In this way, we have obtained a more general completion of vector metric spaces.
📌 3 Maddede Bu Makale
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Amaç & Hipotez: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.
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Temel Bulgular & Veri: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.
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Sonuç & Katkı:In this way, we have obtained a more general completion of vector metric spaces.
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.
Özeken, Ç. C., Çevik, C. (2020). Unbounded Vectorial Cauchy Completion of Vector Metric Spaces. Gazi University Journal of Science, Volume: 33 Issue: 3. https://doi.org/10.35378/gujs.604784
H. Nakano, Ergodic theorems in semi-ordered linear spaces, Ann. Math. 49 (1948), 538-556.
R. DeMarr, Partially ordered linear spaces and locally convex linear topological spaces, Illinois J. Math., 8 (1964), 601--606.
S. Kaplan, On unbounded order convergence, Real Anal. Exchange, 23 (1) (1998-99), 175-184.
N. Gao and F. Xanthos, Unbounded order convergence and application to martingales without probability, J. Math. Anal. Appl., 415 (2) (2014), 931--947.
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.
C. D. Aliprantis and O. Burkinshaw, Positive operators, Springer, Dordrecht, 2006.
W. A. J. Luxemburg and A. C. Zaanen, Riesz Space I, North-Holland, Amsterdam, The Netherland, 1971.
C. Çevik and I. Altun, Vector metric spaces and some properties, Topol. Methods Nonlinear Anal., 34 (2), (2009), 375--382.