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

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Granero A., Sánchez M.

Banach Center Publications

2008

tom 79

nr 1

79-93

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 $d̂(\overline{co}^{w*}(K),X) = d̂(K,X)$.

Distances to convex sets

100%

Granero A., Sánchez M.

Studia Mathematica

2007

tom 182

nr 2

165-181

If X is a Banach space and C a convex subset of X*, we investigate whether the distance $d̂(\overline{co}^{w*}(K),C):= sup{inf{||k-c||: c ∈ C}: k ∈ \overline{co}^{w*}(K)}$ from $\overline{co}^{w*}(K)$ to C is M-controlled by the distance d̂(K,C) (that is, if $d̂(\overline{co}^{w*}(K),C) ≤ M d̂(K,C)$ for some 1 ≤ M < ∞), when K is any weak*-compact subset of X*. We prove, for example, that: (i) C has 3-control if C contains no copy of the basis of ℓ₁(c); (ii) C has 1-control when C ⊂ Y ⊂ X* and Y is a subspace with weak*-angelic closed dual unit ball B(Y*); (iii) if C is a convex subset of X and X is considered canonically embedded into its bidual X**, then C has 5-control inside X**, in general, and 2-control when K ∩ C is weak*-dense in C.

Ograniczanie wyników

1 Banach Center Publications

1 Studia Mathematica

2 Granero A.

2 Sánchez M.

1 2008

1 2007

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

Distances to convex sets