Irregular densifiability produced by James theorem

Abstract

In this paper we point out that in spite of the possibility of defining weak curves “filling” the closed unit ball $B_{X}$ of any normed space $X$ is optimum, the existence of a continuous linear functional which does not attain its sup on the closed unit ball of a non-reflexive Banach space (James’s theorem) allows us to find weak neighborhoods producing an anomalous behavior of the density of any a priori given weak curve in $B_{X}.$ This fact is not an obstacle to define an algorithm to approach solutions to a global optimization problem posed on the closed unit ball of a class of normed spaces which contains the non-reflexive Banach spaces.