Rolle’s Theorem and Negligibility of Points in Infinite Dimensional Banach Spaces
In this note we prove that if a differentiable function oscillates between y« and « on the boundary of the unit ball then there exists a point in the interior of the ball in which the differential of the function has norm equal or less than« . This kind of approximate Rolle’s theorem is interesting...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 1997 |
| País: | España |
| Institución: | Universidad Complutense de Madrid (UCM) |
| Repositorio: | Docta Complutense |
| Idioma: | inglés |
| OAI Identifier: | oai:docta.ucm.es:20.500.14352/57128 |
| Acceso en línea: | https://hdl.handle.net/20.500.14352/57128 |
| Access Level: | acceso abierto |
| Palabra clave: | 517.98 Rolle’s theorem in infinite-dimensional Banach spaces Approximate Rolle’s theorem Continuous norm whose dual norm is locally uniformly rotund C1 bump function Análisis funcional y teoría de operadores |
| Sumario: | In this note we prove that if a differentiable function oscillates between y« and « on the boundary of the unit ball then there exists a point in the interior of the ball in which the differential of the function has norm equal or less than« . This kind of approximate Rolle’s theorem is interesting because an exact Rolle’s theorem does not hold in many infinite dimensional Banach spaces. A characterization of those spaces in which Rolle’s theorem does not hold is given within a large class of Banach spaces. This question is closely related to the existence of C1 diffeomorphisms between a Banach space X and X _ _04 which are the identity out of a ball, and we prove that such diffeomorphisms exist for every C1 smooth Banach space which can be linearly injected into a Banach space whose dual norm is locally uniformly rotund (LUR). |
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