Locally uniformly rotund renorming and decompositions of Banach spaces

Abstract

A norm |·| of a Banach space x is called locally uniformly rotund if lim|xn−x| = 0 whenever xn, x ∈ X, and . It is shown that such an equivalent norm exists on every Banach space x which possesses a projectional resolution {pα} of the identify operator, for which all (pα+1−pα)X admit such norms. This applies, for example, for the dual space of a space with Fréchet differentiable norm.