My question is on the idea behind Von Neumann's formula for the scalar product induced from a norm that satisfies the parallelogram law. $\langle x,y\rangle=\frac 14(\|x+y\|^2-\|x-y\|^2)$. Was it by trial and error or is there some intuitive way that one can guess this formula?
I can prove that this is indeed a scalar product, it is the idea behind it that baffles me.