# Is unitary group paracompact?

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.