Parallelogram Law

Let $\vert\cdot\vert$ denote the Norm of a quantity. Then the quantities $x$ and $y$ satisfy the parallelogram law if

$$\Vert x+y\Vert^2+\Vert x-y\Vert^2 = 2\Vert x\Vert^2+2\Vert y\Vert^2.$$

If the Norm is defined as $\vert f\vert = \sqrt{\left\langle{f\vert f}\right\rangle{}}$ (the so-called L2-Norm), then the law will always hold.

See also L2-Norm, Norm