Self - adjoint and Unitary operator 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.

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., $\langle v_1, w_2\rangle = 0$.