A Banach space which isn't metrizable

A Banach space which isn't metrizable A Banach space which isn't metrizable in weak topology.

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Every Hilbert space has this property (cf. [Hal, problem 21]).

Comment. It is probably true that no infinite dimensional Banach space is metrizable in the weak topology.
Ref.
[Hal] P.R. Halmos, A Hilbert space problem book, Princeton, Van Nostrand, 1967.

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