Exercise 6.A.20

Answers

Proof. We have

\begin{aligned} | | u + v | |^{2} & - | | u - v | |^{2} + | | u + iv | |^{2} i - | | u - iv | |^{2} i \\ = ⟨ u + v, u + v ⟩ - ⟨ u - v, u - v ⟩ + (⟨ u + iv, u + iv ⟩ - ⟨ u - iv, u - iv ⟩) i \\ = 2 ⟨ u, v ⟩ + 2 ⟨ v, u ⟩ + (2 ⟨ u, vi ⟩ + 2 ⟨ iv, u ⟩) i \\ = 2 ⟨ u, v ⟩ + 2 ⟨ v, u ⟩ + (- 2 i ⟨ u, v ⟩ + 2 i ⟨ v, u ⟩) i \\ = 4 ⟨ u, v ⟩ \end{aligned}

Dividing by $4$ yields the desired result. □

celiopassos

2017-10-06 00:00