new type of paranormed sequence space of non-absolute type

Copyright © 2013 by Modern Scientific Press Company, Florida, USA

International Journal of Modern Mathematical Sciences, 2013, 8(3): 196-211

International Journal of Modern Mathematical Sciences

Journal homepage:www.ModernScientificPress.com/Journals/ijmms.aspx

ISSN:2166-286X

Florida, USA

Article

New Type of Paranormed Sequence Space of Non-absolute Type

and a Matrix Transformation

Ab. Hamid Ganie* and Neyaz Ahmad Sheikh

Department of Mathematics, National Institute of Technology Srinagar-190006

* Author to whom correspondence should be addressed; E-Mail: [email protected];

[email protected]

Article history: Received 3 October 2013, Received in revised form 8 November, Accepted 25

November 2013, Published 13 December 2013.

Abstract: In the present paper, we have defined the difference sequence space rq(∆𝑢𝑝

), as

follows:

rq(∆𝑢𝑝

) = { x = (xk) : (∆xk) } ∈rq(u, p)}

where the space rq(u, p) has recently been studied by Neyaz and Hamid (see, [23]).We give

some topological properties and compute α -, β- and γ-duals. Furthermore, construct the basis

of rq(∆𝑢𝑝

). In our final section, we have characterized a matrix class on this sequence space.

Keywords: Sequence space of non-absolute type; paranormed sequence space; α -, β- and

γ-duals; matrix transformations.

Mathematics Subject Classification (2010): 46A45; 40C05; 46J05.

1. Introduction

We denote the set of all sequences (real or complex) by ω. Any subspace of ω iscalled the

sequence space. So the sequence space is the set of scalar sequences(realof complex) which is closed

under co-ordinate wise addition and scalar multiplication.

Int. J. Modern Math. Sci. 2013, 8(3): 196-211


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2. The Riesz Sequence Space rq(∆𝒖𝒑)of Non-absolute Type



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3. Basis and α-, β- and γ-duals of the space rq(∆𝒖𝒑)

In this section, we compute α-, β- and γ-duals of the space rq(∆𝑢𝑝

) and finally we give the basis

for the space rq(∆𝑢𝑝

).



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4. Matrix Mappings on the Space rq(∆𝒖𝒑)



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new type of paranormed sequence space of non-absolute type

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