# Parallogram law

• Mar 1st 2010, 09:43 PM
surjective
Parallogram law
Hello all,

How can I show that for any $\displaystyle \textbf{v},\textbf{w} \in V$ ($\displaystyle V$ is a vector space with an inner product and associated norm) it holds that:

$\displaystyle \Vert \textbf{v} + \textbf{w} \Vert^{2} + \Vert \textbf{v} - \textbf{w} \Vert^{2} = 2 \left( \Vert \textbf{v} \Vert^{2} + \Vert \textbf{w} \Vert^{2} \right)$

I have done the following:

$\displaystyle \Vert \textbf{v} + \textbf{w} \Vert^{2}$= $\displaystyle \langle \textbf{v}+\textbf{w},\textbf{v}+\textbf{w} \rangle$=$\displaystyle \langle \textbf{v},\textbf{v} \rangle + \langle \textbf{w},\textbf{w} \rangle+\langle \textbf{v},\textbf{w} \rangle + \langle \textbf{w},\textbf{v} \rangle$

$\displaystyle \Vert \textbf{v} - \textbf{w} \Vert^{2}$= $\displaystyle \Vert \textbf{v} +(- \textbf{w}) \Vert^{2}$= $\displaystyle \langle \textbf{v}+(-\textbf{w}), \textbf{v}+(-\textbf{w}) \rangle$= $\displaystyle \langle \textbf{v},\textbf{v} \rangle + \langle \textbf{w},\textbf{w} \rangle-\langle \textbf{v},\textbf{w} \rangle - \langle \textbf{w},\textbf{v} \rangle$

$\displaystyle \Vert \textbf{v} + \textbf{w} \Vert^{2} + \Vert \textbf{v} - \textbf{w} \Vert^{2} =$ $\displaystyle \langle \textbf{v},\textbf{v} \rangle + \langle \textbf{w},\textbf{w} \rangle+\langle \textbf{v},\textbf{w} \rangle + \langle \textbf{w},\textbf{v} \rangle + \langle \textbf{v},\textbf{v} \rangle + \langle \textbf{w},\textbf{w} \rangle-\langle \textbf{v},\textbf{w} \rangle - \langle \textbf{w},\textbf{v} \rangle =$ $\displaystyle 2\langle \textbf{v},\textbf{v} \rangle$+ $\displaystyle 2\langle \textbf{w},\textbf{w} \rangle$ = $\displaystyle 2\left ( \Vert \textbf{v}\Vert^{2} + \textbf{w}\Vert^{2} \right)$