Let $V$ be an inner product space. Show that if $||x+y||=||x||+||y||$, then $ax=by$ where $a,b$ are non-negative and not both zero.

Question

Let $V$ be an inner product space. Show that if $||x+y||=||x||+||y||$, then $ax=by$ where $a,b$ are non-negative and not both zero.

I know that the converse is true. I considered the square of the norm sum.

$||x+y||^2=(||x||+||y||)^2=||x||^2+2||x||||y||+||y||^2$

I'm not sure how to conclude that $ax=by$ any hints or solutions are greatly appreciated.

Answer 1

Hint:

Use the equality case of the Cauchy-Schwarz inequality and the fact that in your case the scalar product of $x$ and $y$ is positive.