Polarization Identities

TL;DR

This paper generalizes the polarization identity to complex inner product spaces over associative algebras with involution, providing new characterizations of inner product spaces beyond traditional settings.

Contribution

It introduces a broad generalization of the polarization identity and extends the Jordan-von Neumann theorem to algebraic scalar fields with involution.

Findings

Generalized polarization identity for complex inner product spaces

Characterization of inner product spaces over associative algebras

Extension of classical theorems to algebraic scalar fields

Abstract

We prove a generalization of the polarization identity of linear algebra expressing the inner product of a complex inner product space in terms of the norm, where the field of scalars is extended to an associative algebra equipped with an involution, and polarization is viewed as an averaging operation over a compact multiplicative subgroup of the scalars. Using this we prove a general form of the Jordan-von Neumann theorem on characterizing inner product spaces among normed linear spaces, when the scalars are taken in an associative algebra.