Título inglés | An extension of the Krein-Smulian theorem. |
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Título español | Una extensión del teorema de Krein-Smulian. |
Autor/es | Granero, Antonio S. |
Organización | Dep. Anál. Mat. Fac. Mat. Univ. Complut. Madrid, Madrid, España |
Revista | 0213-2230 |
Publicación | 2006, 22 (1): 93-110, 11 Ref. |
Tipo de documento | articulo |
Idioma | Inglés |
Resumen inglés | Let X be a Banach space, u Î X** and K, Z two subsets of X**. Denote by d(u,Z) and d(K,Z) the distances to Z from the point u and from the subset K respectively. The Krein-Smulian Theorem asserts that the closed convex hull of a weakly compact subset of a Banach space is weakly compact; in other words, every w*-compact subset K Ì X** such that d(K,X) = 0 satisfies d(cow*(K),X) = 0. We extend this result in the following way: if Z Ì X is a closed subspace of X and K Ì X** is a w*-compact subset of X**, then d(cow*(K),Z) ≤ 5d(K,Z). Moreover, if Z ∩ K is w*-dense in K, then d(cow*(K),Z) ≤ 2d(K,Z). However, the equality d(K,X) = d(cow*(K),X) holds in many cases, for instance if l1 Ë X*, if X has w*-angelic dual unit ball (for example, if X is WCG or WLD), if X = L1(I), if K is fragmented by the norm of X**, etc. We also construct under CH a w*-compact subset K Ì B(X**) such that K ∩ X is w*-dense in K, d(K,X) = 1/2 and d(cow*(K),X) = 1. |
Clasificación UNESCO | 120203 |
Palabras clave español | Geometría y estructura de espacios de Banach ; Compacidad |
Código MathReviews | MR2267314 |
Código Z-Math | Zbl pre05077178 |
Acceso al artículo completo |