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

Antonio S. Granero, and Marcos Sánchez. "The extension of the Krein-Šmulian theorem for order-continuous Banach lattices." Banach Center Publications 79.1 (2008): 79-93. <http://eudml.org/doc/286276>.

@article{AntonioS2008,
abstract = {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) = supd(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 $d̂(\overline\{co\}^\{w*\}(K),X) = d̂(K,X)$.},
author = {Antonio S. Granero, Marcos Sánchez},
journal = {Banach Center Publications},
keywords = {Krein-Shmulian theorem; Banach lattices; 1-symmetric spaces},
language = {eng},
number = {1},
pages = {79-93},
title = {The extension of the Krein-Šmulian theorem for order-continuous Banach lattices},
url = {http://eudml.org/doc/286276},
volume = {79},
year = {2008},
}

TY - JOUR
AU - Antonio S. Granero
AU - Marcos Sánchez
TI - The extension of the Krein-Šmulian theorem for order-continuous Banach lattices
JO - Banach Center Publications
PY - 2008
VL - 79
IS - 1
SP - 79
EP - 93
AB - 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) = supd(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 $d̂(\overline{co}^{w*}(K),X) = d̂(K,X)$.
LA - eng
KW - Krein-Shmulian theorem; Banach lattices; 1-symmetric spaces
UR - http://eudml.org/doc/286276
ER -