Research Article
On Generalized Difference Hahn Sequence Spaces
Kuldip Raj
1
and Adem Kiliçman
2
1
School of Mathematics, Shri Mata Vaishno Devi University, Katra 182320, India
2
Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia (UPM),
43400 Serdang, Selangor, Malaysia
Correspondence should be addressed to Adem Kilic¸man; akilic@upm.edu.my
Received 17 March 2014; Accepted 2 May 2014; Published 13 May 2014
Academic Editor: S. A. Mohiuddine
Copyright © 2014 K. Raj and A. Kilic¸man. is is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
We construct some generalized difference Hahn sequence spaces by mean of sequence of modulus functions. e topological
properties and some inclusion relations of spaces ℎ
(F, , Δ
) are investigated. Also we compute the dual of these spaces, and
some matrix transformations are characterized.
1. Introduction and Preliminaries
By a sequence space, we understand a linear subspace of
the space = C
N
of all real or complex-valued sequences,
where C denotes the complex field and N = 0, 1, 2, . . . . For
= (
) ∈ , we write
∞
, , and
0
for the classical
spaces of all bounded, convergent, and null sequences,
respectively. Also by , , and
we denote the space of
all bounded, convergent, and -absolutely convergent series,
which are Banach spaces with the following norms: ‖‖
=
‖‖
= sup
|∑
=1
| and ‖‖
= (∑
|
|
)
1/
, respectively.
Additionally, the spaces V
and ∫ are defined by
V
= { = (
)∈:
∞
∑
=1
−
−1
< ∞} ,
∫ = { = (
) ∈ : (
) ∈ } .
(1)
A coordinate space (or a -space) is a vector space of numer-
ical sequences, where addition and scalar multiplication are
defined pointwise. at is, a sequence space with a linear
topology is called a -space provided that each of the maps
: → C defined by
() =
is continuous for all
∈ N.A -space is a -space, which is also a Banach space
with continuous coordinate functionals
() =
, ( =
1, 2, . . .).A -space is called an -space provided that is
a complete linear metric space. An -space whose topology
is normable is called a -space. If a normed sequence space
contains a sequence (
) with the property that for every
∈ there is a unique sequence of scalars (
) such that
lim
→∞
− (
0
0
+
1
1
+⋅⋅⋅+
)
= 0,
(2)
then (
) is called Schauder basis (or briefly basis) for . e
series ∑
which has the sum is then called the expansion
of with respect to (
) and written as =∑
. An -
space is said to have property, if ⊂ and {
} is a
basis for , where
is a sequence whose only nonzero term
is 1 in th place for each ∈ N and = span{
}, the set of all
finitely nonzero sequences. If is dense in , then is called
an -space, and thus implies .
e notion of difference sequence spaces was introduced
by Kizmaz [1], who defined the sequence spaces as follows:
(Δ) = { = (
) ∈ : (Δ
) ∈ }
for = ,
0
,
∞
,
(3)
where Δ = (Δ
) = (
−
+1
). e notion was further
generalized by Et and C¸ olak [2] by introducing the spaces.
Let be a nonnegative integer; then,
(Δ
) = { = (
) ∈ : (Δ
) ∈ }
for = ,
0
,
∞
,
(4)
Hindawi Publishing Corporation
e Scientific World Journal
Volume 2014, Article ID 398203, 7 pages
http://dx.doi.org/10.1155/2014/398203