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

${\displaystyle U:H_{1}\to H_{2}}$ 

between two inner product spaces, ${\displaystyle H_{1}}$  and ${\displaystyle H_{2},}$  such that

${\displaystyle \langle Ux,Uy\rangle _{H_{2}}=\langle x,y\rangle _{H_{1}}\quad {\text{ for all }}x,y\in H_{1}.}$ 

It is a linear isometry, as one can see by setting ${\displaystyle x=y.}$ 

In the case when ${\displaystyle H_{1}}$  and ${\displaystyle H_{2}}$  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

${\displaystyle U:H_{1}\to H_{2}\,}$ 

between two complex Hilbert spaces such that

${\displaystyle \langle Ux,Uy\rangle ={\overline {\langle x,y\rangle }}=\langle y,x\rangle }$ 

for all ${\displaystyle x}$  and ${\displaystyle y}$  in ${\displaystyle H_{1}}$ , where the horizontal bar represents the complex conjugate.

• Antiunitary
• Orthogonal transformation
• Time reversal
• Unitary group
• Unitary operator
• Unitary matrix
• Wigner's theorem
• Unitary transformations in quantum mechanics