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Hadamard differentiability via Gâteaux differentiability


Author: Luděk Zajíček
Journal: Proc. Amer. Math. Soc. 143 (2015), 279-288
MSC (2010): Primary 46G05; Secondary 26B05, 49J50
DOI: https://doi.org/10.1090/S0002-9939-2014-12228-3
Published electronically: August 29, 2014
MathSciNet review: 3272753
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Abstract: Let $ X$ be a separable Banach space, $ Y$ a Banach space and $ f: X \to Y$ a mapping. We prove that there exists a $ \sigma $-directionally porous set $ A\subset X$ such that if $ x\in X \setminus A$, $ f$ is Lipschitz at $ x$, and $ f$ is Gâteaux differentiable at $ x$, then $ f$ is Hadamard differentiable at $ x$. If $ f$ is Borel measurable
(or has the Baire property) and is Gâteaux differentiable at all points, then $ f$ is Hadamard differentiable at all points except for a set which is a $ \sigma $-directionally porous set (and so is Aronszajn null, Haar null and $ \Gamma $-null). Consequently, an everywhere Gâteaux differentiable $ f: \mathbb{R}^n \to Y$ is Fréchet differentiable except for a nowhere dense $ \sigma $-porous set.


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Additional Information

Luděk Zajíček
Affiliation: Charles University, Faculty of Mathematics and Physics, Sokolovská 83, 186 75 Praha 8-Karlín, Czech Republic
Email: zajicek@karlin.mff.cuni.cz

DOI: https://doi.org/10.1090/S0002-9939-2014-12228-3
Keywords: Hadamard differentiability, G\^ateaux differentiability, Fr\'echet differentiability, $\sigma$-porous set, $\sigma$-directionally porous set, Stepanoff theorem, Aronszajn null set
Received by editor(s): October 10, 2012
Received by editor(s) in revised form: March 27, 2013
Published electronically: August 29, 2014
Additional Notes: This research was supported by the grant GAČR P201/12/0436.
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.