{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,9,29]],"date-time":"2023-09-29T08:15:34Z","timestamp":1695975334750},"reference-count":0,"publisher":"Wiley","issue":"6","license":[{"start":{"date-parts":[[2017,12,29]],"date-time":"2017-12-29T00:00:00Z","timestamp":1514505600000},"content-version":"vor","delay-in-days":28,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematical Logic Qtrly"],"published-print":{"date-parts":[[2017,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We study theorems from Functional Analysis with regard to their relationship with various weak choice principles and prove several results about them: \u201cEvery infinite\u2010dimensional Banach space has a well\u2010orderable Hamel basis\u201d is equivalent to ; \u201c can be well\u2010ordered\u201d implies \u201cno infinite\u2010dimensional Banach space has a Hamel basis of cardinality \u201d, thus the latter statement is true in every Fraenkel\u2010Mostowski model of ; \u201cNo infinite\u2010dimensional Banach space has a Hamel basis of cardinality \u201d is not provable in ; \u201cNo infinite\u2010dimensional Banach space has a well\u2010orderable Hamel basis of cardinality \u201d is provable in ;  (the Axiom of Choice for denumerable families of non\u2010empty finite sets) is equivalent to \u201cno infinite\u2010dimensional Banach space has a Hamel basis which can be written as a denumerable union of finite sets\u201d; Mazur's Lemma (\u201cIf <jats:italic>X<\/jats:italic> is an infinite\u2010dimensional Banach space, <jats:italic>Y<\/jats:italic> is a finite\u2010dimensional vector subspace of <jats:italic>X<\/jats:italic>, and , then there is a unit vector  such that  for all  and all scalars \u03b1\u201d) is provable in ; \u201cA real normed vector space <jats:italic>X<\/jats:italic> is finite\u2010dimensional if and only if its closed unit ball  is compact\u201d is provable in ;  (Principle of Dependent Choices) + \u201c can be well\u2010ordered\u201d does not imply the Hahn\u2010Banach Theorem () in ;  and \u201cno infinite\u2010dimensional Banach space has a Hamel basis of cardinality \u201d are independent from each other in ; \u201cNo infinite\u2010dimensional Banach space can be written as a denumerable union of finite\u2010dimensional subspaces\u201d lies in strength between  (the Axiom of Countable Choice) and ;  implies \u201cNo infinite\u2010dimensional Banach space can be written as a denumerable union of closed proper subspaces\u201d which in turn implies ; \u201cEvery infinite\u2010dimensional Banach space has a denumerable linearly independent subset\u201d is a theorem of , but not a theorem of ; and \u201cEvery infinite\u2010dimensional Banach space has a linearly independent subset of cardinality \u201d implies \u201cevery Dedekind\u2010finite set is finite\u201d.<\/jats:p>","DOI":"10.1002\/malq.201600027","type":"journal-article","created":{"date-parts":[[2017,12,29]],"date-time":"2017-12-29T10:04:39Z","timestamp":1514541879000},"page":"509-535","source":"Crossref","is-referenced-by-count":1,"title":["On infinite\u2010dimensional Banach spaces and weak forms of the axiom of choice"],"prefix":"10.1002","volume":"63","author":[{"given":"Paul","family":"Howard","sequence":"first","affiliation":[{"name":"Department of Mathematics Eastern Michigan University Ypsilanti MI 48197 United States of America"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Eleftherios","family":"Tachtsis","sequence":"additional","affiliation":[{"name":"Department of Mathematics University of the Aegean Karlovassi 83200 Samos Greece"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2017,12,29]]},"container-title":["Mathematical Logic Quarterly"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fmalq.201600027","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/malq.201600027","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,9,29]],"date-time":"2023-09-29T00:33:11Z","timestamp":1695947591000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/malq.201600027"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,12]]},"references-count":0,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2017,12]]}},"alternative-id":["10.1002\/malq.201600027"],"URL":"https:\/\/doi.org\/10.1002\/malq.201600027","archive":["Portico"],"relation":{},"ISSN":["0942-5616","1521-3870"],"issn-type":[{"value":"0942-5616","type":"print"},{"value":"1521-3870","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,12]]}}}