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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.

MSC:
46B40 Ordered normed spaces
46L07 Operator spaces and completely bounded maps
47L25 Operator spaces (= matricially normed spaces)
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