Abstract

Let XX be a separable Banach space, YY a Banach space and f:X→Yf: X \to Y an arbitrary mapping. Then the following implication holds at each point x∈Xx\in X except a σ\sigma-directionally porous set: If the one-sided Hadamard directional derivative fH+′(x,u)f'_{H+}(x,u) exists in all directions uu from a set Sx⊂XS_x \subset X whose linear span is dense in XX, then ff is Hadamard differentiable at xx. This theorem improves and generalizes a recent result of A. D. Ioffe, in which the linear span of SxS_x equals XX and Y=RY = \mathbb{R}. An analogous theorem, in which ff is pointwise Lipschitz, and which deals with the usual one-sided derivatives and Gâteaux differentiability is also proved. It generalizes a result of D. Preiss and the author, in which ff is supposed to be Lipschitz.

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

  • Gâteaux differentiability
  • Hadamard differentiability
  • directional derivatives
  • Hadamard directional derivatives
  • sigma-directionally porous set
  • pointwise Lipschitz mapping

Mathematics Subject Classification

46G05; 26B05, 49J50

References

  1. Y. Benyamini, J. Lindenstrauss: Geometric Nonlinear Functional Analysis. Vol. 1, Colloquium Publications 48, American Mathematical Society, Providence (2000).https://doi.org/10.1090/coll/048Google Scholar
  2. A. D. Ioffe: Typical convexity (concavity) of Dini-Hadamard upper (lower) directional derivatives of functions on separable Banach spaces, J. Convex Analysis 17 (2010) 1019–1032.https://doi.org/10.68381/jca17063Google Scholar
  3. J. Lindenstrauss, D. Preiss, J. Tišer: Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces, Princeton University Press, Princeton (2012).https://doi.org/10.23943/princeton/9780691153551.001.0001Google Scholar
  4. D. Preiss, L. Zajíček: Directional derivatives of Lipschitz functions, Israel J. Math. 125 (2001) 1–27.https://doi.org/10.1007/BF02773371Google Scholar
  5. A. Shapiro: On concepts of directional differentiability, J. Optim. Theory Appl. 66 (1990) 477–487.https://doi.org/10.1007/BF00940933Google Scholar
  6. L. Zajíček: Singular points of order k of Clarke regular and arbitrary functions, Comment. Math. Univ. Carol. 53 (2012) 51–63.Google Scholar
  7. L. Zajíček: Hadamard differentiability via Gâteaux differentiability, preprint (2012), arXiv:1210.4715v1.Google Scholar

Suggested citation

L. Zajíček. “Gâteaux and Hadamard Differentiability via Directional Differentiability.” Journal of Convex Analysis 21 (2014), No. 3, 703–713. https://doi.org/10.68381/jca21037

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Published by Heldermann Verlag, 2014. Rights now held by Banach Press.