= Jordan-von Neumann theorem
{c}
{title2=$\|u+v\|^2+\|u-v\|^2=2\|u\|^2+2\|v\|^2\iff\|u\|^2=\langle u,u\rangle$}
A complex <normed vector space> has an inducing <inner product> exactly when its <norm> satisfies the <parallelogram law>. Define $B(u,v)=[\|u+v\|^2-\|u-v\|^2]/4$. 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 $P(u,u)=\|u\|^2$ 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>.
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