nLab
unitary group

For n, the unitary group U(n) is the group of isometries of an n-dimensional complex Hilbert space.

This is canonically identified with the group of n×n unitary matrices.

More generally, for any Hilbert space, U() is the group of unitary operators on that Hilbert space.

The analog of the unitary group for real metric spaces is the orthogonal group.