Article
| 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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