Hindawi Publishing Corporation
International Journal of Analysis
Volume 2013, Article ID 126163, 7 pages
http://dx.doi.org/10.1155/2013/126163
Research Article
On a New I -Convergent Double-Sequence Space
Vakeel A. Khan and Nazneen Khan
Department of Mathematics, A.M.U., Aligarh 202002, India
Correspondence should be addressed to Vakeel A. Khan; vakhan@math.com
Received 26 November 2012; Revised 17 January 2013; Accepted 18 January 2013
Academic Editor: Wen Xiu Ma
Copyright © 2013 V. A. Khan and N. Khan. his 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.
he sequence space BV
was introduced and studied by Mursaleen (1983) . In this article we introduce the sequence space
2
BV
and study some of its properties and inclusion relations.
1. Introduction and Preliminaries
Let N, R, and C be the sets of all natural, real, and complex
numbers, respectively. We write
={=(
):
∈ C}, (1)
showing the space of all real or complex sequences.
Deinition 1. A double sequence of complex numbers is
deined as a function : N × N → C. We denote a double
sequence as (
) where the two subscripts run through the
sequence of natural numbers independent of each other [1].
A number ∈ C is called a double limit of a double sequence
(
) if for every >0 there exists some =()∈ N such
that
(
)−
<, ∀,≥, (2)
(see [2]).
Let
∞
and denote the Banach spaces of bounded
and convergent sequences, respectively, with norm ‖‖
∞
=
sup
|
|. Let V denote the space of sequences of bounded
variation; that is,
V ={=(
):
∞
∑
=0
−
−1
<∞,
−1
=0}, (3)
where V is a Banach space normed by
‖‖=
∞
∑
=0
−
−1
, (4)
(see [3]).
Deinition 2. Let be a mapping of the set of the positive
integers into itself having no inite orbits. A continuous linear
functional on
∞
is said to be an invariant mean or -mean
if and only if
(i) () ≥ 0 when the sequence =(
) has
≥0 for
all ;
(ii) ()=1, where ={1,1,1,...};
(iii) (
()
)=() for all ∈
∞
.
In case is the translation mapping →+1,a -
mean is oten called a Banach limit (see [4]), and
, the set
of bounded sequences all of whose invariant means are equal,
is the set of almost convergent sequences (see [5]).
If = (
), then = (
) = (
()
). hen it can be
shown that
={=(
):
∞
∑
=1
,
()= uniformly in ,
=− lim },
(5)