
Journal of Convex Analysis 22 (2015), No. 1, 161176 Copyright Heldermann Verlag 2015 Properties of Hadamard Directional Derivatives: DenjoyYoungSaks Theorem for Functions on Banach Spaces Ludek Zajícek Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic zajicek@karlin.mff.cuni.cz [Abstractpdf] \newcommand{\R}{{\mathbb R}} The classical DenjoyYoungSaks theorem on Dini derivatives of arbitrary functions $f: \R \to \R$ was extended by U.S. HaslamJones (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 HaslamJones and Ward to functions on separable Banach spaces. Keywords: Hadamard upper and lower directional derivatives, DenjoyYoungSaks theorem, separable Banach space, Hadamard differentiability, Frechet differentiability, Hadamard subdifferentiability, Frechet subdifferentiability, Gammanull set, Aronszajn null set. MSC: 46G05; 26B05 [ Fulltextpdf (160 KB)] for subscribers only. 