Unbounded Norm Topology in Banach Lattices

2016/08/19 by Marko Kandić, Kandić, M., M. A. A. Marabeh +3 · 1 citation
Mathematics · #Advanced Banach Space Theory #Advanced Topology and Set Theory #Approximation Theory and Sequence Spaces #FOS: Mathematics #Functional Analysis (math.FA) #Primary: 46B42 #Secondary: 46A40

paper · pdf · doi:10.48550/arxiv.1608.05489

openalex publication_date 2016/08/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A net (x_α) in a Banach lattice X is said to un-converge to a vector x if ‖| x_α-x|\wedge u‖→ 0 for every u∈ X₊. In this paper, we investigate un-topology, i.e., the topology that corresponds to un-convergence. We show that un-topology agrees with the norm topology iff X has a strong unit. Un-topology is metrizable iff X has a quasi-interior point. Suppose that X is order continuous, then un-topology is locally convex iff X is atomic. An order continuous Banach lattice X is a KB-space iff its closed unit ball B_X is un-complete. For a Banach lattice X, B_X is un-compact iff X is an atomic KB-space. We also study un-compact operators and the relationship between un-convergence and weak*-convergence.

Citations

• Relative weak compactness of solid hulls in Banach lattices
• Uo-convergence and its applications to Cesàro means in Banach lattices

Cited by

• Strong completeness of Lp-type vector lattices

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