Matrix Transformations in an Incomplete Space

1968 ◽  
Vol 20 ◽  
pp. 727-734 ◽  
Author(s):  
I. J. Maddox
Keyword(s):  
Linear Space ◽  
Cauchy Sequence ◽  
Convergent Sequence ◽  
Convergent Series ◽  
Matrix Transformations ◽  
Cauchy Sequences ◽  
Complex Linear ◽  
Incomplete Space ◽  
Bounded Sequences ◽  
Convergent Sequences

Let X = (X, p) be a seminormed complex linear space with zero θ. Natural definitions of convergent sequence, Cauchy sequence, absolutely convergent series, etc., can be given in terms of the seminorm p. Let us write C = C(X) for the set of all convergent sequences for the set of Cauchy sequences; and L∞ for the set of all bounded sequences.

scholarly journals -Ward Continuity

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Huseyin Cakalli
Keyword(s):  
Cauchy Sequence ◽  
Convergent Sequence ◽  
Lacunary Sequence ◽  
Real Numbers ◽  
Cauchy Sequences ◽  
New Type ◽  
Positive Integers ◽  
Convergent Sequences

A function is continuous if and only if preserves convergent sequences; that is, is a convergent sequence whenever is convergent. The concept of -ward continuity is defined in the sense that a function is -ward continuous if it preserves -quasi-Cauchy sequences; that is, is an -quasi-Cauchy sequence whenever is -quasi-Cauchy. A sequence of points in , the set of real numbers, is -quasi-Cauchy if , where , and is a lacunary sequence, that is, an increasing sequence of positive integers such that and . A new type compactness, namely, -ward compactness, is also, defined and some new results related to this kind of compactness are obtained.

scholarly journals On statistical convergence and strong Cesàro convergence by moduli

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Fernando León-Saavedra ◽  
M. del Carmen Listán-García ◽  
Francisco Javier Pérez Fernández ◽  
María Pilar Romero de la Rosa
Keyword(s):  
Statistical Convergence ◽  
Convergent Sequence ◽  
Modulus Function ◽  
Cesaro Convergence ◽  
Bounded Sequences ◽  
Convergent Sequences

AbstractIn this paper we will establish a result by Connor, Khan and Orhan (Analysis 8:47–63, 1988; Publ. Math. (Debr.) 76:77–88, 2010) in the framework of the statistical convergence and the strong Cesàro convergence defined by a modulus function f. Namely, for every modulus function f, we will prove that a f-strongly Cesàro convergent sequence is always f-statistically convergent and uniformly integrable. The converse of this result is not true even for bounded sequences. We will characterize analytically the modulus functions f for which the converse is true. We will prove that these modulus functions are those for which the statistically convergent sequences are f-statistically convergent, that is, we show that Connor–Khan–Orhan’s result is sharp in this sense.

scholarly journals On the Domain of the Triangle on the Spaces of Null, Convergent, and Bounded Sequences

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Naim L. Braha ◽  
Feyzi Başar
Keyword(s):  
Type Theorem ◽  
Sequence Spaces ◽  
Matrix Transformations ◽  
Topological Properties ◽  
Inclusion Relations ◽  
Bounded Sequences ◽  
Convergent Sequences

We introduce the spaces of -null, -convergent, and -bounded sequences. We examine some topological properties of the spaces and give some inclusion relations concerning these sequence spaces. Furthermore, we compute -, -, and -duals of these spaces. Finally, we characterize some classes of matrix transformations from the spaces of -bounded and -convergent sequences to the spaces of bounded, almost convergent, almost null, and convergent sequences and present a Steinhaus type theorem.

scholarly journals Complete Fuzzy n-normed linear space

2014 ◽  
Vol 3 (1) ◽  
Author(s):  
S. Vijayabalaji ◽  
N. Thillaigovindan
Keyword(s):  
Linear Space ◽  
Cauchy Sequence ◽  
Normed Linear Space ◽  
Convergent Sequence ◽  
Subject Classification

This paper introduces the notion of Cauchy sequence, convergent sequence and completeness in fuzzy n-normed linear space. AMS Mathematics Subject Classification: 46S40, 03E72.

On a Procrustean technique for the numerical transformation of slowly convergent sequences and series

1956 ◽  
Vol 52 (4) ◽  
pp. 663-671 ◽  
Author(s):  
P. Wynn
Keyword(s):  
Convergent Sequence ◽  
Convergent Series ◽  
The Other ◽  
Partial Sums ◽  
Numerical Transformation ◽  
Convergent Sequences

1.The study of the numerical transformation of slowly convergent series and sequences permits of a certain unity of approach, since the partial sums of the former may be regarded as members of a slowly convergent sequence, while the differences of the latter may be treated as terms in a slowly convergent series. In the ensuing discussion, no essential distinction between the two problems is made, and methods devised for one relate with equal facility to the other.

scholarly journals Matrix transformations of strongly convergent sequences into Vσ

Filomat ◽  
2010 ◽  
Vol 24 (3) ◽  
pp. 103-109 ◽  
Author(s):  
S.A. Mohiuddine ◽  
M. Aiyub
Keyword(s):  
Matrix Transformations ◽  
Matrix Classes ◽  
The Matrix ◽  
Bounded Sequences ◽  
Convergent Sequences

In this paper, we define the spaces ?(p, s) and ?p (s), where ?(p, s) = {x:1/n? k=1 K-s |xk -?|pk ? 0 for some ?, s ? 0} and if pk = p for each k, we have ?(p, s)=?p(s). We further characterize the matrix classes (?(p, s), V? ), (?p (s), V? ) and (?p (s), V? )reg , where V? denotes the set of bounded sequences all of whose ?-mean are equal.

scholarly journals SOME RESULTS RELATED WITH FUZZY a −NORMED LINEAR SPACE

2019 ◽  
pp. 11-12
Author(s):  
M. Arunmaran ◽  
K. Kannan
Keyword(s):  
Linear Space ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Normed Linear Space ◽  
Finite Dimension ◽  
Convergent Sequence ◽  
Unit Interval ◽  
Clear Understanding ◽  
Convergent Sequences

Zadeh established the concept of fuzzy set based on the characteristic function. Foundation of fuzzy set theory was introduced by him. Throughout this paper, 𝑀𝑛(𝐹) denotes the set of all fuzzy matrices of order 𝑛 over the fuzzy unit interval [0,1]. Inaddi tion (𝑀𝑛 (𝐹), 𝜃) dis called as fuzzy 𝛼 −normed linear space. The objective of this paper is to investigate the relationships between convergent sequences and fuzzy 𝛼 −normed linear space. The set of all fuzzy points in 𝑀𝑛 (𝐹) is denoted by 𝑃∗(𝑀𝑛(𝐹)). For a fuzzy 𝛼 −normed linear space (𝑀𝑛 (𝐹), 𝜃), we have |𝜃(𝑃𝐴)𝛼 −𝜃(𝑃𝐵)𝛼 | ≤ 𝜃(𝑃𝐴,𝑃𝐵)𝛼. Besides 𝜃 is a continuous function on 𝑀𝑛 (𝐹). That is, if 𝑃𝐴𝑛 → 𝑃𝐴 as 𝑛 → ∞ then 𝜃(𝑃𝐴𝑛 )𝛼 → 𝜃(𝑃𝐴)𝛼 as 𝑛 → ∞, where 𝑃𝐴𝑛 is a sequence in (𝑀𝑛 (𝐹), 𝜃). Hence, 𝜃 is always bounded on 𝑀𝑛(𝐹). Next we introduce the following result: Let 𝑃𝐴𝑛 , 𝑃𝐵𝑛 ∈ 𝑃 ∗ (𝑀𝑛(𝐹)) with 𝑃𝐴𝑛 and 𝑃𝐵𝑛 converge to 𝑃𝐴 and 𝑃𝐵 respectively as 𝑛 → ∞. Then 𝑃𝐴𝑛 + 𝑃𝐵𝑛 converge to 𝑃𝐴 + 𝑃𝐵 as 𝑛 → ∞. Furthermore, we are able to compare two different fuzzy 𝛼 −norms with convergent sequence. The result states that for a fuzzy 𝛼 −normed linear space (𝑀𝑛(𝐹), 𝜃), we have 𝜃(𝑃𝐴)𝛼1 ≥ 𝑀𝜃(𝑃𝐴 )𝛼2 , for some 𝑀 > 0 and 𝑃𝐴 ∈ 𝑃 ∗ (𝑀𝑛(𝐹)). If 𝑃𝐴𝑛 converges to 𝑃𝐴 under fuzzy 𝛼1 −norm then 𝑃𝐴𝑛 converges to 𝑃𝐴 under fuzzy 𝛼2 −norm. Moreover, if (𝑀𝑛 (𝐹), 𝜃) has finite dimension then it should be complete. Through these results, we are able to get clear understanding about the concept fuzzy 𝛼 −normed linear space and its properties.

scholarly journals Matrix transformations from absolutely convergent series to convergent sequences as general weighted mean summability methods

2000 ◽  
Vol 24 (8) ◽  
pp. 533-538
Author(s):  
Jinlu Li
Keyword(s):  
Classical Result ◽  
Sufficient Conditions ◽  
Convergent Series ◽  
Necessary And Sufficient Conditions ◽  
Matrix Transformations ◽  
Weighted Mean ◽  
Summability Methods ◽  
Necessary And Sufficient ◽  
Convergent Sequences

We prove the necessary and sufficient conditions for an infinity matrix to be a mapping, from absolutely convergent series to convergent sequences, which is treated as general weighted mean summability methods. The results include a classical result by Hardy and another by Moricz and Rhoades as particular cases.

scholarly journals On Weak Statistical Convergence

2007 ◽  
Vol 2007 ◽  
pp. 1-9 ◽  
Author(s):  
Vinod K. Bhardwaj ◽  
Indu Bala
Keyword(s):  
Normed Space ◽  
Cauchy Sequence ◽  
Statistical Convergence ◽  
Reflexive Space ◽  
Cauchy Sequences ◽  
Convergent Sequences

The main object of this paper is to introduce a new concept of weak statistically Cauchy sequence in a normed space. It is shown that in a reflexive space, weak statistically Cauchy sequences are the same as weakly statistically convergent sequences. Finally, weak statistical convergence has been discussed inlpspaces.

scholarly journals A Note on The Abel Matrix Transformations

2016 ◽  
Vol 4 (1) ◽  
pp. 48-53
Author(s):  
Mulatu Lemma ◽  
Latrice Tanksley ◽  
Keisha Brown
Keyword(s):  
Matrix Transformations ◽  
Main Research ◽  
Research Questions ◽  
Bounded Sequences ◽  
Convergent Sequences ◽  
Divergent Sequences

The purpose of this research is to investigate the effect of applying At to convergent sequences, bounded sequences, divergent sequences, and absolutely convergent sequences. We considering and answer the following interesting main research questions.

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