| Title:
|
Remarks on Fréchet differentiability of pointwise Lipschitz, cone-monotone and quasiconvex functions (English) |
| Author:
|
Zajíček, Luděk |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
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55 |
| Issue:
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2 |
| Year:
|
2014 |
| Pages:
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203-213 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
cone-monotone function |
| Keyword:
|
Fréchet differentiability |
| Keyword:
|
Gâteaux differentiability |
| Keyword:
|
pointwise Lipschitz function |
| Keyword:
|
$\Gamma$-null set |
| Keyword:
|
quasiconvex function |
| Keyword:
|
separable reduction |
| MSC:
|
46G05 |
| MSC:
|
47H07 |
| MSC:
|
49J50 |
| MSC:
|
58C20 |
| idZBL:
|
Zbl 06391538 |
| idMR:
|
MR3193926 |
| . |
| Date available:
|
2014-06-07T15:36:09Z |
| Last updated:
|
2016-07-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143802 |
| . |
| Reference:
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