Remarks on Fréchet differentiability of pointwise Lipschitz, cone-monotone and quasiconvex functions

Luděk Zajíček

Commentationes Mathematicae Universitatis Carolinae (2014)

  • Volume: 55, Issue: 2, page 203-213
  • ISSN: 0010-2628

Abstract

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We present some consequences of a deep result of J. Lindenstrauss and D. Preiss on Γ -almost everywhere Fréchet differentiability of Lipschitz functions on c 0 (and similar Banach spaces). For example, in these spaces, every continuous real function is Fréchet differentiable at Γ -almost every x at which it is Gâteaux differentiable. Another interesting consequences say that both cone-monotone functions and continuous quasiconvex functions on these spaces are Γ -almost everywhere Fréchet differentiable. In the proofs we use a general observation that each version of the Rademacher theorem for real functions on Banach spaces (i.e., a result on a.e. Fréchet or Gâteaux differentiability of Lipschitz functions) easily implies by a method of J. Malý a corresponding version of the Stepanov theorem (on a.e. differentiability of pointwise Lipschitz functions). Using the method of separable reduction, we extend some results to several non-separable spaces.

How to cite

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Zajíček, Luděk. "Remarks on Fréchet differentiability of pointwise Lipschitz, cone-monotone and quasiconvex functions." Commentationes Mathematicae Universitatis Carolinae 55.2 (2014): 203-213. <http://eudml.org/doc/261857>.

@article{Zajíček2014,
abstract = {We present some consequences of a deep result of J. Lindenstrauss and D. Preiss on $\Gamma $-almost everywhere Fréchet differentiability of Lipschitz functions on $c_0$ (and similar Banach spaces). For example, in these spaces, every continuous real function is Fréchet differentiable at $\Gamma $-almost every $x$ at which it is Gâteaux differentiable. Another interesting consequences say that both cone-monotone functions and continuous quasiconvex functions on these spaces are $\Gamma $-almost everywhere Fréchet differentiable. In the proofs we use a general observation that each version of the Rademacher theorem for real functions on Banach spaces (i.e., a result on a.e. Fréchet or Gâteaux differentiability of Lipschitz functions) easily implies by a method of J. Malý a corresponding version of the Stepanov theorem (on a.e. differentiability of pointwise Lipschitz functions). Using the method of separable reduction, we extend some results to several non-separable spaces.},
author = {Zajíček, Luděk},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {cone-monotone function; Fréchet differentiability; Gâteaux differentiability; pointwise Lipschitz function; $\Gamma $-null set; quasiconvex function; separable reduction; cone-monotone function; Fréchet differentiability; Gâteaux differentiability; pointwise Lipschitz function; -null set; quasiconvex function; separable reduction},
language = {eng},
number = {2},
pages = {203-213},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Remarks on Fréchet differentiability of pointwise Lipschitz, cone-monotone and quasiconvex functions},
url = {http://eudml.org/doc/261857},
volume = {55},
year = {2014},
}

TY - JOUR
AU - Zajíček, Luděk
TI - Remarks on Fréchet differentiability of pointwise Lipschitz, cone-monotone and quasiconvex functions
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2014
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 55
IS - 2
SP - 203
EP - 213
AB - We present some consequences of a deep result of J. Lindenstrauss and D. Preiss on $\Gamma $-almost everywhere Fréchet differentiability of Lipschitz functions on $c_0$ (and similar Banach spaces). For example, in these spaces, every continuous real function is Fréchet differentiable at $\Gamma $-almost every $x$ at which it is Gâteaux differentiable. Another interesting consequences say that both cone-monotone functions and continuous quasiconvex functions on these spaces are $\Gamma $-almost everywhere Fréchet differentiable. In the proofs we use a general observation that each version of the Rademacher theorem for real functions on Banach spaces (i.e., a result on a.e. Fréchet or Gâteaux differentiability of Lipschitz functions) easily implies by a method of J. Malý a corresponding version of the Stepanov theorem (on a.e. differentiability of pointwise Lipschitz functions). Using the method of separable reduction, we extend some results to several non-separable spaces.
LA - eng
KW - cone-monotone function; Fréchet differentiability; Gâteaux differentiability; pointwise Lipschitz function; $\Gamma $-null set; quasiconvex function; separable reduction; cone-monotone function; Fréchet differentiability; Gâteaux differentiability; pointwise Lipschitz function; -null set; quasiconvex function; separable reduction
UR - http://eudml.org/doc/261857
ER -

References

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