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Such pairs are called $K$-based Banach spaces. A based Banach space $X$ is rational if the unit ball of any finite-dimensional subspace spanned by finitely many basic vectors is a polyhedron whose vertices have rational coordinates in the Schauder basis of $X$.\n  Using the technique of Fra\\\"iss\\'e theory, we construct a rational $K$-base"},"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":"1801.10064","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-01-26T22:09:41Z","cross_cats_sorted":[],"title_canon_sha256":"f0d7555fb59d342d7e1197abc1ad0a3e6c9be4005c3b5690a6037ee5c3ae8dd5","abstract_canon_sha256":"9fc9e47d240b567f561d099a3b1c9e8627b4cee83825b44ccbbe6cb50df1b60c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:56:53.629552Z","signature_b64":"1nrMFEa1vq0g0UteYA+dLeevvLZIVEDaKCxr/hparBBmvLfarCE8LGjcZdkDsZowKjnblTZ/n/IG9S9XC+WmAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b4450bd83dd717b4abaef925a38424f538d062805de007060e140afbf33afa47","last_reissued_at":"2026-05-17T23:56:53.628812Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:56:53.628812Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A universal Banach space with a $K$-unconditional basis","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Joanna Garbuli\\'nska-W\\k{e}grzyn, Taras Banakh","submitted_at":"2018-01-26T22:09:41Z","abstract_excerpt":"For a constant $K\\geq 1$ let $\\mathfrak{B}_K$ be the class of pairs $(X,(\\mathbf e_n)_{n\\in\\omega})$ consisting of a Banach space $X$ and an unconditional Schauder basis $(\\mathbf e_n)_{n\\in\\omega}$ for $X$, having the unconditional basic constant $K_u\\leq K$. 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A based Banach space $X$ is rational if the unit ball of any finite-dimensional subspace spanned by finitely many basic vectors is a polyhedron whose vertices have rational coordinates in the Schauder basis of $X$.\n  Using the technique of Fra\\\"iss\\'e theory, we construct a rational $K$-base"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.10064","kind":"arxiv","version":2},"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":"1801.10064","created_at":"2026-05-17T23:56:53.628939+00:00"},{"alias_kind":"arxiv_version","alias_value":"1801.10064v2","created_at":"2026-05-17T23:56:53.628939+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.10064","created_at":"2026-05-17T23:56:53.628939+00:00"},{"alias_kind":"pith_short_12","alias_value":"WRCQXWB524L3","created_at":"2026-05-18T12:33:01.666342+00:00"},{"alias_kind":"pith_short_16","alias_value":"WRCQXWB524L3JK5O","created_at":"2026-05-18T12:33:01.666342+00:00"},{"alias_kind":"pith_short_8","alias_value":"WRCQXWB5","created_at":"2026-05-18T12:33:01.666342+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/WRCQXWB524L3JK5O7ES2HBBE6U","json":"https://pith.science/pith/WRCQXWB524L3JK5O7ES2HBBE6U.json","graph_json":"https://pith.science/api/pith-number/WRCQXWB524L3JK5O7ES2HBBE6U/graph.json","events_json":"https://pith.science/api/pith-number/WRCQXWB524L3JK5O7ES2HBBE6U/events.json","paper":"https://pith.science/paper/WRCQXWB5"},"agent_actions":{"view_html":"https://pith.science/pith/WRCQXWB524L3JK5O7ES2HBBE6U","download_json":"https://pith.science/pith/WRCQXWB524L3JK5O7ES2HBBE6U.json","view_paper":"https://pith.science/paper/WRCQXWB5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1801.10064&json=true","fetch_graph":"https://pith.science/api/pith-number/WRCQXWB524L3JK5O7ES2HBBE6U/graph.json","fetch_events":"https://pith.science/api/pith-number/WRCQXWB524L3JK5O7ES2HBBE6U/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WRCQXWB524L3JK5O7ES2HBBE6U/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WRCQXWB524L3JK5O7ES2HBBE6U/action/storage_attestation","attest_author":"https://pith.science/pith/WRCQXWB524L3JK5O7ES2HBBE6U/action/author_attestation","sign_citation":"https://pith.science/pith/WRCQXWB524L3JK5O7ES2HBBE6U/action/citation_signature","submit_replication":"https://pith.science/pith/WRCQXWB524L3JK5O7ES2HBBE6U/action/replication_record"}},"created_at":"2026-05-17T23:56:53.628939+00:00","updated_at":"2026-05-17T23:56:53.628939+00:00"}