---
tag: 05FKHRT0B4
kind: statement
role: proposition
name: Polarisation identity
msc: [46C05, 15A63]
uses: [0KG4D8AP6G]
sources:
  - url: https://en.wikipedia.org/wiki/Polarization_identity
    locator: Polarization identities
same_as:
  wikidata: Q1817177
  wikipedia: Polarization identity
---

## Statement

Let $V$ be an inner product space with the [[induced norm|0MFJJV97RJ]] $\norm{\cdot}$, 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.