Orthogonality

Proposition

A Banach space $\mathcal{B}$ is a Hilbert space if and only if the parallelogram law holds $\Vert f+g{\Vert }^2+\Vert f-g{\Vert }^2=2(\Vert f{\Vert }^2+\Vert g{\Vert }^2).$ As a consequence, if $L^p({\mathbb{R}}^d)$ with the Lebesgue measure is a Hilbert space, then necessarily $p=2$. $\text{Hint: For the first part. in the real case, let }⟨f,g⟩=\frac{1}{4}(\Vert f+g{\Vert }^2+\Vert f-g{\Vert }^2)$