{"paper":{"title":"A generalization of the Banach-Steinhaus theorem for finite part limits","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Jasson Vindas, Ricardo Estrada","submitted_at":"2014-07-10T16:05:41Z","abstract_excerpt":"It is well known, as follows from the Banach-Steinhaus theorem, that if a sequence $\\left\\{y_{n}\\right\\}_{n=1}^{\\infty}$ of linear continuous functionals in a Fr\\'echet space converges pointwise to a linear functional $Y,$ $Y\\left( x\\right) =\\lim_{n\\rightarrow\\infty}\\left\\langle y_{n},x\\right\\rangle $ for all $x,$ then $Y$ is actually continuous. In this article we prove that in a Fr\\'echet space the continuity of $Y$ still holds if $Y$ is the \\emph{finite part} of the limit of $\\left\\langle y_{n},x\\right\\rangle $ as $n\\rightarrow\\infty.$ We also show that the continuity of finite part limits "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.2842","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"}