{"paper":{"title":"Direct sums and products in topological groups and vector spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.GR"],"primary_cat":"math.GN","authors_text":"Dikran Dikranjan, Dmitri Shakhmatov, Jan Sp\\v{e}v\\'ak","submitted_at":"2015-08-04T05:57:10Z","abstract_excerpt":"We call a subset $A$ of an abelian topological group $G$: (i) $absolutely$ $Cauchy$ $summable$ provided that for every open neighbourhood $U$ of $0$ one can find a finite set $F\\subseteq A$ such that the subgroup generated by $A\\setminus F$ is contained in $U$; (ii) $absolutely$ $summable$ if, for every family $\\{z_a:a\\in A\\}$ of integer numbers, there exists $g\\in G$ such that the net $\\left\\{\\sum_{a\\in F} z_a a: F\\subseteq A\\mbox{ is finite}\\right\\}$ converges to $g$; (iii) $topologically$ $independent$ provided that $0\\not \\in A$ and for every neighbourhood $W$ of $0$ there exists a neighbo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.00667","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"}