[Math] Prove that $U$ is a self adjoint unitary operator

Question

Let $W$ be the finite dimensional subspace of an inner product space $V$ and $V=W\oplus W^\perp $.
Define $U:V \rightarrow V$ by $U(v_1+v_2)=v_1-v_2$ where $v_1\in W$ and $v_2 \in W^\perp$.
Prove that $U$ is a self adjoint unitary operator.

I know I have to show that $\parallel U(x) \parallel=\parallel x \parallel $ but can't proceed from this stage.

Answer 1

$\langle U(x),U(x)\rangle = \langle U(v_1+v_2) , U(v_1+v_2)\rangle = \langle v_1 - v_2, v_1 - v_2\rangle = \langle v_1,v_1\rangle + \langle v_2,v_2\rangle = \langle x,x\rangle$ where last two equalities comes frome the fact that $\langle v_1,v_2\rangle = 0$