# Unbounded norm convergence in Banach lattices

Unbounded norm convergence in Banach lattices A net $$(x_\alpha )$$ ( x α ) in a vector lattice X is unbounded order convergent to $$x \in X$$ x ∈ X if $$|x_\alpha - x| \wedge u$$ | x α - x | ∧ u converges to 0 in order for all $$u\in X_+$$ u ∈ X + . This convergence has been investigated and applied in several recent papers by Gao et al. It may be viewed as a generalization of almost everywhere convergence to general vector lattices. In this paper, we study a variation of this convergence for Banach lattices. A net $$(x_\alpha )$$ ( x α ) in a Banach lattice X is unbounded norm convergent to x if [InlineEquation not available: see fulltext.] for all $$u\in X_+$$ u ∈ X + . We show that this convergence may be viewed as a generalization of convergence in measure. We also investigate its relationship with other convergences. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Unbounded norm convergence in Banach lattices

, Volume 21 (3) – Sep 30, 2016
12 pages

/lp/springer_journal/unbounded-norm-convergence-in-banach-lattices-D6pfjQojZs
Publisher
Springer International Publishing
Copyright © 2016 by Springer International Publishing
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/s11117-016-0446-9
Publisher site
See Article on Publisher Site

### Abstract

A net $$(x_\alpha )$$ ( x α ) in a vector lattice X is unbounded order convergent to $$x \in X$$ x ∈ X if $$|x_\alpha - x| \wedge u$$ | x α - x | ∧ u converges to 0 in order for all $$u\in X_+$$ u ∈ X + . This convergence has been investigated and applied in several recent papers by Gao et al. It may be viewed as a generalization of almost everywhere convergence to general vector lattices. In this paper, we study a variation of this convergence for Banach lattices. A net $$(x_\alpha )$$ ( x α ) in a Banach lattice X is unbounded norm convergent to x if [InlineEquation not available: see fulltext.] for all $$u\in X_+$$ u ∈ X + . We show that this convergence may be viewed as a generalization of convergence in measure. We also investigate its relationship with other convergences.

### Journal

PositivitySpringer Journals

Published: Sep 30, 2016

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