# Ordered normed spaces of functions of bounded variation

## TL;DR

This paper introduces and analyzes the space of functions of bounded variation valued in an ordered vector space, establishing its structure as an absolute order unit space and exploring conditions for completeness.

## Contribution

It defines the space of bounded variation functions into an ordered vector space and investigates its structure as an absolute order unit space, including norm completeness conditions.

## Key findings

- The space forms a nearer absolute order unit space under natural order.
- It can become an absolute order unit space, specifically a unital $AM$-space.
- Certain conditions induce a complete norm on the space.

## 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 $\mathcal{BV}[x,y].$