Unitary operators are usually taken as operating on a Hilbert space, but the same notion serves to define the concept of isomorphism between Hilbert spaces.
A unitary operator is a bounded linear operator U : H → H on a Hilbert space H for which the following hold: The notion of isomorphism in the category of Hilbert spaces is captured if domain and range are allowed to differ in this definition.
Isometries preserve Cauchy sequences; hence the completeness property of Hilbert spaces is preserved[3] The following, seemingly weaker, definition is also equivalent: Definition 3.
The fact that U has dense range ensures it has a bounded inverse U−1.
The linearity requirement in the definition of a unitary operator can be dropped without changing the meaning because it can be derived from linearity and positive-definiteness of the scalar product: Analogously we obtain