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

Orthogonality in $$\ell _p$$ -spaces and its bearing on ordered Banach spaces 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

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

Positivity , Volume 18 (2) – May 16, 2013

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Publisher
Springer Journals
Copyright
Copyright © 2013 by Springer Basel
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-013-0242-8
Publisher site
See Article on Publisher Site

Abstract

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.

Journal

PositivitySpringer Journals

Published: May 16, 2013

References

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