In inner product space $H$ over $\Bbb{C}$, what are the vectors that satisfy $||x+y||^2=||x||^2+||y||^2$?
I have made many attempts over and over again, trying to show that based on the fact that $$\langle x,x\rangle+\langle y,y\rangle=\langle x+y,x+y\rangle=\langle x,x\rangle+\langle y,y\rangle+\langle x,y\rangle+\overline{\langle x,y\rangle}$$ then $\langle x,y\rangle=i\cdot{a}$ for $a\in \Bbb{R}$, not necessarily orthogonal. But is $y=i\cdot r\cdot x$ or $x=i\cdot r\cdot y$? Why can't I seem to go any further as to presenting $x$ as $y$ or $y$ as $x$?
Then I tried to take $\left\langle{ia x\over \|x\|^2}-y,{ia x\over \|x\|^2}-y\right\rangle$ and $\left\langle x-{ia y\over \|y\|^2},x-{ia y\over \|y\|^2}\right\rangle$ with the idea in mind that either $y={xai\over \|x\|^2}$ or $x={yai\over \|y\|^2}$, but there might be too many solutions to that, and I really can't work with the results of this expansion. And I could assume $\|x\|=\|y\|=1$ anyway (I think?).
Could you drop me a hint?