{"paper":{"title":"On uniformly differentiable mappings from $\\ell_\\infty(\\Gamma)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Eva Perneck\\'a, Petr H\\'ajek","submitted_at":"2015-03-12T00:01:57Z","abstract_excerpt":"In 1970 Haskell Rosenthal proved that if $X$ is a Banach space, $\\Gamma$ is an infinite index set, and $T:\\ell_\\infty(\\Gamma)\\to X$ is a bounded linear operator such that $\\inf_{\\gamma\\in\\Gamma}\\|T(e_\\gamma)\\|>0$ then $T$ acts as an isomorphism on $\\ell_\\infty(\\Gamma')$, for some $\\Gamma'\\subset\\Gamma$ of the same cardinality as $\\Gamma$. Our main result is a nonlinear strengthening of this theorem. More precisely, under the assumption of GCH and the regularity of $\\Gamma$, we show that if ${F}:B_{\\ell_\\infty(\\Gamma)}\\to X$ is uniformly differentiable and such that $\\inf_{\\gamma\\in\\Gamma}\\|{F}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.03536","kind":"arxiv","version":2},"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"}