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.

scholarly journals On the closed graph theorem

1976 ◽  
Vol 17 (2) ◽  
pp. 89-97 ◽  
Author(s):  
J. O. Popoola ◽  
I. Tweddle
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Cardinal Number ◽  
Closed Graph ◽  
Graph Theorem ◽  
Closed Graph Theorem ◽  
Infinite Dimensional ◽  
Barrelled Spaces ◽  
Natural Sense ◽  
Locally Convex

Our main purpose is to describe those separated locally convex spaces which can serve as domain spaces for a closed graph theorem in which the range space is an arbitrary Banach space of (linear) dimension at most c, the cardinal number of the real line R. These are the δ-barrelled spaces which are considered in §4. Many of the standard elementary Banach spaces, including in particular all separable ones, have dimension at most c. Also it is known that an infinite dimensional Banach space has dimension at least c (see e.g. [8]). Thus if we classify Banach spaces by dimension we are dealing, in a natural sense, with the first class which contains infinite dimensional spaces.

scholarly journals Observations on the Separable Quotient Problem for Banach Spaces

Axioms ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 7 ◽  
Author(s):  
Sidney A. Morris ◽  
David T. Yost
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Continuous Linear Operator ◽  
Dense Subspace ◽  
Dimensional Banach Space ◽  
Infinite Dimensional ◽  
Infinite Dimensional Banach Space ◽  
Special Cases ◽  
Separable Quotient Problem ◽  
Special Case

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.

scholarly journals S-Barrelled Topological Vector Spaces*

1978 ◽  
Vol 21 (2) ◽  
pp. 221-227 ◽  
Author(s):  
Ray F. Snipes
Keyword(s):  
Banach Spaces ◽  
Vector Spaces ◽  
Fréchet Spaces ◽  
Topological Vector Spaces ◽  
Barrelled Spaces ◽  
Steinhaus Theorem ◽  
Locally Convex

N. Bourbaki [1] was the first to introduce the class of locally convex topological vector spaces called “espaces tonnelés” or “barrelled spaces.” These spaces have some of the important properties of Banach spaces and Fréchet spaces. Indeed, a generalized Banach-Steinhaus theorem is valid for them, although barrelled spaces are not necessarily metrizable. Extensive accounts of the properties of barrelled locally convex topological vector spaces are found in [5] and [8].

scholarly journals Separability of Topological Groups: A Survey with Open Problems

Axioms ◽  
2018 ◽  
Vol 8 (1) ◽  
pp. 3 ◽  
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.

scholarly journals On the separable quotient problem for Banach spaces

2018 ◽  
Vol 59 (2) ◽  
pp. 153-173 ◽  
Author(s):  
Juan C. Ferrando ◽  
Jerzy Kąkol ◽  
Manuel López-Pellicer ◽  
Wiesław Śliwa
Keyword(s):  
Banach Spaces ◽  
Separable Quotient Problem
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