The extension of the Krein-Šmulian theorem for order-continuous Banach lattices

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

Abstrakty

EN

If X is a Banach space and C ⊂ X a convex subset, for x** ∈ X** and A ⊂ X** let d(x**,C) = inf{||x**-x||: x ∈ C} be the distance from x** to C and d̂(A,C) = sup{d(a,C): a ∈ A}. Among other things, we prove that if X is an order-continuous Banach lattice and K is a w*-compact subset of X** we have: (i) $d̂(\overline{co}^{w*}(K),X) ≤ 2d̂(K,X)$ and, if K ∩ X is w*-dense in K, then $d̂(\overline{co}^{w*}(K),X) = d̂(K,X)$; (ii) if X fails to have a copy of ℓ₁(ℵ₁), then $d̂(\overline{co}^{w*}(K),X) = d̂(K,X)$; (iii) if X has a 1-symmetric basis, then $d̂(\overline{co}^{w*}(K),X) = d̂(K,X)$.

Kategorie tematyczne

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

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 2008
• Tom: 79
• Numer: 1
• Strony: 79-93

Daty

wydano

2008

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/bc79-0-6