In mathematics, an antiunitary transformation, is a bijective antilinear map U : H 1 → H 2 {\displaystyle U:H_{1}\to H_{2}\,} between two complex Hilbert spaces such that ⟨ U x, U y ⟩ = ⟨ x, y ⟩ ¯ {\displaystyle \langle Ux,Uy\rangle ={\overline {\langle x,y\rangle }}} for all x {\displaystyle x} and y {\displaystyle y} in H 1 {\displaystyle H_{1}}, where the horizontal bar represents the complex conjugate. Wikipedia