Strictly convex renormings

Abstract

A normed space X is said to be strictly convex if x = y whenever ‖(x + y)/2‖ = ‖x‖ = ‖y, in other words, when the unit sphere of X does not contain non-trivial segments. Our aim in this paper is the study of those normed spaces which admit an equivalent strictly convex norm. We present a characterization in linear topological terms of the normed spaces which are strictly convex renormable. We consider the class of all solid Banach lattices made up of bounded real functions on some set Γ. This class contains the Mercourakis space c1(Σ′ × Γ) and all duals of Banach spaces with unconditional uncountable bases. We characterize the elements of this class which admit a pointwise strictly convex renorming.