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Open AccessArticle

Observations on the Separable Quotient Problem for Banach Spaces

by Sidney A. Morris 1,*,† and David T. Yost 2,†
1
Department of Mathematics and Statistics, La Trobe University, Melbourne, VIC 3086, Australia
2
Centre for Informatics and Applied Optimisation, Federation University Australia, P.O. Box 663, Ballarat, VIC 3353, Australia
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Received: 18 December 2019 / Revised: 8 January 2020 / Accepted: 8 January 2020 / Published: 13 January 2020
(This article belongs to the Collection Topological Groups)
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. View Full-Text
Keywords: Banach space; separable space; quotient space; weakly compactly generated; dual space; separable quotient problem; Markushevich base; biorthogonal system Banach space; separable space; quotient space; weakly compactly generated; dual space; separable quotient problem; Markushevich base; biorthogonal system
MDPI and ACS Style

Morris, S.A.; Yost, D.T. Observations on the Separable Quotient Problem for Banach Spaces. Axioms 2020, 9, 7.

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