Distances to convex sets

• Autorzy: Antonio S. Granero , Marcos Sánchez
• Języki publikacji: EN

Abstrakty

EN

If X is a Banach space and C a convex subset of X*, we investigate whether the distance $d̂(\overline{co}^{w*}(K),C):= sup{inf{||k-c||: c ∈ C}: k ∈ \overline{co}^{w*}(K)}$ from $\overline{co}^{w*}(K)$ to C is M-controlled by the distance d̂(K,C) (that is, if $d̂(\overline{co}^{w*}(K),C) ≤ M d̂(K,C)$ for some 1 ≤ M < ∞), when K is any weak*-compact subset of X*. We prove, for example, that: (i) C has 3-control if C contains no copy of the basis of ℓ₁(c); (ii) C has 1-control when C ⊂ Y ⊂ X* and Y is a subspace with weak*-angelic closed dual unit ball B(Y*); (iii) if C is a convex subset of X and X is considered canonically embedded into its bidual X**, then C has 5-control inside X**, in general, and 2-control when K ∩ C is weak*-dense in C.

Kategorie tematyczne

• 46B20: Geometry and structure of normed linear spaces
• 46B26: Nonseparable Banach spaces

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2007
• Tom: 182
• Numer: 2
• Strony: 165-181

Daty

wydano

2007

Twórcy

autor

Antonio S. Granero

• Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain

autor

Marcos Sánchez

• Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain

Identyfikatory

DOI

10.4064/sm182-2-5