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We introduce a notion of -orthogonality in a general Banach space for . We use this concept to characterize -spaces among Banach spaces and also among complete order smooth -normed spaces as (ordered) Banach spaces with a total -orthonormal set (in the positive cone). We further introduce a notion of -orthogonal decomposition in order smooth -normed spaces. We prove that if the -orthogonal decomposition holds in an order smooth -normed space, then the -orthogonal decomposition holds in the dual space. We also give an example to show that the above said decomposition may not be unique.

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