scholarly journals Ideal Convergence and Completeness of a Normed Space

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 897 ◽  
Author(s):  
Fernando León-Saavedra ◽  
Francisco Javier Pérez-Fernández ◽  
María del Pilar Romero de la Rosa ◽  
Antonio Sala
Keyword(s):  
Normed Space ◽  
Cauchy Sequence ◽  
Convergence Space ◽  
Normed Spaces ◽  
Ideal Convergence ◽  
Underlying Space ◽  
Phd Thesis

We aim to unify several results which characterize when a series is weakly unconditionally Cauchy (wuc) in terms of the completeness of a convergence space associated to the wuc series. If, additionally, the space is completed for each wuc series, then the underlying space is complete. In the process the existing proofs are simplified and some unanswered questions are solved. This research line was originated in the PhD thesis of the second author. Since then, it has been possible to characterize the completeness of a normed spaces through different convergence subspaces (which are be defined using different kinds of convergence) associated to an unconditionally Cauchy sequence.

scholarly journals Strongly Lacunary Ward Continuity in 2-Normed Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Hüseyin Çakalli ◽  
Sibel Ersan
Keyword(s):  
Normed Space ◽  
Cauchy Sequence ◽  
Normed Spaces ◽  
Cauchy Sequences

A functionfdefined on a subsetEof a 2-normed spaceXis strongly lacunary ward continuous if it preserves strongly lacunary quasi-Cauchy sequences of points inE; that is,(f(xk))is a strongly lacunary quasi-Cauchy sequence whenever (xk) is strongly lacunary quasi-Cauchy. In this paper, not only strongly lacunary ward continuity, but also some other kinds of continuities are investigated in 2-normed spaces.

scholarly journals New Fuzzy Normed Spaces

2012 ◽  
Vol 9 (3) ◽  
pp. 559-564 ◽  
Author(s):  
Baghdad Science Journal
Keyword(s):  
Normed Space ◽  
Cauchy Sequence ◽  
Normed Spaces ◽  
Fuzzy Normed Space ◽  
Continuous Convergence ◽  
Definition Of

In this paper the research introduces a new definition of a fuzzy normed space then the related concepts such as fuzzy continuous, convergence of sequence of fuzzy points and Cauchy sequence of fuzzy points are discussed in details.

scholarly journals On Lacunary Mean Ideal Convergence in Generalized Randomn-Normed Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Awad A. Bakery ◽  
Mustafa M. Mohammed
Keyword(s):  
Normed Space ◽  
Complex Numbers ◽  
Normed Spaces ◽  
Ideal Convergence ◽  
Cauchy Sequences ◽  
Limit Points ◽  
Positive Integers ◽  
Cluster Points

An idealIis a hereditary and additive family of subsets of positive integersℕ. In this paper, we will introduce the concept of generalized randomn-normed space as an extension of randomn-normed space. Also, we study the concept of lacunary mean (L)-ideal convergence andL-ideal Cauchy for sequences of complex numbers in the generalized randomn-norm. We introduceIL-limit points andIL-cluster points. Furthermore, Cauchy andIL-Cauchy sequences in this construction are given. Finally, we find relations among these concepts.

scholarly journals Ideally Statistical Convergence in n-normed Space

2020 ◽  
pp. 75-84
Author(s):  
Nazneen Khan ◽  
Amani Shatarah
Keyword(s):  
Normed Space ◽  
Cauchy Sequence ◽  
Statistical Convergence ◽  
Topological Properties ◽  
Normed Spaces

The aim of the article is to extend the concept of Ideally statistical convergence from 2 normed spaces to n-normed space. We have also study and prove some important algebraic and topological properties of Ideally-statistical convergence of real sequences in n-normed space. In the last part of this article we obtain a criterion for I-statistically Cauchy sequence in n-normed space to be I-statistically Cauchy with respect to ∥.∥∞.

scholarly journals On Lacunary Ideal Convergence in Random -Normed Space

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
U. Yamancı ◽  
M. Gürdal
Keyword(s):  
Normed Space ◽  
Normed Spaces ◽  
Ideal Convergence ◽  
Random Normed Space ◽  
Random Normed Spaces

We introduce the notions of lacunary -convergence and lacunary -Cauchy in the topology induced by random -normed spaces and prove some important results.

On I-convergence in random 2-normed spaces

2011 ◽  
Vol 61 (6) ◽  
Author(s):  
M. Mursaleen ◽  
Abdullah Alotaibi
Keyword(s):  
Banach Spaces ◽  
Normed Space ◽  
Statistical Convergence ◽  
Normed Spaces ◽  
Ideal Convergence

AbstractRecently the concepts of statistical convergence and ideal convergence have been studied in 2-normed and 2-Banach spaces by various authors. In this paper we define and study the notion of ideal convergence in random 2-normed space and construct some interesting examples.

On ideal convergence in probabilistic normed spaces

2012 ◽  
Vol 62 (1) ◽  
Author(s):  
M. Mursaleen ◽  
S. Mohiuddine
Keyword(s):  
Normed Space ◽  
Statistical Convergence ◽  
Normed Spaces ◽  
Probabilistic Normed Space ◽  
Ideal Convergence ◽  
Probabilistic Normed Spaces ◽  
Real Anal ◽  
The Relationship ◽  
Cluster Points ◽  
Interesting Generalization

AbstractAn interesting generalization of statistical convergence is I-convergence which was introduced by P.Kostyrko et al [KOSTYRKO,P.—ŠALÁT,T.—WILCZYŃSKI,W.: I-Convergence, Real Anal. Exchange 26 (2000–2001), 669–686]. In this paper, we define and study the concept of I-convergence, I*-convergence, I-limit points and I-cluster points in probabilistic normed space. We discuss the relationship between I-convergence and I*-convergence, i.e. we show that I*-convergence implies the I-convergence in probabilistic normed space. Furthermore, we have also demonstrated through an example that, in general, I-convergence does not imply I*-convergence in probabilistic normed space.

scholarly journals ON $\mathcal{I}_2$-CONVERGENCE AND $\mathcal{I}_2$-CAUCHY DOUBLE SEQUENCES OF FUNCTIONS IN 2-NORMED SPACES

2020 ◽  
pp. 801
Author(s):  
Sevim Yegül ◽  
Erdinç Dündar
Keyword(s):  
Normed Space ◽  
Cauchy Sequence ◽  
Double Sequences ◽  
Normed Spaces

In this study, firstly, we studied some properties of $\mathcal{I}_2$-convergence. Then, we introduced $\mathcal{I}_2$-Cauchy and $\mathcal{I}_2^*$-Cauchy sequence of double sequences of functions in $2$-normed space. Also, were investigated relationships between them for double sequences of functions in $2$-normed spaces.

scholarly journals A variation on strongly lacunary delta ward continuity in 2-normed spaces

2019 ◽  
Vol 38 (7) ◽  
pp. 195-202
Author(s):  
Sibel Ersan
Keyword(s):  
Normed Space ◽  
Cauchy Sequence ◽  
Normed Spaces ◽  
Cauchy Sequences

A sequence $(x_{k})$ of points in a subset E of a 2-normed space $X$ is called strongly lacunary $\delta$-quasi-Cauchy, or $N_\theta$-$\delta$-quasi-Cauchy if $(\Delta x_k)$ is $N_\theta$-convergent to 0, that is $\lim_{r\rightarrow\infty}\frac{1}{h_r}\sum_{k\in I_r}||\Delta^2 x_k, z||=0$ for every fixed $z\in X$. A function defined on a subset $E$ of $X$ is called strongly lacunary $\delta$-ward continuous if it preserves  $N_{\theta}$-$\delta$-quasi-Cauchy sequences, i.e. $(f(x_{k}))$ is an $N_{\theta}$-$\delta$-quasi-Cauchy sequence whenever $(x_{k})$ is. In this study we obtain some theorems related to strongly lacunary $\delta$-quasi-Cauchy sequences.

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