Parallelogram law
= Parallelogram law
{title2=$\|u+v\|^2+\|u-v\|^2=2\|u\|^2+2\|v\|^2$}
{wiki}
= Parallelogram identity
{synonym}
The parallelogram law for a quadratic form $Q$ is
$$
Q(x+y)+Q(x-y)=2Q(x)+2Q(y).
$$
Conversely, under the usual divisibility assumptions, a function satisfying this identity and the appropriate homogeneity comes from a symmetric bilinear form by polarization.
For an <inner product space>, taking $Q(v)=\|v\|^2$ gives the displayed squared-norm identity by expanding and cancelling the mixed <inner products>. It is the geometric ingredient that makes minimizing sequences converge in the <Hilbert projection theorem>.