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Mazur Intersection Properties and Differentiability of Convex Functions in Banach Spaces
Author(s) -
Georgiev P. G.,
Granero A. S.,
Jiménez Sevilla M.,
Moreno J. P.
Publication year - 2000
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610799008625
Subject(s) - mathematics , banach space , intersection (aeronautics) , pure mathematics , codimension , countable set , regular polygon , property (philosophy) , baire space , space (punctuation) , differentiable function , subspace topology , mathematical analysis , philosophy , linguistics , geometry , engineering , aerospace engineering , epistemology
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 l 1 (Γ) and l ∞ (Γ) are Fréchet differentiable on a dense G δ 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.