scholarly journals On some spaces of summable sequences and their duals

1986 ◽  
Vol 9 (1) ◽  
pp. 71-79 ◽  
Author(s):  
Geraldo Soares de Souza ◽  
G. O. Golightly
Keyword(s):  
Bounded Variation ◽  
Dual Space ◽  
Sequence Spaces ◽  
Simple Argument

Suppose thatSis the space of all summable sequencesαwith‖α‖S=supn≥0|∑j=n∞αj|andJthe space of all sequencesβof bounded variation with‖β‖J=|β0|+∑j=1∞|βj−βj−1|. Then forαinSandβinJ |∑j=0∞αjβj|≤‖α‖S‖β‖J; this inequality leads to the description of the dual space ofSasJ. It, related inequalities, and their consequences are the content of this paper. In particular, the inequality cited above leads directly to the Stolz form of Abel's theorem and provides a very simple argument. Also, some other sequence spaces are discussed.

scholarly journals On Domain of Nörlund Sequences

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 268 ◽  
Author(s):  
Kuddusi Kayaduman ◽  
Fevzi Yaşar
Keyword(s):  
Banach Space ◽  
Bounded Variation ◽  
Sequence Space ◽  
Convergent Series ◽  
Sequence Spaces ◽  
Matrix Transformations ◽  
Nörlund Matrix ◽  
Inclusion Relations ◽  
The Matrix ◽  
New Space

In 1978, the domain of the Nörlund matrix on the classical sequence spaces lp and l∞ was introduced by Wang, where 1 ≤ p < ∞. Tuğ and Başar studied the matrix domain of Nörlund mean on the sequence spaces f0 and f in 2016. Additionally, Tuğ defined and investigated a new sequence space as the domain of the Nörlund matrix on the space of bounded variation sequences in 2017. In this article, we defined new space and and examined the domain of the Nörlund mean on the bs and cs, which are bounded and convergent series, respectively. We also examined their inclusion relations. We defined the norms over them and investigated whether these new spaces provide conditions of Banach space. Finally, we determined their α­, β­, γ­duals, and characterized their matrix transformations on this space and into this space.

scholarly journals Some new sequence spaces defined by lacunary sequences

2007 ◽  
Vol 57 (4) ◽  
Author(s):  
Ekrem Savas ◽  
Vatan Karakaya
Keyword(s):  
Bounded Variation ◽  
Sequence Spaces ◽  
Almost Convergence ◽  
Lacunary Sequences

AbstractIt is natural to expect that lacunary almost convergence must be related to the some concept of lacunary almost bounded variations in the some view as almost convergence is related to almost bounded variation. The purpose of this paper is to examine this new concept in some details. Some inclusion theorems have been established.

An Abstract Concept of the Sum of a Numerical Series

1970 ◽  
Vol 22 (4) ◽  
pp. 863-874 ◽  
Author(s):  
William H. Ruckle
Keyword(s):  
Topological Space ◽  
Sequence Space ◽  
Dual Space ◽  
Linear Topological Space ◽  
Sequence Spaces ◽  
Abstract Concept ◽  
Numerical Series ◽  
Locally Convex ◽  
Theory Of Duality

Our aim in this paper, generally stated, is to formulate an abstract concept of the notion of the sum of a numerical series. More particularly, it is a study of the type of sequence space called “sum space”. The idea of sum space arose in connection with two distinct problems.1.1 The Köthe-Toeplitz dual of a sequence space T consists of all sequences t such that st ∈ l1 (absolutely summable sequences) for each s∈T. It is known that if cs or bs is used in place of l1, an analogous theory of duality for sequence spaces can be developed (cf. [2]). What other spaces of sequences can play a rôle analogous to l1? This problem is treated in [6].1.2. Let {xn, fn} be a complete biorthogonal sequence in (X, X*), where X is a locally convex linear topological space and X* is its topological dual space.

scholarly journals Onβ-dual of vector-valued sequence spaces of Maddox

2002 ◽  
Vol 30 (7) ◽  
pp. 383-392 ◽  
Author(s):  
Suthep Suantai ◽  
Winate Sanhan
Keyword(s):  
Sequence Space ◽  
Dual Space ◽  
Sequence Spaces ◽  
Bk Space ◽  
Vector Valued

Theβ-dual of a vector-valued sequence space is defined and studied. We show that if anX-valued sequence spaceEis a BK-space having AK property, then the dual space ofEand itsβ-dual are isometrically isomorphic. We also give characterizations ofβ-dual of vector-valued sequence spaces of Maddoxℓ(X,p),ℓ∞(X,p),c0(X,p), andc(X,p).

scholarly journals The spectra of weighted mean operators on bν0

1992 ◽  
Vol 52 (2) ◽  
pp. 242-250 ◽  
Author(s):  
B. E. Rhoades
Keyword(s):  
Bounded Variation ◽  
Sequence Spaces ◽  
Bounded Operators ◽  
Weighted Mean ◽  
Weighted Mean Matrices ◽  
Weighted Mean Operators

AbstractIn a series of papers, the author has previously investigated the spectra and fine spectra for weighted mean matrices, considered as bounded operators over various sequence spaces. This paper examines the spectra of weighted mean matrices as operators over bνv0, the space of null sequences of bounded variation.

scholarly journals A Study of Bounded Variation Sequence Spaces

2019 ◽  
Author(s):  
Vakeel Ahmad Khan ◽  
Hira Fatima ◽  
Mobeen Ahmad
Keyword(s):  
Bounded Variation ◽  
Sequence Spaces

scholarly journals On the Essential Norm of Multiplications Operators Acting on Cesàro Sequence Spaces

2019 ◽  
Vol 2019 ◽  
pp. 1-5
Author(s):  
Julio C. Ramos-Fernández ◽  
María A. Rivera-Sarmiento ◽  
Margot Salas-Brown
Keyword(s):  
Dual Space ◽  
Measure Of Noncompactness ◽  
Multiplication Operator ◽  
Essential Norm ◽  
Sequence Spaces

In this article, we establish an important property about the growth of sequences in the dual space of Cesàro sequence spaces. As a consequence of this fact, we calculate the measure of noncompactness or the essential norm of the multiplication operator Mu acting on Cesàro sequence spaces cesp.

scholarly journals Some new sequence spaces defined by bounded variation in 2-normed spaces

Filomat ◽  
2019 ◽  
Vol 33 (2) ◽  
pp. 483-492
Author(s):  
Ekrem Savaş
Keyword(s):  
Bounded Variation ◽  
Sequence Space ◽  
Orlicz Function ◽  
Sequence Spaces ◽  
Normed Spaces ◽  
Algebraic Properties

In this paper, by using Orlicz function and almost lacunary bounded variation we introduce and examine a new sequence space in 2-normed spaces. We also study some basic topological and algebraic properties of these spaces. In addition, we shall established inclusion theorems between these sequence spaces.

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