Consistency theory for SM-methods

Boos, Johann; Leiger, Toivo; Zeller, Karl

doi:10.1007/bf02907056

Cited by 18 publications

(11 citation statements)

References 15 publications

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“…Since there are various convergence rules for double sequences (see [10] for other types of convergence), we give a new definition for β−space. Definition 2.4.…”

Section: Dual Spaces Of L(t)mentioning

confidence: 99%

On the paranormed space L(t) of double sequences

Çapan

Başar

2018

Filomat

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In this paper, we introduce the paranormed sequence space L(t) which is the generalization of the space L q of all absolutely q−summable double sequences. We examine some topological properties of the space L(t) and determine its alpha-, beta-and gamma-duals. Finally, we characterize some classes of four-dimensional matrix transformations from the space L(t) into some spaces of double sequences.

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“…Since there are various convergence rules for double sequences (see [10] for other types of convergence), we give a new definition for β−space. Definition 2.4.…”

Section: Dual Spaces Of L(t)mentioning

confidence: 99%

On the paranormed space L(t) of double sequences

Çapan

Başar

2018

Filomat

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“…[7]): let ᐂ be a double sequence space, that is a linear subspace of the linear space Ω of all double sequences y = (y nr ), and let Ꮽ = (A (r ) ) be a sequence of infinite matrices A (r ) = (a (r ) nk ) n,k . We put…”

Section: Applicationsmentioning

confidence: 99%

Dual pairs of sequence spaces

Boos

Leiger

2001

International Journal of Mathematics and Mathematical Sciences

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Abstract. The paper aims to develop for sequence spaces E a general concept for reconciling certain results, for example inclusion theorems, concerning generalizations of the Köthe-Toeplitz duals E × (× ∈ {α, β}) combined with dualities (E, G), G ⊂ E × , and the SAKproperty (weak sectional convergence). TakingE cs , where cs denotes the set of all summable sequences, as a starting point, then we get a general substitute of E cs by replacing cs by any locally convex sequence space S with sum s ∈ S (in particular, a sum space) as defined by Ruckle (1970). This idea provides a dual pair (E, E S ) of sequence spaces and gives rise for a generalization of the solid topology and for the investigation of the continuity of quasi-matrix maps relative to topologies of the duality (E, E β ). That research is the basis for general versions of three types of inclusion theorems: two of them are originally due to Bennett and Kalton (1973) and generalized by the authors (see Leiger (1993 and), and the third was done by Große-Erdmann (1992). Finally, the generalizations, carried out in this paper, are justified by four applications with results around different kinds of Köthe-Toeplitz duals and related section properties.2000 Mathematics Subject Classification. 46A45, 46A20, 46A30, 40A05.

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“…The summability of double sequences defining by the product of the complex sequences x = (x n ) and y = (y k ), Jardas and Sarapa [6] prove the Silverman-Toeplitz and Steinhaus type theorems for threedimensional matrices. Boos, Leiger and Zeller [3] define the concept of V-SM-method by the application domain of a matrix sequence A = (A (υ) ) of infinite matrices and give the consistency theory for such type methods and introduce the notion of C e -convergence for double sequences. By C e and C be , we denote the spaces of all C e -convergent and of all bounded C e -convergent double sequences, respectively.…”

Section: Introductionmentioning

confidence: 99%

Some new spaces of double sequences

Altay

Başar

2005

Journal of Mathematical Analysis and Applications

119

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In this study, we define the double sequence spaces BS, BS(t), CS p , CS bp , CS r and BV, and also examine some properties of those sequence spaces. Furthermore, we show that these sequence spaces are complete paranormed or normed spaces under some certain conditions. We determine the α-duals of the spaces BS, BV, CS bp and the β(ϑ)-duals of the spaces CS bp and CS r of double series. Finally, we give the conditions which characterize the class of four-dimensional matrix mappings defined on the spaces CS bp , CS r and CS p of double series.  2004 Elsevier Inc. All rights reserved.

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Consistency theory for SM-methods

Cited by 18 publications

References 15 publications

On the paranormed space L(t) of double sequences

On the paranormed space L(t) of double sequences

Dual pairs of sequence spaces

Some new spaces of double sequences

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