Unbounded Norm Convergence in Banach Lattices

Abstract

A net $(x_{\alpha })$ in a vector lattice $X$ is unbounded order convergent to $x\in X$ if $|x_{\alpha }-x|\wedge u$ converges to $0$ in order for all $u\in X_{+}$. 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 $(x_{\alpha })$ in a Banach lattice $X$ is unbounded norm convergent to $x$ if $\Vert |x_{\alpha }-x|\wedge u\Vert \to 0$ for all $u\in X_{+}$. We show that this convergence may be viewed as a generalization of convergence in measure. We also investigate its relationship with other convergences.

References in corpus (1)

• Uo-convergence and its applications to Cesàro means in Banach lattices

Cited by in corpus (8)

• Order convergence in infinite-dimensional vector lattices is not topological
• Compact-Like Operators in Lattice-Normed Spaces
• Some loose ends on unbounded order convergence
• Unbounded convergence in Banach lattices and applications
• Statistical unbounded convergence in Banach lattices
• Multiplicative weak convergence in Banach $f$-algebras
• Continuous functionals for unbounded convergence in Banach lattices
• Unbounded Norm Topology in Banach Lattices