More generally, for a Hilbert space , is the group of unitary operators on that Hilbert space. For the purposes of studying unitary representations of Lie groups, the topology is chosen to be the strong operator topology, although other topologies on are frequently considered for other purposes.
More specifically, is a maximal compact subgroup of .
By the Gram-Schmidt process.
See also Kuiper's theorem.
This in contrast to the finite dimensional situation. For (), is not contractible.
or sometimes . Notice that this is very different from for an infinite-dimensional Hilbert space. See topological K-theory for more on this.
Hence it is a semidirect product group
Actually it is already the intersection of any two of these three, a fact also known as the “2 out of 3-property” of the unitary group.
This intersection property makes a G-structure for (an almost Hermitian structure) precisely a joint orthogonal structure, almost symplectic structure and almost complex structure. In the first-order integrable case this is precisely a joint orthogonal structure (Riemannian manifold structure), symplectic structure and complex structure.
is the circle group.
The subgroup of unitary matrices with determinant equal to 1 is the special unitary group. The quotient by the center is the projective unitary group. The space of equivalence classes of unitary matrices under conjugation is the symmetric product of circles.
The analog of the unitary group for real metric spaces is the orthogonal group.