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On the range of the derivatives of a smooth function between Banach spacesy

dc.contributor.authorAzagra Rueda, Daniel
dc.contributor.authorJiménez Sevilla, María Del Mar
dc.contributor.authorDeville, Robert
dc.date.accessioned2023-06-20T09:30:41Z
dc.date.available2023-06-20T09:30:41Z
dc.date.issued2003-01
dc.description.abstractWe 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.departmentDepto. de Análisis Matemático y Matemática Aplicada
dc.description.facultyFac. de Ciencias Matemáticas
dc.description.refereedTRUE
dc.description.sponsorshipDGICYT
dc.description.sponsorshipBFM
dc.description.statuspub
dc.eprint.idhttps://eprints.ucm.es/id/eprint/13961
dc.identifier.doi10.1017/S0305004102006278
dc.identifier.issn1469-8064
dc.identifier.officialurlhttp://journals.cambridge.org/action/displayJournal?jid=PSP
dc.identifier.urihttps://hdl.handle.net/20.500.14352/49769
dc.issue.number1
dc.journal.titleMathematical Proceedings of the Cambridge Philosophical Society
dc.language.isoeng
dc.page.final185
dc.page.initial163
dc.publisherCambridge University Press
dc.relation.projectIDPB96-0607
dc.relation.projectID2000-0609
dc.rights.accessRightsopen access
dc.subject.cdu517.98
dc.subject.keywordSmooth bump
dc.subject.keywordSmooth surjection
dc.subject.keywordBanach space
dc.subject.keywordSmooth body
dc.subject.keywordDerivative
dc.subject.ucmAnálisis funcional y teoría de operadores
dc.titleOn the range of the derivatives of a smooth function between Banach spacesy
dc.typejournal article
dc.volume.number134
dspace.entity.typePublication
relation.isAuthorOfPublication6696556b-dc2e-4272-8f5f-fa6a7a2f5344
relation.isAuthorOfPublication36c2a4e7-ac6d-450d-b64c-692a94ff6361
relation.isAuthorOfPublication.latestForDiscovery6696556b-dc2e-4272-8f5f-fa6a7a2f5344

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