Some further results on ideal summability of nets in ( $$\ell $$ ℓ ) groups

Some further results on ideal summability of nets in ( $$\ell $$ ℓ ) groups In this paper we continue in the line of recent investigation of order summability of nets using ideals by Boccuto et al. (Czechoslovak Math. J. 62(137):1073–1083 2012; J. Appl. Anal. 20(1), 2014) where they had introduced the notions of $$\mathcal {I}$$ I and $$\mathcal {I}^*$$ I ∗ order convergence, $$\mathcal {I}$$ I and $$\mathcal {I}^*$$ I ∗ divergence of nets and its further extensions, namely the notions of $$\mathcal {I}^{\mathcal {K}}$$ I K -order convergence and $$\mathcal {I}^{\mathcal {K}}$$ I K -divergence of nets in a $$(\ell )$$ ( ℓ ) -group and investigate the relation between $$\mathcal {I}$$ I and $$\mathcal {I}^{\mathcal {K}}$$ I K -concepts where a special class of ideals called $$\Lambda P(\mathcal {I},\mathcal {K})-$$ Λ P ( I , K ) - ideals plays very important role. We also introduce, for the first time, the notion of $$\mathcal {I}^{\mathcal {K}}$$ I K -order Cauchy condition and $$\mathcal {I}$$ I -order cluster points of nets in ( $$\ell $$ ℓ ) groups and examine some of its characterizations and its consequences. In particular the role of $$\mathcal {I}$$ I -order cluster points in making the above mentioned Cauchy nets convergent is studied. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Some further results on ideal summability of nets in ( $$\ell $$ ℓ ) groups

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Publisher
Springer Basel
Copyright
Copyright © 2014 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/s11117-014-0282-8
Publisher site
See Article on Publisher Site

Abstract

In this paper we continue in the line of recent investigation of order summability of nets using ideals by Boccuto et al. (Czechoslovak Math. J. 62(137):1073–1083 2012; J. Appl. Anal. 20(1), 2014) where they had introduced the notions of $$\mathcal {I}$$ I and $$\mathcal {I}^*$$ I ∗ order convergence, $$\mathcal {I}$$ I and $$\mathcal {I}^*$$ I ∗ divergence of nets and its further extensions, namely the notions of $$\mathcal {I}^{\mathcal {K}}$$ I K -order convergence and $$\mathcal {I}^{\mathcal {K}}$$ I K -divergence of nets in a $$(\ell )$$ ( ℓ ) -group and investigate the relation between $$\mathcal {I}$$ I and $$\mathcal {I}^{\mathcal {K}}$$ I K -concepts where a special class of ideals called $$\Lambda P(\mathcal {I},\mathcal {K})-$$ Λ P ( I , K ) - ideals plays very important role. We also introduce, for the first time, the notion of $$\mathcal {I}^{\mathcal {K}}$$ I K -order Cauchy condition and $$\mathcal {I}$$ I -order cluster points of nets in ( $$\ell $$ ℓ ) groups and examine some of its characterizations and its consequences. In particular the role of $$\mathcal {I}$$ I -order cluster points in making the above mentioned Cauchy nets convergent is studied.

Journal

PositivitySpringer Journals

Published: Mar 9, 2014

References

  • On $$I$$ I -convergence of nets in locally solid Riesz spaces
    Das, P; Savas, E

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