Journal of Convex Analysis 22 (2015), No. 1, 161--176
Copyright Heldermann Verlag 2015
Properties of Hadamard Directional Derivatives: Denjoy-Young-Saks 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
[Abstract-pdf]
\newcommand{\R}{{\mathbb R}} 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.
Keywords: Hadamard upper and lower directional derivatives, Denjoy-Young-Saks theorem, separable Banach space, Hadamard differentiability, Frechet differentiability, Hadamard subdifferentiability, Frechet subdifferentiability, Gamma-null set, Aronszajn null set.
MSC: 46G05; 26B05
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