2 The Double Sequence Spaces L

(1)

www.i-csrs.org

Available free online at http://www.geman.in

Some Double Sequence Spaces Defined by a Modulus Function

G¨ulsen Kılın¸c¹ and ˙Ihsan Solak²

1Adıyaman University, Faculty of Education Department of Primary Mathematics, Adıyaman, Turkey

E-mail: [email protected].

2Nev¸sehir University, Faculty of Art and Sciences Department of Mathematics, Nev¸sehir, Turkey

E-mail: [email protected].

(Received: 15-4-14 / Accepted: 23-9-14) Abstract

In the present paper we construct some new double sequence spaces defined by a modulus function. Further we give its some topological and algebraic properties.

Keywords: Double sequence space, Modulus Function, vector valued double sequence space.

1 Introduction

Some works on double sequences were studied by Hardy[5], Moricz[6], Moricz and Rhoades[7]. Hardy introduced the notion of regular convergence for double sequences. Hill [8] was the first who applied methods of functional analysis to doubles sequences. He described the topological dual of the space of all regularly convergent double sequences and perfectness of matrices with respect to the regular convergence. T¨urkmeno˘glu [10] showed under which conditions that C_0p(t), L_u and C_0bp(t) are paranormed double sequence spaces, deter- mined their duals and gave some inclusion relations between those spaces.

Boos, Leiger and Zeller [11] defined the concept of ν-SM method by the ap- plication domain of a matrix sequence A= A^(ν)

of infinite matrices, gave the consistency theory for such type methods and introduced the notion of

(2)

C_e-convergence for double sequences. By using gliding hump method, Zeltser [12] characterized the class of four-dimensional matrix mappings from λ into µ, where λ, µ ∈ {Ce, Cbe}. By Ce and Cbe, we denote the spaces of all Ce- convergent and of all bounded C_e-convergent double sequences, respectively.

Also employing the same arguments, Zeltser [13] gave the theorems determin- ing the necessary and sufficient conditions forCe-SM andCbe-SM methods to be concervative and coercive.

Modulus function introduced by Nakano [1] and used to solve some struc- tural problems of the scalar FK-spaces theory. Ruckle [2] constructed a class of scalar FK-spacesL(f), wheref is a modulus function. L(f) is a general- ization of the spaces l_p,(0< p≤1).

Yılmaz [3] introduced and investigated the sequence space λ(Xk, r, f, s) defined by a modulus function and constructed its FK-structure under some conditions.

In the present paper we introduce some new double sequence spaces by using a modulus functionf and investigate some properties of these sequence spaces.

Let Ω (X) be the space of allX-valued double sequences and X be a Banach space. The topology of Ω (X) is a locally convex topology produced by the family of all seminorms defined by

p_ij(x) = kx_ijk.

Since the family of {p_ij :i, j ∈ N×N} are countable, this topology is metriz- able, wherek,k is the norm on X.

Also, total paranorm generating this metric, which is constructed by the Frechet combination of seminorms {p_ij :i, j ∈N×N}, is given by

g(x) = X 1 2^i+j

kx_ijk 1 +kxijk.

A locally convex double sequence space (E, τ) is said to be a DK-space, if all the seminorms defined by

r_kl : E →R x = (xij)→ |xkl|,

are continuous. A DK-space with a Frechet topology is called an FDK-space.

A normed FDK-space is called BDK-space [4].

Forλ(X)⊂Ω (X),a Frechet sequence spaceλ(X) is called an FDK-space, if the coordinate maps

fkl:λ(X) → X

x → f_kl(x) = x_kl are continuous.

(3)

2 The Double Sequence Spaces L

_u

(f )

Letf be a modulus and let us define the double squence space L_u(f) by L_u(f) ={x∈Ω : X

m,n∈N

f(|x_mn|)<∞}.

It is easy to show that L_u(f) is a linear space, where Ω denotes the space of all real or complex valued double sequences.

Theorem 2.1 L_u(f) is a paranormed space with the function g(x) = X

m,n∈N

f(|xmn|).

Proof: It is obvious that g is well-defined by the definition of L_u(f).Let us verify that g provides the paranorm conditions:

i)g(θ) = P

m,n∈N

f(|θ |) = 0.

ii) For eachx∈ Lu(f), it is clear that g(x) =g(−x). iii) For all x, y ∈ L_u(f),we have

g(x+y) = X

m,n∈N

f(|x_mn+y_mn|)≤ X

m,n∈N

f(|x_mn|+|y_mn|)

≤ X

m,n∈N

f(|x_mn|) + X

m,n∈N

f(|y_mn|) = g(x) +g(y). iv) (a) Suppose that λ is a scalar and g(x)→0. Then we get

g(λx) = X

m,n∈N

f(|λ| |x_mn|)≤K. X

m,n∈N

f(|x_mn|), whereK is a positive integer such that |λ| ≤K. So g(λx)→0.

(b) Suppose that λ^r → 0 and x ∈ L_u(f). Then there exist positive numbersε and k such that

∞

X

m,n=k+1

f(|x_mn|)< ε 2,

by virtue of the the factf(|x_mn|)<∞. Now let us write h(t) =

k

X

m,n=1

f(t|x_mn|).

(4)

Thenh is continuous at 0. Therefore there exists a number δ such that 0< δ <1, for each 0< t < δ

|h(t)|< ε 2.

Then there is a numberN such that for each r > N, we get

|λ^r|< δ.

Sinceλ^r→0, for each r > N, then we have g(λ^rx) =

k

X

m,n=1

f(|λ^rxmn|) +

∞

X

m,n=k+1

f(|λ^rxmn|)

≤ ε 2+

∞

X

m,n=k+1

f(|x_mn|)

≤ ε 2+ ε

2 =ε, which completes the proof.

It can be verified easily that g is total, i.e., for each x∈ L_u(f), then g(x) = 0 ⇒x=θ.

Theorem 2.2 The double sequence space L_u(f) is a DK-space.

Proof: For each k, l∈N, the functions P_kl :L_u(f) → C

x → P_kl(x) = x_kl

are continuous. For all ε > 0, there exists a δ > 0 such that g(x) < δ.

Choosingδ =f(ε),we have

∞

X

k,l

f(|x_kl|)< f(ε)⇒f(|x_kl|)< f (ε). Sincef is increasing, we obtain

|x_kl|=|P_kl(x)|< ε.

Theorem 2.3 L_u(f) is complete.

(5)

Proof: Let x^l

be a Cauchy sequence in L_u(f). For eachε > 0,∃n₀ ∈N 3for each l, r > n₀, g x^l−x^r

< ε. Since, for each i, j, coordinate functions fij(x) = xij are continuous on Lu(f), then x^l_ij

is a Cauchy sequence in C, (l > n₀).Because of the completeness of C, the Cauchy sequence converges to a point. Letx_ij be such a point. Construct the double sequencex= (x_ij) with these limit points. Then we have

g x^l−x^r

< ε⇒X

i,j

f

x^l_ij −x^r_ij

< ε⇒

m

X

i=0 n

X

j=0

f

x^l_ij −x^r_ij

< ε

⇒ lim

r m

X

i=0 n

X

j=0

f

x^l_ij −x^r_ij

=

m

X

i=0 n

X

j=0

f

x^l_ij −x_ij

< ε

⇒ g x^l−x

< ε(∀ l > n0)⇒x^l →x.

Also, we get

∞

X

i=0

∞

X

j=0

f(|x_ij|) =

m

X

i=0 n

X

j=0

f

xij −x^l_ij −x^l_ij

≤

∞

X

i=0

∞

X

j=0

f

x_ij −x^l_ij

+

m

X

i=0 n

X

j=0

f x^l_ij

< ε,

which impliesx∈ L_u(f).

Corollary 2.4 Lu(f) is an FDK-space.

Theorem 2.5 L_u(f) ⊆ L_u, where L_u ={x∈Ω :

∞

P

i,j=1

|x_ij |<∞}.

Proof: Suppose that x∈ Lu(f) but x /∈ Lu. Then we have

∞

X

k,l=1

f(|x_kl|)<∞

and ∞

X

k,l=1

(|x_kl|) = ∞.

From the last equation above, one can see that there is a double sequence of natural numbers (n_i, m_j) such that,

ni+1

X

k=ni

mj+1

X

l=mj

(|x_kl|)>1⇒f(1) < f





ni+1

X

k=ni

mj+1

X

l=mj

(|x_kl|)



=

ni+1

X

k=ni

mj+1

X

l=mj

f(|x_kl|).

(6)

Then we obtain

∞

X

k,l=1

f(|x_kl|)<∞

and ni+1

X

k=ni

mj+1

X

l=mj

f(|x_kl|)→0,(i, j → ∞),

which implies a contradiction such that f(1) = 0. Hence, we get x ∈ L_u. This completes the proof.

We note that

f(x) =x⇒ L_u(f) =L_u. Iff is unbounded, L_u(f)⊂ L_u.

Theorem 2.6 The double sequence of unit vectors is bounded in L_u(f). Proof: Let δ be the double sequence of unit vectors, i.e.,

δ=





e₁₁ e₁₂ ...

e₂₁ e₂₂ ...

. . ...



, eij =







0 0 ... 0 ...

. . . 0 ...

0 0 ... 1 ...

. . ... 0 ...





 ,

where 1 is term of (i, j).

If a∈C,then g(ae₁₁) =...=g(ae_ij) =f(|a|). Let us choose aλ ∈C such that f(|λ|)< .Then, for 0 ≤a≤λ, we write

g(ae11) =f(|a|)≤ f(|λ|)< . This implies thatae₁₁∈ {x:g(x)< ε}. Since we have g(ae_ij) = g(ae_kl) = f(|a|),

for (i, j) 6= (k, l), then we get aδ ∈ {x : g(x) < ε}, which shows that each sphere centered at origin contains the sequenceδ. Thusδ is a bounded double sequnce inL_u(f).

3 The Double Sequence Space L

_u

(X, f )

LetX be a Banach Space. We define

L_u(X, f) ={x∈Ω (X) :X

m,n

f(kx_mnk)<∞},

wherek,k is the norm of X.It is obvious that L_u(X, f) is a linear space.

(7)

Theorem 3.1 L_u(X, f) is paranormed space with p(x) = X

i,j

f(kxmnk).

Proof: i) If x=θ, for eachi, j, x_ij = 0 and p(θ) = 0.

ii) It is obvious that p(−x) =p(x). iii) For x, y ∈ Lu(X, f), we write

p(x+y) = X

i,j

f(kx_ij +y_ijk)≤X

i,j

f(kx_ijk) +f(ky_ijk)

= X

i,j

f(kxijk) +X

i,j

fkyijk).

iv) Let x = x^l

∈ L_u(X, f) be a sequence and λ = λ^l

be a sequence of scalars. Assume that λ → λ⁰and p x^l−x⁰

→ 0, (l→ ∞). So scalar sequence λ = λ^l

is convergent, for each l ∈N. Then there exists a positive numberK such that |λ^l |≤K. Therefore we have

p λ^lx^l−λ⁰x⁰

= X

i,j

f

λ^lx^l_ij −λ⁰x⁰_ij

= X

i,j

f

λ^lx^l_ij −λ^lx⁰_ij +λ^lx⁰_ij −λ⁰x⁰_ij

≤ X

i,j

f(

λ^lx^l_ij −λ^lx⁰_ij

) +f

λ^lx⁰_ij −λ⁰x⁰_ij

≤ X

i,j

f( λ^l

x^l_ij −x⁰_ij

) +X

i,j

f(

λ^l−λ⁰

x⁰_ij )

= K.p x^l−x⁰

+X

i,j

f(

λ^l−λ⁰

x⁰_ij

). (1) Since K is a constant and p x^l−x⁰

→0 (l→ ∞), the first term in the right hand side of inequality in (1) tends to zero. Also, for each l ∈N, there exists a T ≥0 such that

T x⁰ = T x⁰_ij

∈ L_u(X, f).

Forε >0, there exist numbersi₀, j₀ such that for each l ∈N, we get





 P

i>i0

P

j>j0f(

λ^l−λ⁰

x⁰_ij )

+

i0

P

i=0

P∞

j=j0+1f(

λ^l−λ⁰

x⁰_ij )

+

∞

P

i=i0+1

Pj0

j=0f(

λ^l−λ⁰ kx_ijk)







≤





 P

i>i0

P

j>j0f(T x⁰_ij

)

+

i0

P

i=0

P∞

j=j0+1f(T x⁰_ij

)

+

∞

P

i=i0+1

Pj0

j=0f(T x⁰_ij

)







< ε 6 +ε

6 +ε 6 = ε

2

(8)

On the other hand, since

l→∞lim

i0

X

i=0 j0

X

j=0

f(

λ^l−λ⁰

x⁰_ij ) =

i0

X

i=0 j0

X

j=0

f( lim

l→∞

λ^l−λ⁰

x⁰_ij ) = 0.

For the sameε, there exists an l₀ such that for each l > l₀

i0

X

i=0 j0

X

j=0

f(

λ^l−λ⁰

x⁰_ij )< ε

2. Then we get

X

i,j

f(

λ^l−λ⁰

x⁰_ij )≤







P

i>i0

P

j>j0f(T x⁰_ij

)

+

i0

P

i=0

P∞

j=j0+1f(T x⁰_ij

)

+

∞

P

i=i0+1

Pj0

j=0f(T x⁰_ij

)

+

i0

P

i=0

Pj0

j=0f(

λ^l−λ⁰

x⁰_ij )







< ε 6+ε

6+ε 6+ε

2 =ε,

for each ε > 0 and l > l₀. Thus the second term in inequality (1) tends to zero, i.e.,

p λ^lx^l−λ⁰x⁰

→0 (l → ∞). So the proof is completed.

Theorem 3.2 L_u(X, f) is an FDK-space.

Proof: i) We know thatL_u(X, f) is a linear space and paranormed space.

ii) Let us proof continuity of coordinate mapsp_kl which are defined by p_kl : L_u(X, f)→X

pkl(x) = xkl.

For eachε >0, let us choose δ =f(ε) such that p(x) =X

k,l

f(kx_klk)< δ =f(ε). We get

f(kx_klk)< f(ε) and

kx_klk=kp_kl(x)k< ε.

(9)

iii) We will show thatL_u(X, f) is complete. Let x^l

be a Cauchy sequence inL_u(X, f).For each ε >0, there exists an n₀ ∈Nsuch that, for l, r > n₀

p x^l−x^r

< ε.

Since functions

p_ij(x) = x_ij are continuous for each (i, j) ∈ N ×N, x^l_ij

is a Cauchy sequence in X, (for each (i, j)∈ N×N and l > n0). Since X is a Banach space, the Cauchy sequence converges to a pointx_ij. Let’s set the double sequence x= (x_ij) by this limit points. Now we have

x^l → x⇒p x^l−x^r

< ε

⇒ X

i,j

f

x^l_ij −x^r_ij

< ε

⇒

m

X

i=0 n

X

j=0

f

x^l_ij −x^r_ij

< ε

limr m

X

i=0 n

X

j=0

f

x^l_ij−x^r_ij

= X

i,j

f

x^l_ij −x^r_ij

< ε

⇒ p x^l−x

< ε, for each l > n0.

On the other hand we obtain X

i,j

f kx^l_ij k

= X

i

X

j

f kx_ij −x^l_ij +x^l_ij k

≤

∞

X

i=0

∞

X

j=0

f kx_ij −x^l_ij k +

∞

X

i=0

∞

X

j=0

f kx^l_ij k

< ε

which shows thatx∈ L_u(X, f).

4 Double Sequence Spaces L

_u

(X ) and M

_u

(X )

Let’s define double sequence spaces Lu(X) and Mu(X) as follow:

M_u(X) = {x= (x_mn)∈Ω (X) : supkx_mnk

m,n∈N

<∞}, L_u(X) = {x= (x_mn)∈Ω (X) :X

m,n

kx_mnk<∞}.

(10)

Theorem 4.1 The space sL_u(X) and M_u(X) are BDK-spaces.

Proposition 4.2 For each modulus function f, L_u(X, f) ⊂ L_u(X) . Proof: We assume that x ∈ L_u(X, f) and x /∈ L_u(X). Then increasing sequences (k_n) and (l_n) can be found such that

kn−1

X

i=kn−1

ln−1

X

j=ln−1

kxijk ≥ 1

⇒ f(1)≤f





kn−1

X

i=kn−1

ln−1

X

j=ln−1

kx_ijk





≤

kn−1

X

i=kn−1

ln−1

X

j=ln−1

f(kx_ijk)

⇒

kn−1

X

i=kn−1

ln−1

X

j=ln−1

f(kx_ijk)<∞

limn kn−1

X

i=kn−1

ln−1

X

j=ln−1

f(kx_ijk) = 0

It means f(1) = 0. This contradicts with f being a modulus function. Then x∈ L_u(X).

Proposition 4.3 For each x∈ L_u(X), x can be written as follows:

x= X

i,j∈N×N

I_ij(x_ij), where

Iij : X → Lu(X)

I_ij(t) = y3y_ij =t and y_kl = 0,if (k, l)6= (i, j).

Note that the meaning of the above expression is that the net{S_F (x) :F ∈ F } converges to a point x according to norm topology of Lu(X), where F is a family of all finite subsets ofN×N. This family is directed with relation ⊆.

Theorem 4.4 For each x∈ L_u(X, f), x can be represented as follows:

x= X

(i,j)∈F

I_ij(x_ij), where

Iij : X → Lu(X, f)

I_ij(t) = y3y_ij =t and y_kl = 0,if (k, l)6= (i, j).

(11)

Proof: We will show that for all givenε >0, there exists anF₀ =F₀(ε)∈ F such that whenF₀ ⊆F, p(x−S_F(x))< ε.Due to the definition of I_ij, we can define a function as follow:

SF : Lu(X, f)→ Lu(X, f) S_F(x_ij) = x_ij, ((i, j)∈F) S_F(x_ij) = 0. ((i, j)∈/ F) Then we write

p(x−S_F (x)) = X

i,j∈N×N

f(kx_ij − {S_F (x)}_i,jk)

= X

i,j∈N×N\F

f(kx_ijk), which impliesx∈ L_u(X, f). So P

i,j

f(kx_ijk)<∞.

Thus forε > 0, there exists an F0(ε)3P

i,j

f(kxijk)< ε. Therefore for all F ⊇F₀, we have

X

i,j∈N×N\F

f(kx_ijk) =p(S_F(x)−x)< ε.

It means that the net (S_F(x) :F ∈ F) converges to x. This completes the proof.

References

[1] H. Nakano, Concave modulares, J. Math. Loc. Japan, 5(1953), 29-49.

[2] W.H. Ruckle, FK-spaces in which the sequence of coordinate vectors is bounded,Can. J. Math., 25(5) (1973), 973-978.

[3] Y. Yılmaz and I. Solak, Vector valued FK-spaces defined by a modulus function,Demonstratıo Mathematıca, XXXVIII(3) (2005), 633-640.

[4] M. Zeltser, Investigation of double sequence spaces by soft and hard an- alytical methods, Dissertationes Mathematicae, Universitatis Tartuensis, Tartu, (2001).

[5] G.H. Hardy, On the convergence of certain multiple series, Proc. Camb.

Phil. Soc., 19(1917), 86-95.

[6] F. Moricz, Extension of the spaces c and c₀ from single to double sequences,Acta Math. Hungarica, 57(1991), 129-136.

(12)

[7] F. Moricz and B.E. Rhoades, Almost convergence of double sequences and strong regularity of summabilty matrices,Math. Proc. Camb. Phil. Soc., 104(1988), 283-294.

[8] J.D. Hill, On perfect summability of double sequences,Bull. Amer. Math.

Soc., 46(1940), 327-331.

[9] T.J. Bromwich,An Introduction to the Theory of Infinite Series, Macmil- lan and Co. Ltd. Newyork, (1965).

[10] A. T¨urkmeno˘glu, Bazı ¸cift indisli dizi uzayları , Fırat ¨Univ. Fen Bil. Enst., PhD Thesis, (1993).

[11] J. Boos, T. Leiger and K. Zeller, Consistency theory for SM-methods, Acta Math. Hungar., 76(1997), 109-142.

[12] M. Zeltser, Matrix transformations of double sequences, Acta Comment.

Univ. Tartu Math., 4(2000), 39-51.

[13] M. Zeltser, On conservative and coercive SM-methods, Proc. Est. Acad.

Sci. Phys. Math., 50(2001), 76-85.