-metrizable?
I suspect the answer is no. Is there an interesting counter-example?
Thank you in advance.

A:

The unit ball in $E$ is metrizable with respect to the weak topology if and only if $E^*$ is separable. You can find the proof of this theorem in Brezis Functional Analysis, Theorem 3.29

