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).

Some vector valued sequence spaces of Musielak-Orlicz functions and their operator ideals

2018 ◽  
Vol 68 (1) ◽  
pp. 115-134 ◽  
Author(s):  
Mohammad Mursaleen ◽  
Kuldip Raj
Keyword(s):  
Sequence Space ◽  
Sequence Spaces ◽  
Operator Ideals ◽  
Orlicz Sequence Space ◽  
Geometric Properties ◽  
Orlicz Functions ◽  
Opial Property ◽  
Uniform Opial Property ◽  
Vector Valued

AbstractIn the present paper we introduce generalized vector-valued Musielak-Orlicz sequence spacel(A,𝓜,u,p,Δr,∥·,… ,·∥)(X) and study some geometric properties like uniformly monotone, uniform Opial property for this space. Further, we discuss the operators ofs-type and operator ideals by using the sequence ofs-number (in the sense of Pietsch) under certain conditions on matrixA.

scholarly journals On matrix transformations concerning the Nakano vector-valued sequence space

2001 ◽  
Vol 26 (11) ◽  
pp. 671-678
Author(s):  
Suthep Suantai
Keyword(s):  
Sequence Space ◽  
Sequence Spaces ◽  
Matrix Transformations ◽  
Real Numbers ◽  
Positive Real ◽  
The Matrix ◽  
Vector Valued ◽  
Bounded Sequences

We give the matrix characterizations from Nakano vector-valued sequence spaceℓ(X,p)andFr(X,p)into the sequence spacesEr,ℓ∞,ℓ¯∞(q),bs, andcs, wherep=(pk)andq=(qk)are bounded sequences of positive real numbers such thatPk>1for allk∈ℕandr≥0.

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 Vector valued multiple sequence spaces defined by Orlicz function

2014 ◽  
Vol 33 (1) ◽  
pp. 67 ◽  
Author(s):  
Binod Chandra Tripathy ◽  
Rupanjali Goswami
Keyword(s):  
Sequence Space ◽  
Orlicz Function ◽  
Sequence Spaces ◽  
Multiple Sequence ◽  
Vector Valued ◽  
Inclusion Results

In this article we define some vector valued multiple sequence space defined by Orlicz function. We study some of their properties like solidness, symmetry, completeness etc and prove some inclusion results.

On topological sequence spaces

1967 ◽  
Vol 63 (4) ◽  
pp. 997-1019 ◽  
Author(s):  
D. J. H. Garling
Keyword(s):  
Boolean Algebra ◽  
Sequence Space ◽  
Linear Subspace ◽  
Sequence Spaces ◽  
Vector Spaces ◽  
Order Structure ◽  
Complex Numbers ◽  
Vector Valued ◽  
Considerable Detail ◽  
Partially Ordered Spaces

We denote by ω the linear space of all sequences of real or complex numbers. A linear subspace of ω is called a sequence space. A subset A of ω is solid if whenever x ∈ A and |yi| ≤ |xi| for each i, then y ∈ A. The theory of solid sequence spaces, topologized in a variety of ways, has been developed in considerable detail, in particular by Köthe and Toeplitz (13) and subsequently by Köthe (see, for example (12)). These results have been generalized to function spaces by Dieudonné(6), to vector-valued sequence spaces by Pietsch (18), to vector spaces with a Boolean algebra of projections by Cooper ((4), (5)), and in the real case, to partially-ordered spaces by Luxemburg and Zaanen (see, for example (14)) and Fremlin (8). This last generalization shows that many of the properties of solid sequence spaces depend upon their order structure, rather than upon their structure as sequence spaces.

scholarly journals Topological properties and matrix transformations of certain ordered generalized sequence spaces

1995 ◽  
Vol 18 (2) ◽  
pp. 341-356 ◽  
Author(s):  
Manjul Gupta ◽  
Kalika Kaushal
Keyword(s):  
Sequence Space ◽  
Topological Structure ◽  
Necessary Conditions ◽  
Riesz Space ◽  
Linear Operators ◽  
Sequence Spaces ◽  
Matrix Transformations ◽  
Topological Properties ◽  
Vector Valued ◽  
Locally Convex

In this note, we carry out investigations related to the mixed impact of ordering and topological structure of a locally convex solid Riesz space(X,τ)and a scalar valued sequence spaceλ, on the vector valued sequence spaceλ(X)which is formed and topologized with the help ofλandX, and vice versa. Besides,we also characterizeo-matrix transformations fromc(X),ℓ∞(X)to themselves,cs(X)toc(X)and derive necessary conditions for a matrix of linear operators to transformℓ1(X)into a simple ordered vector valued sequence spaceΛ(X).

scholarly journals On Vector-Valued Generalized Lorentz Difference Sequence Space

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Birsen Sağır ◽  
Oğuz Oğur
Keyword(s):  
Sequence Space ◽  
Sequence Spaces ◽  
Difference Sequence ◽  
Inclusion Relations ◽  
Difference Sequence Space ◽  
Vector Valued ◽  
Difference Sequence Spaces

We introduce generalized Lorentz difference sequence spaces d(v,Δ,p). Also we study some topologic properties of this space and obtain some inclusion relations.

scholarly journals Approximation numbers of matrix transformations and inclusion maps

2011 ◽  
Vol 42 (2) ◽  
pp. 193-203
Author(s):  
M. Gupta ◽  
L. R. Acharya
Keyword(s):  
Banach Space ◽  
Sequence Space ◽  
Sequence Spaces ◽  
Matrix Transformations ◽  
Diagonal Operator ◽  
Approximation Numbers ◽  
Vector Valued

In this paper we establish relationships of the approximation numbers of matrix transformations acting between the vector-valued sequence spaces spaces of the type $\lambda(X)$ defined corresponding to a scalar-valued sequence space $\lambda$ and a Banach space $(X,\|.\|)$ as $$\lambda(X)=\{\overline x=\{x_i\}: x_i\in X, \forall~i\in \mathbb{N},~\{\|x_i\|_X\}\in \lambda\};$$ with those of their component operators. This study leads to a characterization of a diagonal operator to be approximable. Further, we compute the approximation numbers of inclusion maps acting between $\ell^p(X)$ spaces for $1\leq p\leq \infty$.

scholarly journals On some new sequence spaces of non-absolute type related to the spaces ℓp and ℓ∞ I

Filomat ◽  
2011 ◽  
Vol 25 (2) ◽  
pp. 33-51 ◽  
Author(s):  
M. Mursaleen ◽  
Abdullah Noman
Keyword(s):  
Sequence Space ◽  
Normed Space ◽  
Sequence Spaces ◽  
Inclusion Relations ◽  
Bk Space

In the present paper, we introduce the sequence space l?p of non-absolute type and prove that the spaces ??p and lp are linearly isomorphic for 0 < p ? ?. Further, we show that ??p is a p-normed space and a BK-space in the cases of 0 < p < 1 and 1 ? p ? ?, respectively. Furthermore, we derive some inclusion relations concerning the space ??p. Finally, we construct the basis for the space ??p, where 1 ? p < ?.

scholarly journals On some vector valued generalized difference modular sequence spaces

Filomat ◽  
2011 ◽  
Vol 25 (3) ◽  
pp. 15-27 ◽  
Author(s):  
Vatan Karakaya ◽  
Hemen Dutta
Keyword(s):  
Sequence Space ◽  
Sequence Spaces ◽  
Topological Structures ◽  
Vector Valued

In this paper we generalize the modular sequence space ?{Mk} by introducing the sequence space ?{Mk,p,q,s,?n/vm}. We give various properties relevant to algebraic and topological structures of this space and derived some other spaces.

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