Some New Type of Difference Sequence Space of Non-absolute Type

Transcription

1 Article International Journal of Modern Mathematical Sciences, 2016, 14(1): International Journal of Modern Mathematical Sciences Journal homepage: ISSN: X Florida, USA Some New Type of Difference Sequence Space of Non-absolute Type A.H.Ganie 1,*, S.A.Mobin 2, N.A.Sheikh 3, T.Jalal 4 & S.A.Gupkari 5 1 Department of Applied Sciences and Humanities, SSM College of Engineering & Technology Pattan, Jammu and Kashmir 2 Department of Mathematics, Faculty of Science Jazan University, Jazan-Saudi Arabia 3,4 Department of Mathematics, National Institute of Technology Srinagar, , India 5 Jammu Kashmir Institute of Math. Sciences, Kashmir * Author to whom correspondence should be addressed; ashamidg@rediffmail.com, profmobin@yahoo.com, neyaznit@yahoo.co.in, tanweerjalal@yahoo.co.uk and sameergupkari@rediffmail.com Article history: Received 19 January 2016; Received in revised form 21 February 2016; Accepted 2 March 2016; Published 8 March Abstract: In the present paper, we introduce the spaces bv 0 (u, p) and bv c (u, p). We show their completeness property and prove that the spaces bv 0 (u, p) and bv c (u, p) are respectively linearly isomorphic to the spaces c 0 (p) and c(p), where 0 < p k H = sup x k <. k Keywords: Sequence space of non-absolute type; difference sequence space; infinite matrices. Mathematics Subject Classification Code (2010): 46A45; 46CO5, 46B Introduction, Preliminary and Background We denote the set of all sequences (real or complex) by ω. Any subspace of ω is called the sequence space. So the sequence space is the set of scalar sequences (real of complex) which is closed under co-ordinate wise addition and scalar multiplication. Throughout the paper N, R and C denotes the set of non-negative integers, the set of real numbers and the set of complex numbers, respectively.

2 117 Let l, c and c 0, respectively, denotes the space of all bounded sequences, the space of convergent sequences and the sequences converging to zero. Also, by l(p), we denote the space of all p-absolutely convergent series [1, 4, 7, 8, 10, 22, 23]. Throughout the text, for terms with suffix negative will be taken as zero. Let X be a real or complex linear space, h be a function from X to R. Then, the pair (X, h) is called a paranormed space and h is a paranorm for X, if the following axioms are satisfied for all elements x, y X and for all scalars α:

3 Main Results

4 119

5 The Basis for the Spaces bv 0 (u, p) and bv c (u, p) In the present section, we give two sequences of points of the spaces bv 0 (u, p) and bv c (u, p) which form the basis for these spaces.

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