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Characterizations of almost transitive superreflexive Banach spaces

Julio Becerra Guerrero, Angel Rodriguez Palacios (2001)

Commentationes Mathematicae Universitatis Carolinae

Almost transitive superreflexive Banach spaces have been considered in [7] (see also [4] and [6]), where it is shown that such spaces are uniformly convex and uniformly smooth. We prove that convex transitive Banach spaces are either almost transitive and superreflexive (hence uniformly smooth) or extremely rough. The extreme roughness of a Banach space X means that, for every element u in the unit sphere of X , we have lim sup ∥ h ∥ → 0 ∥ u + h ∥ + ∥ u - h ∥ - 2 ∥ h ∥ = 2 . We note that, in general, the property of convex transitivity for a Banach...

Continuous version of the Choquet integral representation theorem

Piotr Puchała (2005)

Studia Mathematica

Let E be a locally convex topological Hausdorff space, K a nonempty compact convex subset of E, μ a regular Borel probability measure on E and γ > 0. We say that the measure μ γ-represents a point x ∈ K if s u p | | f | | ≤ 1 | f ( x ) - ∫ K f d μ | < γ for any f ∈ E*. In this paper a continuous version of the Choquet theorem is proved, namely, if P is a continuous multivalued mapping from a metric space T into the space of nonempty, bounded convex subsets of a Banach space X, then there exists a weak* continuous family ( μ t ) of regular Borel...

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