, 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


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.


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 


  1. 1.
    Aliprantis, C.D., Burkinshaw, O.: Locally Solid Riesz Spaces with Applications to Economics. Mathematical Surveys and Monographs, vol. 105. American Mathematical Society, Providence (2003)CrossRefzbMATHGoogle Scholar
  2. 2.
    Aliprantis, C.D., Burkinshaw, O.: Positive Operators, 2nd edn. Springer, Berlin (2006)CrossRefzbMATHGoogle Scholar
  3. 3.
    Aydın, A., Emelyanov, E.Y., Özcan, N.E., Marabeh, M.A.A.: Unbounded \(p\)-convergence in lattice-normed vector lattices. Preprint, arXiv:1609.05301
  4. 4.
    Aydın, A., Emelyanov, E.Y., Özcan, N.E., Marabeh, M.A.A.: Compact-like operators in lattice-normed spaces. Preprint, arXiv:1701.03073v2
  5. 5.
    Dabboorasad, Y., Emelyanov, E.Y., Marabeh, M.A.A.: Order convergence in infinite-dimensional vector lattices is not topological. Preprint, arXiv:1705.09883v1
  6. 6.
    Dabboorasad, Y., Emelyanov, E.Y., Marabeh, M.A.A.: \(u\tau \)-Convergence in locally solid vector lattices. Preprint, arXiv:1706.02006v3
  7. 7.
    Deng, Y., O’Brien, M., Troitsky, V.G.: Unbounded norm convergence in Banach lattices. Positivity 21, 963–974 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Emelyanov, E.Y., Marabeh, M.A.A.: Two measure-free versions of the Brezis–Lieb Lemma. Vladikavkaz Math. J. 18(1), 21–25 (2016)MathSciNetGoogle Scholar
  9. 9.
    Gao, N.: Unbounded order convergence in dual spaces. J. Math. Anal. Appl. 419, 347–354 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gao, N., Troitsky, V.G., Xanthos, F.: Uo-convergence and its applications to Cesàro means in Banach lattices. Isr. J. Math. 220, 649–689 (2017)CrossRefzbMATHGoogle Scholar
  11. 11.
    Gao, N., Leung, D.H., Xanthos, F.: Duality for unbounded order convergence and applications. Preprint, arXiv:1705.06143
  12. 12.
    Gao, N., Leung, D.H., Xanthos, F.: The dual representation problem of risk measures. Ppreprint, arXiv:1610.08806
  13. 13.
    Gao, N., Xanthos, F.: Unbounded order convergence and application to martingales without probability. J. Math. Anal. Appl. 415, 931–947 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Jarchow, H.: Locally Convex Spaces. Mathematische Leitfden. B. G. Teubner, Stuttgart (1981)CrossRefzbMATHGoogle Scholar
  15. 15.
    Kandić, M., Marabeh, M.A.A., Troitsky, V.G.: Unbounded norm topology in Banach lattices. J. Math. Anal. Appl. 451, 259–279 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kandić, M., Li, H., Troitsky, V.G.: Unbounded norm topology beyond normed lattices. Preprint, arXiv:1703.10654
  17. 17.
    Kandić, M., Vavpetić, A.: Topological aspects of order in \(C(X)\). Preprint, arXiv:1612.05410
  18. 18.
    Kusraev, A.G.: Dominated Operators. Mathematics and its Applications, vol. 519. Kluwer Academic Publishers, Dordrecht (2000)CrossRefGoogle Scholar
  19. 19.
    Li, H., Chen, Z.: Some loose ends on unbounded order convergence. Positivity (2017). doi: 10.1007/s11117-017-0501-1 Google Scholar
  20. 20.
    Schaefer, H.H.: Banach Lattices and Positive Operators. Springer, Berlin (1974)CrossRefzbMATHGoogle Scholar
  21. 21.
    Taylor, M.A.: Unbounded topologies and uo-convegence in locally solid vector lattices. Preprint, arXiv:1706.01575
  22. 22.
    Troitsky, V.G.: Measures of non-compactness of operators on Banach lattices. Positivity 8(2), 165–178 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Vulikh, B.Z.: Introduction to the Theory of Partially Ordered Spaces. Wolters-Noordhoff Scientific Publications Ltd., Groningen (1967)zbMATHGoogle Scholar
  24. 24.
    Wickstead, A.W.: Weak and unbounded order convergence in Banach lattices. J. Aust. Math. Soc. Ser. A 24(3), 312–319 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Zabeti, O.: Unbounded absolute weak convergence in Banach lattices. Positivity (2017). doi: 10.1007/s11117-017-0524-7 MathSciNetGoogle Scholar

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

Personalised recommendations