On the polar representation of an inner product.

Question

Take $H$ an inner product space. $x,y \in H$.

Take $b = |<x,y>|$ . Then the polar representation of $<x,y>$ is:

$$<x,y> = be^{i\theta}$$ for some $\theta \in (-\pi, \pi]$.

Why is this? If the answer is too long a reference will suffice, thank you.

Answer 1

For any complex number $z$ we have

$$z=|z|e^{i\theta}$$ where $\theta\in(-\pi,\pi]$ is the principal argument of $z$.