Geodesic
Unbounded order convergence and the Gordon theorem
Vladikavkazskij matematičeskij žurnal, Tome 21 (2019) no. 4, pp. 56-62 Cet article a éte moissonné depuis la source Math-Net.Ru

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The celebrated Gordon's theorem is a natural tool for dealing with universal completions of Archimedean vector lattices. Gordon's theorem allows us to clarify some recent results on unbounded order convergence. Applying the Gordon theorem, we demonstrate several facts on order convergence of sequences in Archimedean vector lattices. We present an elementary Boolean-Valued proof of the Gao–Grobler–Troitsky–Xanthos theorem saying that a sequence $x_n$ in an Archimedean vector lattice $X$ is $uo$-null ($uo$-Cauchy) in $X$ if and only if $x_n$ is $o$-null ($o$-convergent) in $X^u$. We also give elementary proof of the theorem, which is a result of contributions of several authors, saying that an Archimedean vector lattice is sequentially $uo$-complete if and only if it is $\sigma$-universally complete. Furthermore, we provide a comprehensive solution to Azouzi's problem on characterization of an Archimedean vector lattice in which every $uo$-Cauchy net is $o$-convergent in its universal completion. 
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E. Y. Emelyanov; S. G. Gorokhova; S. S. Kutateladze. Unbounded order convergence and the Gordon theorem. Vladikavkazskij matematičeskij žurnal, Tome 21 (2019) no. 4, pp. 56-62. http://geodesic.mathdoc.fr/item/VMJ_2019_21_4_a4/

[1] Aliprantis C. D., Burkinshaw O., Locally Solid Riesz Spaces with Applications to Economics, Mathematical Surveys and Monographs, 105, 2nd ed., Amer. Math. Soc., Providence, RI, 2003 | DOI | MR | Zbl

[2] Kusraev A. G., Dominated Operators, Kluwer, Dordrecht, 2000 | MR | Zbl

[3] Gao N., Troitsky V. G., Xanthos F., “Uo-Convergence and its Applications to Cesáro Means in Banach Lattices”, Israel Journal of Mathematics, 220:2 (2017), 649–689 | DOI | MR | Zbl

[4] Kaplan S., “On Unbounded Order Convergence”, Real Anal. Exchange, 23:1 (1997/98), 175–184 | DOI | MR

[5] Kusraev A. G., Kutateladze S. S., Boolean-valued Analysis, Kluwer Academic, Dordrecht, 1999 | MR | Zbl

[6] Abramovich Y. A., Sirotkin G. G., “On Order Convergence of Nets”, Positivity, 9:3 (2005), 287–292 | DOI | MR | Zbl

[7] Dabboorasad Y. A., Emelyanov E. Y., Marabeh M. A. A., “Order Convergence in Infinite-dimensional Vector Lattices is not Topological”, Uzbek Math. J., 3 (2019), 4–10

[8] Gorokhova S. G., “Intrinsic Characterization of the Space $c_0(A)$ in the Class of Banach Lattices”, Math. Notes, 60:3 (1996), 330–333 | DOI | MR | Zbl

[9] Nakano H., “Ergodic Theorems in Semi-ordered Linear Spaces”, Ann. Math., 49:2 (1948), 538–556 | DOI | MR | Zbl

[10] Wickstead A. W., “Weak and Unbounded Order Convergence in Banach Lattices”, J. Austral. Math. Soc. Ser. A, 24:3 (1977), 312–319 | DOI | MR | Zbl

[11] Deng Y., O'Brien M., Troitsky V. G., “Unbounded Norm Convergence in Banach Lattices”, Positivity, 21:3 (2017), 963–974 | DOI | MR | Zbl

[12] Emelyanov E. Y., Marabeh M. A. A., “Two Measure-free Versions of the Brezis–Lieb Lemma”, Vladikavkaz Math. J., 18:1 (2016), 21–25 | DOI | MR

[13] Li H., Chen Z., “Some Loose Ends on Unbounded Order Convergence”, Positivity, 22:1 (2018), 83–90 | DOI | MR | Zbl

[14] Emelyanov E. Y., Erkurşun-Özcan N., Gorokhova S. G., “Komlós Properties in Banach Lattices”, Acta Math. Hungarica, 155:2 (2018), 324–331 | DOI | MR | Zbl

[15] Azouzi Y., “Completeness for Vector Lattices”, J. Math. Anal. Appl., 472:1 (2019), 216–230 | DOI | MR | Zbl

[16] Taylor M. A., “Completeness of Unbounded Convergences”, Proc. Amer. Math. Soc., 146:8 (2018), 3413–3423 | DOI | MR | Zbl

[17] Ayd{\i}n A., Emelyanov E. Y., Erkurşun-Özcan N., Marabeh M. A. A., “Unbounded $p$-Convergence in Lattice-normed Vector Lattices”, Siberian Adv. Math., 29:3 (2019), 164–182 | DOI | MR

[18] Taylor M. A., Master Thesis, University of Alberta, Edmonton, 2019

[19] Gordon E. I., “Real Numbers in Boolean-Valued Models of Set Theory and $K$-Spaces”, Dokl. Akad. Nauk SSSR, 237:4 (1977), 773–775 | MR | Zbl

[20] Kantorovich L. V., “On Semi-Ordered Linear Spaces and their Applications to the Theory of Linear Operations”, Dokl. Akad. Nauk SSSR, 4:1–2 (1935), 11–14

[21] Gordon E. I., “$K$-spaces in Boolean-valued Models of Set Theory”, Dokl. Akad. Nauk SSSR, 258:4 (1981), 777–780 | MR | Zbl

[22] Gordon E. I., “Theorems on Preservation of Relations in $K$-spaces”, Sib. Math. J., 23:3 (1982), 336–344 | DOI | MR | Zbl

[23] Dales H. G., Woodin W. H., An Introduction to Independence for Analysts, Cambridge University Press, Cambridge, 1987, 241 pp. | MR | Zbl

[24] Gordon E. I., Kusraev A. G., Kutateladze S. S., Infinitesimal Analysis, Kluwer Academic Publishers, Dordrecht, 2002 | MR | Zbl

[25] Gutman A. E., Kusraev A. G., Kutateladze S. S., “The Wickstead Problem”, Sib. Elektron. Mat. Izv., 5 (2008), 293–333 | MR | Zbl