Abstract
LetD([0, 1]) be the space of left continuous real valued functions on [0, 1] which have a right limit at each point. We show thatD([0, 1]) has no equivalent norm which is Gâteau differentiable. Hence the class of spaces which can be renormed by a Gâteau differentiable norm fails the three spaces property. We show that there is no norm onℒ([0, Ω]) such that its dual is strictly convex. However, there is an equivalent Fréchet differentiable norm on this space.
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Références
J. Diestel,Geometry of Banach Spaces, Selected Topics, Lecture Notes in Math.485, Springer-Verlag, Berlin, 1975.
M. Talagrand,Deux exemples de fonctions convexes, C. R. Acad. Sci. Paris288 (1979), 461–464.
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Talagrand, M. Renormages de Quelquesℒ(K). Israel J. Math. 54, 327–334 (1986). https://doi.org/10.1007/BF02764961
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DOI: https://doi.org/10.1007/BF02764961