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Positivity

, Volume 22, Issue 2, pp 653–667 | Cite as

um-Topology in multi-normed vector lattices

  • Y. A. Dabboorasad
  • E. Y. Emelyanov
  • M. A. A. Marabeh
Article

Abstract

Let \({\mathcal {M}}=\{m_\lambda \}_{\lambda \in \Lambda }\) be a separating family of lattice seminorms on a vector lattice X, then \((X,{\mathcal {M}})\) is called a multi-normed vector lattice (or MNVL). We write \(x_\alpha \xrightarrow {\mathrm {m}} x\) if \(m_\lambda (x_\alpha -x)\rightarrow 0\) for all \(\lambda \in \Lambda \). A net \(x_\alpha \) in an MNVL \(X=(X,{\mathcal {M}})\) is said to be unbounded m-convergent (or um-convergent) to x if \(|x_\alpha -x |\wedge u \xrightarrow {\mathrm {m}} 0\) for all \(u\in X_+\). um-Convergence generalizes un-convergence (Deng et al. in Positivity 21:963–974, 2017; Kandić 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 (Aydın et al. in Unbounded p-convergence in lattice-normed vector lattices. arXiv:1609.05301) and \(u\tau \)-convergence (Dabboorasad et al. in \(u\tau \)-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 \((X,{\mathcal {M}})\), 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 \(\sigma \)-Lebesgue and \(\sigma \)-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.

Keywords

Vector lattice Banach lattice Multi-normed vector lattice um-Convergence um-Topology uo-Convergence un-Convergence 

Mathematics Subject Classification

Primary 46A03 46A40 Secondary 32F45 46A50 

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsIslamic University of GazaGaza CityPalestine
  2. 2.Department of MathematicsMiddle East Technical UniversityAnkaraTurkey
  3. 3.Department of Applied MathematicsPalestine Technical University-KadoorieTulkaremPalestine

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