The uniform bounded deciding property and the separable quotient problem

2018 ◽  
Vol 113 (2) ◽  
pp. 1223-1230 ◽  
Author(s):  
S. López-Alfonso ◽  
S. Moll
Keyword(s):  
Separable Quotient Problem

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.

scholarly journals Metrizable (LF)-spaces, (db)-spaces, and the separable quotient problem

1981 ◽  
Vol 23 (1) ◽  
pp. 65-80 ◽  
Author(s):  
Stephen A. Saxon ◽  
P.P. Narayanaswami
Keyword(s):  
Locally Convex Spaces ◽  
Separable Quotient Problem ◽  
Locally Convex ◽  
Convex Spaces ◽  
Vector Subspaces

The existence of metrizable (LF)-spaces was announced by Stephen A. Saxon (“Metrizable generalized (LF)-spaces”, 701–46–14), in Notices Amer. Math. Soc. 20 (1973), A–143. Elsewhere, the authors have discovered an abundant existence of metrizable and normable (generalized) (LF)-spaces, while observing that an (LF)-space is metrizable if and only if it is Baire-like. Recently, W. Robertson, I. Tweddle and F.E. Yeomans introduced the class of locally convex spaces E having the property(db) if E is the union of an increasing sequence (En) of vector subspaces, then some En is dense and barrelled.

scholarly journals The separable quotient problem for topological groups

2019 ◽  
Vol 234 (1) ◽  
pp. 331-369 ◽  
Author(s):  
Arkady G. Leiderman ◽  
Sidney A. Morris ◽  
Mikhail G. Tkachenko
Keyword(s):  
Topological Groups ◽  
Separable Quotient Problem

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.

A REMARK ON THE SEPARABLE QUOTIENT PROBLEM FOR TOPOLOGICAL GROUPS

2019 ◽  
Vol 100 (3) ◽  
pp. 453-457 ◽  
Author(s):  
SIDNEY A. MORRIS
Keyword(s):  
Banach Space ◽  
Topological Group ◽  
Vector Space ◽  
Topological Groups ◽  
Abelian Topological Group ◽  
Dimensional Banach Space ◽  
Infinite Dimensional ◽  
Infinite Dimensional Banach Space ◽  
Special Cases ◽  
Separable Quotient Problem

The Banach–Mazur separable quotient problem asks whether every infinite-dimensional Banach space $B$ has a quotient space that is an infinite-dimensional separable Banach space. The question has remained open for over 80 years, although an affirmative answer is known in special cases such as when $B$ is reflexive or even a dual of a Banach space. Very recently, it has been shown to be true for dual-like spaces. An analogous problem for topological groups is: Does every infinite-dimensional (in the topological sense) connected (Hausdorff) topological group $G$ have a quotient topological group that is infinite dimensional and metrisable? While this is known to be true if $G$ is the underlying topological group of an infinite-dimensional Banach space, it is shown here to be false even if $G$ is the underlying topological group of an infinite-dimensional locally convex space. Indeed, it is shown that the free topological vector space on any countably infinite $k_{\unicode[STIX]{x1D714}}$-space is an infinite-dimensional toplogical vector space which does not have any quotient topological group that is infinite dimensional and metrisable. By contrast, the Graev free abelian topological group and the Graev free topological group on any infinite connected Tychonoff space, both of which are connected topological groups, are shown here to have the tubby torus $\mathbb{T}^{\unicode[STIX]{x1D714}}$, which is an infinite-dimensional metrisable group, as a quotient group.

Sign in / Sign up

Export Citation Format

Share Document