proof of parallelogram law

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.

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