More precisely, a unitary transformation is an isometric isomorphism between two inner product spaces (such as Hilbert spaces).
In other words, a unitary transformation is a bijective function between two inner product spaces,
are the same space, a unitary transformation is an automorphism of that Hilbert space, and then it is also called a unitary operator.
A closely related notion is that of antiunitary transformation, which is a bijective function between two complex Hilbert spaces such that for all
, where the horizontal bar represents the complex conjugate.