Orthogonality in ${\ell }_p$-spaces and its bearing on ordered Banach 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.