The Tubby Torus as a Quotient Group
Let <i>E</i> be any metrizable nuclear locally convex space and <inline-formula> <math display="inline"> <semantics> <mover accent="true"> <mi>E</mi> <mo>^</mo> </mover> </semantics> </math> </inline...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-01-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/9/1/11 |
| _version_ | 1818479573400027136 |
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| author | Sidney A. Morris |
| author_facet | Sidney A. Morris |
| author_sort | Sidney A. Morris |
| collection | DOAJ |
| description | Let <i>E</i> be any metrizable nuclear locally convex space and <inline-formula> <math display="inline"> <semantics> <mover accent="true"> <mi>E</mi> <mo>^</mo> </mover> </semantics> </math> </inline-formula> the Pontryagin dual group of <i>E</i>. Then the topological group <inline-formula> <math display="inline"> <semantics> <mover accent="true"> <mi>E</mi> <mo>^</mo> </mover> </semantics> </math> </inline-formula> has the tubby torus (that is, the countably infinite product of copies of the circle group) as a quotient group if and only if <i>E</i> does not have the weak topology. This extends results in the literature related to the Banach−Mazur separable quotient problem. |
| first_indexed | 2024-12-10T11:12:32Z |
| format | Article |
| id | doaj.art-ed01a5c548e143a19537b7dc9218ff7a |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-12-10T11:12:32Z |
| publishDate | 2020-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-ed01a5c548e143a19537b7dc9218ff7a2022-12-22T01:51:22ZengMDPI AGAxioms2075-16802020-01-01911110.3390/axioms9010011axioms9010011The Tubby Torus as a Quotient GroupSidney A. Morris0School of Science, Engineering and Information Technology, Federation University Australia, P.O.B. 663, Ballarat, VIC 3353, AustraliaLet <i>E</i> be any metrizable nuclear locally convex space and <inline-formula> <math display="inline"> <semantics> <mover accent="true"> <mi>E</mi> <mo>^</mo> </mover> </semantics> </math> </inline-formula> the Pontryagin dual group of <i>E</i>. Then the topological group <inline-formula> <math display="inline"> <semantics> <mover accent="true"> <mi>E</mi> <mo>^</mo> </mover> </semantics> </math> </inline-formula> has the tubby torus (that is, the countably infinite product of copies of the circle group) as a quotient group if and only if <i>E</i> does not have the weak topology. This extends results in the literature related to the Banach−Mazur separable quotient problem.https://www.mdpi.com/2075-1680/9/1/11torustubby torusseparable quotient problemlocally convex spacenuclear spacebanach spacepontryagin dualityweak topology |
| spellingShingle | Sidney A. Morris The Tubby Torus as a Quotient Group Axioms torus tubby torus separable quotient problem locally convex space nuclear space banach space pontryagin duality weak topology |
| title | The Tubby Torus as a Quotient Group |
| title_full | The Tubby Torus as a Quotient Group |
| title_fullStr | The Tubby Torus as a Quotient Group |
| title_full_unstemmed | The Tubby Torus as a Quotient Group |
| title_short | The Tubby Torus as a Quotient Group |
| title_sort | tubby torus as a quotient group |
| topic | torus tubby torus separable quotient problem locally convex space nuclear space banach space pontryagin duality weak topology |
| url | https://www.mdpi.com/2075-1680/9/1/11 |
| work_keys_str_mv | AT sidneyamorris thetubbytorusasaquotientgroup AT sidneyamorris tubbytorusasaquotientgroup |