Ordered normed spaces of functions of bounded variation

Abstract

In this paper, we define and study the space of all the functions of bounded variation $f:[x,y]\to \mathbb{Y}$ denoted by $\mathcal{BV}[x,y],$ where $[x,y]$ is an ordered interval and $\mathbb{Y}$ is an absolute order unit space having vector lattice structure. By default, under the order structure of $\mathbb{Y},$ the space $\mathcal{BV}[x,y]$ forms a nearer absolute order unit space structure and in some cases it turns out to be an absolute order unit space (in fact, a unital $AM$-space). By help of variation function, we also define a different kind of order structure on the space $\mathcal{BV}[x,y]$ that also makes $\mathcal{BV}[x,y]$ a nearer absolute order unit space structure. Later, we also show that under certain conditions this ordering induces a complete norm on $\mathcal{BV}[x,y].$