Wyniki wyszukiwania

1

The Lindelöf property and σ-fragmentability

100%

Cascales B., Namioka I.

Fundamenta Mathematicae

|

2003

|

tom 180

|

nr 2

161-183

EN

In the previous paper, we, together with J. Orihuela, showed that a compact subset X of the product space $[-1,1]^{D}$ is fragmented by the uniform metric if and only if X is Lindelöf with respect to the topology γ(D) of uniform convergence on countable subsets of D. In the present paper we generalize the previous result to the case where X is K-analytic. Stated more precisely, a K-analytic subspace X of $[-1,1]^{D}$ is σ-fragmented by the uniform metric if and only if (X,γ(D)) is Lindelöf, and if this is the case then $(X,γ(D))^{ℕ}$ is also Lindelöf. We give several applications of this theorem in areas of topology and Banach spaces. We also show by examples that the main theorem cannot be extended to the cases where X is Čech-analytic and Lindelöf or countably K-determined.

2

James boundaries and σ-fragmented selectors

81%

Cascales B., Muñoz M., Orihuela J.

Studia Mathematica

|

2008

|

tom 188

|

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.

3

Fragmentability and compactness in C(K)-spaces

81%

Cascales B., Manjabacas G., Vera G.

Studia Mathematica

|

1998

|

tom 131

|

nr 1

73-87

EN

Let K be a compact Hausdorff space, $C_p(K)$ the space of continuous functions on K endowed with the pointwise convergence topology, D ⊂ K a dense subset and $t_p(D)$ the topology in C(K) of pointwise convergence on D. It is proved that when $C_p(K)$ is Lindelöf the $t_p(D)$-compact subsets of C(K) are fragmented by the supremum norm of C(K). As a consequence we obtain some Namioka type results and apply them to prove that if K is separable and $C_p(K)$ is Lindelöf, then K is metrizable if, and only if, there is a countable and dense subset D ⊂ K such that $(C(K),t_p(D))$ is analytic. We also show that if K is a separable Rosenthal compact space, then K is metrizable if, and only if, $C_p(K)$ is Lindelöf. We complete our study by showing that if K does not contain a copy of βℕ, then convex $t_p(D)$-compact subsets of C(K) have the weak Radon-Nikodym property.

4

The Lindelöf property in Banach spaces

81%

Cascales B., Namioka I., Orihuela J.

Studia Mathematica

|

2003

|

tom 154

|

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.

Ograniczanie wyników

3 Studia Mathematica

1 Fundamenta Mathematicae

4 Cascales B.

2 Namioka I.

2 Orihuela J.

1 Manjabacas G.

1 Muñoz M.

1 Vera G.

1 2008

2 2003

1 1998

The Lindelöf property and σ-fragmentability

James boundaries and σ-fragmented selectors

Fragmentability and compactness in C(K)-spaces

The Lindelöf property in Banach spaces