On a question of H.H. Corson and some related problems

Pol, Roman

doi:10.4064/fm-109-2-143-154

Cited by 34 publications

(16 citation statements)

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“…PROPOSITION Si done Γespace ((^(JBΓ) /JΓ )*«-*) est separable, Γespace (X 1 , «**) qui s'y identifie est separable. L'espace X est done une intersection denombrable d'hyperplans fermes de ^{K) 9 …”

Section: • Espace Des Fonctions Continues Sur Un Compact Deunclassified

“…Tout espace de Banach separable a clairement la propriete (^). La propriete (^) a ete etudiee par E. Pol dans [9], qui a montre en particulier que Γespace des functions continues sur Γintervalle eclate a la propriete (^), et qui a pose la question suivante: Γespace des functions continues sur le "compact de Helly" α-t'il la propriete (^)? On va repondre positivement a cette question, en montrant que la propriete est veriίiee pour tout compact de Rosenthal.…”

Section: • Espace Des Fonctions Continues Sur Un Compact Deunclassified

“…Soit E ne sous-algebre unitaire fermee de s°°(N) 9 (E, Cyl(E)) est un espace mesurable qui n'est pas a base denombrable.…”

Section: Sous-algebres De S°°{n)unclassified

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Compacts de Rosenthal

Godefroy¹

1980

Pacific J. Math.

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Section: • Espace Des Fonctions Continues Sur Un Compact Deunclassified

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Compacts de Rosenthal

Godefroy¹

1980

Pacific J. Math.

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“…Since K is a James boundary for K A Banach space X or more generally a convex subset M of X is said to have property C (after Corson) if each collection of relatively closed convex subsets of M with empty intersection has a countable subcollection with empty intersection. Since closed convex sets in X are also weakly closed, if (M, w) is Lindelöf then M has property C. A good reference for property C is [43].…”

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confidence: 99%

James boundaries and σ-fragmented selectors

2008

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Abstract. We study the boundary structure for w * -compact subsets of dual Banach spaces. To be more precise, for a Banach space X, 0 < ε < 1 and a subset T of the dual space X * such that S {B(t, ε) : t ∈ T } contains a James boundary for BX * we study different kinds of conditions on T , besides T being countable, which ensure thatWe analyze two different non-separable cases where the equality (SP) holds: (a) if J : X → 2 B X * is the duality mapping and there exists a σ-fragmented map f : X → X * such that B(f (x), ε) ∩ J(x) = ∅ for every x ∈ X, then (SP) holds for T = f (X) and in this case X is Asplund; (b) if T is weakly countably K-determined then (SP) holds, X * is weakly countably K-determined and moreover for every James boundary B of BX * we have BX * = co(B)· . Both approaches use Simons' inequality and ideas exploited by Godefroy in the separable case (i.e., when T is countable). While proving (a) we show that X is Asplund if, and only if, the duality mapping has an ε-selector, 0 < ε < 1, that sends separable sets into separable ones. A consequence is that the dual unit ball BX * is norm fragmented if, and only if, it is norm ε-fragmented for some fixed 0 < ε < 1. Our analysis is completed by a characterization of those Banach spaces (not necessarily separable) without copies of 1 via the structure of the boundaries of w * -compact sets of their duals. Several applications and complementary results are proved. Our results extend to the non-separable case results by Godefroy, Contreras-Payá and Rodé.

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“…[13]). In such a setting, it is perhaps more natural to consider property (C) of Corson rather than weak Lindelöfness itself (see [17]). …”

Section: Introductionmentioning

confidence: 99%