{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,11]],"date-time":"2026-05-11T10:26:52Z","timestamp":1778495212817,"version":"3.51.4"},"reference-count":12,"publisher":"Wiley","issue":"3","license":[{"start":{"date-parts":[[2010,11,17]],"date-time":"2010-11-17T00:00:00Z","timestamp":1289952000000},"content-version":"vor","delay-in-days":4338,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematical Logic Qtrly"],"published-print":{"date-parts":[[1999,1]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We work in set theory ZF without axiom of choice. Though the Hahn\u2010Banach theorem cannot be proved in ZF, we prove that every Gateaux\u2010differentiable uniformly convex Banach space <jats:italic>E<\/jats:italic> satisfies the following continuous Hahn\u2010Banach property: if <jats:italic>p<\/jats:italic> is a continuous sublinear functional on <jats:italic>E<\/jats:italic>, if <jats:italic>F<\/jats:italic> is a subspace of <jats:italic>E<\/jats:italic>, and if <jats:italic>f<\/jats:italic>: <jats:italic>F<\/jats:italic> \u2192 \u211d is a linear functional such that <jats:italic>f<\/jats:italic> \u2264 p|F then there exists a linear functional <jats:italic>g<\/jats:italic> : <jats:italic>E<\/jats:italic> \u2192 \u211d such that <jats:italic>g<\/jats:italic> extends <jats:italic>f<\/jats:italic> and <jats:italic>g<\/jats:italic> \u2264 <jats:italic>p<\/jats:italic>. We also prove that the continuous Hahn\u2010Banach property on a topological vector space <jats:italic>E<\/jats:italic> is equivalent to the classical geometrical forms of the Hahn\u2010Banach theorem on <jats:italic>E<\/jats:italic>. We then prove that the axiom of Dependent choices DC is equivalent to Ekeland's variational principle, and that it implies the continuous Hahn\u2010Banach property on Gateaux\u2010differentiable Banach spaces. Finally, we prove that, though separable normed spaces satisfy the continuous Hahn\u2010Banach property, they do not satisfy the whole Hahn\u2010Banach property in ZF+DC.<\/jats:p>","DOI":"10.1002\/malq.19990450303","type":"journal-article","created":{"date-parts":[[2010,11,17]],"date-time":"2010-11-17T16:30:38Z","timestamp":1290011438000},"page":"299-314","source":"Crossref","is-referenced-by-count":21,"title":["The Hahn\u2010Banach Property and the Axiom of Choice"],"prefix":"10.1002","volume":"45","author":[{"given":"Juliette","family":"Dodu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Marianne","family":"Morillon","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2010,11,17]]},"reference":[{"key":"e_1_2_1_2_2","volume-title":"Geometry of Banach Spaces","author":"Beauzamy B.","year":"1985"},{"key":"e_1_2_1_3_2","first-page":"933","article-title":"The Baire category theorem implies the principle of dependent choice","volume":"25","author":"Blair C. 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Soc."},{"key":"e_1_2_1_5_2","volume-title":"Espaces vectoriels topologiques","author":"Bourbaki N.","year":"1981"},{"key":"e_1_2_1_6_2","volume-title":"Smoothness and Renormings in Banach Spaces","author":"Deville R.","year":"1993"},{"key":"e_1_2_1_7_2","doi-asserted-by":"publisher","DOI":"10.1112\/S0024610798005675"},{"key":"e_1_2_1_8_2","doi-asserted-by":"publisher","DOI":"10.2307\/2274230"},{"key":"e_1_2_1_9_2","doi-asserted-by":"publisher","DOI":"10.1112\/blms\/21.1.79"},{"key":"e_1_2_1_10_2","volume-title":"The Axiom of Choice","author":"Jech T.","year":"1973"},{"key":"e_1_2_1_11_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-66557-8"},{"key":"e_1_2_1_12_2","doi-asserted-by":"publisher","DOI":"10.2307\/2272118"},{"key":"e_1_2_1_13_2","volume-title":"Foundations of Analysis","author":"Schechter D.","year":"1997"}],"container-title":["Mathematical Logic Quarterly"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fmalq.19990450303","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/malq.19990450303","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,29]],"date-time":"2023-10-29T17:37:57Z","timestamp":1698601077000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/malq.19990450303"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1999,1]]},"references-count":12,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1999,1]]}},"alternative-id":["10.1002\/malq.19990450303"],"URL":"https:\/\/doi.org\/10.1002\/malq.19990450303","archive":["Portico"],"relation":{},"ISSN":["0942-5616","1521-3870"],"issn-type":[{"value":"0942-5616","type":"print"},{"value":"1521-3870","type":"electronic"}],"subject":[],"published":{"date-parts":[[1999,1]]}}}