Unbounded Vectorial Cauchy Completion of Vector Metric Spaces

Yayın Aralığı: Yılda 4 Sayı
Başlangıç: 1988
Yayıncı: Gazi Üniversitesi, Fen Bilimleri Enstitüsü

A sequence (an) in a Riesz space E is called uo-convergent (or unbounded order convergent) to a in E if inf{|an-a|,u} is order convergent to 0 for all u in E+ and unbounded order Cauchy (uo-Cauchy) if |an-an+p|is uo-convergent to 0. In the first part of this study we define ud,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,

PDF

___

[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.