The proof supplied here for the parallelogram law uses the properties of norms and inner products. See the entries about these objects for more details regarding the following calculations.

Proof.

$\Vert x+y \Vert^2+\Vert x-y \Vert^2$	$=\langle x+y,x+y \rangle + \langle x-y,x-y \rangle$
	$=\langle x,x+y \rangle + \langle y,x+y \rangle + \langle x,x-y \rangle - \langle y,x-y \rangle$
	$=\overline{\langle x+y,x \rangle}+\overline{\langle x+y,y \rangle}+\overline{\langle x-y,x \rangle}-\overline{\langle x-y,y \rangle}$
	$\displaystyle =\overline{\langle x,x \rangle + \langle y,x \rangle}+\overline{\langle x,y \rangle + \langle y,y \rangle}+\overline{\langle x,x \rangle - \langle y,x \rangle}-\left( \overline{\langle x,y \rangle - \langle y,y \rangle} \right)$
	$=\overline{\langle x,x \rangle}+\overline{\langle y,x \rangle}+\overline{\langle x,y \rangle}+\overline{\langle y,y \rangle}+\overline{\langle x,x \rangle}-\overline{\langle y,x \rangle}-\overline{\langle x,y \rangle}+\overline{\langle y,y \rangle}$
	$=\langle x,x \rangle + \langle y,y \rangle + \langle x,x \rangle + \langle y,y \rangle$
	$=2\langle x,x \rangle +2 \langle y,y \rangle$
	$=2 \Vert x \Vert^2+2 \Vert y \Vert^2$