The extension of the Krein–Šmulian theorem for Orlicz sequence spaces and convex sets

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 d(A,C) = sup{d(a,C): a ∈ A}. In this paper we prove that if ϕ is an Orlicz function, I an infinite set and X = ϕ(I ) the corresponding Orlicz space, equipped with either the Luxemburg or the Orlicz norm, then for every w∗-compact subset K ⊂ X∗∗ we have d(cow∗(K),X) = d(K,X) if and only if ϕ satisfies the Δ2-condition at 0. We also prove that for every Banach space X, every nonempty convex subset C ⊂ X and every w∗-compact subset K ⊂ X∗∗ then d(cow∗(K),C) 9d(K,C) and, if K ∩ C is w∗-dense in K, then d(cow∗(K),C) 4d(K,C). © 2006 Elsevier Inc. All rights reserved.