Abstract
In this paper we continue the investigation of topological properties of the unbounded norm (un-)topology in normed lattices. We characterize separability and second countability of the un-topology in terms of properties of the underlying normed lattice. We apply our results to prove that an order continuous Banach function space X over a semi-finite measure space is separable if and only if it has a \(\sigma \)-finite carrier and is separable with respect to the topology of local convergence in measure. We also address the question when a normed lattice is a normal space with respect to the un-topology.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abramovich, Y.A., Aliprantis, C.D.: An invitation to operator theory. Graduate Studies in Mathematics, vol. 50. American Mathematical Society, Providence, RI (2002)
Albiac, F., Kalton, N.J.: Topics in Banach Space Theory. Graduate Texts in Mathematics, vol. 233. Springer, New York (2006)
Aliprantis, C.D., Burkinshaw, O.: Positive Operators. Springer, Dordrecht, Reprint of the 1985 original (2006)
Bogachev, V.I.: Measure Theory, vol. I. Springer, Berlin (2007)
Dabboorasad, Y.A., Emelyanov, E.Y., Marabeh, M.A.A.: \(u\tau \)-convergence in locally solid vector lattices. Positivity 22(4), 1065–1080 (2018)
Deng, Y., de Jeu, M.: Vector lattices with a Hausdorff uo-Lebesgue topology. J. Math. Anal. Appl 505(1), 12545530 (2022)
Deng, Y., O’Brien, M., Troitsky, V.G.: Unbounded norm convergence in Banach lattices. Positivity 21(3), 963–974 (2017)
DeMarr, R.: Partially ordered linear spaces and locally convex linear topological spaces. Illinois J. Math. 8, 601–606 (1964)
Drnovšek, R.: Common invariant subspaces for collections of operators. Integr. Equ. Oper. Theory 39(3), 253–266 (2001)
Dugundji, J.: Topology. Allyn and Bacon, Inc., Boston, Mass., (1966)
Fremlin, D.H.: Measure theory. Vol. 3. Torres Fremlin, Colchester, 2004. Measure algebras, Corrected second printing of the original (2002)
Gao, N.: Unbounded order convergence in dual spaces. J. Math. Anal. Appl. 419(1), 347–354 (2014)
Gao, N., Troitsky, V.G., Xanthos, F.: Uo-convergence and its applications to Cesàro means in Banach lattices. Israel J. Math. 220(2), 649–689 (2017)
Gao, N., Xanthos, F.: Unbounded order convergence and application to martingales without probability. J. Math. Anal. Appl. 415(2), 931–947 (2014)
Husain, T.: Introduction to Topological Groups. W. B. Saunders Co., Philadelphia (1966)
Kandić, M., Li, H., Troitsky, V.G.: Unbounded norm topology beyond normed lattices. Positivity 22(3), 745–760 (2018)
Kandić, M., Marabeh, M.A.A., Troitsky, V.G.: Unbounded norm topology in Banach lattices. J. Math. Anal. Appl. 451(1), 259–279 (2017)
Kandić, M., Taylor, M.A.: Metrizability of minimal and unbounded topologies. J. Math. Anal. Appl. 466(1), 144–159 (2018)
Kaplan, S.: On unbounded order convergence. Real Anal. Exchange 23(1), 175–184 (1997)
Kelley, J. L., Namioka, I.: Linear topological spaces. Graduate Texts in Mathematics, No. 36, Springer-Verlag, New York-Heidelberg, (1976)
Lozanovskiĭ, G. Y., A. A. Mekler. Completely linear functionals and reflexivity in normed linear lattices. Izv. Vysš Učebn. Zaved. Matematika, 66(11) 47–53, (1967)
Luxemburg, W.A.J.: Notes on Banach function spaces. XIVb. Nederl. Akad. Wetensch. Proc. Ser. A 68=Indag. Math., 27:240–248, (1965)
Luxemburg, W.A.J.: Notes on Banach function spaces. XVIb. Nederl. Akad. Wetensch. Proc. Ser. A 68=Indag. Math., 27:658–667, (1965)
Luxemburg, W.A.J., Zaanen, A.C.: Riesz Spaces. Vol. I. North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., New York, (1971)
Meyer-Nieberg, P.: Charakterisierung einiger topologischer und ordnungstheoretischer Eigenschaften von Banachverbänden mit Hilfe disjunkter Folgen. Arch. Math. 24, 640–647 (1973)
Meyer-Nieberg, P.: Banach Lattices. Universitext. Springer-Verlag, Berlin (1991)
Nakano, H.: Ergodic theorems in semi-ordered linear spaces. Ann. Math. 2(49), 538–556 (1948)
Ross, K.A., Stone, A.H.: Products of separable spaces. Am. Math. Monthly 71, 398–403 (1964)
Schaefer, H.H.: Topological vector spaces. Springer-Verlag, New York-Berlin, Third printing corrected, Graduate Texts in Mathematics, Vol. 3 (1971)
Taylor, M.A.: Completeness of unbounded convergences. Proc. Am. Math. Soc. 146(8), 3413–3423 (2018)
Taylor, M.A.: Unbounded topologies and \(uo\)-convergence in locally solid vector lattices. J. Math. Anal. Appl. 472(1), 981–1000 (2019)
Willard, S.: General Topology. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., (1970)
Zaanen, A.C.: Introduction to Operator Theory in Riesz Spaces. Springer, Berlin (1997)
Zabeti, O.: Unbounded absolute weak convergence in Banach lattices. Positivity 22(2), 501–505 (2018)
Acknowledgements
The authors thank the anonymous referee for the helpful comments that improved the quality of the manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The first author acknowledges financial support from the Slovenian Research Agency, Grants Nos. P1-0222, J1-2453 and J1-2454. The second author acknowledges financial support from the Slovenian Research Agency, Grant No. P1-0292.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Kandić, M., Vavpetič, A. On separability of the unbounded norm topology. Positivity 27, 43 (2023). https://doi.org/10.1007/s11117-022-00967-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11117-022-00967-1