On weak convergence in Köthe-Bochner function spaces
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866913122644656128 |
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| author | Rodríguez, José |
| author_facet | Rodríguez, José |
| contents | Let $E$ be an order continuous Köthe function space over a non purely atomic probability measure $μ$ and let $X$ be a Banach space, with topological duals $E^*$ and $X^*$, respectively. Let $E(X)$ and $E^*(X^*)$ be the corresponding Köthe-Bochner function spaces and consider $E^*(X^*)$ as a subspace of $E(X)^*$. We prove that if $X^*$ fails the Radon-Nikodým property, then there is a bounded, non weakly null sequence $(f_n)$ in $E(X)$ such that $\langle φ,f_n\rangle \to 0$ for every $φ\in E^*(X^*)$; 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. Pallarés [Collect. Math. 45 (1994), 263--270] on the case $E=L_1(μ)$ 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]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_13240 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On weak convergence in Köthe-Bochner function spaces Rodríguez, José Functional Analysis Let $E$ be an order continuous Köthe function space over a non purely atomic probability measure $μ$ and let $X$ be a Banach space, with topological duals $E^*$ and $X^*$, respectively. Let $E(X)$ and $E^*(X^*)$ be the corresponding Köthe-Bochner function spaces and consider $E^*(X^*)$ as a subspace of $E(X)^*$. We prove that if $X^*$ fails the Radon-Nikodým property, then there is a bounded, non weakly null sequence $(f_n)$ in $E(X)$ such that $\langle φ,f_n\rangle \to 0$ for every $φ\in E^*(X^*)$; 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. Pallarés [Collect. Math. 45 (1994), 263--270] on the case $E=L_1(μ)$ 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]. |
| title | On weak convergence in Köthe-Bochner function spaces |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2605.13240 |