Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2009, Article ID 696971, 12 pages
doi:10.1155/2009/696971
Research Article
Some p-Type New Sequence Spaces and
Their Geometric Properties
Ekrem Savaş,1 Vatan Karakaya,2 and Necip Şimşek3
1
Department of Mathematics, İstanbul Commerce University, Uskudar 36472, İstanbul, Turkey
Department of Mathematical Engineering, Yildiz Technical University, Davutpasa Campus,
34210, Esenler, İstanbul, Turkey
3
Department of Mathematics, Faculty of Arts and Science, Adıyaman University, 02040, Adıyaman, Turkey
2
Correspondence should be addressed to Ekrem Savaş, ekremsavas@yahoo.com
Received 17 March 2009; Accepted 4 August 2009
Recommended by Agacik Zafer
We introduce an p-type new sequence space and investigate its some topological properties
including AK and AD properties. Besides, we examine some geometric properties of this space
concerning Banach-Saks type p and Gurarii’s modulus of convexity.
Copyright q 2009 Ekrem Savaş et al. This 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.
1. Introduction
In general, the p-type spaces have many useful applications because of the properties of
the spaces p and p . In 1, it was shown that the subspaces of Orlicz spaces, which have
rich geometric properties, are isomorphic to the space p . Also since the space p is reflexive
and convex, it is natural to consider the geometric structure of these spaces.
Recently there has been a lot of interest in investigating geometric properties of
sequence spaces besides topological and some other usual properties. In literature, there are
many papers concerning the geometric properties of different sequence spaces. For example;
in 2, Mursaleen et al. studied some geometric properties of normed Euler sequence space.
Sanhan and Suantai 3 investigated the geometric properties of Cesáro sequence space cesp
equipped with Luxemburg norm. Further information on geometric properties of sequence
spaces can be found in 4–7.
The main purpose of our work is to introduce an p -type new sequence space together
with matrix domain and its summability methods. Also we investigate some topological
properties of this new space as the paranorm, AK and AD properties, and furthermore
characterize geometric properties concerning Banach-Saks type p and Gurarii’s modulus of
convexity.
2
Abstract and Applied Analysis
2. Preliminaries and Notations
Let w be the space of all real-valued sequences. Each linear subspace of w is called a sequence
space denoted by λ. We denote by 1 and p absolutely and p-absolutely convergent series,
respectively.
A sequence space λ with a linear topology is called a K-space provided that each of
the maps pi : λ → C defined by pi x xi is continuous for all i ∈ N, where C denotes the
complex field, and N {0, 1, 2, 3, . . .}. A K-space λ is called FK-space provided that λ is a
complete linear metric space. An FK-space whom topology is normable is called BK-space.
An FK-space λ is said to have AK property, if φ ⊂ λ and {ek } is a basis for λ, where ek is a
sequence whose only nonzero term is 1, kth place for each k ∈ N, and φ span{ek }, the set
of all AD-space, thus AK implies AD.
A linear topological space X over the real field R is said to be a paranormed space if
there is a subadditivity function g : X → R such that gθ 0, g−x gx and scalar
multiplication is continuous. It is well known that the space p is AK-space where 1 ≤ p < ∞.
Throughout this work, we suppose that pk is a bounded sequence of strictly positive
real numbers with sup pk H and M max{1, H}. Also the summation without limits runs
from 0 to ∞. In 8, the linear space p was defined by Maddox see also Simons 9 and
Nakano 10 as follows:
p x xk ∈ w :
|xn |pn < ∞
2.1
n
which is a complete space paranormed by
1/M
pn
gx .
|xn |
2.2
n
Let λ, μ be any two sequence spaces, and let A ank be an infinite matrix of real
numbers ank , where n, k ∈ N. Then we write Ax Axn , the A-transform of x, if Axn k ank xk converges for each n ∈ N.
A matrix A ank is called a triangle if ank 0 for k > n and ann / 0 for all n ∈ N. It
is trivial that ABx ABx holds for the triangle matrices A, B and a sequence x. Further,
a triangle matrix P uniquely has an invert P −1 Q which is also a triangle matrix. Then if
P x y,
x P Qx QP x,
x Qy
2.3
hold for all x ∈ w.
By λ, μ, we denote the class of all infinite matrices A such that A : λ → μ. The
matrix domain λA of an infinite matrix A in a sequence space λ is defined by λA {x xk ∈ w : Ax ∈ λ} which is a sequence space. It is well known that the new sequence space
λA generated by the limitation matrix A from a sequence space λ is the expansion or the
contraction of original space λ.
If A is triangle, then one can easily observe that the sequence spaces λA and λ are
λ. Let λ be a sequence space. Then the continuous dual λA
linearly isomorphic, that is, λA Abstract and Applied Analysis
3
of the space λA is defined by λA {f : f g ◦ A; g ∈ λ }. Let X be a seminormed space. A
set Y ⊂ X is called fundamental set if the span of Y is dense in X. An application of HahnBanach theorem on fundamental set is as follows: if Y is the subset of a seminormed space X
and fY 0 implies f 0 for f ∈ X , then Y is a fundamental set see 11.
By the idea mentioned above, let us give the definitions of some matrices to construct
a new sequence space in sequel to this work. We denote Δ δnk and S snk by
δnk ⎧
⎨−1n−k ,
if n − 1 ≤ k ≤ n,
⎩0,
otherwise,
snk ⎧
⎨1,
if 0 ≤ k ≤ n,
⎩0,
otherwise.
2.4
Malkowsky and Savas 12, Choudhary and Mishra 13, and Altay and Basar 14 have
defined the sequence spaces Zu, v; X, p, and u, v; p, respectively. By using the
matrix domain, the spaces Zu, v; X, p, and u, v; p may be redefined by Zu, v; X XGu,v , p p S , and u, v; p pGu,v , respectively.
If λ ⊂ w is a sequence space and x xk ∈ λ, Sx-transform with 2.4 corresponds
to nth partial sum of the series n xn and it is denoted by s sn .
By using 2.4 and any infinite lower triangular matrix A, we can define two infinite
as follows: A AS and A
ΔA. Let x xk be a
lower triangular matrices A and A
sequence in λ. By considering the multiplication of infinite lower triangular matrices, we
have ASx Ax, that is,
tn n
n
anv xv .
anv sv v0
2.5
v0
ΔA, we have Ax
ΔAx, that is,
Also since A
tn − tn−1 n
anv xv .
2.6
v0
Now let us write the following equality:
n
zn Ax
anv xv .
n
2.7
v0
yn Ax
n Ay
n and
It can be seen that for any sequences x, y and scalar α ∈ R, Ax
Aαx
n αAxn . We now define new sequence space as follows:
p A;
pn
x xk ∈ w :
Ax
<∞ .
n
n
2.8
For some special cases of the infinite lower triangular matrix A and the sequence pk , we
obtain the following spaces.
4
Abstract and Applied Analysis
p reduces to the normed space p A
denoted
i If pk p for all k ∈ N, the space A;
by
p
n
x xk ∈ w :
p A
anv xv < ∞ .
n v0
2.9
p corresponds to
ii If A C, 1, which is Cesáro matrix order 1, then the space A;
the space C; p denoted by
p C;
pn
x xk ∈ w :
Cx < ∞ ,
n
n
2.10
n 1/nn 1 n kxk for n ≥ 1 and Cx
0 x0 .
where Cx
k1
p reduces to
iii If A N, pn , which is Nörlund type matrix, then the space A;
p |N, pn |r see 15, 16 denoted by
the space N;
pn
x xk ∈ w :
Nx
<∞ ,
p N;
n
n
2.11
n pn /Pn Pn−1 n Pk−1 xk for n ≥ 1 and Nx
0 x0 .
where Nx
k1
p |N, pn |r reduce to
Also if pk p for all k ∈ N, then the spaces C; p and N;
and p N
|N p | see 17, respectively.
the spaces p C
Now let us introduce some definitions of geometric properties of sequence spaces.
Let X, · be a normed linear space, and let SX and BX be the unit sphere and
unit ball of X for the brevity X X, · , respectively. Consider Clarkson’s modulus of
convexity Clarkson 18 and Day 19 defined by
x y
; x, y ∈ SX, x − y ε ,
δX ε inf 1 −
2
2.12
where 0 ≤ ε ≤ 2. The inequality δX ε > 0 for all ε ∈ 0, 2 characterizes the uniformly convex
spaces.
In 20, Gurariı̆’s modulus of convexity is defined by
βX ε inf 1 − inf αx 1 − αy ; x, y ∈ SX, x − y ε ,
α∈0,1
2.13
where 0 ≤ ε ≤ 2. It is easily shown that δX ε ≤ βX ε ≤ 2δX ε for any 0 ≤ ε ≤ 2. Also if
0 < βX ε < 1, then X is uniformly convex, and if βX ε < 1, then X is strictly convex.
Abstract and Applied Analysis
5
A Banach space X is said to have the Banach-Saks property if every bounded sequence
xn in X admits a subsequence zn such that the sequence {tk z} is convergent in the norm
in X see 21, where
tk z 1
z0 z1 z2 · · · zk k1
2.14
k ∈ N.
Let 1 < p < ∞. A Banach space is said to have the Banach-Saks type p or property BSp ,
if every weakly null sequence xk has a subsequence xkl such that for some C > 0
n
xkl < Cn 11/p
l0 2.15
for all n ∈ N see 22.
p
3. Some Topological Properties of the Space A;
p as
In this section, we investigate some topological properties of the sequence space A;
the paranorm AK property and AD property. Let us begin the following theorem.
p is complete linear metric space with respect to the paranorm
Theorem 3.1. i The space A;
defined by
hx |Axn |pn
1/M
3.1
.
n
is a Banach space normed
ii If the sequence pn is constant sequence and p ≥ 1, then p A
by
1/p
z
p x
p A
n
p
3.2
.
|Axn |
Proof. The proof of ii is routine verification by using standard techniques and hence it is
omitted.
p with respect to the coordinatewise
The proof of i is that the linearity of A;
addition and scalar multiplication follows from the following inequalities which are satisfied
p:
for x, y ∈ A;
pn
xy
A
n
n
1/M
≤
pn
Ax
n
n
1/M
pn
Ay
n
n
1/M
3.3
and |α|pn ≤ max{1, |α|M } for any α ∈ R see 23. After this step, we must show that the space
p holds the paranorm property and the completeness with respect to given paranorm.
A;
6
Abstract and Applied Analysis
p. Besides, from 3.3 we
It is easy to show that hθ 0, and hx h−x for all x ∈ A;
p. To complete the paranorm conditions
obtain hx y ≤ hx hy for all x, y ∈ A;
for the space A; p, it remains to show the continuity of the scalar multiplication. Let xm p such that hxm − x → 0, and let αm be also any sequence of
be any sequence in A;
scalars such that |αm − α| → 0 m → ∞. From subadditivity of h, we give the inequality
hxm ≤ hx hxm − x. Hence {hxm } is bounded and we have
m
hαm x − αx pn 1/M
n
n
m
m
αm − α anv xv α anv xv − xv n
v0
v0
3.4
which tends to zero as m → ∞. Consequently we obtain that h is a paranorm over the space
p. To prove the completeness of the space A;
p, let us take any Cauchy sequence xi A;
in the space A; p. Then for a given ε > 0, there exists a positive integer n0 ε such that
hxi − xj < ε for all i, j ≥ n0 ε. By using the definitions of the Cauchy sequence and the
paranorm, we have, for each fixed n,
i
j ≤
− Ax
Ax
n
n
pn 1/M
i
j
− A x <ε
A x
n
n
n
3.5
0 n , Ax
1 n , Ax
2 n , . . .} is a
for every i, j ≥ n0 ε. Hence we obtain that the sequence {Ax
Cauchy sequence of real numbers for every fixed n ∈ N. Since R is complete, it converges,
n as j → ∞, where {Ax
n } {Ax
0 , Ax
1 , Ax
2 , . . .}. Now let us
j n → Ax
that is, Ax
m
pn
i
j
M
choose m ∈ N such that n0 |Ax n − Ax n | < ε for each m ∈ N and i, j ≥ n0 ε. By
taking j → ∞ and for every i ≥ n0 ε, we get
m pn
i
− Ax
Ax
< εM .
n
n0
3.6
n
Again taking m → ∞ and for every i ≥ n0 ε, it is obtained that hxi − x < ε. We write the
following equality:
i − Ax
i .
Ax
Ax Ax
n
n
n
n
3.7
By using 3.7 and Minkowski’s inequality, we get
pn
Ax
n
n
1/M
≤ h xi h xi − x
3.8
p. It follows xi → x as i → ∞. Consequently, since xi is any
which implies x ∈ A;
p is complete. This completes the proof.
Cauchy sequence, we obtain that the space A;
p is linearly isomorphic to the space p.
Theorem 3.2. The space A;
Abstract and Applied Analysis
7
p and p such that x → z Ax.
Proof. Let us define A-transform
between the spaces A;
is linear, injective and surjective. The linearity of
We have to show that the transformation A
θ. For the surjective
A is obvious. Moreover it is injective because of x θ whenever Ax
n . We
property, let y ∈ p. From 2.3 and 2.7, there exists a matrix B such that xn By
have
pn 1/M
pn 1/M
yn By
g y < ∞.
hx A
n
n
3.9
n
is surjective. Consequently, the spaces A;
p and
Hence we obtain that the transformation A
p are linearly isomorphic spaces.
has AD property.
Theorem 3.3. The space p A
. Then fx gAx
for some g ∈ p . Since p has AK property and
Proof. Let f ∈ p A
∼
p q where 1/p 1/q 1,
fx an Ax
n
3.10
n
∼
for some a an ∈ q . Also since p A
p and the inclusion φ ⊂ p holds, we have
k
φ ⊂ p A. For any f ∈ p A and e ∈ φ, we have
k Ha
f ek an Ae
,
n
n
k
3.11
is dense
is transpose of the matrix A.
Hence from Hahn-Banach theorem, φ ⊂ p A
where H
if and only if Ha
θ for a ∈ q implies a θ. Besides, since the null space of the
in p A
has AD property. Hence the proof is completed.
operator H on w is {θ}, p A
4. Some Geometric Properties of the space p A
In this section, we give some geometric properties for the space p A.
has the Banach-Saks of type p.
Theorem 4.1. The space p A
Proof. Let εn be a sequence of positive numbers for which ∞
n1 εn ≤ 1/2. Let xn be a
Set z0 x0 0 and z1 xn1 x1 . Then there exists s1 ∈ N
weakly null sequence in Bp A.
8
Abstract and Applied Analysis
such that
∞
i z1 ie is 1
< ε1.
p A
1
4.1
Since xn is a weakly null sequence implies that xn → 0 with respect to the coordinatwise,
there is an n2 ∈ N such that
s
1
i xn ie i0
< ε1 ,
p A
4.2
where n ≥ n2 . Set z2 xn2 . Then there exists an s2 > s1 such that
∞
z2 iei is 1
< ε2 .
p A
2
4.3
By using the fact that xn → 0 coordinatwise, there exists an n3 > n2 such that
s
2
i xn ie i0
< ε2 ,
p A
4.4
where n ≥ n3 .
If we continue this process, we can find two increasing subsequences si and ni such
that
s
j
i xn ie i0
< εj
4.5
< εj ,
4.6
p A
for each n ≥ nj1 and
∞
i zj ie isj 1
p A
Abstract and Applied Analysis
9
where zj xnj . Hence,
n zj j0 p A
⎛
⎞
sj−1
sj
∞
n i
i
i ⎠
⎝
zj ie zj ie zj ie
j0 i0
isj−1 1
isj 1
⎛
⎞
sj
n
i ⎠
⎝
≤
z
ie
j
j0 isj−1 1
p A
⎛
⎞
∞
n
i ⎠
⎝
zj ie
j0 isj 1
⎛
⎞
sj
n
i ⎠
⎝
≤
z
ie
j
j0 isj−1 1
sj−1
n i z
ie
j
j0 i0
p A
n
2 εj .
j0
and x
On the other hand, since xn ∈ Bp A
p A
p
p A
4.7
p A
p A
that x
p A
< 1. Therefore x
p A
∞ i
p 1/p
iv xv | , it can be seen
i0 |
v0 a
< 1. We have
⎛
⎞p
sj
n
i ⎠
⎝
z
ie
j
j0 isj−1 1
p A
p
sj
n
i
aiv zj v
j0 is 1 v0
j−1
p
n i
∞ ≤
aiv zj v
j0 i0 v0
4.8
≤ n 1.
Hence we obtain
⎛
⎞
sj
n
i
⎝
⎠
z
ie
j
j0 isj−1 1
≤ n 11/p .
4.9
p A
By using the fact 1 ≤ n 11/p for all n ∈ N and 1 ≤ p < ∞, we have
n zj j0 ≤ n 11/p 1 ≤ 2n 11/p .
p A
has the Banach-Saks type p. This completes the proof.
Hence p A
4.10
10
Abstract and Applied Analysis
is
Theorem 4.2. Gurarii’s modulus of convexity for the normed space p A
ε p 1/p
βp A ε ≤ 1 − 1 −
,
2
4.11
where 0 ≤ ε ≤ 2.
Then we have
Proof. We have x ∈ p A.
x
p A
Ax
lp
p
Ax
n
n
1/p
.
4.12
Let 0 ≤ ε ≤ 2 and consider the following sequences:
ε
ε p 1/p
, B
x xn B
, 0, 0, . . . ,
1−
2
2
ε
ε p 1/p
, B − , 0, 0, . . . ,
y yn B
1−
2
2
4.13
Since zn Ax
n and tn Ay
n , we have
where B is the inverse of the matrix A.
ε p 1/p ε , 0, 0, . . . ,
,
1−
2
2
z zn t tn ε p 1/p ε , − , 0, 0, . . . .
1−
2
2
By using sequences given above, we obtain the following equalities:
p
x
p A
p ε p 1/p p ε p
Ax 1 −
lp
2
2
1−
1,
ε p
2
ε p
2
4.14
Abstract and Applied Analysis
p
y
p A
11
p ε p 1/p p ε p
Ay 1 −
− lp
2
2
1−
ε p
2
ε p
2
1,
x − y − Ay
Ax
p A
lp
ε p 1/p ε p 1/p p ε ε p 1/p
1−
− 1−
− − 2
2
2
2
ε.
4.15
To complete the conditions of βp A
ε for Gurarii’s modulus of convexity, it remains to show
the infimum of αx 1 − αt
p A
for 0 ≤ α ≤ 1. We have
inf αx 1 − αyp A
0≤α≤1
1 − αAy
inf αAx
0≤α≤1
lp
ε p 1/p
ε p 1/p
ε p 1/p p ε 1 − α − inf α 1 −
1 − α 1 −
α
0≤α≤1 2
2
2
2
p 1/p
ε p
p ε
inf 1 −
|2α − 1|
0≤α≤1
2
2
ε p 1/p
1−
.
2
4.16
Consequently we get for p ≥ 1
ε p 1/p
βp A
.
ε ≤ 1 − 1 −
2
This is the desired result. Hence the proof is completed.
Corollary 4.3. i If ε 2, then βp A
ε ≤ 1 and hence p A is strictly convex.
is uniformly convex.
ii If 0 < ε < 2, then 0 < β ε < 1 and hence p A
p A
Corollary 4.4. If α 1/2, then δp A
ε βp A
ε.
4.17
12
Abstract and Applied Analysis
References
1 P. K. Kamthan and M. Gupta, Sequence Spaces and Series, vol. 65 of Lecture Notes in Pure and Applied
Mathematics, Marcel Dekker, New York, NY, USA, 1981.
2 M. Mursaleen, F. Başar, and B. Altay, “On the Euler sequence spaces which include the spaces p and
∞ . II,” Nonlinear Analysis: Theory, Methods & Applications, vol. 65, no. 3, pp. 707–717, 2006.
3 W. Sanhan and S. Suantai, “Some geometric properties of Cesàro sequence space,” Kyungpook
Mathematical Journal, vol. 43, no. 2, pp. 191–197, 2003.
4 Y. Cui, C. Meng, and R. Płuciennik, “Banach-Saks property and property β in Cesàro sequence
spaces,” Southeast Asian Bulletin of Mathematics, vol. 24, no. 2, pp. 201–210, 2000.
5 V. Karakaya, “Some geometric properties of sequence spaces involving lacunary sequence,” Journal of
Inequalities and Applications, vol. 2007, Article ID 81028, 8 pages, 2007.
6 S. Suantai, “On the H-property of some Banach sequence spaces,” Archivum Mathematicum, vol. 39,
no. 4, pp. 309–316, 2003.
7 N. Şimşek and V. Karakaya, “On some geometrical properties of generalized modular spaces of
Cesàro type defined by weighted means,” Journal of Inequalities and Applications, vol. 2009, Article
ID 932734, 13 pages, 2009.
8 I. J. Maddox, “Spaces of strongly summable sequences,” The Quarterly Journal of Mathematics. Oxford.
Second Series, vol. 18, pp. 345–355, 1967.
9 S. Simons, “The sequence spaces lpv and mpv ,” Proceedings of the London Mathematical Society, vol.
15, no. 1, pp. 422–436, 1965.
10 H. Nakano, “Modulared sequence spaces,” Proceedings of the Japan Academy, vol. 27, pp. 508–512, 1951.
11 A. Wilansky, Summability through Functional Analysis, vol. 85 of North-Holland Mathematics Studies,
North-Holland, Amsterdam, The Netherlands, 1984.
12 E. Malkowsky and E. Savas, “Matrix transformations between sequence spaces of generalized
weighted means,” Applied Mathematics and Computation, vol. 147, no. 2, pp. 333–345, 2004.
13 B. Choudhary and S. K. Mishra, “On Köthe-Toeplitz duals of certain sequence spaces and their matrix
transformations,” Indian Journal of Pure and Applied Mathematics, vol. 24, no. 5, pp. 291–301, 1993.
14 B. Altay and F. Başar, “Generalization of the sequence space p derived by weighted mean,” Journal
of Mathematical Analysis and Applications, vol. 330, no. 1, pp. 174–185, 2007.
15 V. K. Bhardwaj and N. Singh, “Some sequence spaces defined by |N, pn | summability,” Demonstratio
Mathematica, vol. 32, no. 3, pp. 539–546, 1999.
16 V. K. Bhardwaj and N. Singh, “Some sequence spaces defined by |N, pn | summability and a modulus
function,” Indian Journal of Pure and Applied Mathematics, vol. 32, no. 12, pp. 1789–1801, 2001.
17 D. G. Bourgin, “Linear topological spaces,” American Journal of Mathematics, vol. 65, pp. 637–659, 1943.
18 J. A. Clarkson, “Uniformly convex spaces,” Transactions of the American Mathematical Society, vol. 40,
no. 3, pp. 396–414, 1936.
19 M. M. Day, “Uniform convexity in factor and conjugate spaces,” Annals of Mathematics, vol. 45, pp.
375–385, 1944.
20 V. I. Gurariı̆, “Differential properties of the convexity moduli of Banach spaces,” Matematicheskie
Issledovaniya, vol. 2, pp. 141–148, 1967.
21 J. Diestel, Sequences and Series in Banach Spaces, vol. 92 of Graduate Texts in Mathematics, Springer, New
York, NY, USA, 1984.
22 H. Knaust, “Orlicz sequence spaces of Banach-Saks type,” Archiv der Mathematik, vol. 59, no. 6, pp.
562–565, 1992.
23 I. J. Maddox, “Paranormed sequence spaces generated by infinite matrices,” Proceedings of the
Cambridge Philosophical Society, vol. 64, pp. 335–340, 1968.