Abstract

Banach spaces XX with an equivalent σ(X,F)\sigma(X,F)-lower semicontinuous and locally uniformly rotund norm, for a norming subspace F⊂X∗F\subset X^*, are those spaces XX that admit countably many families of convex and σ(X,F)\sigma(X,F)-lower semicontinuous functions {φin:X→R+;i∈In}n=1∞\{\varphi_i^n:X \rightarrow {\mathbb R}^+; i \in I_n\}_{n=1}^\infty such that there are open subsets Gin⊂{φin>0}∩{φjn=0:j≠i,j∈In}G_i^n \subset \{\varphi_i^n >0\} \cap\{\varphi_j^n =0: j\neq i, j \in I_n\} with {Gin:i∈In,n∈N}\{G_i^n: i\in I_n, n\in {\mathbb N}\} a basis for the norm topology of XX.

Author information

Contact details are reproduced from the original publication and may be historical.

José Orihuela

Dep. de Matemáticas, Universidad de Murcia, 30.100 Espinardo - Murcia, Spain

joseori@um.es

Stanimir Troyanski

Dep. de Matemáticas, Universidad de Murcia, 30.100 Espinardo - Murcia, Spain

stroya@um.es

Keywords

  • Banach space
  • local uniform rotundity
  • slicely-isolatedness
  • network
  • convex biorthogonal system

Mathematics Subject Classification

46B03, 46B20, 46B26, 54E35

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Suggested citation

J. Orihuela, S. Troyanski. “Deville's Master Lemma and Stone's Discreteness in Renorming Theory.” Journal of Convex Analysis 16 (2009), No. 3&4, 959–972. https://doi.org/10.68381/jca16058

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Published by Heldermann Verlag, 2009. Rights now held by Banach Press LLC.