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In this paper Martin Schottenloher notices that the unitary group $U(H)$ of a separable Hilbert space $H$ is metrizable in the strong operator topology. As a corollary (see R.Engelking, 5.1.3), it is paracompact (when $H$ is separable). I wonder

if $U(H)$ is paracompact for an arbitrary (not necessarily separable) Hilbert space $H$ in the strong operator topology.

P.S. This continues the discussions here, here, here and here, but I must say that I don't even understand why $U(H)$ is not locally compact in the infinite dimensional case.

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