Orthogonality in $\ell_p$-spaces and its bearing on ordered Banach spaces

TL;DR

This paper introduces a new concept of p-orthogonality in Banach spaces, characterizes ℓ_p-spaces using this notion, and explores orthogonal decompositions in order smooth p-normed spaces, revealing their properties and limitations.

Contribution

It defines p-orthogonality in Banach spaces, characterizes ℓ_p-spaces within this framework, and investigates orthogonal decompositions and their duality in order smooth p-normed spaces.

Findings

p-orthogonality characterizes ℓ_p-spaces among Banach spaces

In order smooth ∞-normed spaces, ∞-orthogonal decompositions imply 1-orthogonal decompositions in duals

Orthogonal decompositions may not be unique in these spaces

Abstract

We introduce a notion of p-orthogonality in a general Banach space $1 \le p \le \infty$. We use this concept to characterize $\ell_p$-spaces among Banach spaces and also among complete order smooth p-normed spaces. We further introduce a notion of $p$-orthogonal decomposition in order smooth p-normed spaces. We prove that if the $\infty$-orthogonal decomposition holds in an order smooth $\infty$-normed space, then the 1-orthogonal decomposition holds in the dual space. We also give an example to show that the above said decomposition may not be unique.