Quadratic functional equation

Summary

The quadratic functional equation is the functional equation $f(x + y) + f(x - y) = 2 f(x) + 2 f(y)$. The unknown $f$ maps an abelian group, usually a real vector space, to the real numbers. Every quadratic form solves it, and on $\R$ the continuous solutions are the functions $f(x) = cx^2$. It is the functional form of the parallelogram law, and it lies behind the Jordan–von Neumann theorem.