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Fundamentals of Linear and Topological Spaces
Published in Eberhard Malkowsky, Vladimir Rakočević, Advanced Functional Analysis, 2019
Eberhard Malkowsky, Vladimir Rakočević
A sequence (xn) in a semimetric space (X, d) is said to be a Cauchy sequence, if, for every ε > 0, there exists n0∈N such that d(xm,xn)<ε for all m, n ≥ n0. Obviously every convergent sequence is a Cauchy sequence, but the converse implication is not true, in general. A semimetric space X with the property that every Cauchy sequence converges to an element x ∈ X is said to be complete.
Topological and Metric Spaces
Published in J. Tinsley Oden, Leszek F. Demkowicz, Applied Functional Analysis, 2017
J. Tinsley Oden, Leszek F. Demkowicz
Complete Metric Spaces. A metric space (X, d) is said to be complete if every Cauchy sequence in (X, d) is convergent. Order completeness of R $ \mathbb R $ implies (comp. Theorem 4.8.1) that R $ \mathbb R $ equipped with the standard metric d(x,y)=|x-y| $ d(x,y) = \vert x - y \vert $ is a complete metric space. We will show now how the completeness of R $ \mathbb R $ implies completeness of the most commonly used metric spaces discussed in this chapter.
Subsequences and Cauchy Sequences
Published in John D. Ross, Kendall C. Richards, Introductory Analysis, 2020
John D. Ross, Kendall C. Richards
We are now in a position to answer our question from the beginning of this chapter: yes, all Cauchy sequences in R are convergent. This result, along with our results from earlier in the chapter, will imply that a sequence in R converges if and only if it is Cauchy.
Partial quasi-metrics and fixed point theory: an aggregation viewpoint
Published in International Journal of General Systems, 2021
Pilar Fuster-Parra, Juan José Miñana, Óscar Valero
In the case of partial metrics, the aforesaid fixed point methods were based on the so-called Matthews fixed point theorem. In order to recall such a fixed point theorem, let us introduce a few pertinent notions. According to Matthews (1994), a mapping from a partial metric space into itself is said to be a contraction if there exists such that for all . The preceding constant c is said to be the contractive constant of the contraction f. Moreover, a sequence in a partial metric space is said to be a Cauchy sequence if exists in . Thus, a partial metric space is called complete provided that for every Cauchy sequence in X there exists a point such that in , i.e. .
A contribution to best proximity point theory and an application to partial differential equation
Published in Optimization, 2023
Sakan Termkaew, Parin Chaipunya, Dhananjay Gopal, Poom Kumam
Let and ζ be a non-triangular metric on X. We will use the following notations The notions of convergent and Cauchy sequences, as well as completeness, are defined similarly as in a metric space. We say that a subset is closed if the limit of a convergent sequence in U is always an element in U. The collection of all closed subsets in is denoted by . Note that closed subsets of a complete non-triangular metric space is again complete.
Remarks on asymptotic regularity in A-metric spaces
Published in Optimization, 2022
Temel Ermiş, Özcan Gelişgen, Mujahid Abbas
Let be an A-metric space. A sequence in X is said to be; A-convergent to if for each there exists an such that for all , we have We denote this by or . Thus, A-Cauchy if ; that is, for each , there exists an such that for all we have An A-metric space is said to be A-complete if every A-Cauchy sequence is an A-convergent sequence.Let be an A-metric space. A map is called A-continuous at if for every there exists such that if , then . In other words, f is continuous at if and only if such that .