@article{article_604784, title={Unbounded Vectorial Cauchy Completion of Vector Metric Spaces}, journal={Gazi University Journal of Science}, volume={33}, pages={761–765}, year={2020}, DOI={10.35378/gujs.604784}, author={Özeken, Çetin Cemal and Çevik, Cüneyt}, keywords={Unbounded order convergence,vector metric spaces,unbounded vectorial convergence,unbounded Cauchy completion,Riesz space}, abstract={<p> <span lang="en-us" style="font-size:12pt;line-height:107%;font-family:’Times New Roman’, serif;" xml:lang="en-us">A sequence ( <i>a <sub>n </sub> </i> </span> <span lang="en-us" style="font-size:12pt;line-height:107%;font-family:’Times New Roman’, serif;" xml:lang="en-us">) in a Riesz space <i>E </i> is called uo-convergent (or unbounded order convergent) to <i>a </i> in <i>E </i> </span> <span style="font-size:11pt;line-height:107%;font-family:Calibri, sans-serif;"> </span> <span lang="en-us" style="font-size:12pt;line-height:107%;font-family:’Times New Roman’, serif;" xml:lang="en-us"> if </span> <span style="font-size:11pt;line-height:107%;font-family:Calibri, sans-serif;"> </span> <span lang="en-us" style="font-size:12pt;line-height:107%;font-family:’Times New Roman’, serif;" xml:lang="en-us"> inf{| <i>a <sub>n </sub>-a </i>|, <i>u </i>} is order convergent to 0  </span> <span lang="en-us" style="font-size:12pt;line-height:107%;font-family:’Times New Roman’, serif;" xml:lang="en-us">for all <i>u </i> in <i>E <sub>+ </sub> </i> </span> <span style="font-size:11pt;line-height:107%;font-family:Calibri, sans-serif;"> </span> <span lang="en-us" style="font-size:12pt;line-height:107%;font-family:’Times New Roman’, serif;" xml:lang="en-us"> and
unbounded order Cauchy (uo-Cauchy) if  </span> <span style="font-family:’Times New Roman’, serif;font-size:16px;">| </span> <i style="font-family:’Times New Roman’, serif;font-size:16px;">a <sub style="font-size:12px;">n </sub>-a <sub>n+p </sub> </i> <span style="font-family:’Times New Roman’, serif;font-size:16px;">| </span> <span lang="en-us" style="font-size:12pt;line-height:107%;font-family:’Times New Roman’, serif;" xml:lang="en-us">is
uo-convergent to 0. In the first part of this study we define <i>u <sub>d,E </sub> </i> </span> <span style="font-size:11pt;line-height:107%;font-family:Calibri, sans-serif;"> </span> <span lang="en-us" style="font-size:12pt;line-height:107%;font-family:’Times New Roman’, serif;" xml:lang="en-us">-convergence </span> <span lang="en-us" style="font-size:12pt;line-height:107%;font-family:’Times New Roman’, serif;" xml:lang="en-us">
(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. </span> </p> <span style="font-size:11pt;line-height:107%;font-family:Calibri, sans-serif;"> </span> <p> <span lang="en-us" style="font-size:12pt;line-height:107%;font-family:’Times New Roman’, serif;" xml:lang="en-us"> <br /> </span> </p>}, number={3}, publisher={Gazi University}