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In addition it is shown that some spreading model of a sequence in $(X, |\\!|\\!|\\cdot |\\!|\\!|)$ is 1-equivalent to the unit vector basis of $\\ell_1$ (respectively, $c_0$) implies that $X$ con"},"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":"math/9709217","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.FA","submitted_at":"1997-09-18T00:00:00Z","cross_cats_sorted":[],"title_canon_sha256":"d2c6b250cd0a91afae30f756cfc3e8119a011f353b865f7814888c2c72e3f659","abstract_canon_sha256":"f00eb611b99005a0c65e250f2a688c2391df11e3076a04abcbc0a047ba668cc3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:34.825815Z","signature_b64":"bv9ywJzZOhVv5EF5WiMn5xlud+q+VQazUM+MYhIS1Ex0vn5Dx76GSzn8pDyYDDIZosvagK8+XSD4StIQZCMjAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fcf53fc74d89e90a3a6800424ec3f913055434cc954c5863e7b813adf4efcdb7","last_reissued_at":"2026-05-18T01:05:34.825081Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:34.825081Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On asymptotic properties of Banach spaces under renormings","license":"","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Edward Odell, Thomas Schlumprecht","submitted_at":"1997-09-18T00:00:00Z","abstract_excerpt":"It is shown that a separable Banach space $X$ can be given an equivalent norm $|\\!|\\!|\\cdot |\\!|\\!|$ with the following properties:\\quad If $(x_n)\\subseteq X$ is relatively weakly compact and $\\lim_{m\\to\\infty} \\lim_{n\\to\\infty}\\break |\\!|\\!| x_m + x_n |\\!|\\!| = 2\\lim_{m\\to\\infty} |\\!|\\!| x_m|\\!|\\!|$ then $(x_n)$ converges in norm. 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