Let $H$ be an infinite-dimensional complex Hilbert space. Then the set of unitary operators on $H$ forms a group, known as the unitary group or Hilbert group. My question is, is this group a Lie group? That is, is there a standard Lie group structure for this group?
I ask because in quantum mechanics, at least subgroups of this group seem to be treated as Lie groups.
EDIT: The second page of this paper off-handedly mentions "the Frechet Lie group $U(H)$ consisting of all unitary operators on H, equipped with the strong operator topology". What that means is that the Lie group structure on $U(H)$ with the strong operator topology is a Frechet manifold, i.e. a manifold which is locally isomorphic to a Frechet space rather than a finite-dimensional Euclidean space. But does anyone know the details of this Frechet manifold structure?