Jordan-von Neumann theorem ($\Vert u+v{\Vert }^2+\Vert u-v{\Vert }^2=2\Vert u{\Vert }^2+2\Vert v{\Vert }^2⟺\Vert u{\Vert }^2=⟨u,u⟩$)

A complex normed vector space has an inducing inner product exactly when its norm satisfies the parallelogram law. Define $B(u,v)=[\Vert u+v{\Vert }^2-\Vert u-v{\Vert }^2]/4$. Subtract two parallelogram identities to obtain $B(u+w,v)+B(u-w,v)=2B(u,v)$. Taking $w=u$ gives doubling, and then replacing $u,w$ by the half-sum and half-difference gives additivity. Integer scaling and division give rational homogeneity; continuity gives real homogeneity. Thus $B$ is a symmetric real-bilinear form. The complex formula $P(u,v)=B(u,v)+iB(u,iv)$ satisfies $P(iu,v)=iP(u,v)$, because $B(iu,v)=-B(u,iv)$ and $B(iu,iv)=B(u,v)$. Symmetry gives conjugate symmetry of $P$, and $P(u,u)=\Vert u{\Vert }^2$ gives positive definiteness. The reverse implication follows by expanding squared norms. Completeness is needed only for the resulting inner product space to be a Hilbert space.