An almost nowhere Fréchet smooth norm on superreflexive spaces
Every separable infinite-dimensional superreflexive Banach space admits an equivalent norm which is Fréchet differentiable only on an Aronszajn null set.
Concerning weak-extreme points
Every separable nonreflexive Banach space admits an equivalent norm such that the set of the weak-extreme points of the unit ball is discrete.
The Banach-Saks property and Haar null sets
A characterization of Haar null sets in the sense of Christensen is given. Using it, we show that if the dual of a Banach space has the Banach-Saks property, then closed and convex subsets of with empty interior are Haar null.
One counterexample concerning the Fréchet differentiability of convex functions on closed sets
How small are the sets where the metric projection fails to be continuous
Reflexivity and approximate fixed points
A Banach space X is reflexive if and only if every bounded sequence xₙ in X contains a norm attaining subsequence. This means that it contains a subsequence for which is attained at some f in the dual unit sphere . A Banach space X is not reflexive if and only if it contains a normalized sequence xₙ with the property that for every , there exists such that . Combining this with a result of Shafrir, we conclude that every infinite-dimensional Banach space contains an unbounded closed convex...
A note on intersections of non-Haar null sets
We show that in every Polish, abelian, non-locally compact group G there exist non-Haar null sets A and B such that the set {g ∈ G; (g+A) ∩ B is non-Haar null} is empty. This answers a question posed by Christensen.
Second order differentiability and Lipschitz smooth points of convex functionals
Remarks on continuous images of Radon-Nikodým compacta
A family of compact spaces containing continuous images of Radon-Nikod’ym compacta is introduced and studied. A family of Banach spaces containing subspaces of Asplund generated (i.e., GSG) spaces is introduced and studied. Further, for a continuous image of a Radon-Nikod’ym compact we prove: If is totally disconnected, then it is Radon-Nikod’ym compact. If is adequate, then it is even Eberlein compact.
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