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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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