On the Fréchet differentiability of distance functions
Zajíček, L.
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Zajíček, L.
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Marek Cúth (2012)
Fundamenta Mathematicae
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We simplify the presentation of the method of elementary submodels and we show that it can be used to simplify proofs of existing separable reduction theorems and to obtain new ones. Given a nonseparable Banach space X and either a subset A ⊂ X or a function f defined on X, we are able for certain properties to produce a separable subspace of X which determines whether A or f has the property in question. Such results are proved for properties of sets: of being dense, nowhere dense,...
Pandelis Dodos (2008)
Fundamenta Mathematicae
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It is proved that the class of separable Rosenthal compacta on the Cantor set having a uniformly bounded dense sequence of continuous functions is strongly bounded.
Elżbieta Pol (1978)
Colloquium Mathematicae
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Robert Moore (1926)
Fundamenta Mathematicae
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The purpose of this paper is to establish two theorems: Theoreme: In order that every subclass of a given class D of Fréchet should be separable it is necessary and sufficient that every uncountable subclass of that class D should have a limit point. Theoreme: If D_s is a separable class D then every uncountable subclass of D_s contains a point of condensation.
Stefan Rolewicz (2005)
Control and Cybernetics
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Mohammed Yahdi (1998)
Revista Matemática Complutense
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Let C(X) be the set of all convex and continuous functions on a separable infinite dimensional Banach space X, equipped with the topology of uniform convergence on bounded subsets of X. We show that the subset of all convex Fréchet-differentiable functions on X, and the subset of all (not necessarily equivalent) Fréchet-differentiable norms on X, reduce every coanalytic set, in particular they are not Borel-sets.
Ljubiša Kočinac (1989)
Publications de l'Institut Mathématique
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Wee-Kee Tang (1995)
Commentationes Mathematicae Universitatis Carolinae
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Equivalent conditions for the separability of the range of the subdifferential of a given convex Lipschitz function defined on a separable Banach space are studied. The conditions are in terms of a majorization of by a -smooth function, separability of the boundary for or an approximation of by Fréchet smooth convex functions.