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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":true,"formal_links_present":false},"canonical_record":{"source":{"id":"2605.13240","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2026-05-13T09:26:14Z","cross_cats_sorted":[],"title_canon_sha256":"1c0b69d7ff1d11563cfa407bfdbdb78268c03f0b7f1b1e6f0193aa1ef477bc3d","abstract_canon_sha256":"8280a233cf1b65b8083ac40c612b4149f47ff7d9e6237ffa38d8b29aa1f34b08"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:44:49.512494Z","signature_b64":"FqOhrfd3QS/TL8iIJcHKu42nWA1NJQJr+NYKkYxVVe0CzfnMUNncN142axaP/TC7AupDDa9ikW+gNJ12zChhBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"01d0c4945aae8afbbac3158f5a40db8e752a91d7c568c4378048fd8085346ec3","last_reissued_at":"2026-05-18T02:44:49.512028Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:44:49.512028Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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