polarization identity
The norm seems to forget direction — it only sees length. The polarization identity is the surprising reversal: in an inner product space you can rebuild the full inner product, angles and all, from lengths alone. Nothing is actually lost when you pass from <-,-> to ||.||.
Over the reals it reads <u,v> = (||u + v||^2 - ||u - v||^2) / 4. Expand both norms via the inner product and the diagonal terms cancel, leaving 4 <u,v>. The complex case needs more: you must also probe with i, giving <u,v> = (1/4) sum_{k=0..3} i^k ||u + i^k v||^2, the so-called four-term formula.
The reason the complex formula is longer is conjugate-linearity. A single real combination of norms cannot separate the real and imaginary parts of <u,v>; rotating v by i, -1, and -i and weighting by powers of i untangles them.
Conceptually this is the partner of the parallelogram law. Parallelogram tells you WHEN a norm comes from an inner product; polarization tells you HOW to recover that product. Together they show the norm and the inner product carry identical information.
Two faces of the same idea: the real case needs two norms, the complex case four, because i must be probed to recover the imaginary part.
Watch the convention: with first-slot-linear inner products the complex formula above gives <u,v>; with second-slot-linear it gives <v,u>. A wrong convention silently conjugates your answer.