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For $K=([0,1],| {\\cdot} |^{\\alpha})$, $0<\\alpha<1$, we prove that $\\lip(K)$ is a proper $M$-ideal in a certain subspace of $\\Lip(K)$ containing a copy of $\\ell^{\\infty}$."},"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/0201144","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.FA","submitted_at":"2002-01-16T14:00:18Z","cross_cats_sorted":[],"title_canon_sha256":"f0b8c7a9e30cb4b086d725b8c9d290f1d4e00359fee1457adaa4d85599b551ad","abstract_canon_sha256":"171f2fff8da4f6e088bdc8a7ec292507ec5b866f58a0a7742f961ec9d7985074"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:26:44.106120Z","signature_b64":"OD8y/Q8kYb6Gbnls+uWEzUVcAQxoH1njMi9nF6PlrfUZ5zPpbG5UxMdIWt5UW8cl3/+MMJXgt+TfPc67cpB8BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"67d7309a5d0ce6eae7306d0c005d9455d495c59e1cab1c1caeb338ef3c77ab92","last_reissued_at":"2026-05-18T04:26:44.105611Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:26:44.105611Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Lipschitz spaces and M-ideals","license":"","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Dirk Werner, Heiko Berninger","submitted_at":"2002-01-16T14:00:18Z","abstract_excerpt":"For a metric space $(K,d)$ the Banach space $\\Lip(K)$ consists of all scalar-valued bounded Lipschitz functions on $K$ with the norm $\\|f\\|_{L}=\\max(\\|f\\|_{\\infty},L(f))$, where $L(f)$ is the Lipschitz constant of $f$. 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For $K=([0,1],| {\\cdot} |^{\\alpha})$, $0<\\alpha<1$, we prove that $\\lip(K)$ is a proper $M$-ideal in a certain subspace of $\\Lip(K)$ containing a copy of $\\ell^{\\infty}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0201144","kind":"arxiv","version":1},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"math/0201144","created_at":"2026-05-18T04:26:44.105680+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0201144v1","created_at":"2026-05-18T04:26:44.105680+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0201144","created_at":"2026-05-18T04:26:44.105680+00:00"},{"alias_kind":"pith_short_12","alias_value":"M7LTBGS5BTTO","created_at":"2026-05-18T12:25:51.375804+00:00"},{"alias_kind":"pith_short_16","alias_value":"M7LTBGS5BTTOVZZQ","created_at":"2026-05-18T12:25:51.375804+00:00"},{"alias_kind":"pith_short_8","alias_value":"M7LTBGS5","created_at":"2026-05-18T12:25:51.375804+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/M7LTBGS5BTTOVZZQNUGAAXMUKX","json":"https://pith.science/pith/M7LTBGS5BTTOVZZQNUGAAXMUKX.json","graph_json":"https://pith.science/api/pith-number/M7LTBGS5BTTOVZZQNUGAAXMUKX/graph.json","events_json":"https://pith.science/api/pith-number/M7LTBGS5BTTOVZZQNUGAAXMUKX/events.json","paper":"https://pith.science/paper/M7LTBGS5"},"agent_actions":{"view_html":"https://pith.science/pith/M7LTBGS5BTTOVZZQNUGAAXMUKX","download_json":"https://pith.science/pith/M7LTBGS5BTTOVZZQNUGAAXMUKX.json","view_paper":"https://pith.science/paper/M7LTBGS5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0201144&json=true","fetch_graph":"https://pith.science/api/pith-number/M7LTBGS5BTTOVZZQNUGAAXMUKX/graph.json","fetch_events":"https://pith.science/api/pith-number/M7LTBGS5BTTOVZZQNUGAAXMUKX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/M7LTBGS5BTTOVZZQNUGAAXMUKX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/M7LTBGS5BTTOVZZQNUGAAXMUKX/action/storage_attestation","attest_author":"https://pith.science/pith/M7LTBGS5BTTOVZZQNUGAAXMUKX/action/author_attestation","sign_citation":"https://pith.science/pith/M7LTBGS5BTTOVZZQNUGAAXMUKX/action/citation_signature","submit_replication":"https://pith.science/pith/M7LTBGS5BTTOVZZQNUGAAXMUKX/action/replication_record"}},"created_at":"2026-05-18T04:26:44.105680+00:00","updated_at":"2026-05-18T04:26:44.105680+00:00"}