Polarization identity

In 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. Let ${\displaystyle \|x\|\,}$ denote the norm of vector x and ${\displaystyle \langle x,\ y\rangle \,}$ the inner product of vectors x and y. Then the underlying theorem, attributed to Fréchet, von Neumann and Jordan, is stated as:

The various forms given below are all related by the parallelogram law:

The polarization identity can be generalized to various other contexts in abstract algebra, linear algebra, and functional analysis.

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