The strong WCD property for Banach spaces
In this paper, we introduce weakly compact version of the weakly countably determined (WCD) property, the strong WCD (SWCD) property. A Banach space X is said to be SWCD if there s a sequence (An) of weak ∗ compact subsets of X∗∗ such that if K⊂X is weakly compact, there is an (nm)⊂N such that K⊂⋂m=...
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| Format: | Article |
| Language: | English |
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Wiley
1995-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171295000081 |
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| author | Dave Wilkins |
| author_facet | Dave Wilkins |
| author_sort | Dave Wilkins |
| collection | DOAJ |
| description | In this paper, we introduce weakly compact version of the weakly countably
determined (WCD) property, the strong WCD (SWCD) property. A Banach space X is said to
be SWCD if there s a sequence (An) of weak ∗ compact subsets of X∗∗ such that if K⊂X is
weakly compact, there is an (nm)⊂N such that K⊂⋂m=1∞Anm⊂X. In this case, (An) is called a
strongly determining sequence for X. We show that SWCG⇒SWCD and that the converse does
not hold in general. In fact, X is a separable SWCD space if and only if (X, weak) is an ℵ0-space.
Using c0 for an example, we show how weakly compact structure theorems may be used to
construct strongly determining sequences. |
| format | Article |
| id | doaj-art-e1beccade700458487cb7a5eb7f8b46d |
| institution | DOAJ |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1995-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-e1beccade700458487cb7a5eb7f8b46d2025-08-20T03:20:59ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251995-01-01181677010.1155/S0161171295000081The strong WCD property for Banach spacesDave Wilkins0Department of Mathematics, Lewis Universty, Romeoville 60441, IL, USAIn this paper, we introduce weakly compact version of the weakly countably determined (WCD) property, the strong WCD (SWCD) property. A Banach space X is said to be SWCD if there s a sequence (An) of weak ∗ compact subsets of X∗∗ such that if K⊂X is weakly compact, there is an (nm)⊂N such that K⊂⋂m=1∞Anm⊂X. In this case, (An) is called a strongly determining sequence for X. We show that SWCG⇒SWCD and that the converse does not hold in general. In fact, X is a separable SWCD space if and only if (X, weak) is an ℵ0-space. Using c0 for an example, we show how weakly compact structure theorems may be used to construct strongly determining sequences.http://dx.doi.org/10.1155/S0161171295000081Banach spacesWCGWCDSWCG. |
| spellingShingle | Dave Wilkins The strong WCD property for Banach spaces International Journal of Mathematics and Mathematical Sciences Banach spaces WCG WCD SWCG. |
| title | The strong WCD property for Banach spaces |
| title_full | The strong WCD property for Banach spaces |
| title_fullStr | The strong WCD property for Banach spaces |
| title_full_unstemmed | The strong WCD property for Banach spaces |
| title_short | The strong WCD property for Banach spaces |
| title_sort | strong wcd property for banach spaces |
| topic | Banach spaces WCG WCD SWCG. |
| url | http://dx.doi.org/10.1155/S0161171295000081 |
| work_keys_str_mv | AT davewilkins thestrongwcdpropertyforbanachspaces AT davewilkins strongwcdpropertyforbanachspaces |