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.
Best Answer
You could also just check, for $v = v_1 + v_2$ and $w = w_1 + w_2$ where $v_1, w_1 \in W$ and $v_2, w_2 \in W'$, that \[ \langle Uv, w\rangle = \langle v_1 - v_2, w_1 + w_2\rangle \] is equal to $\langle v, Uw\rangle$. Use the linearity of the inner product and that, e.g., $\langle v_1, w_2\rangle = 0$.