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On the Kunen-Shelah properties in Banach spaces



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Granero, A. S. and Jiménez Sevilla, María del Mar and Montesinos, Alejandro and Moreno, José Pedro and Plichko, Anatolij (2003) On the Kunen-Shelah properties in Banach spaces. Studia Mathematica, 157 (2). pp. 97-120. ISSN 0039-3223

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We introduce and study the Kunen-Shelah properties KSi, i = 0, 1,..., 7. Let us highlight for a Banach space X some of our results: (1) X ∗ has a w ∗-nonseparable equivalent dual ball iff X has an ω1-polyhedron (i.e., a bounded family {xi}i<ω1 such that xj / ∈ co({xi: i ∈ ω1 \ {j}}) for every j ∈ ω1) iff X has an uncountable bounded almost biorthonal system (UBABS) of type η, for some η ∈ [0, 1), (i.e., a bounded family {(xα, fα)}1≤α<ω1 ⊂ X × X ∗ such that fα(xα) = 1 and |fα(xβ) | ≤ η, if α = β); (2) if X has an uncountable ω-independent system then X has an UBABS of type η for every η ∈ (0, 1); (3) if X has not the property (C) of Corson, then X has an ω1-polyhedron; (4) X has not an ω1-polyhedron iff X has not a convex right-separated ω1-family (i.e., a bounded family {xi}i<ω1 such that xj / ∈ co({xi: j < i < ω1}) for every j ∈ ω1) iff every w ∗-closed convex subset of X ∗ is w ∗-separable iff every convex subset of X ∗ is w ∗-separable iff µ(X) = 1, µ(X) being the Finet-Godefroy index of X (see [1]).

Item Type:Article
Uncontrolled Keywords:Uncountable basic sequences, Biorthogonal and Markuschevich systems, W-independence, Kunen-Shelah properties.
Subjects:Sciences > Mathematics > Topology
ID Code:22119
Deposited On:26 Jun 2013 17:59
Last Modified:25 Jun 2018 07:05

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