The separable quotient problem for barrelled spaces

1993 ◽  
pp. 289-308
Author(s):  
P. P. Narayanaswami
Keyword(s):  
Barrelled Spaces ◽  
Separable Quotient Problem

BARRELLED SPACES WITH(OUT) SEPARABLE QUOTIENTS

2014 ◽  
Vol 90 (2) ◽  
pp. 295-303 ◽  
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.

Optimal Cardinals for Metrizable Barrelled Spaces

1995 ◽  
Vol 51 (1) ◽  
pp. 137-147 ◽  
Author(s):  
S.A. Saxon ◽  
L.M. Sánchez Ruiz
Keyword(s):  
Barrelled Spaces

scholarly journals Some remarks on countably barrelled and countably quasibarrelled spaces

1967 ◽  
Vol 15 (4) ◽  
pp. 295-296 ◽  
Author(s):  
Sunday O. Iyahen
Keyword(s):  
Convex Space ◽  
Locally Convex Space ◽  
Locally Convex Spaces ◽  
Countable Union ◽  
Bounded Subset ◽  
Linear Subspaces ◽  
Barrelled Spaces ◽  
Locally Convex ◽  
Barrelled Space ◽  
Convex Spaces

Barrelled and quasibarrelled spaces form important classes of locally convex spaces. In (2), Husain considered a number of less restrictive notions, including infinitely barrelled spaces (these are the same as barrelled spaces), countably barrelled spaces and countably quasibarrelled spaces. A separated locally convex space E with dual E' is called countably barrelled (countably quasibarrelled) if every weakly bounded (strongly bounded) subset of E' which is the countable union of equicontinuous subsets of E' is itself equicontinuous. It is trivially true that every barrelled (quasibarrelled) space is countably barrelled (countably quasibarrelled) and a countably barrelled space is countably quasibarrelled. In this note we give examples which show that (i) a countably barrelled space need not be barrelled (or even quasibarrelled) and (ii) a countably quasibarrelled space need not be countably barrelled. A third example (iii)shows that the property of being countably barrelled (countably quasibarrelled) does not pass to closed linear subspaces.

scholarly journals The Tubby Torus as a Quotient Group

Axioms ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 11
Author(s):  
Sidney A. Morris
Keyword(s):  
Topological Group ◽  
Convex Space ◽  
Weak Topology ◽  
Quotient Group ◽  
Infinite Product ◽  
Locally Convex Space ◽  
Dual Group ◽  
Separable Quotient Problem ◽  
Countably Infinite ◽  
Locally Convex

Let E be any metrizable nuclear locally convex space and E ^ the Pontryagin dual group of E. Then the topological group E ^ has the tubby torus (that is, the countably infinite product of copies of the circle group) as a quotient group if and only if E does not have the weak topology. This extends results in the literature related to the Banach–Mazur separable quotient problem.

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