{"paper":{"title":"Galerkin approximation of linear problems in Banach and Hilbert spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Isabelle Chalendar (LAMA), Robert Eymard (LAMA), Wolfgang Arendt","submitted_at":"2019-08-09T06:32:57Z","abstract_excerpt":"In this paper we study the conforming Galerkin approximation of the problem: find u $\\in$ U such that a(u, v) = <L, v> for all v $\\in$ V, where U and V are Hilbert or Banach spaces, a is a continuous bilinear or sesquilinear form and L $\\in$ V' a given data. The approximate solution is sought in a finite dimensional subspace of U, and test functions are taken in a finite dimensional subspace of V. We provide a necessary and sufficient condition on the form a for convergence of the Galerkin approximation, which is also equivalent to convergence of the Galerkin approximation for the adjoint prob"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.03326","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.03326/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}