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Journal of Functional Analysis 182, 207 226 (2001) The Failure of Rolle's Theorem in Infinite-Dimensional
 

Summary: Journal of Functional Analysis 182, 207 226 (2001)
The Failure of Rolle's Theorem in Infinite-Dimensional
Banach Spaces
Daniel Azagra and Mar Jimenez-Sevilla
Departamento de Analisis Matematico, Facultad de Ciencias Matematicas,
Universidad Complutense, 28040 Madrid, Spain; and Equipe d'Analyse,
Universite Pierre et Marie Curie Paris 6, 4, Place Jussieu, 75005 Paris, France
E-mail: daniel─sunam1.mat.ucm.es, azagra─ccr.jussieu.fr, marjim─sunam1.mat.ucm.es,
marjim─ccr.jussieu.fr
Communicated by G. Pisier
Received July 18, 2000; revised October 10, 2000; accepted October 25, 2000
We prove the following new characterization of C p
(Lipschitz) smoothness in
Banach spaces. An infinite-dimensional Banach space X has a C p
smooth (Lipschitz)
bump function if and only if it has another C p
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

  

Source: Azagra Rueda, Daniel - Facultad de Ciencias Matemßticas, Universidad Complutense de Madrid

 

Collections: Mathematics