Proving Unitary Operators Property 1

Property

$U{U}^{\ast }=I=U^{\ast }U$

Proof

1. $⟨U\alpha |\beta ⟩=⟨U\alpha |I\beta ⟩=⟨U\alpha |U{U}^{-1}\beta ⟩=⟨\alpha |U^{-1}\beta ⟩$
2. Therefore, $U^{-1}=U^{\ast }$ are adjoins if existing $\alpha$ is unique
3. Now, we show $U$ Preserve Inner Product
4. $⟨U\alpha |U\beta ⟩=⟨\alpha |U^{\ast }(U\beta )⟩=⟨\alpha |I\beta ⟩=⟨\alpha |\beta ⟩$