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**Deveno** i will use U* to denote the conjugate-transpose or adjoint of U.

by definition, this is the linear transformation for which: <Uv,w> = <v,U*w> for any vectors (or "kets" as physicists like to call them) v,w.

if U is unitary, which means U*U = UU* = I (the identity transformation), then using Uw in place of w gives:

<Uv,Uw> = <v,U*(Uw)> = <v,(U*U)w> = <v,Iw> = <v,w>

note that in a complex inner-product space, <v,_> is often defined as the linear functional v*(_), that is: <v|w> = (|v>)*|w> (a row consisting of the conjugates of the coordinates of v times a column consisting of the coordinates of w).

this insures sesquilinearity (most mathematical treatments do the opposite and define <v,w> = (v*w)*, in other words we have linearity in the first argument, rather than the second. this is your typical "left-handed" versus "right-handed" choice...that is, in physics we have: <x|y> = <y,x> according to the "usual" mathematical definition).