2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/111524We consider the unitary group $\U$ of complex, separable, infinite-dimensional Hilbert space as a discrete group. It is proved that, whenever $\U$ acts by isometries on a metric space, every orbit is bounded. Equivalently, $\U$ is not the union of a countable chain of proper subgroups, and whenever $\E\subseteq \U$ generates $\U$, it does so by words of a fixed finite length.Functional AnalysisGroup Theorytext