# 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

, Volume 18 (2) – May 16, 2013
12 pages

/lp/springer_journal/orthogonality-in-ell-p-spaces-and-its-bearing-on-ordered-banach-spaces-eim7CxOPW9
Publisher
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

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month ### Explore the DeepDyve Library ### Search Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly ### Organize Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place. ### Access Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve ### Freelancer DeepDyve ### Pro Price FREE$49/month
\$360/year

Save searches from
PubMed

Create lists to

Export lists, citations