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A 'unitary transformation' is an isomorphism between two Hilbert spaces. In other words, a ''unitary transformation'' is a bijective function
:$U:H\_1\; o\; H\_2,$
where $H\_1$ and $H\_2$ are Hilbert spaces, such that
:$langle\; Ux,\; Uy$
angle = langle x, y
angle
for all $x$ and $y$ in $H\_1$. A unitary transformation is an isometry, as one can see by setting $x=y$ in this formula.
In the case when $H\_1$ and $H\_2$ 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
:$U:H\_1\; o\; H\_2,$
between two complex Hilbert spaces such that
:$langle\; Ux,\; Uy$
angle = overline{langle x, y
angle}=langle y, x
angle
for all $x$ and $y$ in $H\_1$, where the horizontal bar represents the complex conjugate.

Contents

See also

See also

★ Time reversal

★ Unitary group

★ Unitary operator