Does $ \langle x, x \rangle = \langle y,y \rangle $ imply $ \| x \| = \| y \|. $
Since $ \langle x, x \rangle = \| x \|^2 $ we have $$ \langle x, x \rangle = \langle y,y \rangle $$ is equivalent to $$ \| x \|^2 = \| y\|^2. $$ Am I able to claim that this implies $$ \| x \| = \| y \|$$ since $\| x \| \geq 0?$