On the range of the derivative of a smooth mapping between Banach spaces
We survey recent results on the structure of the range of the derivative of a smooth mapping f between two Banach spaces X and Y. We recall some necessary conditions and some sufficient conditions on a subset A of ℒ(X,Y) for the existence of a Fréchet differentiable mapping f from X into Y so that f...
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Hindawi Limited
2005-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/AAA.2005.499 |
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doaj-0ff7fa63388e433eb584fa096300d74b2020-11-24T22:01:17ZengHindawi LimitedAbstract and Applied Analysis1085-33751687-04092005-01-012005549950710.1155/AAA.2005.499On the range of the derivative of a smooth mapping between Banach spacesRobert Deville0Laboratoire Bordelais d'Analyse et Geométrie, Institut de Mathématiques de Bordeaux, Université de Bordeaux 1, 351 cours de la Libération, Talence Cedex 33405, FranceWe survey recent results on the structure of the range of the derivative of a smooth mapping f between two Banach spaces X and Y. We recall some necessary conditions and some sufficient conditions on a subset A of ℒ(X,Y) for the existence of a Fréchet differentiable mapping f from X into Y so that f′(X)=A. Whenever f is only assumed Gâteaux differentiable, new phenomena appear: for instance, there exists a mapping f from ℓ1(ℕ) into ℝ2, which is bounded, Lipschitz-continuous, and so that for all x,y∈ℓ1(ℕ), if x≠y, then ‖f′(x)−f′(y)‖>1.http://dx.doi.org/10.1155/AAA.2005.499 |
| collection |
DOAJ |
| language |
English |
| format |
Article |
| sources |
DOAJ |
| author |
Robert Deville |
| spellingShingle |
Robert Deville On the range of the derivative of a smooth mapping between Banach spaces Abstract and Applied Analysis |
| author_facet |
Robert Deville |
| author_sort |
Robert Deville |
| title |
On the range of the derivative of a smooth mapping between Banach spaces |
| title_short |
On the range of the derivative of a smooth mapping between Banach spaces |
| title_full |
On the range of the derivative of a smooth mapping between Banach spaces |
| title_fullStr |
On the range of the derivative of a smooth mapping between Banach spaces |
| title_full_unstemmed |
On the range of the derivative of a smooth mapping between Banach spaces |
| title_sort |
on the range of the derivative of a smooth mapping between banach spaces |
| publisher |
Hindawi Limited |
| series |
Abstract and Applied Analysis |
| issn |
1085-3375 1687-0409 |
| publishDate |
2005-01-01 |
| description |
We survey recent results on the structure of the range of the derivative of a smooth mapping f between two Banach spaces X and Y. We recall some necessary conditions and some sufficient conditions on a subset A of ℒ(X,Y) for the existence of a Fréchet differentiable mapping f from X into Y so that f′(X)=A. Whenever f is only assumed Gâteaux differentiable, new phenomena appear: for instance,
there exists a mapping f from ℓ1(ℕ) into ℝ2, which is bounded, Lipschitz-continuous, and so that for all x,y∈ℓ1(ℕ), if x≠y, then ‖f′(x)−f′(y)‖>1. |
| url |
http://dx.doi.org/10.1155/AAA.2005.499 |
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AT robertdeville ontherangeofthederivativeofasmoothmappingbetweenbanachspaces |
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