Journal of Convex Analysis 16 (2009), No. 3, 959--972
Copyright Heldermann Verlag 2009
Deville's Master Lemma and Stone's Discreteness in Renorming Theory
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
[Abstract-pdf]
Banach spaces $X$ with an equivalent $\sigma(X,F)$-lower semicontinuous and locally uniformly rotund norm, for a norming subspace $F\subset X^*$, are those spaces $X$ that admit countably many families of convex and $\sigma(X,F)$-lower semicontinuous functions $\{\varphi_i^n:X \rightarrow {\mathbb R}^+ ; i \in I_n\}_{n=1}^\infty$ such that there are open subsets $$G_i^n \subset \{\varphi_i^n >0\} \cap\{\varphi_j^n =0: j\neq i, j \in I_n\}$$ with $\{G_i^n: i\in I_n, n\in {\mathbb N}\}$ a basis for the norm topology of $X$.
Keywords: Banach space, local uniform rotundity, slicely-isolatedness, network, convex biorthogonal system.
MSC: 46B03, 46B20, 46B26, 54E35
[ Fulltext-pdf (164 KB)] for subscribers only.