If x and y are complex numbers then $x \perp y$ means $||x+iy||^2= ||x||^2 + ||y||^2$

Question

I don't see how if $x$ and $y$ are elements of an inner product space of the complex numbers then saying $x \perp y$ is equivalent to saying $||x+y||^2= ||x||^2 +||y||^2$ and $||x+iy||^2= ||x||^2 +||y||^2$

I can see that $||x+y||^2= ||x||^2 +||y||^2$ is true. However how is it that $||x+iy||^2= ||x||^2 +||y||^2$ ?

Answer 1

Recall that

$$\|x+\lambda y\|^2 = \langle x+\lambda y, x+\lambda y\rangle = \langle x, x\rangle + \lambda \langle y,x\rangle + \bar{\lambda}\langle x,y\rangle + \lambda\bar{\lambda} \langle y,y\rangle.$$

Can you take it from here?