On Locally Uniformly Rotund Renormings in C(K) Spaces

Abstract

A characterization of the Banach spaces of type C(K) which admit an equivalent locally uniformly rotund norm is obtained, and a method to apply it to concrete spaces is developed. As an application the existence of such renorming is deduced when K is a Namioka-Phelps compact or for some particular class of Rosenthal compacta, results recently obtained in [3] and [6] that were originally proved with methods developed ad hoc.2006 Mathematics Subject Classification: 46B03, 46B20.

Locally uniformly rotund renormings of the spaces of continuous functions on Fedorchuk compacts

“…In this way is proved that C(X) is LUR renormable when X is Helly compact. A generalization of this result when X is a particular case of Rosenthal compacts can be found in [10], see also [12].…”

LUR renormings through Deville’s Master Lemma

“…The σ-discrete basis for the norm topology of a normed space X can be refined to obtain the basis described in Theorem 1. More recent contributions show an interplay between this method and the one based on Deville's lemma, [7,11,12]. It is our intention here to give a straightforward proof of the main renorming construction in [13,18].…”