Abstract
We study the equivalence under renorming of several geometric and topological properties of the unit sphere of a Banach space with respect to weak topologies. The geometric properties considered are stronger forms of rotundity and the topological properties are generalizations of metrizability. In the case of dual Banach spaces endowed with the weak\(^*\) topology our results provide a full understanding of the rotund case and complement our previous work on \(w^*\)-LUR renorming.
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Cascales, B., Orihuela, J.: A biased view of topology as a tool in functional analysis. In: Hart, K.P., van Mill, J., Simon, P. (eds.) Recent Progress in General Topology III, pp. 93–164. Atlantis Press, Paris (2014)
Ceder, J.G.: Some generalizations of metric spaces. Pac. J. Math. 11, 105–125 (1961)
Deville, R., Godefory, G., Zizler, V.: Smoothness and Renormings in Banach Spaces, Pitman Monographs 64. Longman Scientific and Technical. Wiley, New York (1993)
Fabian, M., Habala, P., Hájek, P., Montesinos, V., Zizler, V.: Banach Space Theory. CMS Books in Mathematics, The Basis for Linear and Nonlinear Analysis. Springer, New York (2011)
Ferrari, S., Orihuela, J., Raja, M.: Weakly metrizability of spheres and renormings of Banach spaces. Q. J. Math. 67, 15–27 (2016)
Gruenhage, G.: Generalized metric spaces. In: Kunen, K., Vaughan, J.E. (eds.) Handbook of Set-Theoretic Topology, pp. 423–501. North-Holland, Amsterdam (1984)
Hansell, R.W.: Descriptive sets and the topology of nonseparable Banach spaces. Serdica Math. J. 27, 1–66 (2001)
Johnson, W.B., Lindenstrauss, J. (eds.): Handbook of the Geometry of Banach Spaces I and II, p. 2003. Elsevier, Amsterdam (2001)
Orihuela, J.: On \({{\cal{T}}}_p\)-locally uniformly rotund norms. Set-Valued Var. Anal. 21(4), 691–709 (2013)
Orihuela, J., Troyanski, S.: LUR renormings through Deville’s master lemma. RACSAM Rev. R. Acad. Cien. Serie A. Mat. 103(1), 75–85 (2009)
Orihuela, J., Smith, R., Troyanski, S.: Strictly convex norms and topology. Proc. Lond. Math. Soc. 104(1), 197–222 (2012)
Raja, M.: Weak\({}^*\) locally uniformly rotund norms and descriptive compact spaces. J. Funct. Anal. 197, 1–13 (2003)
Raja, M.: Dentability indices with respect to measures of non-compactness. J. Funct. Anal. 253, 273–286 (2007)
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Dedicated to the memory of our friend Bernardo Cascales.
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This work was supported by the Grants of Ministerio de Economía, Industria y Competitividad MTM2017-83262-C2-2-P; and Fundación Séneca Región de Murcia 19368/PI/14.
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Ferrari, S., Orihuela, J. & Raja, M. Generalized metric properties of spheres and renorming of Banach spaces. RACSAM 113, 2655–2663 (2019). https://doi.org/10.1007/s13398-019-00652-1
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DOI: https://doi.org/10.1007/s13398-019-00652-1