{"paper":{"title":"From weak to strong types of $L_E^1$-convergence by the Bocce-criterion","license":"","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Erik J. Balder, Maria Girardi, Vincent Jalby","submitted_at":"1994-02-17T18:28:30Z","abstract_excerpt":"Necessary and sufficient oscillation conditions are given for a weakly convergent sequence (resp. relatively weakly compact set) in the Bochner-Lebesgue space $\\l1$ to be norm convergent (resp. relatively norm compact), thus extending the known results for $\\rl1$. Similarly, necessary and sufficient oscillation conditions are given to pass from weak to limited (and also to Pettis-norm) convergence in $\\l1$. It is shown that tightness is a necessary and sufficient condition to pass from limited to strong convergence. Other implications between several modes of convergence in $\\l1$ are also stud"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9402210","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","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"}