Polarization Identity/Complex Vector Space

Theorem
Let $\struct {V, \innerprod \cdot \cdot}$ be an inner product space over $\C$.

Let $\norm \cdot$ be the inner product norm on $V$.

Then, we have:


 * $4 \innerprod x y = \norm {x + y}^2 - \norm {x - y}^2 + i \norm {x + i y}^2 - i \norm {x - iy}^2$

for each $x, y \in V$.

Proof
We write:

We then compute:

We then have:

and:

Then we have: