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Title: Fréchet differentiability via partial Fréchet differentiability (English)
Author: Zajíček, Luděk
Language: English
Journal: Commentationes Mathematicae Universitatis Carolinae
ISSN: 0010-2628 (print)
ISSN: 1213-7243 (online)
Volume: 64
Issue: 2
Year: 2023
Pages: 185-207
Summary lang: English
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Category: math
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Summary: 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 $f\colon X \to Y$ 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. (English)
Keyword: Fréchet differentiability
Keyword: partial Fréchet differentiability
Keyword: first category set
Keyword: Asplund space
Keyword: $\sigma$-porous set
MSC: 46G05
MSC: 46T20
idZBL: Zbl 07790591
idMR: MR4658999
DOI: 10.14712/1213-7243.2023.025
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Date available: 2023-12-13T13:38:14Z
Last updated: 2024-02-13
Stable URL: http://hdl.handle.net/10338.dmlcz/151856
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