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The unitary group $\mathrm U(\mathcal H)$ on an infinite dimensional complexHilbert space $\mathcal H$ in its strong topology is a topological group andhas some further nice properties, e.g. it is metrizable and contractible if$\mathcal H$ is separable. As an application Hilbert bundles are classified byhomotopy.

The Unitary Group In Its Strong Topology