Is the ball of a reflexive space weakly metrizable?

Question

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.

Try looking at a non-separable Hilbert space. – Aweygan Jun 14 '17 at 22:15 — Aweygan, Jun 14 '17 at 22:15

score 1 · Accepted Answer · answered Jun 17 '17 at 05:03

1

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

answered Jun 17 '17 at 05:03

Gio67

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Is the ball of a reflexive space weakly metrizable?

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