Geometry of Banach spaces with property β
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1999
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Hebrew University Magnes Press
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Suárez Granero, A., Jiménez Sevilla, M. M., Moreno, J. P. «Geometry of Banach Spaces with Property β». Israel Journal of Mathematics, vol. 111, n.o 1, diciembre de 1999, pp. 263-73. DOI.org (Crossref), https://doi.org/10.1007/BF02810687.
Abstract
We prove that every Banach space can be isometrically and 1-complementably embedded into a Banach space which satisfies property β and has the same character of density. Then we show that, nevertheless, property β satisfies a hereditary property. We study strong subdifferentiability of norms with property β to characterize separable polyhedral Banach spaces as those admitting a strongly subdifferentiable β norm. In general, a Banach space with such a norm is polyhedral. Finally, we provide examples of non-reflexive spaces whose usual norm fails property β and yet it can be approximated by norms with this property, namely (L 1[0,1], ‖·‖1) and (C(K), ‖·‖∗) whereK is a separable Hausdorff compact space
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The authors wish to thank S. Troyanski for valuable discussions. Also, they are very grateful to the referee for many helpful suggestions and, in particular, for observing the validity of Proposition 3.2.