The extension of the Krein-Šmulian theorem for Orlicz sequence spaces and convex sets
The Krein-Šmulian theorem, which states that the closed convex hull of weakly compact subset of a Banach space is weakly compact, may be stated equivalently as follows: If \(X\) is a Banach space and \(K\subset X^{**}\) is a \(w^*\)-compact subset such that the distance \(d(K,X) = 0\), then \(d(\overline{co}^{w^*}(K), X) =0\). The natural question then arises, for what Banach spaces holds \(d(K,X) = d(\overline{co}^{w^*}(K), X)\) for any \(w^*\)-compact set \(K\subset X^{**}\)? Although the equality is not satisfied for all Banach spaces, it still holds for a fairly large class of spaces. The author shows here that in the case when \(X=\ell_\varphi(I)\), equipped with either the Luxemburg or Orlicz norm, equality holds if and only if \(\varphi\) satisfies the \(\Delta_2\)-condition. The second main result states that if \(C\subset X\) is nonempty and convex, and \(K\subset X^{**}\) is \(w^*\)-compact, then \(d(\overline{co}^{w^*}(K),C) \leq 9d(K,C)\); and if \(K\cap C\) is \(w^*\)-dense in \(K\), then \(d(\overline{co}^{w^*}(K), C) \leq 4 d(K,C)\). This is a generalization of the author's previous results stated for \(C=X\) [Rev. Mat. Iberoam. 22, No. 1, 93--110 (2006; Zbl 1117.46002)].
scientific article; zbMATH DE number 5125980
scientific article