| 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 |
| . |
| Category:
|
math |
| . |
| 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 |
| . |
| Date available:
|
2023-12-13T13:38:14Z |
| Last updated:
|
2025-07-07 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/151856 |
| . |
| Reference:
|
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