Fréchet differentiability via partial Fréchet differentiability

Luděk Zajíček

Commentationes Mathematicae Universitatis Carolinae (2023)

  • Volume: 64, Issue: 2, page 185-207
  • ISSN: 0010-2628

Abstract

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Let X 1 , ⋯ , X n be Banach spaces and f a real function on X = X 1 × ⋯ × X n . Let A f be the set of all points x ∈ X at which f is partially Fréchet differentiable but is not Fréchet differentiable. Our results imply that if X 1 , ⋯ , X n - 1 are Asplund spaces and f is continuous (respectively Lipschitz) on X , then A f is a first category set (respectively a σ -upper porous set). We also prove that if X , Y are separable Banach spaces and f : X → Y is a Lipschitz mapping, then there exists a σ -upper porous set A ⊂ X such that f is Fréchet differentiable at every point x ∈ X ∖ A at which it is Fréchet differentiable along a closed subspace of finite codimension and Gâteaux differentiable. A number of related more general results are also proved.

How to cite

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Zajíček, Luděk. "Fréchet differentiability via partial Fréchet differentiability." Commentationes Mathematicae Universitatis Carolinae 64.2 (2023): 185-207. <http://eudml.org/doc/299154>.

@article{Zajíček2023,
abstract = {Let $X_1, \dots , X_n$ be Banach spaces and $f$ a real function on $X=X_1 \times \dots \times X_n$. Let $A_f$ be the set of all points $x \in X$ at which $f$ is partially Fréchet differentiable but is not Fréchet differentiable. Our results imply that if $X_1, \dots , X_\{n-1\}$ are Asplund spaces and $f$ is continuous (respectively Lipschitz) on $X$, then $A_f$ is a first category set (respectively a $\sigma $-upper porous set). We also prove that if $X$, $Y$ are separable Banach spaces and $f\colon X \rightarrow Y$ is a Lipschitz mapping, then there exists a $\sigma $-upper porous set $A \subset X$ such that $f$ is Fréchet differentiable at every point $x \in X \setminus A$ at which it is Fréchet differentiable along a closed subspace of finite codimension and Gâteaux differentiable. A number of related more general results are also proved.},
author = {Zajíček, Luděk},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Fréchet differentiability; partial Fréchet differentiability; first category set; Asplund space; $\sigma $-porous set},
language = {eng},
number = {2},
pages = {185-207},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Fréchet differentiability via partial Fréchet differentiability},
url = {http://eudml.org/doc/299154},
volume = {64},
year = {2023},
}

TY - JOUR
AU - Zajíček, Luděk
TI - Fréchet differentiability via partial Fréchet differentiability
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2023
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 64
IS - 2
SP - 185
EP - 207
AB - Let $X_1, \dots , X_n$ be Banach spaces and $f$ a real function on $X=X_1 \times \dots \times X_n$. Let $A_f$ be the set of all points $x \in X$ at which $f$ is partially Fréchet differentiable but is not Fréchet differentiable. Our results imply that if $X_1, \dots , X_{n-1}$ are Asplund spaces and $f$ is continuous (respectively Lipschitz) on $X$, then $A_f$ is a first category set (respectively a $\sigma $-upper porous set). We also prove that if $X$, $Y$ are separable Banach spaces and $f\colon X \rightarrow Y$ is a Lipschitz mapping, then there exists a $\sigma $-upper porous set $A \subset X$ such that $f$ is Fréchet differentiable at every point $x \in X \setminus A$ at which it is Fréchet differentiable along a closed subspace of finite codimension and Gâteaux differentiable. A number of related more general results are also proved.
LA - eng
KW - Fréchet differentiability; partial Fréchet differentiability; first category set; Asplund space; $\sigma $-porous set
UR - http://eudml.org/doc/299154
ER -

References

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