Polarization identity

In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space. If a norm arises from an inner product then the polarization identity can be used to express this inner product entirely in terms of the norm. The polarization identity shows that a norm can arise from at most one inner product; however, there exist norms that do not arise from any inner product.

The norm associated with any inner product space satisfies the parallelogram law: ${\displaystyle \|x+y\|^{2}+\|x-y\|^{2}=2\|x\|^{2}+2\|y\|^{2}.}$ In fact, as observed by John von Neumann,^[1] the parallelogram law characterizes those norms that arise from inner products. Given a normed space ${\displaystyle (H,\|\cdot \|)}$, the parallelogram law holds for ${\displaystyle \|\cdot \|}$ if and only if there exists an inner product ${\displaystyle \langle \cdot ,\cdot \rangle }$ on ${\displaystyle H}$ such that ${\displaystyle \|x\|^{2}=\langle x,\ x\rangle }$ for all ${\displaystyle x\in H,}$ in which case this inner product is uniquely determined by the norm via the polarization identity.^[2][3]