Barrelled Spaces With(out) Separable Quotients

Ka̧kol, Jerzy; Saxon, Stephen A.; Todd, Aaron R.

doi:10.1017/s0004972714000422

Cited by 15 publications

(11 citation statements)

References 23 publications

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“…An extension of the previous corollary to all barrelled spaces C k (X) with the compactopen topology has been obtained in [41].…”

Section: Weak* Compactness Of B X * and Separable Quotientsmentioning

confidence: 87%

“…For these classes of spaces we fix particular separable quotients by means of the literature on the subject and the previous results. This line of research has been also continued in a more general setting for the class of topological vector spaces, particularly for spaces C(X) of real-valued continuous functions endowed with the pointwise and compact-open topology, see [40], [41] and [42].…”

Section: Problem 2 Does Any Infinite Dimensional Banach Space Admitsmentioning

confidence: 99%

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On the separable quotient problem for Banach spaces

Ferrando¹,

Ka̧kol²,

Pellicer³

et al. 2018

Funct. Approx. Comment. Math.

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While the classic separable quotient problem remains open, we survey general results related to this problem and examine the existence of a particular infinitedimensional separable quotient in some Banach spaces of vector-valued functions, linear operators and vector measures. Most of the results presented are consequence of known facts, some of them relative to the presence of complemented copies of the classic sequence spaces c 0 and ℓ p , for 1 ≤ p ≤ ∞. Also recent results of Argyros, Dodos, Kanellopoulos [1] andŚliwa [66] are provided. This makes our presentation supplementary to a previous survey (1997) due to Mujica. 1991 Mathematics Subject Classification. 46B28, 46E27, 46E30.

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“…An extension of the previous corollary to all barrelled spaces C k (X) with the compactopen topology has been obtained in [41].…”

Section: Weak* Compactness Of B X * and Separable Quotientsmentioning

confidence: 87%

Section: Problem 2 Does Any Infinite Dimensional Banach Space Admitsmentioning

confidence: 99%

On the separable quotient problem for Banach spaces

Ferrando¹,

Ka̧kol²,

Pellicer³

et al. 2018

Funct. Approx. Comment. Math.

Self Cite

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show abstract

“…[26] Every strict inductive limit of a strictly increasing sequence (E m ) of Fréchet spaces with at least one E m non-normable has a properly separable quotient locally convex space. Jerzy Kakol, Steve Saxon and Aaron Todd [17] answered Problem 1.12 in the negative. Theorem 1.…”

Section: Introductionmentioning

confidence: 99%

“…16. [17] There exist infinite-dimensional barrelled locally convex spaces which do not have any infinite-dimensional separable quotient locally convex spaces.…”

Section: Introductionmentioning

confidence: 99%

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The separable quotient problem for topological groups

2019

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The famous Banach-Mazur problem, which asks if every infinitedimensional Banach space has an infinite-dimensional separable quotient Banach space, has remained unsolved for 85 years, though it has been answered in the affirmative for reflexive Banach spaces and even Banach spaces which are duals. The analogous problem for locally convex spaces has been answered in the negative, but has been shown to be true for large classes of locally convex spaces including all non-normable Fréchet spaces. In this paper the analogous problem for topological groups is investigated. Indeed there are four natural analogues: Does every non-totally disconnected topological group have a separable quotient group which is (i) non-trivial; (ii) infinite; (iii) metrizable; (iv) infinite metrizable. All four questions are answered here in the negative. However, positive answers are proved for important classes of topological groups including (a) all compact groups; (b) all locally compact abelian groups; (c) all σ-compact locally compact groups; (d) all abelian pro-Lie groups; (e) all σ-compact pro-Lie groups; (f) all pseudocompact groups. Negative answers are proved for precompact groups.Problem 1.1. Separable Quotient Problem for Banach Spaces. Does every infinite-dimensional Banach space have a quotient Banach space which is separable and infinite-dimensional? The related Quotient Schauder Basis Problem for Banach Spaces is due to Aleksander (Olek) Pe lczyński [24]. Problem 1.2. Quotient Schauder Basis Problem for Banach Spaces. Does every infinite-dimensional Banach space have a quotient Banach space which is infinite-dimensional and has a Schauder basis?

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