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um-Topology in multi-normed vector lattices
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Date
2018-04-01
Author
Dabboorasad, Y. A.
Emelyanov, Eduard
Marabeh, M. A. A.
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Let be a separating family of lattice seminorms on a vector lattice X, then is called a multi-normed vector lattice (or MNVL). We write if for all . A net in an MNVL is said to be unbounded m-convergent (or um-convergent) to x if for all . um-Convergence generalizes un-convergence (Deng et al. in Positivity 21:963-974, 2017; KandiAc et al. in J Math Anal Appl 451:259-279, 2017) and uaw-convergence (Zabeti in Positivity, 2017. doi:10.1007/s11117-017-0524-7), and specializes up-convergence (AydA +/- n et al. in Unbounded p-convergence in lattice-normed vector lattices. arXiv:1609.05301) and -convergence (Dabboorasad et al. in -Convergence in locally solid vector lattices. arXiv:1706.02006v3). um-Convergence is always topological, whose corresponding topology is called unbounded m-topology (or um-topology). We show that, for an m-complete metrizable MNVL , the um-topology is metrizable iff X has a countable topological orthogonal system. In terms of um-completeness, we present a characterization of MNVLs possessing both Lebesgue's and Levi's properties. Then, we characterize MNVLs possessing simultaneously the -Lebesgue and -Levi properties in terms of sequential um-completeness. Finally, we prove that every m-bounded and um-closed set is um-compact iff the space is atomic and has Lebesgue's and Levi's properties.
Subject Keywords
Vector lattice
,
Banach lattice
,
Multi-normed vector lattice
,
Um-Convergence
,
Um-Topology
,
Uo-Convergence
,
Un-Convergence
URI
https://hdl.handle.net/11511/48027
Journal
POSITIVITY
DOI
https://doi.org/10.1007/s11117-017-0533-6
Collections
Department of Mathematics, Article
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Y. A. Dabboorasad, E. Emelyanov, and M. A. A. Marabeh, “um-Topology in multi-normed vector lattices,”
POSITIVITY
, pp. 653–667, 2018, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/48027.