On the range of the derivatives of a smooth function between
Banach spacesy
dc.contributor.author | Azagra Rueda, Daniel | |
dc.contributor.author | Jiménez Sevilla, María Del Mar | |
dc.contributor.author | Deville, Robert | |
dc.date.accessioned | 2023-06-20T09:30:41Z | |
dc.date.available | 2023-06-20T09:30:41Z | |
dc.date.issued | 2003-01 | |
dc.description.abstract | We study the size of the range of the derivatives of a smooth function between Banach spaces. We establish conditions on a pair of Banach spaces X and Y to ensure the existence of a C-p smooth (Frechet smooth or a continuous G (a) over cap teaux smooth) function f from X onto Y such that f vanishes outside a bounded set and all the derivatives of f are surjections. In particular we deduce the following results. For the Gateaux case, when X and Y are separable and X is infinite-dimensional, there exists a continuous G (a) over cap teaux smooth function f from X to Y, with bounded support, so that f'(X) = L (X, Y). In the Frechet case, we get that if a Banach space X has a Frechet smooth bump and dens X = dens L(X, Y), then there is a Frechet smooth function f: X --> Y with bounded support so that f'(X) = L(X, Y). Moreover, we see that if X has a C-p smooth bump with bounded derivatives and dens X = dens L-s(m) (X; Y) then there exists another C-p smooth function f : X --> Y so that f((k)) (X) = L-s(k) (X; Y) for all k = 0,1,...,m. As an application, we show that every bounded starlike body on a separable Banach space X with a (Frechet or G (a) over cap teaux) smooth bump can be uniformly approximated by smooth bounded starlike bodies whose cones of tangent hyperplanes fill the dual space X-*. In the non-separable case, we prove that X has such property if X has smooth partitions of unity. | |
dc.description.department | Depto. de Análisis Matemático y Matemática Aplicada | |
dc.description.faculty | Fac. de Ciencias Matemáticas | |
dc.description.refereed | TRUE | |
dc.description.sponsorship | DGICYT | |
dc.description.sponsorship | BFM | |
dc.description.status | pub | |
dc.eprint.id | https://eprints.ucm.es/id/eprint/13961 | |
dc.identifier.doi | 10.1017/S0305004102006278 | |
dc.identifier.issn | 1469-8064 | |
dc.identifier.officialurl | http://journals.cambridge.org/action/displayJournal?jid=PSP | |
dc.identifier.uri | https://hdl.handle.net/20.500.14352/49769 | |
dc.issue.number | 1 | |
dc.journal.title | Mathematical Proceedings of the Cambridge Philosophical Society | |
dc.language.iso | eng | |
dc.page.final | 185 | |
dc.page.initial | 163 | |
dc.publisher | Cambridge University Press | |
dc.relation.projectID | PB96-0607 | |
dc.relation.projectID | 2000-0609 | |
dc.rights.accessRights | open access | |
dc.subject.cdu | 517.98 | |
dc.subject.keyword | Smooth bump | |
dc.subject.keyword | Smooth surjection | |
dc.subject.keyword | Banach space | |
dc.subject.keyword | Smooth body | |
dc.subject.keyword | Derivative | |
dc.subject.ucm | Análisis funcional y teoría de operadores | |
dc.title | On the range of the derivatives of a smooth function between Banach spacesy | |
dc.type | journal article | |
dc.volume.number | 134 | |
dspace.entity.type | Publication | |
relation.isAuthorOfPublication | 6696556b-dc2e-4272-8f5f-fa6a7a2f5344 | |
relation.isAuthorOfPublication | 36c2a4e7-ac6d-450d-b64c-692a94ff6361 | |
relation.isAuthorOfPublication.latestForDiscovery | 6696556b-dc2e-4272-8f5f-fa6a7a2f5344 |
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