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James boundaries and σ-fragmented selectors

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Cascales B., Muñoz M., Orihuela J.

Studia Mathematica

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2008

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tom 188

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nr 2

97-122

EN

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 ⋃ {B(t,ϵ): t ∈ T} contains a James boundary for $B_{X*}$ we study different kinds of conditions on T, besides T being countable, which ensure that $X* = \overline{span T}^{||·||}$. (SP) We 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 $B_{X*}$ we have $B_{X*} = \overline{co(B)}^{||·||}$. 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 $B_{X*}$ 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 ℓ¹ 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.

2

The Lindelöf property in Banach spaces

100%

Cascales B., Namioka I., Orihuela J.

Studia Mathematica

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2003

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tom 154

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nr 2

165-192

EN

A topological space (T,τ) is said to be fragmented by a metric d on T if each non-empty subset of T has non-empty relatively open subsets of arbitrarily small d-diameter. The basic theorem of the present paper is the following. Let (M,ϱ) be a metric space with ϱ bounded and let D be an arbitrary index set. Then for a compact subset K of the product space $M^{D}$ the following four conditions are equivalent: (i) K is fragmented by $d_{D}$, where, for each S ⊂ D, $d_{S}(x,y) = sup{ϱ(x(t),y(t)): t∈ S}$. (ii) For each countable subset A of D, $(K,d_{A})$ is separable.i (iii) The space (K,γ(D)) is Lindelöf, where γ(D) is the topology of uniform convergence on the family of countable subsets of D. (iv) $(K,γ(D))^{{ℕ}}$ is Lindelöf.The rest of the paper is devoted to applications of the basic theorem. Here are some of them. A compact Hausdorff space K is Radon-Nikodým compact if, and only if, there is a bounded subset D of C(K) separating the points of K such that (K,γ(D)) is Lindelöf. If X is a Banach space and H is a weak*-compact subset of the dual X* which is weakly Lindelöf, then $(H,{weak})^{ℕ}$ is Lindelöf. Furthermore, under the same condition $\overline{span(H)}^{|| ||}$ and $\overline{co(H)}^{w*}$ are weakly Lindelöf. The last conclusion answers a question by Talagrand. Finally we apply our basic theorem to certain classes of Banach spaces including weakly compactly generated ones and the duals of Asplund spaces.

3

Every Radon-Nikodym Corson compact space is Eberlein compact

87%

Orihuela J., Schachermayer W., Valdivia M.

Studia Mathematica

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1991

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tom 98

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nr 2

157-174

Ograniczanie wyników

3 Studia Mathematica

3 Orihuela J.

2 Cascales B.

1 Muñoz M.

1 Namioka I.

1 Schachermayer W.

1 Valdivia M.

1 2008

1 2003

1 1991

James boundaries and σ-fragmented selectors

The Lindelöf property in Banach spaces

Every Radon-Nikodym Corson compact space is Eberlein compact