A Riesz representation theory for completely regular Hausdorff spaces and its applications
Let X be a completely regular Hausdorff space, E and F be Banach spaces. Let Cb(X, E) be the space of all E-valued bounded, continuous functions on X, equipped with the strict topology β. We develop the Riemman-Stieltjes-type Integral representation theory of (β, || · ||F) -continuous operators T :...
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| Format: | Article |
| Language: | English |
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De Gruyter
2016-01-01
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| Series: | Open Mathematics |
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| Online Access: | https://doi.org/10.1515/math-2016-0043 |
| _version_ | 1819122024796127232 |
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| author | Nowak Marian |
| author_facet | Nowak Marian |
| author_sort | Nowak Marian |
| collection | DOAJ |
| description | Let X be a completely regular Hausdorff space, E and F be Banach spaces. Let Cb(X, E) be the space of all E-valued bounded, continuous functions on X, equipped with the strict topology β. We develop the Riemman-Stieltjes-type Integral representation theory of (β, || · ||F) -continuous operators T : Cb(X, E) → F with respect to the representing Borel operator measures. For X being a k-space, we characterize strongly bounded (β, || · ||F)-continuous operators T : Cb(X, E) → F. As an application, we study (β, || · ||F)-continuous weakly compact and unconditionally converging operators T : Cb(X, E) → F. In particular, we establish the relationship between these operators and the corresponding Borel operator measures given by the Riesz representation theorem. We obtain that if X is a k-spaceand E is reflexive, then (Cb(X, E), β) has the V property of Pełczynski. |
| first_indexed | 2024-12-22T06:45:53Z |
| format | Article |
| id | doaj.art-24fb371edc7f43189634df04e4421117 |
| institution | Directory Open Access Journal |
| issn | 2391-5455 |
| language | English |
| last_indexed | 2024-12-22T06:45:53Z |
| publishDate | 2016-01-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Open Mathematics |
| spelling | doaj.art-24fb371edc7f43189634df04e44211172022-12-21T18:35:17ZengDe GruyterOpen Mathematics2391-54552016-01-0114147449610.1515/math-2016-0043math-2016-0043A Riesz representation theory for completely regular Hausdorff spaces and its applicationsNowak Marian0Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, ul. Szafrana 4A, 65-516 Zielona Góra, PolandLet X be a completely regular Hausdorff space, E and F be Banach spaces. Let Cb(X, E) be the space of all E-valued bounded, continuous functions on X, equipped with the strict topology β. We develop the Riemman-Stieltjes-type Integral representation theory of (β, || · ||F) -continuous operators T : Cb(X, E) → F with respect to the representing Borel operator measures. For X being a k-space, we characterize strongly bounded (β, || · ||F)-continuous operators T : Cb(X, E) → F. As an application, we study (β, || · ||F)-continuous weakly compact and unconditionally converging operators T : Cb(X, E) → F. In particular, we establish the relationship between these operators and the corresponding Borel operator measures given by the Riesz representation theorem. We obtain that if X is a k-spaceand E is reflexive, then (Cb(X, E), β) has the V property of Pełczynski.https://doi.org/10.1515/math-2016-0043spaces of vector-valued continuous functionsstrict topologiesoperator measuresstrongly bounded operatorsunconditionally converging operatorsweakly compact operators46g1046e4046a7028a32 |
| spellingShingle | Nowak Marian A Riesz representation theory for completely regular Hausdorff spaces and its applications Open Mathematics spaces of vector-valued continuous functions strict topologies operator measures strongly bounded operators unconditionally converging operators weakly compact operators 46g10 46e40 46a70 28a32 |
| title | A Riesz representation theory for completely regular Hausdorff spaces and its applications |
| title_full | A Riesz representation theory for completely regular Hausdorff spaces and its applications |
| title_fullStr | A Riesz representation theory for completely regular Hausdorff spaces and its applications |
| title_full_unstemmed | A Riesz representation theory for completely regular Hausdorff spaces and its applications |
| title_short | A Riesz representation theory for completely regular Hausdorff spaces and its applications |
| title_sort | riesz representation theory for completely regular hausdorff spaces and its applications |
| topic | spaces of vector-valued continuous functions strict topologies operator measures strongly bounded operators unconditionally converging operators weakly compact operators 46g10 46e40 46a70 28a32 |
| url | https://doi.org/10.1515/math-2016-0043 |
| work_keys_str_mv | AT nowakmarian arieszrepresentationtheoryforcompletelyregularhausdorffspacesanditsapplications AT nowakmarian rieszrepresentationtheoryforcompletelyregularhausdorffspacesanditsapplications |