A Reverse Theorem on the ·-w* Continuity of the Dual Map
Given a Banach space X, x∈𝖲X, and 𝖩Xx=x*∈𝖲X*:x*x=1, we define the set 𝖩X*x of all x*∈𝖲X* for which there exist two sequences xnn∈N⊆𝖲X∖{x} and xn*n∈N⊆𝖲X* such that xnn∈N converges to x, xn*n∈N has a subnet w*-convergent to x*, and xn*xn=1 for all n∈N. We prove that if X is separable and reflexive and...
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| Format: | Article |
| Language: | English |
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Wiley
2015-01-01
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| Series: | Journal of Function Spaces |
| Online Access: | http://dx.doi.org/10.1155/2015/864173 |
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| author | Mienie de Kock Francisco Javier García-Pacheco |
| author_facet | Mienie de Kock Francisco Javier García-Pacheco |
| author_sort | Mienie de Kock |
| collection | DOAJ |
| description | Given a Banach space X, x∈𝖲X, and 𝖩Xx=x*∈𝖲X*:x*x=1, we define the set 𝖩X*x of all x*∈𝖲X* for which there exist two sequences xnn∈N⊆𝖲X∖{x} and xn*n∈N⊆𝖲X* such that xnn∈N converges to x, xn*n∈N has a subnet w*-convergent to x*, and xn*xn=1 for all n∈N. We prove that if X is separable and reflexive and X* enjoys the Radon-Riesz property, then 𝖩X*x is contained in the boundary of 𝖩Xx relative to 𝖲X*. We also show that if X is infinite dimensional and separable, then there exists an equivalent norm on X such that the interior of 𝖩Xx relative to 𝖲X* is contained in 𝖩X*x. |
| format | Article |
| id | doaj-art-99a27e7b4992419fb4a3ab150c0dc91c |
| institution | OA Journals |
| issn | 2314-8896 2314-8888 |
| language | English |
| publishDate | 2015-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Function Spaces |
| spelling | doaj-art-99a27e7b4992419fb4a3ab150c0dc91c2025-08-20T02:18:25ZengWileyJournal of Function Spaces2314-88962314-88882015-01-01201510.1155/2015/864173864173A Reverse Theorem on the ·-w* Continuity of the Dual MapMienie de Kock0Francisco Javier García-Pacheco1Department of Mathematics and Physics, Texas A&M University Central Texas, Killeen, TX 76548, USADepartment of Mathematics, University of Cadiz, 11519 Puerto Real, SpainGiven a Banach space X, x∈𝖲X, and 𝖩Xx=x*∈𝖲X*:x*x=1, we define the set 𝖩X*x of all x*∈𝖲X* for which there exist two sequences xnn∈N⊆𝖲X∖{x} and xn*n∈N⊆𝖲X* such that xnn∈N converges to x, xn*n∈N has a subnet w*-convergent to x*, and xn*xn=1 for all n∈N. We prove that if X is separable and reflexive and X* enjoys the Radon-Riesz property, then 𝖩X*x is contained in the boundary of 𝖩Xx relative to 𝖲X*. We also show that if X is infinite dimensional and separable, then there exists an equivalent norm on X such that the interior of 𝖩Xx relative to 𝖲X* is contained in 𝖩X*x.http://dx.doi.org/10.1155/2015/864173 |
| spellingShingle | Mienie de Kock Francisco Javier García-Pacheco A Reverse Theorem on the ·-w* Continuity of the Dual Map Journal of Function Spaces |
| title | A Reverse Theorem on the ·-w* Continuity of the Dual Map |
| title_full | A Reverse Theorem on the ·-w* Continuity of the Dual Map |
| title_fullStr | A Reverse Theorem on the ·-w* Continuity of the Dual Map |
| title_full_unstemmed | A Reverse Theorem on the ·-w* Continuity of the Dual Map |
| title_short | A Reverse Theorem on the ·-w* Continuity of the Dual Map |
| title_sort | reverse theorem on the · w continuity of the dual map |
| url | http://dx.doi.org/10.1155/2015/864173 |
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