# An extension of the Krein-Smulian theorem.

Revista Matemática Iberoamericana (2006)

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

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topGranero, 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 -