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.

  • 4
    $\begingroup$ Try looking at a non-separable Hilbert space. $\endgroup$ – Aweygan Jun 14 '17 at 22:15

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


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