2026-06-29T07:24:08Zhttps://docta.ucm.es/rest/oai/request

oai:docta.ucm.es:20.500.14352/571292025-01-20T16:57:37Zcom_20.500.14352_14col_20.500.14352_15

The Failure of Rolle's Theorem in Infinite-Dimensional Banach Spaces Azagra Rueda, Daniel Jiménez Sevilla, María Del Mar 517.98 Negligibility Rolle theorem Smooth norm Brouwer fixed point theorem Bump Análisis funcional y teoría de operadores We prove the following new characterization of Cp Lipschitz) smoothness in Banach spaces. An infinite-dimensional Banach space X has a Cp smooth (Lipschitz) bump function if and only if it has another Cp smooth (Lipschitz) bump function f such that its derivative does not vanish at any point in the interior of the support of f (that is, f does not satisfy Rolle's theorem). Moreover, the support of this bump can be assumed to be a smooth starlike body. The ``twisted tube'' method we use in the proof is interesting in itself, as it provides other useful characterizations of Cp smoothness related to the existence of a certain kind of deleting diffeomorphisms, as well as to the failure of Brouwer's fixed point theorem even for smooth self-mappings of starlike bodies in all infinite-dimensional spaces. Depto. de Análisis Matemático y Matemática Aplicada Fac. de Ciencias Matemáticas TRUE pub 2023-06-20T16:49:13Z 2023-06-20T16:49:13Z 2001-05-10 journal article https://hdl.handle.net/20.500.14352/57129 0022-1236 10.1006_jfan.2000.3709 eng open access application/pdf Elsevier