Abstract

The classical Denjoy-Young-Saks theorem on Dini derivatives of arbitrary functions f:R→Rf: \mathbb{R} \to \mathbb{R} was extended by U.S. Haslam-Jones (1932) and A.J. Ward (1935) to arbitrary functions on R2\mathbb{R}^2. This extension gives the strongest relation among upper and lower Hadamard directional derivatives fH+(x,v)f^+_H (x,v), fH−(x,v)f^-_H (x,v) (v∈Xv \in X) which holds almost everywhere for an arbitrary function f:R2→Rf:\mathbb{R}^2\to \mathbb{R}. Our main result extends the theorem of Haslam-Jones and Ward to functions on separable Banach spaces.

Author information

Contact details are reproduced from the original publication and may be historical.

Luděk Zajíček

Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic

zajicek@karlin.mff.cuni.cz

Keywords

  • Hadamard upper and lower directional derivatives
  • Denjoy-Young-Saks theorem
  • separable Banach space
  • Hadamard differentiability
  • Fréchet differentiability
  • Hadamard subdifferentiability
  • Fréchet subdifferentiability
  • Gamma-null set
  • Aronszajn null set

Mathematics Subject Classification

46G05; 26B05

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Suggested citation

L. Zajíček. “Properties of Hadamard Directional Derivatives: Denjoy-Young-Saks Theorem for Functions on Banach Spaces.” Journal of Convex Analysis 22 (2015), No. 1, 161–176. https://doi.org/10.68381/jca22009

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Published by Heldermann Verlag, 2015. Rights now held by Banach Press LLC.