Let V {\displaystyle {}V} be a K {\displaystyle {}{\mathbb {K} }} -vector space, endowed with an inner product ⟨ − , − ⟩ {\displaystyle {}\left\langle -,-\right\rangle } . Show that, for u , v ∈ V {\displaystyle {}u,v\in V} , the equality ‖ v + u ‖ = ‖ v − u ‖ {\displaystyle {}\Vert {v+u}\Vert =\Vert {v-u}\Vert } holds if and only if ⟨ v , u ⟩ = 0 {\displaystyle {}\left\langle v,u\right\rangle =0} holds.
