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Generalized Metric Spaces and Locally Uniformly Rotund Renormings

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1951)

Abstract

A class of generalized metric spaces is a class of spaces defined by a property shared by all metric αspaces which is close to metrizability in some sense [Gru84]. The s-spaces are defined by replacing the base by network in the Bing-Nagata-Smirnov metrization theorem; i.e. a topological space is a αspace if it has a αdiscrete network. Here we shall deal with a further re- finement replacing discrete by isolated or slicely isolated. Indeed we will see that the identity map from a subset A of a normed space is A of a normedslicely continuous if, and only if, the weak topology relative to A has a s-slicely isolated network. If A is also a radial set then we have that the identity map Id : (X, weak) ǁ (X,→) is A of a normedslicely continuous and Theorem 1.1 in the Introduction says that this is the case if, and only if, X has an equivalent LUR norm. After our study of this class of maps we can now formulate the following theorem and its corollaries summarizing different characterizations f LUR renormability for a Banach space.

Keywords

  • Banach Space
  • Normed Space
  • Unit Sphere
  • Weak Topology
  • Norm Topology

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Correspondence to Aníbal Moltó .

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© 2009 Springer-Verlag Berlin Heidelberg

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Moltó, A., Orihuela, J., Troyanski, S., Valdivia, M. (2009). Generalized Metric Spaces and Locally Uniformly Rotund Renormings. In: A Nonlinear Transfer Technique for Renorming. Lecture Notes in Mathematics, vol 1951. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85031-1_3

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