Properties of Hadamard directional derivatives: Denjoy-Young-Saks theorem for functions on Banach spaces

TL;DR

This paper extends the Denjoy-Young-Saks theorem, originally for real functions, to functions on separable Banach spaces, establishing the strongest relations among Hadamard directional derivatives almost everywhere.

Contribution

It generalizes the classical theorem to infinite-dimensional Banach spaces, providing new insights into the behavior of Hadamard derivatives in this setting.

Findings

Established the strongest relations among Hadamard derivatives almost everywhere

Extended classical theorems from real functions to Banach spaces

Provided a framework for analyzing directional derivatives in infinite dimensions

Abstract

The classical Denjoy-Young-Saks theorem on Dini derivatives of arbitrary functions $f: \R \to \R$ was extended by U.S. Haslam-Jones (1932) and A.J. Ward (1935) to arbitrary functions on $\R^2$. This extension gives the strongest relation among upper and lower Hadamard directional derivatives $f^+_H (x,v)$, $f^-_H (x,v)$ ($v \in X$) which holds almost everywhere for an arbitrary function $f:\R^2\to \R$. Our main result extends the theorem of Haslam-Jones and Ward to functions on separable Banach spaces.