Unbounded p-Convergence in Lattice-Normed Vector Lattices

Aydın, A.
Emelyanov, Eduard
Erkurşun-Özcan, N.
Marabeh, M.
A net xα in a lattice-normed vector lattice (X, p, E) is unbounded p-convergent to x ∈ X if p(| xα− x| ∧ u) → o 0 for every u ∈ X+. This convergence has been investigated recently for (X, p, E) = (X, |·|, X) under the name of uo-convergence, for (X, p, E) = (X, ‖·‖, ℝ) under the name of un-convergence, and also for (X, p, ℝX ′) , where p(x)[f]:= |f|(|x|), under the name uaw-convergence. In this paper we study general properties of the unbounded p-convergence.
Siberian Advances in Mathematics


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Unbounded order convergence and the Gordon theorem#
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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 xn in an Archimedean vector lattice X is uo-null (uo-Cauchy) in ...
Citation Formats
A. Aydın, E. Emelyanov, N. Erkurşun-Özcan, and M. Marabeh, “Unbounded p-Convergence in Lattice-Normed Vector Lattices,” Siberian Advances in Mathematics, vol. 29, no. 3, pp. 164–182, 2019, Accessed: 00, 2021. [Online]. Available: https://hdl.handle.net/11511/94914.