On the separable quotient problem for Banach spaces

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.

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BARRELLED SPACES WITH(OUT) SEPARABLE QUOTIENTS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972714000422 ◽

2014 ◽

Vol 90 (2) ◽

pp. 295-303 ◽

Cited By ~ 10

Author(s):

JERZY KĄKOL ◽

STEPHEN A. SAXON ◽

AARON R. TODD

Keyword(s):

Positive Solution ◽

Banach Spaces ◽

Barrelled Spaces ◽

Separable Quotient Problem

AbstractWhile the separable quotient problem is famously open for Banach spaces, in the broader context of barrelled spaces we give negative solutions. Obversely, the study of pseudocompact$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}X$and Warner bounded$X$allows us to expand Rosenthal’s positive solution for Banach spaces of the form$ C_{c}(X) $to barrelled spaces of the same form, and see that strong duals of arbitrary$C_{c}(X) $spaces admit separable quotients.

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Separability of Topological Groups: A Survey with Open Problems

Axioms ◽

10.3390/axioms8010003 ◽

2018 ◽

Vol 8 (1) ◽

pp. 3 ◽

Cited By ~ 2

Author(s):

Arkady Leiderman ◽

Sidney Morris

Keyword(s):

Banach Space ◽

Banach Spaces ◽

Quotient Space ◽

Topological Groups ◽

Open Problems ◽

Infinite Products ◽

Matrix Groups ◽

Compact Spaces ◽

Finite Dimensional ◽

Separable Quotient Problem

Separability is one of the basic topological properties. Most classical topological groups and Banach spaces are separable; as examples we mention compact metric groups, matrix groups, connected (finite-dimensional) Lie groups; and the Banach spaces C ( K ) for metrizable compact spaces K; and ℓ p , for p ≥ 1 . This survey focuses on the wealth of results that have appeared in recent years about separable topological groups. In this paper, the property of separability of topological groups is examined in the context of taking subgroups, finite or infinite products, and quotient homomorphisms. The open problem of Banach and Mazur, known as the Separable Quotient Problem for Banach spaces, asks whether every Banach space has a quotient space which is a separable Banach space. This paper records substantial results on the analogous problem for topological groups. Twenty open problems are included in the survey.

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