Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 783731, 8 pages
http://dx.doi.org/10.1155/2013/783731
Research Article
Almost Sequence Spaces Derived by the Domain of the Matrix
Ali Karaisa and ÜmJt KarabJyJk
Department of Mathematics-Computer Science, Faculty of Sciences, Necmettin Erbakan University, Meram Yerles ¸kesi, Meram,
42090 Konya, Turkey
Correspondence should be addressed to Ali Karaisa; alikaraisa@hotmail.com
Received 9 May 2013; Revised 26 August 2013; Accepted 26 September 2013
Academic Editor: Feyzi Bas ¸ar
Copyright © 2013 A. Karaisa and
¨
U. Karabıyık. 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.
By using
, we introduce the sequence spaces
,
0
, and
of normed space and -space and prove that
,
0
, and
are
linearly isomorphic to the sequence spaces ,
0
, and , respectively. Further, we give some inclusion relations concerning the
spaces
,
0
, and the nonexistence of Schauder basis of the spaces and
is shown. Finally, we determine the - and -duals
of the spaces
and
. Furthermore, the characterization of certain matrix classes on new almost convergent sequence and series
spaces has exhaustively been examined.
1. Preliminaries, Background and Notation
By , we will denote the space of all real or complex valued
sequences. Any vector subspace of is called sequence space.
We will write ℓ
∞
,
0
, , and ℓ
for the spaces of all bounded,
null, convergent, and absolutely -summable sequences, re-
spectively, which are -space with the usual sup-norm de-
ined by ‖‖
∞
= sup
|
| and ‖‖
ℓ
= (∑
|
|
)
1/
, for 1<
<∞, where, here and in what follows, the summation with-
out limits runs from 0 to ∞. Further, we will write , for
the spaces of all sequences associated with bounded and con-
vergent series, respectively, which are -spaces with their
natural norm [1].
Let and be two sequence spaces and = (
) an
ininite matrix of real or complex numbers
, where ,∈
N. hen, we say that deines a matrix mapping from into
and we denote it by writing that :→ and if for
every sequence =(
)∈ the sequence = ()
, the
-transform of is in , where
()
=∑
, (∈ N).
(1)
he notation ( : ) denotes the class of all matrices
such that :→. hus, ∈(:) if and only if the series
on the right hand side of (1) converges for each ∈ N and
every ∈ and we have = {()
}
∈N
∈ for all ∈.
he matrix domain
of an ininite matrix in a sequence
space is deined by
={=(
)∈:∈}. (2)
he approach constructing a new sequence space by
means of the matrix domain of a particular triangle has re-
cently been employed by several authors in many research
papers. For example, they introduced the sequence spaces
()
1
= in [2], (ℓ
)
=
and (ℓ
∞
)
=
∞
in [3],
=
(, V;) in [4], (
0
)
Λ
=
0
and
Λ
=
in [5], and (ℓ
)
=
and (ℓ
∞
)
=
∞
in [6]. Recently, matrix domains of the gen-
eralized diference matrix (,) and triple band matrix
(,,) in the sets of almost null and almost convergent se-
quences have been investigated by Bas ¸ar and Kiris ¸c ¸i [7] and
S¨ onmez [8], respectively. Later, Kayaduman and S ¸eng¨ on¨ ul
introduced some almost convergent spaces which are the
matrix domains of the Riesz matrix and Ces` aro matrix of
order 1 in the sets of almost null and almost convergent
sequences (see [9, 10]).
We now focus on the sets of almost convergent sequences.
A continuous linear functional on ℓ
∞
is called a Banach
limit if (i) () ⩾ 0 for =(
) and
⩾0 for every , (ii)
(
()
) = (
), where is shit operator which is deined
on by ()=+1, and (iii) ()=1, where =(1,1,1,...).
A sequence =(
)∈ℓ
∞
is said to be almost convergent to