On weak convergence in K\"{o}the-Bochner function spaces

TL;DR

This paper investigates weak convergence properties in K"othe-Bochner function spaces, linking the Radon-Nikodým property of dual spaces to the existence of specific bounded sequences and boundary behaviors.

Contribution

It extends previous results by characterizing weak convergence in K"othe-Bochner spaces when the dual space lacks the Radon-Nikodým property, answering a recent open question.

Findings

Existence of bounded, non weakly null sequences under certain conditions.

The closed unit ball of $E^*(X^*)$ is not a James boundary for $E(X)$.

Extension of known results from the case $E=L_1(u)$.

Abstract

Let $E$ be an order continuous K\"{o}the function space over a non purely atomic probability measure $\mu$ and let $X$ be a Banach space, with topological duals $E^*$ and $X^*$, respectively. Let $E(X)$ and $E^*(X^*)$ be the corresponding K\"{o}the-Bochner function spaces and consider $E^*(X^*)$ as a subspace of $E(X)^*$. We prove that if $X^*$ fails the Radon-Nikod\'{y}m property, then there is a bounded, non weakly null sequence $(f_n)$ in $E(X)$ such that $\langle \varphi,f_n\rangle \to 0$ for every $\varphi\in E^*(X^*)$; in particular, the closed unit ball of $E^*(X^*)$ is not a James boundary for $E(X)$. This extends a result by B. Cascales and A.J. Pallar\'{e}s [Collect. Math. 45 (1994), 263--270] on the case $E=L_1(\mu)$ 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].