Characterisation of $L_p$-norms via H\"older's inequality

TL;DR

This paper characterizes $L_p$-norms on integrable step functions using a form of H"older's inequality, establishing conditions for optimality in a probabilistic setting.

Contribution

It provides a novel characterization of $L_p$-norms through H"older's inequality with an optimality condition, specifically for integrable step functions.

Findings

$L_p$-norms are uniquely characterized by H"older's inequality.

The characterization applies to functions on a probabilistic space.

Optimality conditions are established for the inequality.

Abstract

We characterise $L_p$-norms on the space of integrable step functions, defined on a probabilistic space, via H\"older's type inequality with an optimality condition.