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MAZUR INTERSECTION PROPERTIES AND DIFFERENTIABILITY OF CONVEX FUNCTIONS IN BANACH SPACES

Published online by Cambridge University Press:  01 April 2000

P. G. GEORGIEV
Affiliation:
Faculty of Mathematics and Informatics, Sofia University, St. Kl. Ohridski, Sofia, Bulgaria; [email protected]
A. S. GRANERO
Affiliation:
Departamento de Análisis Matemático, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Madrid 28040, Spain; [email protected], [email protected]
M. JIMÉNEZ SEVILLA
Affiliation:
Departamento de Análisis Matemático, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Madrid 28040, Spain; [email protected], [email protected]
J. P. MORENO
Affiliation:
Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, Madrid 28049, Spain; [email protected]
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Abstract

It is proved that the dual of a Banach space with the Mazur intersection property is almost weak* Asplund. Analogously, the predual of a dual space with the weak* Mazur intersection property is almost Asplund. Through the use of these arguments, it is found that, in particular, almost all (in the Baire sense) equivalent norms on [lscr ]1(Γ) and [lscr ](Γ) are Fréchet differentiable on a dense Gδ subset. Necessary conditions for Mazur intersection properties in terms of convex sets satisfying a Krein–Milman type condition are also discussed. It is also shown that, if a Banach space has the Mazur intersection property, then every subspace of countable codimension can be equivalently renormed to satisfy this property.

Type
Research Article
Copyright
The London Mathematical Society 2000

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