{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T15:14:04Z","timestamp":1760368444646,"version":"build-2065373602"},"reference-count":13,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2020,1,13]],"date-time":"2020-01-13T00:00:00Z","timestamp":1578873600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The longstanding Banach\u2013Mazur separable quotient problem asks whether every infinite-dimensional Banach space has a quotient (Banach) space that is both infinite-dimensional and separable. Although it remains open in general, an affirmative answer is known in many special cases, including (1) reflexive Banach spaces, (2) weakly compactly generated (WCG) spaces, and (3) Banach spaces which are dual spaces. Obviously (1) is a special case of both (2) and (3), but neither (2) nor (3) is a special case of the other. A more general result proved here includes all three of these cases. More precisely, we call an infinite-dimensional Banach space X dual-like, if there is another Banach space E, a continuous linear operator T from the dual space     E *     onto a dense subspace of X, such that the closure of the kernel of T (in the relative weak* topology) has infinite codimension in     E *    . It is shown that every dual-like Banach space has an infinite-dimensional separable quotient.<\/jats:p>","DOI":"10.3390\/axioms9010007","type":"journal-article","created":{"date-parts":[[2020,1,15]],"date-time":"2020-01-15T03:20:22Z","timestamp":1579058422000},"page":"7","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Observations on the Separable Quotient Problem for Banach Spaces"],"prefix":"10.3390","volume":"9","author":[{"given":"Sidney A.","family":"Morris","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, La Trobe University, Melbourne, VIC 3086, Australia"}]},{"given":"David T.","family":"Yost","sequence":"additional","affiliation":[{"name":"Centre for Informatics and Applied Optimisation, Federation University Australia, P.O. Box 663, Ballarat, VIC 3353, Australia"}]}],"member":"1968","published-online":{"date-parts":[[2020,1,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"217","DOI":"10.4064\/cm-37-2-217-226","article-title":"The equivalence of some Banach space problems","volume":"37","author":"Saxon","year":"1977","journal-title":"Colloq. Math."},{"key":"ref_2","first-page":"299","article-title":"Separable quotients of Banach Spaces","volume":"10","author":"Mujica","year":"1997","journal-title":"Rev. Matem\u00e1tica Univ. Complut. Madr."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"153","DOI":"10.7169\/facm\/1704","article-title":"On the separable quotient problem for Banach spaces","volume":"59","author":"Ferrando","year":"2018","journal-title":"Funct. Approx. Comment. 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Saturated extensions, the attractors method and hereditarily James tree spaces. Methods in Banach Space Theory, Cambridge University Press.","DOI":"10.1017\/CBO9780511721366.003"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1007\/s00208-007-0179-y","article-title":"Unconditional families in Banach spaces","volume":"341","author":"Argyros","year":"2008","journal-title":"Math. Ann."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1017\/S0305004100053706","article-title":"Weakly compactly generated locally convex spaces","volume":"82","author":"Hunter","year":"1977","journal-title":"Math. Proc. Camb. Phil. Soc."},{"key":"ref_11","unstructured":"Robertson, A.P., and Robertson, W.J. (1964). Topological vector spaces. 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CMS Books in Mathematics\/Ouvrages de Math\u00e9matiques de la SMC, Springer."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/9\/1\/7\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T14:33:03Z","timestamp":1760365983000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/9\/1\/7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,1,13]]},"references-count":13,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,3]]}},"alternative-id":["axioms9010007"],"URL":"https:\/\/doi.org\/10.3390\/axioms9010007","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2020,1,13]]}}}