- Title
- Observations on the separable quotient problem for banach spaces
- Creator
- Morris, Sidney; Yost, David
- Date
- 2020
- Type
- Text; Journal article
- Identifier
- http://researchonline.federation.edu.au/vital/access/HandleResolver/1959.17/171788
- Identifier
- vital:14393
- Identifier
-
https://doi.org/10.3390/axioms9010007
- Identifier
- ISBN:2075-1680 (ISSN)
- Abstract
- The longstanding Banach-Mazur 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. © 2020 by the authors.
- Publisher
- MDPI AG
- Relation
- Axioms Vol. 9, no. 1 (2020), p.
- Rights
- http://creativecommons.org/licenses/by/4.0/
- Rights
- Copyright @ 2020 by the authors.s. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
- Rights
- Open access
- Rights
- This metadata is freely available under a CCO license
- Subject
- Banach space; Biorthogonal system; Dual space; Markushevich base; Quotient space; Separable quotient problem; Separable space; Weakly compactly generated
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