Deville's Master Lemma and Stone's Discreteness in Renorming Theory

J. Orihuela ; S. Troyanski

Journal of convex analysis, Tome 16 (2009) no. 3, pp. 959-972

Résumé

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 $X$.

Détail

Classification : 46B03, 46B20, 46B26, 54E35
Mots-clés : Banach space, local uniform rotundity, slicely-isolatedness, network, convex biorthogonal system