Explicit characterizations of orthogonality in \(\ell_S^p(C)\)
type: Item
instance of: scholarly article
title: Explicit characterizations of orthogonality in \(\ell_S^p(C)\)
zbMATH Open document ID: 1017.46013
DOI: 10.1016/S0022-247X(02)00144-0
author: Fathi B. Saidi
published in: Journal of Mathematical Analysis and Applications
publication date: 2002-11-28T00:00:00Z
review text: A family \((x_\alpha)_{\alpha\in A}\) of vectors in a Banach space \(E\) is said to be orthogonal if \[ \Big\|\sum_{\alpha\in A} c_\alpha x_\alpha\Big\|= \Big\|\sum_{\alpha\in A} |c_\alpha|x_\alpha\Big\| \] whenever \(\sum_{\alpha\in A} c_\alpha x_\alpha\) converges in \(E\). For a basic sequence, that means that it is \(1\)-unconditional. The author studies orthogonality in the complex spaces \(\ell_p(I)\). When \(p\) is not an even integer, he asks whether two elements \(a\) and \(b\) of \(\ell_p(I)\) have disjoint supports when they are orthogonal and gives some partial answers. When \(p\) is an even integer, \(a=(a_i)_{i\in I}\) and \(b=(b_i)_{i\in I}\) can be orthogonal without having disjoint supports and if \(a_i\neq 0\), \(b_i\not= 0\) for every \(i\in I\), the author shows that \(I\) must have at least \(p/2+1\) elements.
reviewed by: Daniel Li
Mathematics Subject Classification ID: 46B45
zbMATH DE Number: 1836767
zbMATH Keywords: orthogonality
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1836767