|
This article is cited in 2 scientific papers (total in 2 papers)
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.
Received: 04.07.2019
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
Linking options:
https://www.mathnet.ru/eng/vmj706 https://www.mathnet.ru/eng/vmj/v21/i4/p56
|
Statistics & downloads: |
Abstract page: | 220 | Full-text PDF : | 73 | References: | 31 |
|