Best approximation from the positive cone of an inner product lattice

TL;DR

This paper characterizes when an inner product ordered vector space is a vector lattice with a lattice norm, linking it to the existence and properties of metric projections onto the positive cone, and shows best approximation coincides with positive part.

Contribution

It extends previous results by establishing equivalences involving the structure of the space, the positive cone, and approximation properties in a non-complete setting.

Findings

Equivalence between vector lattice structure and properties of metric projection.

Best approximation from the positive cone equals the positive part of an element.

Extension of prior results to non-complete inner product lattice spaces.

Abstract

Let $E$ be a directed (i.e., positively generated) ordered vector space endowed with an inner product. In this note, we prove that the following statements are equivalent: i) $E$ is a vector lattice and its norm induced by its inner product is a lattice norm. ii) The metric projection onto the positive cone $E^+$ of $E$ exists and it is both isotone and subadditive. Moreover, in this case, the best approximation to any $x \in E$ from $E^+$ coincides with its positive part $x^+$. This result extends previous work to the non-complete setting.