The multiplicative property characterizes $\ell_p$ and $L_p$ norms

TL;DR

This paper characterizes the $\, ext{l}_p$ and $L_p$ norms as unique norms satisfying specific invariance and multiplicative properties, using large deviation theory for the proof.

Contribution

It provides a novel characterization of $\, ext{l}_p$ and $L_p$ norms based on invariance and multiplicativity, with a proof leveraging Cramér's large deviation theorem.

Findings

$ ext{l}_p$ norms are uniquely invariant under coordinate permutation and multiplicative under tensor products.

$L_p$ norms are the only rearrangement-invariant norms satisfying multiplicativity for independent variables.

The characterization uses large deviation principles to establish the uniqueness of these norms.

Abstract

We show that $\ell_p$ norms are characterized as the unique norms which are both invariant under coordinate permutation and multiplicative with respect to tensor products. Similarly, the $L_p$ norms are the unique rearrangement-invariant norms on a probability space such that $\|X Y\|=\|X\|\cdot\|Y\|$ for every pair $X,Y$ of independent random variables. Our proof relies on Cram\'er's large deviation theorem.