Bidual octahedral renormings and strong regularity in Banach spaces
We prove that every separable Banach space containing \(\ell_1\) can be equivalently renormed so that its bidual space is octahedral, which answers, in the separable case, a question by Godefroy in 1989. As a direct consequence, we obtain that every dual Banach space, with a separable predual, faili...
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Published in | arXiv.org |
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Main Authors | , |
Format | Paper Journal Article |
Language | English |
Published |
Ithaca
Cornell University Library, arXiv.org
11.04.2019
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Subjects | |
Online Access | Get full text |
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Summary: | We prove that every separable Banach space containing \(\ell_1\) can be equivalently renormed so that its bidual space is octahedral, which answers, in the separable case, a question by Godefroy in 1989. As a direct consequence, we obtain that every dual Banach space, with a separable predual, failing to be strongly regular (that is, without convex combinations of slices with diameter arbitrarily small for some closed, convex and bounded subset) can be equivalently renormed with a dual norm to satisfy the strong diameter two property (that is, such that every convex combination of slices in its unit ball has diameter two). |
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ISSN: | 2331-8422 |
DOI: | 10.48550/arxiv.1902.04170 |