Question

Lex X be an inner product space over R. If x ,y∈ X are such that ||x + y||=||x-y||then show that x ⊥ y.

Answer 1

Let $\langle \cdot\ , \cdot\rangle$ be inner product on $X$ .

Elements $x,y\in X$ are such that $\|x+y\|=\|x-y\|$ .

If $\|x+y\|=\|x-y\|$ , then $\|x+y\|^2=\|x-y\|^2$ .

$\|x+y\|^2=\langle x+y,x+y\rangle=\langle x,x\rangle +\langle x,y\rangle+\langle y,x\rangle +\langle y, y\rangle =\|x\|^2+\|y\|^2+2\langle x,y\rangle$

$\|x-y\|^2=\langle x-y,x-y\rangle=\langle x,x\rangle -\langle x,y\rangle-\langle y,x\rangle +\langle y, y\rangle =\|x\|^2+\|y\|^2-2\langle x,y\rangle$

Then we have that $0=\|x+y\|^2-\|x-y\|^2=4\langle x,y\rangle$ .

$\langle x,y\rangle =0$ if and only if $x\perp y$ .