nLab
unitary group

For $n\in \mathbb{N}$, 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 $n\times n$ unitary matrices.

More generally, for $\mathscr{H}$ any Hilbert space, $U(\mathscr{H})$ is the group of unitary operators on that Hilbert space.

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