The "polarization identity" constructs an inner product from a norm provided the parallelogram law is met by the norm. How does the term originate, i.e. what has this to do with "polarization" ?
(For reference:
- Polarization identity (real space): $\langle v, w \rangle = ¼(||v + w||^2 - ||v - w||^2 )$
- Polarization identity (complex space): $\langle v, w \rangle = ¼(||v + w||^2 - ||v - w||^2 + i||v + iw||^2 - i||v - iw||^2)$
- Parallelogram law: $2||v||^2 + 2||w||^2 = ||v + w||^2 + ||v - w||^2$
