{"paper":{"title":"Universal stability of Banach spaces for $\\varepsilon$-isometries","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Duanxu Dai, Lixin Cheng, Yunbai Dong, Yu Zhou","submitted_at":"2013-01-15T14:58:32Z","abstract_excerpt":"Let $X$, $Y$ be two real Banach spaces and $\\varepsilon>0$. A standard $\\varepsilon$-isometry $f:X\\rightarrow Y$ is said to be $(\\alpha,\\gamma)$-stable (with respect to $T:L(f)\\equiv\\overline{{\\rm span}}f(X)\\rightarrow X$ for some $\\alpha, \\gamma>0$) if $T$ is a linear operator with $\\|T\\|\\leq\\alpha$ so that $Tf-Id$ is uniformly bounded by $\\gamma\\varepsilon$ on $X$. The pair $(X,Y)$ is said to be stable if every standard $\\varepsilon$-isometry $f:X\\rightarrow Y$ is $(\\alpha,\\gamma)$-stable for some $\\alpha,\\gamma>0$. $X (Y)$ is said to be universally left (right)-stable, if $(X,Y)$ is always "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.3374","kind":"arxiv","version":4},"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"}