We introduce a notion of $$p$$ -orthogonality in a general Banach space for $$1 \le p \le \infty $$ . We use this concept to characterize $$\ell _p$$ -spaces among Banach spaces and also among complete order smooth $$p$$ -normed spaces 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.
Positivity – Springer Journals
Published: May 16, 2013
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