Some l(p)-type new sequence spaces and their geometric properties

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Volume 2009, Article ID 696971,12pages doi:10.1155/2009/696971

Research Article

Some p-Type New Sequence Spaces and Their Geometric Properties

Ekrem Savas¸,

¹

Vatan Karakaya,

²

and Necip S¸ims¸ek

³

1Department of Mathematics, ˙Istanbul Commerce University, Uskudar 36472, ˙Istanbul, Turkey

2Department of Mathematical Engineering, Yildiz Technical University, Davutpasa Campus, 34210, Esenler, ˙Istanbul, Turkey

3Department of Mathematics, Faculty of Arts and Science, Adıyaman University, 02040, Adıyaman, Turkey

Correspondence should be addressed to Ekrem Savas¸,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.

Copyrightq 2009 Ekrem Savas¸ 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. In1, it was shown that the subspaces of Orlicz spaces, which have rich geometric properties, are isomorphic to the space p. Also since the space _pis 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;

in2, Mursaleen et al. studied some geometric properties of normed Euler sequence space.

Sanhan and Suantai3 investigated the geometric properties of Ces´aro sequence space cesp

equipped with Luxemburg norm. Further information on geometric properties of sequence spaces can be found in4–7.

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.

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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 _pabsolutely 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 p_i : λ → C defined by pix xiis continuous for all i ∈ N, where C denotes the complex field, andN {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 {e^k} is a basis for λ, where e^kis a sequence whose only nonzero term is 1, kth place for each k∈ N, and φ span{e^k}, the set of all AD-space, thus AK implies AD.

A linear topological space X over the real fieldR 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 pis AK-space where 1≤ p < ∞.

Throughout this work, we suppose thatpk 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 Nakano10 as follows:

p

x xk ∈ w :

n |xn|^pⁿ < ∞

2.1

which is a complete space paranormed by

gx 

n |xn|^pⁿ

1/M

. 2.2

Let λ, μ be any two sequence spaces, and let A  ank be an infinite matrix of real numbers a_nk, where n, k∈ N. Then we write Ax Ax_n, the A-transform of x, if Ax_n

kankxkconverges 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 Px y,

x PQx QPx, 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 λ_A 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 linearly isomorphic, that is, λ_Aλ. Let λ be a sequence space. Then the continuous dual λ_A

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of the space λAis defined by λ_A  {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 Hahn- Banach 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 setsee 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 denoteΔ δnk and S snk by

δnk

⎧⎨

⎩

−1^n−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  X_Gu,v, p p_S, and u, v; p p_Gu,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

nx_nand it is denoted by s sn.

By using2.4 and any infinite lower triangular matrix A, we can define two infinite lower triangular matrices A and A as follows: A AS and A ΔA. Let x xk be a sequence in λ. By considering the multiplication of infinite lower triangular matrices, we have ASx Ax, that is,

tnⁿ

v0

anvsvⁿ

v0

anvxv. 2.5

Also since A ΔA, we have Ax ΔAx, that is,

t_n− t_n−1ⁿ

v0

anvx_v. 2.6

Now let us write the following equality:

z_n Ax

nⁿ

v0

anvx_v. 2.7

It can be seen that for any sequences x, y and scalar α∈ R, Ax y_n Ax_n Ay_nand

Aαx_n α Ax_n. We now define new sequence space as follows:

A; p

x xk ∈ w :

n

Ax

n

^pⁿ < ∞

. 2.8

For some special cases of the infinite lower triangular matrix A and the sequencepk, we obtain the following spaces.

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i If pk p for all k ∈ N, the space A; p reduces to the normed space p A denoted by

_p

A

x xk ∈ w :

n

n v0

anvx_v

p

< ∞

. 2.9

ii If A C, 1, which is Ces´aro matrix order 1, then the space A; p corresponds to the space C; p denoted by

C; p

x xk ∈ w :

n

Cx

n

^pⁿ < ∞

, 2.10

where Cx_n 1/nn 1 _n

k1kxkfor n≥ 1 and Cx₀ x0.

iii If A N, pn, which is N¨orlund type matrix, then the space A; p reduces to the space N; p |N, pn|r see 15,16 denoted by

N; p

x xk ∈ w :

n

Nx

n

^pⁿ < ∞

, 2.11

whereNx_n  pn/PnP_n−1 _n

k1P_k−1xkfor n≥ 1 and Nx₀ x0.

Also if p_k  p for all k ∈ N, then the spaces C; p and N; p |N, pn|r reduce to the spaces p C and pN |Np| see 17, respectively.

Now let us introduce some definitions of geometric properties of sequence spaces.

Let unit ball of X

convexityClarkson 18 and Day 19 defined by

δ_Xε inf

1−x y

2 ; x, y∈ SX,x − y ε

, 2.12

where 0≤ ε ≤ 2. The inequality δXε > 0 for all ε ∈ 0, 2 characterizes the uniformly convex spaces.

In20, Gurari˘ı’s modulus of convexity is defined by

βXε inf

1− inf

α∈0,1αx 1 − αy; x,y ∈ SX, x − y ε

, 2.13

where 0 ≤ ε ≤ 2. It is easily shown that δXε ≤ βXε ≤ 2δXε for any 0 ≤ ε ≤ 2. Also if 0 < β_Xε < 1, then X is uniformly convex, and if βXε < 1, then X is strictly convex.

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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 {tkz} is convergent in the norm in Xsee 21, where

t_kz 1

k 1z0 z1 z2 · · · zk k ∈ N. 2.14

Let 1 < p <∞. A Banach space is said to have the Banach-Saks type p or property BSp, if every weakly null sequencexk has a subsequence xkl such that for some C > 0

n l0

xkl

< Cn 1^1/p 2.15

for all n∈ N see 22.

3. Some Topological Properties of the Space A; p

In this section, we investigate some topological properties of the sequence space A; p as the paranorm AK property and AD property. Let us begin the following theorem.

Theorem 3.1. i The space A; p is complete linear metric space with respect to the paranorm defined by

hx 

n |Ax_n|^pⁿ

1/M

. 3.1

ii If the sequence pn is constant sequence and p ≥ 1, then p A is a Banach space normed by

p p A

n

|Ax_n|^p

_1/p

. 3.2

Proof. The proof ofii is routine verification by using standard techniques and hence it is omitted.

The proof of i is that the linearity of A; p with respect to the coordinatewise addition and scalar multiplication follows from the following inequalities which are satisfied for x, y∈ A; p:

n

A

x y

n

^pⁿ

1/M

≤

n

Ax

n

^pⁿ

1/M

n

Ay

n

^pⁿ

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

A; p holds the paranorm property and the completeness with respect to given paranorm.

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It is easy to show that hθ 0, and hx h−x for all x ∈ A; p. Besides, from 3.3 we obtain hx y ≤ hx hy for all x, y ∈ A; p. To complete the paranorm conditions for the space A; p, it remains to show the continuity of the scalar multiplication. Let x^m be any sequence in A; p such that hx^m− x → 0, and let αm be also any sequence of scalars such that|αm− α| → 0 m → ∞. From subadditivity of h, we give the inequality hx^m ≤ hx hx^m− x. Hence {hx^m} is bounded and we have

hαmx^m− αx

n

α_m− αⁿ

v0

anvx_v^m αⁿ

v0

anvx^m_v − xv

pn1/M

3.4

which tends to zero as m → ∞. Consequently we obtain that h is a paranorm over the space

A; p. To prove the completeness of the space A; p, let us take any Cauchy sequence xⁱ in the space A; p. Then for a given ε > 0, there exists a positive integer n0ε such that hxⁱ− x^j < ε for all i, j ≥ n0ε. By using the definitions of the Cauchy sequence and the paranorm, we have, for each fixed n,

Ax ⁱ

n− Ax ^j

n

≤

n

A xⁱ

n− A x^j

n

^pⁿ

_1/M

< ε 3.5

for every i, j ≥ n0ε. Hence we obtain that the sequence { Ax⁰_n, Ax¹_n, Ax²_n, . . .} is a Cauchy sequence of real numbers for every fixed n ∈ N. Since R is complete, it converges, that is, Ax^j_n → Ax_nas j → ∞, where { Ax_n} { Ax₀, Ax₁, Ax₂, . . .}. Now let us choose m ∈ N such that _m

n0| Axⁱ_n− Ax^j_n|^pⁿ < ε^M for each m ∈ N and i, j ≥ n0ε. By taking j → ∞ and for every i ≥ n0ε, we get

m n0

Ax ⁱ

n− Ax

n

^pⁿ < ε^M. 3.6

Again taking m → ∞ and for every i ≥ n0ε, it is obtained that hxⁱ− x < ε. We write the following equality:

Ax

n

Ax

n Ax ⁱ

n− Ax ⁱ

n

. 3.7

By using3.7 and Minkowski’s inequality, we get

n

Ax

n

^pⁿ

_1/M

≤ h xⁱ

h xⁱ− x

3.8

which implies x ∈ A; p. It follows xⁱ → x as i → ∞. Consequently, since xⁱ is any Cauchy sequence, we obtain that the space A; p is complete. This completes the proof.

Theorem 3.2. The space A; p is linearly isomorphic to the space p.

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Proof. Let us define A-transform between the spaces A; p and p such that x → z Ax.

We have to show that the transformation A is linear, injective and surjective. The linearity of A is obvious. Moreover it is injective because of x θ whenever  Ax θ. For the surjective property, let y∈ p. From 2.3 and 2.7, there exists a matrix B such that xn  By_n. We have

hx 

n

A

By

n

^pⁿ

_1/M

n

yn^pⁿ_1/M

 g y

< ∞. 3.9

Hence we obtain that the transformation A is surjective. Consequently, the spaces A; p and

p are linearly isomorphic spaces.

Theorem 3.3. The space p A has AD property.

Proof. Let f ∈ p A. Then fx g Ax for some g ∈ _p. Since p has AK property and

_p ∼ qwhere 1/p 1/q 1,

fx 

n

a_n Ax

n 3.10

for some a  an ∈ q. Also since p A ∼ p and the inclusion φ ⊂ p holds, we have φ ⊂ p A. For any f ∈ p Aand e^k ∈ φ, we have

f e^k

n

a_n Ae ^k

n Ha

k, 3.11

where H is transpose of the matrix A. Hence from Hahn-Banach theorem, φ ⊂ p A is dense in p A if and only if Ha θ for a ∈ q implies a  θ. Besides, since the null space of the operator H on w is {θ}, p A has AD property. Hence the proof is completed.

4. Some Geometric Properties of the space

_p

A

In this section, we give some geometric properties for the space _p A.

Theorem 4.1. The space p A has the Banach-Saks of type p.

Proof. Let εn be a sequence of positive numbers for which _∞

n1ε_n ≤ 1/2. Let xn be a weakly null sequence in Bp A. Set z0 x0 0 and z1  xn1  x1. Then there exists s₁ ∈ N

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such that

∞ is1 1

z₁ieⁱ

p A

< ε_1. 4.1

Sincexn is a weakly null sequence implies that xn → 0 with respect to the coordinatwise, there is an n₂ ∈ N such that

s1

i0

xnieⁱ

p A

< ε1, 4.2

where n≥ n2. Set z2 xn2. Then there exists an s2> s1such that

∞ is2 1

z2ieⁱ

p A

< ε2. 4.3

By using the fact that x_n → 0 coordinatwise, there exists an n3> n₂such that

s2

i0

xnieⁱ

p A

< ε2, 4.4

where n≥ n3.

If we continue this process, we can find two increasing subsequencessi and ni such that

sj

i0

xnieⁱ

p A

< εj 4.5

for each n≥ nj 1and

∞ isj 1

z_jieⁱ

p A

< ε_j, 4.6

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where zj xnj. Hence,

n j0

zj

p A

n j0

⎛

⎝^s^j−1

i0

zjieⁱ

sj

isj−1 1

zjieⁱ ^∞

isj 1

zjieⁱ

⎞

⎠

p A

≤

n j0

⎛

⎝ ^s^j

isj−1 1

zjieⁱ

⎞

⎠

p A

n j0

_s_j−1

i0

zjieⁱ

p A

n j0

⎛

⎝ ^∞

isj 1

zjieⁱ

⎞

⎠

p A

≤

n j0

⎛

⎝ ^s^j

isj−1 1

zjieⁱ

⎞

⎠

p A

2ⁿ

j0

εj.

4.7

On the other hand, since xn ∈ Bp _p_A _∞

i0| _i

v0aivxv|^p^1/p, it can be seen

that _p_A ^p

p A< 1. We have

n j0

⎛

⎝ ^s^j

isj−1 1

zjieⁱ

⎞

⎠

p

p A

ⁿ

j0 sj

isj−1 1

i v0

aivzjv

p

≤ⁿ

j0

∞ i0

i v0

aivz_jv

p

≤ n 1.

4.8

Hence we obtain

n j0

⎛

⎝ ^s^j

isj−1 1

z_jieⁱ

⎞

⎠

p A

≤ n 1^1/p. 4.9

By using the fact 1≤ n 1^1/pfor all n∈ N and 1 ≤ p < ∞, we have

n j0

zj

p A

≤ n 1^1/p 1 ≤ 2n 1^1/p. 4.10

Hence _p A has the Banach-Saks type p. This completes the proof.

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Theorem 4.2. Gurarii’s modulus of convexity for the normed space p A is

β_p^Aε ≤ 1 −

1− ε

2

p1/p

, 4.11

where 0≤ ε ≤ 2.

Proof. We have x∈ p A. Then we have

p A Ax_l

p

n

Ax

n

^p

_1/p

. 4.12

Let 0≤ ε ≤ 2 and consider the following sequences:

x xn

B

1− ε 2

p_1/p , B ε

2

, 0, 0, . . .

,

y  yn

B

1− ε 2

p_1/p , B

−ε 2

, 0, 0, . . .

,

4.13

where B is the inverse of the matrix A. Since zn Ax_nand tn Ay_n, we have

z zn

1− ε 2

p_1/p , ε

2

, 0, 0, . . .

,

t tn

1− ε 2

p_1/p ,

−ε 2

, 0, 0, . . .

.

4.14

By using sequences given above, we obtain the following equalities:

p

p A Ax^p_l

p

1− ε

2

p_1/p

p

ε 2

^p

1 − ε 2

p

 ε 2

p

 1,

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y^p

p A Ay^p_l

p

1− ε

2

p1/p

p

−ε 2

^p

1 − ε 2

p

 ε 2

p

 1,

x − y_p_A Ax − Ay_l

p

1− ε

2

p1/p

−

1− ε

2

p1/p

p

ε 2 −

−ε 2

^p

1/p

 ε.

4.15

To complete the conditions of β_p_Aε for Gurarii’s modulus of convexity, it remains to show the infimum of _p_Afor 0≤ α ≤ 1. We have

0≤α≤1inf αx 1 − αy_p_A

inf

0≤α≤1

α Ax 1 − α Ay

lp

inf

0≤α≤1

α

1− ε

2

p_1/p

1 − α

1− ε

2

p_1/p

p

α ε 2

1 − α

−ε 2

^p

1/p

inf

0≤α≤1

1− ε

2

p

|2α − 1|^p ε 2

p_1/p

1− ε

2

p1/p

.

4.16

Consequently we get for p≥ 1

β_p_Aε ≤ 1 −

1− ε

2

p_1/p

. 4.17

This is the desired result. Hence the proof is completed.

Corollary 4.3. i If ε 2, then β_p_Aε ≤ 1 and hence p A is strictly convex.

ii If 0 < ε < 2, then 0 < β_p_Aε < 1 and hence p A is uniformly convex.

Corollary 4.4. If α 1/2, then δ_p_Aε β_p_Aε.

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Some l(p)-type new sequence spaces and their geometric properties

Research Article

Some p-Type New Sequence Spaces and Their Geometric Properties

Ekrem Savas¸,

Vatan Karakaya,

and Necip S¸ims¸ek

1. Introduction

2. Preliminaries and Notations

3. Some Topological Properties of the Space  A; p

4. Some Geometric Properties of the space

 A

References

Submit your manuscripts at http://www.hindawi.com

Mathematics

Complex Analysis

Optimization

Combinatorics

Function Spaces

The Scientific World Journal

Discrete Mathematics

Stochastic Analysis

3. Some Topological Properties of the Space A; p

A