Dear Mathematician,

I need to prove that:

norm ||x||=||y|| implies that inproduct < x+y,x-y > = 0

I thought about it for a long time, but I don't come any further.

I've got that:

< x+y,x-y > = <x,x> + <x,y> - <y,x> - <y,y>

= ||x||^2 + <x,y> - <y,x> - ||y||^2

So somehow ||x||=||y|| implies that <x,y> = <y,x>

I thought of using Cauchy-Schwartz, the triangle inequality or the parallelogram law..

Can someone finish the prove?

Cheers, Eva