{"paper":{"title":"The general linear equation on open connected sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.FA","authors_text":"Jens Schwaiger, Paolo Leonetti","submitted_at":"2019-05-31T12:12:49Z","abstract_excerpt":"Fix non-zero reals $\\alpha_1,\\ldots,\\alpha_n$ with $n\\ge 2$ and let $K$ be a non-empty open connected set in a topological vector space such that $\\sum_{i\\le n}\\alpha_iK\\subseteq K$ (which holds, in particular, if $K$ is an open convex cone and $\\alpha_1,\\ldots,\\alpha_n>0$). Let also $Y$ be a vector space over $\\mathbb{F}:=\\mathbb{Q}(\\alpha_1,\\ldots,\\alpha_n)$. We show, among others, that a function $f: K\\to Y$ satisfies the general linear equation $$ \\textstyle \\forall x_1,\\ldots,x_n \\in K,\\,\\,\\,\\,\\, f\\left(\\sum_{i\\le n}\\alpha_i x_i\\right)=\\sum_{i\\le n}\\alpha_i f(x_i) $$ if and only if there "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.13541","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"}