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Metric Spaces
Published in James K. Peterson, Basic Analysis III, 2020
Cauchy Sequence: A sequence (xn) in X is said to be a Cauchy Sequence if ∀ϵ>0,∃N∋d(xn,xm)<ϵifn,m>N From our earlier work, it is clear convergence sequences are Cauchy sequences as this proof is an easy modification of arguments in the first two volumes. If every Cauchy sequence in X converges to some object x in X, we say the metric space (X, d) is a Complete Metric Space.
Basics of Discrete Volterra Series
Published in Andrzej Borys, Nonlinear Aspects of Telecommunications, 2018
Now, using the above definition, we define the complete space. That is, the complete metric space X is a space in which every Cauchy sequence possesses its limit, which belongs to this space. In this book, we deal with metric spaces, which are complete. Furthermore, knowing what the notion of completeness of a metric space means, we are now able to define the set compactness in the following way. Let X be a complete metric space. Then we say that a set BX X is compact if every infinite sequence xn of elements xn BX possesses a subsequence { x n } convergent to some element x* BX . We begin with the proof of Lemma 1.1. For this purpose, let us denote by xn a sequence in B . Afterwards, observe that, xn(0) M1 according to assumption 1.199. To proceed further, recall the well-known Bolzano-Weierstrass21 theorem, that every finite segment a,b on the -line is a compact set. According to this theorem, we are able to find a subsequence of xn, which converges at point k 0 to some element we call x0(0). That is,
Density Aspects in Semi-Infinite Vector Optimization
Published in Anurag Jayswal, Tadeusz Antczak, Continuous Optimization and Variational Inequalities, 2023
We recall that a topological space is a Baire space, if the countable intersection of open dense subsets is dense. Alternatively, a space is a Baire space, if the countable union of closed sets with empty interior has empty interior. Baire category theorem states that a compact Hausdorff space or a complete metric space is a Baire space. A subset of a topological space is a Gδ set, if it is a countable intersection of open sets. For more details refer to the book by Munkres [19].
Convex feasibility problems on uniformly convex metric spaces
Published in Optimization Methods and Software, 2020
Byoung Jin Choi, Un Cig Ji, Yongdo Lim
Let be a complete metric space and be a bounded sequence. For a given point , set The asymptotic radius of is defined by and the asymptotic centre of is defined as Note that Now, we recall the notion of Δ-convergence in complete metric space. Let . A sequence is said to Δ-converge to x if for any subsequence of , the point x is the unique asymptotic centre of In this case we call x the Δ-limit of . A point x in M is called a Δ-cluster point of a sequence if there exists a subsequence of such that Δ-converges to x.
Optimal contraception control for a size-structured population model with extra mortality
Published in Applicable Analysis, 2020
Obviously, if is a fixed point of the map , then it is a solution of Equation (11) and vice versa. Define a new norm in by for some , which is equivalent to the usual norm. It is clear that is a Banach space; that is, is a complete metric space. Here for , . It is easy to see that maps into itself. Now, we show that is a contraction mapping on the complete metric space . For any , , we have Choose λ such that . Then is a contraction mapping on the complete metric space . By the Banach fixed-point theorem, owns a unique fixed point , which is the solution of Equation (11).
Path-based incremental target level algorithm on Riemannian manifolds
Published in Optimization, 2020
Let be a sequence in the complete metric space . If is quasi-Fejér convergent to a set then is bounded and converges for all . If furthermore, a cluster point y of belongs to W, then .