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Vladikavkaz. Mat. Zh., 2019, Volume 21, Number 4, Pages 56–62 (Mi vmj706)  

Unbounded order convergence and the Gordon theorem

E. Y. Emelyanovab, S. G. Gorokhovac, S. S. Kutateladzeb

a Middle East Technical University, 1 Dumlupinar Bulvari, Ankara 06800, Turkey
b Sobolev Institute of Mathematics, 4 Koptyug prospect, Novosibirsk 630090, Russia
c Southern Mathematical Institute VSC RAS, 22 Marcus St., Vladikavkaz 362027, Russia

Abstract: 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.

Key words: unbounded order convergence, universally complete vector lattice, Boolean valued analysis.

Funding Agency Grant Number
Siberian Branch of Russian Academy of Sciences I.1.2, Project № 0314-2019-0005
The research was partially supported by the Science Support Foundation Program of the Siberian Branch of the Russian Academy of Sciences; № I.1.2, Project № 0314-2019-0005.


DOI: https://doi.org/10.23671/VNC.2019.21.44624

Full text: PDF file (216 kB)
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UDC: 510.898, 517.98
MSC: 03H05, 46S20, 46A40
Received: 04.07.2019
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Citation: E. Y. Emelyanov, S. G. Gorokhova, S. S. Kutateladze, “Unbounded order convergence and the Gordon theorem”, Vladikavkaz. Mat. Zh., 21:4 (2019), 56–62

Citation in format AMSBIB
\Bibitem{EmeGorKut19}
\by E.~Y.~Emelyanov, S.~G.~Gorokhova, S.~S.~Kutateladze
\paper Unbounded order convergence and the Gordon theorem
\jour Vladikavkaz. Mat. Zh.
\yr 2019
\vol 21
\issue 4
\pages 56--62
\mathnet{http://mi.mathnet.ru/vmj706}
\crossref{https://doi.org/10.23671/VNC.2019.21.44624}


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