Polarisation identity

Statement

Let $V$ be an inner product space with the induced norm $\norm{\cdot}$, and let $x, y \in V$. If $V$ is a real space, then $$\langle x, y \rangle = \frac{1}{4} \left( \norm{x + y}^2 - \norm{x - y}^2 \right).$$ If $V$ is a complex space, with the inner product linear in the first argument, then $$\langle x, y \rangle = \frac{1}{4} \sum_{k=0}^{3} i^k \norm{x + i^k y}^2.$$