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Orthogonality in \(\ell _p\)-spaces and its bearing on ordered Banach spaces. (English) Zbl 1314.46026
Summary: We introduce a notion of \(p\)-orthogonality in a general Banach space for \(1 \leq p \leq \infty \). We use this concept to characterize \(\ell _p\)-spaces among Banach spaces and also among complete order smooth \(p\)-normed spaces as (ordered) Banach spaces with a total \(p\)-orthonormal set (in the positive cone). 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.

46B40 Ordered normed spaces
46L07 Operator spaces and completely bounded maps
47L25 Operator spaces (= matricially normed spaces)
Full Text: DOI arXiv
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