On the set of G\^ateaux differentiability of the $L^1$ norm

TL;DR

This paper characterizes the points where the $L^1$ norm is Gâteaux differentiable in measure spaces and provides explicit derivative formulas at those points.

Contribution

It precisely identifies the Gâteaux differentiability set for the $L^1$ norm and derives the corresponding derivative formulas.

Findings

Characterization of the Gâteaux differentiability set for the $L^1$ norm.

Explicit formulas for the Gâteaux derivatives at differentiable points.

Enhanced understanding of the differentiability structure of $L^1$ spaces.

Abstract

Let $(\Omega, \mathcal{M}, \mu)$ be a measure space. In this paper we establish the set of G\^ateaux differentiability for the usual norm of $L^{1}(\Omega, \mu)$ and the corresponding derivative formulae at each point in this set.