{"paper":{"title":"Some natural subspaces and quotient spaces of $L^1$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Gilles Godefroy, Nicolas Lerner","submitted_at":"2017-02-20T16:19:07Z","abstract_excerpt":"We show that the space $\\text{Lip}_0(\\mathbb R^n)$ is the dual space of $L^{1}({\\mathbb R}^{n}; {\\mathbb R}^{n})/N$ where $N$ is the subspace of $L^{1}({\\mathbb R}^{n}; {\\mathbb R}^{n})$ consisting of vector fields whose divergence vanishes. We prove that although the quotient space $L^{1}({\\mathbb R}^{n}; {\\mathbb R}^{n})/N$ is weakly sequentially complete, the subspace $N$ is not nicely placed - in other words, its unit ball is not closed for the topology $\\tau_m$ of local convergence in measure. We prove that if $\\Omega$ is a bounded open star-shaped subset of $\\mathbb {R}^n$ and $X$ is a c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.06049","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"}