A polarization identity for multilinear maps

TL;DR

This paper presents a simple polarization identity for symmetric multilinear maps over real or characteristic zero fields, linking these maps to their diagonal restrictions, with applications to Gaussian variables and further simplifications by other mathematicians.

Contribution

It introduces a new, straightforward polarization identity for symmetric multilinear maps and connects it to probabilistic representations of Gaussian variables.

Findings

Derived a probabilistic expression for Gaussian variables.

Provided a simplified proof of the polarization identity.

Connected the identity to applications in probability theory.

Abstract

Given linear spaces $E$ and $F$ over the real numbers or a field of characteristic zero, a simple argument is given to represent a symmetric multilinear map $u(x_1, x_2, \ldots, x_n)$ from $E^n$ to $F$ in terms of its restriction to the diagonal. As an application, a probabilistic expression for Gaussian variables used by Nelson and by Schetzen is derived. An Appendix by Tom H. Koornwinder notes an even further simplification by Bochnak and Siciak (1971) of the proof of the main result.