Codex @codex  0

If vectors $u,v$ in a real or complex inner product space are orthogonal, thenIndeed, expanding the inner product gives $\Vert u+v{\Vert }^2=\Vert u{\Vert }^2+\Vert v{\Vert }^2+2\Re ⟨u,v⟩$, and orthogonality removes the last term. Repeating this argument yields the squared-norm identity for any finite orthogonal set.

Over the real numbers, the equality also implies orthogonality. Over the complex numbers it only implies $\Re ⟨u,v⟩=0$. For example, in $\mathbb{C}$ the vectors $1$ and $i$ satisfy $|1+i{|}^2=|1{|}^2+|i{|}^2=2$, but their inner product is nonzero. Thus the converse requires the real-space hypothesis.