Suppose $(H,(\cdot,\cdot))$ is a scalar product space and $||\cdot||$ is the norm induced by the scalar product.
I am trying to describe all vectors that satisfy $||x+y||^2 = ||x||^2+||y||^2$.
Clearly, if $(x,y) = 0$ i.e. (x,y) are orthogonal, then the equality is satisfied thanks to Pythagoras. I am wondering if the converse is true. That is, if $x,y$ satisfy the equality, are they necessarily orthogonal? I know this to be true in the reals, however in general Hilbert spaces, I am not sure.
I attempted a solution:$$||x+y||^2 = ||x||^2+||y||^2 \iff (x+y,x+y) = (x,x)+(y,y)$$
$$\iff (x,y) + (y,x) = 0$$
$$\iff (x,y) = 0 \text{ (real Hilbert space), or } (x,y)=-\overline{(x,y)}.$$
So if we are in a real Hilbert space, then $(x,y)$ are orthogonal. If we are in a complex Hilbert space, $(x,y) = \lambda i$ for some real $\lambda$. So I am think that it does not hold true in general?
Your calculations are correct: in a complex Hilbert space, all you can conclude is that $(x,y)$ is purely imaginary, not necessarily that it is $0$. As a sanity check, here is a very simple example: if $x$ is a nonzero vector and $y=ix$, then $x+y=(1+i)x$ and $\|y\|=\|x\|$ so $$\|x+y\|^2=|1+i|^2\|x\|^2=2\|x\|^2=\|x\|^2+\|y\|^2,$$ while $(x,y)=-i(x,x)\neq 0$.