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 ⊆ U generates U, it does so by words of a fixed finite length.
@article{urn:eudml:doc:41805,
title = {On the algebraic structure of the unitary group.},
journal = {Collectanea Mathematica},
volume = {58},
year = {2007},
pages = {181-192},
zbl = {1129.22012},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41805}
}
Ricard, Éric; Rosendal, Christian. On the algebraic structure of the unitary group.. Collectanea Mathematica, Tome 58 (2007) pp. 181-192. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41805/