A polarization identity for multilinear maps

Understand

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.