Jordan Decomposition for WBV-functions in Ordered Normed Spaces
Amit Kumar
TL;DR
The work addresses extending Jordan-type decompositions for functions of bounded variation to WBV-functions valued in normed spaces by introducing a weak relation induced by extensible cones. It defines strong and weak relations, builds extensible cones via Hahn–Banach extensions, and proves a weak-type Jordan decomposition: every WBV-function $f:[a,b]\to \mathbb{X}$ can be written as $f \sim_{x_0}(f_{x_{0,1}}-f_{x_{0,2}})$ with $f_{x_{0,i}}$ increasing in the cone-ordered space $(\mathbb{X}, \mathbb{X}_{x_0}^+)$. The results connect order-structure in vector lattices to normed-space analysis, enabling WBV-function decompositions that generalize classical BV theory. This provides a framework for generalized decompositions in ordered normed spaces with potential applications to generalized solutions in analysis and physics.
Abstract
In this paper, we define two relations one by orthogonality in vector lattices named as strong relation and the other by bounded linear functionals in normed spaces named as weak relation. It turns out that strong relation is an equivalence relation. We study some of the characterizations of these relations. Given a non-zero element in a normed space, we construct an extensible cone which makes that normed space, an ordered normed space. This extensible cone induces the weak relation in the normed space. Later, we prove a Jordan Decomposition Theorem in a normed space by the weak relation induced by the extensible cone.