Conditional weak compactness and weak sequential completeness in vector-valued function spaces
Let \((\Omega, \Sigma, \mu)\) be a finite measure space, \(X\) be a Banach space. For a function \(g: \Omega \to X^*\) and \(x \in X\) denote \(g_x(\omega):=\langle g(\omega), x \rangle\). The function \(g\) is called weak* measurable if all \(g_x\), \(x \in X\), are measurable, and \(L_0(X^*,X)\) denotes the set of equivalence classes of weak* measurable functions \(g: \Omega \to X^*\). Denote also by \(\vartheta(g)\) the order-supremum of the family \(\{ |g_x|: x \in B_X \}\) in \(L_0:=L_0(\Omega, \Sigma, \mu)\). For an order ideal \(E \subset L_0\) its Köthe dual is denoted by \(E'\). \(L_0(X)\) stands for the set of equivalence classes of strongly measurable functions \(f: \Omega \to X\), \(E(X):= \{ f \in L_0(X): \|f(\cdot)\| \in E\}\), \(E'(X^*,X):= \{ g \in L_0(X^*,X): \vartheta(g) \in E'\}\). The spaces \(E'(X^*,X)\) and \(E(X)\) form a dual pair under the natural duality \(\langle g, f \rangle = \int_\Omega \langle g(\cdot), f(\cdot)\rangle d\mu\). In the paper under review for a countably generated \(\Sigma\) a conditional \(\sigma(E'(X^{*},X), E(X))\)-compactness criterion in \(E'(X^{*},X)\) is given. It is proved that \(L^1(X^{*},X)\) is \(\sigma(L^1(X^{*},X), L^{\infty} (X))\)-sequentially complete, and as a corollary a criterion for \(\sigma(L^1(X^{*},X), L^{\infty} (X))\)-sequential compactness in the case of countably generated \(\Sigma\) is deduced.
scientific article; zbMATH DE number 5949313
scientific article