On ${{\cal T}_{\!p}}$ -Locally Uniformly Rotund Norms

Abstract

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 Molto et al. (2009) is answered and some connections with Kenderov’s School of Optimization is presented in this paper.