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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1301.3656","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-01-16T11:06:47Z","cross_cats_sorted":[],"title_canon_sha256":"767e419b3e4064c5017aa5f734d4c22667c4069be7b6cbdf4ed8246c6d52db57","abstract_canon_sha256":"ff8ea60ba6ba8a5e4838e4c48b03674fde297963e2f446ee2cc517e908d62a97"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:57:33.811428Z","signature_b64":"qm4SjmhSBZFMajUcO2S4p9XQgWiTCzUByXkwVs0IX+KKYYB5Ken1ySJlI/X5FLST9HfkeeT3PQxVB5rCpAnMCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1fb5103630f94c8107a79c0df42d331d9d2e7c1a28ab5dfeddea727ffde05806","last_reissued_at":"2026-05-18T02:57:33.810895Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:57:33.810895Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $\\eps\\geq0$. 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