{"paper":{"title":"On universal left-stability of $\\epsilon$-isometries","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Duanxu Dai, Lingxin Bao, Lixin Cheng, Qingjin Cheng","submitted_at":"2013-01-16T11:06:47Z","abstract_excerpt":"Let $X$, $Y$ be two real Banach spaces, and $\\eps\\geq0$. A map $f:X\\rightarrow Y$ is said to be a standard $\\eps$-isometry if $|\\|f(x)-f(y)\\|-\\|x-y\\||\\leq\\eps$ for all $x,y\\in X$ and with $f(0)=0$. We say that a pair of Banach spaces $(X,Y)$ is stable if there exists $\\gamma>0$ such that for every such $\\eps$ and every standard $\\eps$-isometry $f:X\\rightarrow Y$ there is a bounded linear operator $T:L(f)\\equiv\\overline{{\\rm span}}f(X)\\rightarrow X$ such that $\\|Tf(x)-x\\|\\leq\\gamma\\eps$ for all $x\\in X$. $X (Y)$ is said to be left (right)-universally stable, if $(X,Y)$ is always stable for ever"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.3656","kind":"arxiv","version":3},"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"}