On the Nakano Individual Convergence

Abstract

We have recently defined the notion of individual convergence for a sequence of positive elements of an Archimedean Riesz space $E$. In the note we complete the definition (i.e., we define the individual convergence for sequences of not necessarily positive elements of $E$), and we prove that our notion of individual convergence is a natural extension of the individual convergence as defined by Nakano: we will prove that if a sequence of elements of $E$ has an individual limit in the Nakano sense, then it converges individually with respect to our definition.