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 $$\langle x, y \rangle = \frac{1}{4} \sum_{k=0}^{3} i^k \norm{x + i^k y}^2.$$

Remarks

So the norm determines the inner product. In particular, a linear map that keeps norms also keeps inner products. With the convention of linearity in the second argument, the factor $i^k$ moves to the other vector, and the complex formula changes accordingly.

Explanations

L1 human

If we know the lengths of all vectors, then we also know all inner products. In the real case, we only need the lengths of the two diagonals $x + y$ and $x - y$. So lengths determine angles.