Aníbal Moltó, José Orihuela, Stanimir Troyanski, Manuel Valdivia (eds.)3540850309, 9783540850304
Abstract topological tools from generalized metric spaces are applied in this volume to the construction of locally uniformly rotund norms on Banach spaces. The book offers new techniques for renorming problems, all of them based on a network analysis for the topologies involved inside the problem.
Maps from a normed space X to a metric space Y, which provide locally uniformly rotund renormings on X, are studied and a new frame for the theory is obtained, with interplay between functional analysis, optimization and topology using subdifferentials of Lipschitz functions and covering methods of metrization theory. Any one-to-one operator T from a reflexive space X into c0 (T) satisfies the authors’ conditions, transferring the norm to X. Nevertheless the authors’ maps can be far from linear, for instance the duality map from X to X* gives a non-linear example when the norm in X is Fréchet differentiable.
This volume will be interesting for the broad spectrum of specialists working in Banach space theory, and for researchers in infinite dimensional functional analysis.
Table of contents :
Front Matter….Pages I-XI
Introduction….Pages 1-11
σ -Continuous and Co- σ -continuous Maps….Pages 13-47
Generalized Metric Spaces and Locally Uniformly Rotund Renormings….Pages 49-72
σ -Slicely Continuous Maps….Pages 73-99
Some Applications….Pages 101-116
Some Open Problems….Pages 117-129
Back Matter….Pages 131-148
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