On \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${{\cal T}_{\!p}}$\end{document}-Locally Uniformly Rotund Norms

Linear topological characterizations of Banach spaces E ⊂ ℓ ∞ (Γ) which admit pointwise locally uniformly rotund norms are obtained. We introduce a new way to construct the norm with families of sliced sets. The topological properties described are related with the theory of generalized metric spaces, in particular with Moore spaces and σ-spaces. A non liner transfer is obtained, Question 6.16 in Moltó et al. (2009) is answered and some connections with Kenderov’s School of Optimization is presented in this paper.