For W be a finite dimensional subspace of an inner product space V. Given V is the direct sum of W and its orthogonal complement W'.
For a map U defined on V as U (v + v') = v - v' , for all v in W , v' in W'.
I have to show that U is a self - adjoint and unitary operator.
The part that U is unitary operator is clear as U preserves length.
I am trying to show that U is self- adjoint. Please suggest.