Unbounded norm convergence in Banach lattices

By Y. Deng, M. O’Brien, V. G. Troitsky. 2016

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A net $$(x_\alpha )$$(xα) in a vector lattice X is unbounded order convergent to $$x \in X$$x∈X if $$|x_\alpha - x| \wedge u$$|xα-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 $$(x_\alpha )$$(xα) in a Banach lattice X is unbounded norm convergent to x if 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.

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