# On the algebraic structure of the unitary group

Research paper by Eric Ricard, Christian Rosendal

Indexed on: 11 Apr '06Published on: 11 Apr '06Published in: Mathematics - Functional Analysis

#### Abstract

We 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.