# An extension of the Krein-Smulian theorem.

• Volume: 22, Issue: 1, page 93-110
• ISSN: 0213-2230

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## Abstract

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Let X be a Banach space, u ∈ 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 ⊂ X** such that d(K,X) = 0 satisfies d(cow*(K),X) = 0.We extend this result in the following way: if Z ⊂ X is a closed subspace of X and K ⊂ X** is a w*-compact subset of X**, thend(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⊄ 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 ⊂ B(X**) such that K ∩ X is w*-dense in K, d(K,X) = 1/2 and d(cow*(K),X) = 1.

## How to cite

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Granero, Antonio S.. "An extension of the Krein-Smulian theorem.." Revista Matemática Iberoamericana 22.1 (2006): 93-110. <http://eudml.org/doc/41966>.

@article{Granero2006,
abstract = {Let X be a Banach space, u ∈ 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 ⊂ X** such that d(K,X) = 0 satisfies d(cow*(K),X) = 0.We extend this result in the following way: if Z ⊂ X is a closed subspace of X and K ⊂ X** is a w*-compact subset of X**, thend(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⊄ 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 ⊂ B(X**) such that K ∩ X is w*-dense in K, d(K,X) = 1/2 and d(cow*(K),X) = 1.},
author = {Granero, Antonio S.},
journal = {Revista Matemática Iberoamericana},
keywords = {Geometría y estructura de espacios de Banach; Compacidad; closed convex hull; Krein-Šmulian theorem},
language = {eng},
number = {1},
pages = {93-110},
title = {An extension of the Krein-Smulian theorem.},
url = {http://eudml.org/doc/41966},
volume = {22},
year = {2006},
}

TY - JOUR
AU - Granero, Antonio S.
TI - An extension of the Krein-Smulian theorem.
JO - Revista Matemática Iberoamericana
PY - 2006
VL - 22
IS - 1
SP - 93
EP - 110
AB - Let X be a Banach space, u ∈ 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 ⊂ X** such that d(K,X) = 0 satisfies d(cow*(K),X) = 0.We extend this result in the following way: if Z ⊂ X is a closed subspace of X and K ⊂ X** is a w*-compact subset of X**, thend(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⊄ 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 ⊂ B(X**) such that K ∩ X is w*-dense in K, d(K,X) = 1/2 and d(cow*(K),X) = 1.
LA - eng
KW - Geometría y estructura de espacios de Banach; Compacidad; closed convex hull; Krein-Šmulian theorem
UR - http://eudml.org/doc/41966
ER -

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