Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Convex Analysis 16 (2009), No. 3, 959--972Copyright 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.