{"paper":{"title":"On weak convergence in K\\\"{o}the-Bochner function spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"If X* lacks the Radon-Nikodým property, the closed unit ball of E*(X*) is not a James boundary for E(X).","cross_cats":[],"primary_cat":"math.FA","authors_text":"Jos\\'e Rodr\\'iguez","submitted_at":"2026-05-13T09:26:14Z","abstract_excerpt":"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 boun"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"If X^* fails the Radon-Nikodým property, then there is a bounded, non weakly null sequence (f_n) in E(X) such that ⟨ϕ,f_n⟩→0 for every ϕ∈E^*(X^*); in particular, the closed unit ball of E^*(X^*) is not a James boundary for E(X).","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"E is an order continuous Köthe function space over a non purely atomic probability measure μ (stated in the opening sentence of the abstract).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"If X* fails the Radon-Nikodým property then the closed unit ball of E*(X*) is not a James boundary for the Köthe-Bochner space E(X) when E is order continuous over a non-purely-atomic measure.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"If X* lacks the Radon-Nikodým property, the closed unit ball of E*(X*) is not a James boundary for E(X).","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"8fee91f8c3256efbe975495ff407ec44c8256994d976c4106ddeec74017bec5d"},"source":{"id":"2605.13240","kind":"arxiv","version":1},"verdict":{"id":"adc36da2-4a49-4d58-9f4f-3ad252f6e6ec","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T19:10:50.231645Z","strongest_claim":"If X^* fails the Radon-Nikodým property, then there is a bounded, non weakly null sequence (f_n) in E(X) such that ⟨ϕ,f_n⟩→0 for every ϕ∈E^*(X^*); in particular, the closed unit ball of E^*(X^*) is not a James boundary for E(X).","one_line_summary":"If X* fails the Radon-Nikodým property then the closed unit ball of E*(X*) is not a James boundary for the Köthe-Bochner space E(X) when E is order continuous over a non-purely-atomic measure.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"E is an order continuous Köthe function space over a non purely atomic probability measure μ (stated in the opening sentence of the abstract).","pith_extraction_headline":"If X* lacks the Radon-Nikodým property, the closed unit ball of E*(X*) is not a James boundary for E(X)."},"references":{"count":17,"sample":[{"doi":"","year":2017,"title":"P. A. H. Brooker,Non-Asplund Banach spaces and operators, J. Funct. Anal.273(2017), no. 12, 3831–3858","work_id":"0cde7fe6-3682-423a-a7cf-f5187b9c8423","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1994,"title":"B. Cascales and A. J. Pallar´ es,La propiedad de Radon-Nikodym en espacios de Banach duales, Collect. Math.45(1994), no. 3, 263–270","work_id":"4363f35c-d185-440b-a2af-c2197c7f0f56","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1977,"title":"J. Diestel and J. J. Uhl, Jr.,Vector measures, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977","work_id":"48e32e95-18fb-4f38-9b0d-1f50a1e9dc39","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2026,"title":"Dwivedi,Weak ∗-weak points of continuity on the state spaces, Rev","work_id":"88951f29-4130-4445-8805-1a9aafc81f20","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1980,"title":"G. A. Edgar,Asplund operators and a.e. convergence, J. Multivariate Anal.10(1980), no. 3, 460–466","work_id":"16729177-ddfe-4e88-8032-b2972b3a4dc4","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":17,"snapshot_sha256":"2bd08cf8299546decfe481f601ab6be8a1301e8c6ed70ddfcecaf48b42f42bbc","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}