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?
Hi, thnx for your reaction.
I forgot to mention that it is about a complex space. Otherwise it would be very obvious.
The inproduct is not defined yet, so this statement has to be proved for every possible inproduct in the complex space.
Does anyone have a clue?