To be specific: I need to prove that $||\mathbf{x}-\mathbf{y}||=\sqrt{(||\mathbf{x}||^2+||\mathbf{y}||^2)}$, where we know that $\mathbf{x}\perp\mathbf{y}$ in an inner product space (hence not necessarily in $\mathbb{R}^n$).
I reach the point where $||\mathbf{x}-\mathbf{y}||=\sqrt{\langle x,x\rangle -2\langle x,y\rangle+\langle y,y\rangle}$. But my question is: Why does the term $-2\langle x,y\rangle$ cancel? I know that in $\mathbb{R}^n$ this is because of the Pythagorean Law (at least when $2\langle x,y\rangle$ is positive), but I am a bit more confused in this case, since it's not specific in what space we are.
Thank you!
By assumption, $x \perp y$ which is defined as $\langle x, y\rangle =0$. Further, $||x||^2 = \langle x, x\rangle$.