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A topological group observation on the Banach-Mazur separable quotient problem

- Gabriyelyan, Saak, Morris, Sidney

**Authors:**Gabriyelyan, Saak , Morris, Sidney**Date:**2019**Type:**Text , Journal article**Relation:**Topology and Its Applications Vol. 259, no. (2019), p. 283-286**Full Text:****Reviewed:****Description:**The Separable Quotient Problem of Banach and Mazur asks if every infinite-dimensional Banach space has an infinite-dimensional separable quotient Banach space. It has remained unsolved for 85 years but has been answered in the affirmative for special cases such as reflexive Banach spaces. An affirmative answer to the Separable Quotient Problem would obviously imply that every infinite-dimensional Banach space has a quotient topological group which is separable, metrizable, and infinite-dimensional in the sense of topology. In this paper it is proved that every infinite-dimensional Banach space has as a quotient group the separable metrizable infinite-dimensional topological group, T

**Authors:**Gabriyelyan, Saak , Morris, Sidney**Date:**2019**Type:**Text , Journal article**Relation:**Topology and Its Applications Vol. 259, no. (2019), p. 283-286**Full Text:****Reviewed:****Description:**The Separable Quotient Problem of Banach and Mazur asks if every infinite-dimensional Banach space has an infinite-dimensional separable quotient Banach space. It has remained unsolved for 85 years but has been answered in the affirmative for special cases such as reflexive Banach spaces. An affirmative answer to the Separable Quotient Problem would obviously imply that every infinite-dimensional Banach space has a quotient topological group which is separable, metrizable, and infinite-dimensional in the sense of topology. In this paper it is proved that every infinite-dimensional Banach space has as a quotient group the separable metrizable infinite-dimensional topological group, T

The tubby torus as a quotient group

**Authors:**Morris, Sidney**Date:**2020**Type:**Text , Journal article , Article**Relation:**Axioms Vol. 9, no. 1 (2020), p.**Full Text:****Reviewed:****Description:**Let E be any metrizable nuclear locally convex space and E the Pontryagin dual group of E. Then the topological group bE 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. © 2020 by the author.

**Authors:**Morris, Sidney**Date:**2020**Type:**Text , Journal article , Article**Relation:**Axioms Vol. 9, no. 1 (2020), p.**Full Text:****Reviewed:****Description:**Let E be any metrizable nuclear locally convex space and E the Pontryagin dual group of E. Then the topological group bE 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. © 2020 by the author.

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