On weak convergence in Köthe-Bochner function spaces

Abstract

Let $E$ be an order continuous Köthe function space over a non purely atomic probability measure $\hat{I}¼$ and let $X$ be a Banach space, with topological duals $E^{\ast }$ and $X^{\ast }$, respectively. Let $E(X)$ and $E^{\ast }(X^{\ast })$ be the corresponding Köthe-Bochner function spaces and consider $E^{\ast }(X^{\ast })$ as a subspace of $E(X{)}^{\ast }$. We prove that if $X^{\ast }$ fails the Radon-Nikodým property, then there is a bounded, non weakly null sequence $(f_n)$ in $E(X)$ such that $⟨\ddot{I}†,f_n⟩\to 0$ for every $\ddot{I}†\in E^{\ast }(X^{\ast })$; in particular, the closed unit ball of $E^{\ast }(X^{\ast })$ is not a James boundary for $E(X)$. This extends a result by B. Cascales and A.J. Pallarés [Collect. Math. 45 (1994), 263--270] on the case $E=L_1(\hat{I}¼)$ and allows us to answer a question posed recently by S. Dwivedi [Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. RACSAM 120 (2026), 71].