Unbounded Vectorial Cauchy Completion of Vector Metric Spaces

Year 2020, Volume: 33 Issue: 3, 761 - 765, 01.09.2020

Çetin Cemal Özeken Cüneyt Çevik

https://doi.org/10.35378/gujs.604784

Cited By: 2

Abstract

A sequence (a_n) in a Riesz space E is called uo-convergent (or unbounded order convergent) to a in E if  inf{|a_n-a|,u} is order convergent to 0 for all u in E₊ and
unbounded order Cauchy (uo-Cauchy) if |a_n-a_n+p|is
uo-convergent to 0. In the first part of this study we define u_d,E-convergence
(or unbounded vectorial convergence) in vector metric spaces, which is more
general than usual metric spaces, and examine relations between unbounded order
convergence, unbounded vectorial convergence, vectorial convergence and order
convergence. In the last part we construct the unbounded Cauchy completion of
vector metric spaces by the motivation of the fact that every metric space has
Cauchy completion. In this way, we have obtained a more general completion of
vector metric spaces.

Keywords

Unbounded order convergence, vector metric spaces, unbounded vectorial convergence, unbounded Cauchy completion, Riesz space

References

• [1] H. Nakano, Ergodic theorems in semi-ordered linear spaces, Ann. Math. 49 (1948), 538-556.
• [2] R. DeMarr, Partially ordered linear spaces and locally convex linear topological spaces, Illinois J. Math., 8 (1964), 601--606.
• [3] S. Kaplan, On unbounded order convergence, Real Anal. Exchange, 23 (1) (1998-99), 175-184.
• [4] N. Gao and F. Xanthos, Unbounded order convergence and application to martingales without probability, J. Math. Anal. Appl., 415 (2) (2014), 931--947.
• [5] N. Gao, V. G. Troitsky, and F. Xanthos, Uo-convergence and its applications to cesáro means in banach lattices, Israel Journal of Math., 220 (2017), 649-689.
• [6] C. D. Aliprantis and O. Burkinshaw, Positive operators, Springer, Dordrecht, 2006.
• [7] W. A. J. Luxemburg and A. C. Zaanen, Riesz Space I, North-Holland, Amsterdam, The Netherland, 1971.
• [8] C. Çevik and I. Altun, Vector metric spaces and some properties, Topol. Methods Nonlinear Anal., 34 (2), (2009), 375--382.