Math. J. Okayama Univ. 51 (2009), 149–157

THE SPACE Lq OF DOUBLE SEQUENCESFeyzi BAŞAR and Yurdal SEVERAbstract. The spaces BS, BS(t), CS p , CS bp , CS r and BV of double
sequences have recently been studied by Altay and Başar [J. Math. Anal.
Appl. 309(1)(2005), 70–90]. In this work, following Altay and Başar [1],
we introduce the Banach space Lq of double sequences corresponding to
the well-known space ℓq of single sequences and examine some propertiesof the space Lq . Furthermore, we determine the β(υ)-dual of the spaceand establish that the α- and γ-duals of the space Lq coincide with theβ(υ)-dual; where 1 ≤ q < ∞ and υ ∈ {p, bp, r}.

1. Introduction
By w and Ω, we denote the set of all real valued single and double sequences which are the vector spaces with coordinatewise addition and scalarmultiplication. Any vector subspaces of w and Ω are called as the single anddouble sequence spaces, respectively. The space Mu of all bounded double
sequences is defined by()Mu :=x = (xmn ) ∈ Ω : kxk∞ = sup |xmn | < ∞ ,m,n∈N

which is a Banach space with the norm k · k∞ ; where N denotes the set of
all positive integers. Consider a sequence x = (xmn ) ∈ Ω. If for every ε > 0there exists n0 = n0 (ε) ∈ N and l ∈ R such that|xmn − l| < εfor all m, n > n0 then we call that the double sequence x is convergent in thePringsheim’s sense to the limit l and write p − lim xmn = l; where R denotes
the real field. By Cp we denote the space of all convergent double sequencesin the Pringsheim’s sense. It is well-known that there are such sequences inthe space Cp but not in the space Mu . So, we can consider the space Cbp ofthe double sequences which are both convergent in the Pringsheim’s senseand bounded, i.e., Cbp = Cp ∩ Mu . A sequence in the space Cp is said tobe regularly convergent if it is a single convergent sequence with respect toeach index and the set of all such sequences denoted by Cr . Also by Cbp0 andCr0 , we denote the spaces of all double sequences converging to 0 containedMathematics Subject Classification. Primary: 46A45; Secondary: 40C05.
Key words and phrases. Double sequence space, paranormed sequence space, α-, β-,γ-duals of a double sequence space.
149

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FEYZİ BAŞAR AND YURDAL SEVER

in the sequence spaces Cbp and Cr , respectively. Móricz [7] proved that Cbp ,Cbp0 , Cr and Cr0 are Banach spaces with the norm k · k∞ .
Let us consider the isomorphism T which plays an essential role for thepresent study, defined by Zeltser [11, p. 36] as(1.1)

T: Ω −→ wx 7−→ z = (zi ) := (xψ−1 (i) ),where ψ : N × N → N is a bijection defined byψ[(1, 1)] = 1,ψ[(1, 2)] = 2, ψ[(2, 2)] = 3, ψ[(2, 1)] = 4,..
.
ψ[(1, n)] = (n − 1)2 + 1, ψ[(2, n)] = (n − 1)2 + 2, . . . ,ψ[(n, n)] = (n − 1)2 + n, ψ[(n, n − 1)] = n2 − n + 2, . . . , ψ[(n, 1)] = n2 ,..
.
Let us consider a double sequence x = (xmn ) and define the sequence s =(smn ) via x by(1.2)

smn :=m,nX

xij ; (m, n ∈ N),i,j=1

which will be used throughout. For the sake Pof brevity,herePand P
in what fol∞ P∞
mnlows, we abbreviate the summations
i=1
j=1 ,
i=1
j=1 andPn
Pn Pn
Pm,nP
i,j=1 andi,j=1 , respectively. Then the pair (x, s)i=1i,j ,j=1 byand the sequence s = (smn ) are called as a double series and the sequenceof partial sums of the double series, respectively. Let λ be the space of
double sequences, converging with respect to some
P linear convergence rule
υ − lim : λ → R. The sum of a double series i,j xij with respect to thisPrule is defined by υ − i,j xij := υ − lim smn .
Let us define the following sets of double sequences:()
Mu (t) :=(xmn ) ∈ Ω : sup |xmn |tmn < ∞ ,m,n∈Ntmn
Cp (t) := (xmn ) ∈ Ω : p − lim |xmn − l|
= 0 for some l ∈ C ,m,n→∞C0p (t) := (xmn ) ∈ Ω : p − lim |xmn |tmn = 0 ,m,n→∞

Lu (t) :=((xmn ) ∈ Ω :Xm,n)|xmn |tmn < ∞ ,THE SPACE Lq OF DOUBLE SEQUENCES

151Cbp (t) := Cp (t) ∩ Mu (t) and C0bp (t) := C0p (t) ∩ Mu (t);where t = (tmn ) is the sequence of strictly positive reals tmn for all m, n ∈ N.
In the case tmn = 1 for all m, n ∈ N; Mu (t), Cp (t), C0p (t), Lu (t), Cbp (t) andC0bp (t) reduce to the sets Mu , Cp , C0p , Lu , Cbp and C0bp , respectively. Now,
we may summarize the knowledge given in some document related to thedouble sequence spaces. Gökhan and Çolak [4, 5] have proved that Mu (t)and Cp (t), Cbp (t) are complete paranormed spaces of double sequences andgave the α-, β-, γ-duals of the spaces Mu (t) and Cbp (t). Quite recently,
in her PhD thesis, Zeltser [11] has essentially studied both the theory of
topological double sequence spaces and the theory of summability of double
sequences. Mursaleen and Edely [8] have recently introduced the statistical convergence and Cauchy for double sequences and given the relation
between statistical convergent and strongly Cesàro summable double sequences. Nextly, Mursaleen [9] and Mursaleen and Edely [10] have defined
the almost strong regularity of matrices for double sequences and appliedthese matrices to establish a core theorem and introduced the M -core fordouble sequences and determined those four dimensional matrices transforming every bounded double sequence x = (xjk ) into one whose core is asubset of the M -core of x. More recently, Altay and Başar [1] have defined
the spaces BS, BS(t), CS p , CS bp , CS r and BV of double sequences consistingof all double series whose sequence of partial sums are in the spaces Mu ,Mu (t), Cp , Cbp , Cr and Lu , respectively, and also examined some propertiesof those sequence spaces and determined the α-duals of the spaces BS, BV,CS bp and the β(ϑ)-duals of the spaces CS bp and CS r of double series.
In the present paper, we introduce the space LqXLq := (xij ) ∈ Ω :|xij |q < ∞ , (1 ≤ q < ∞)i,jof double sequences corresponding to the space ℓq of single sequences andexamine some properties of the space.
2. The Double Sequence Space Lq

In this section, we give the theorem which states that Lq is a sequencespace and is a Banach space with the norm k · kq , firstly. Subsequent togiving some inclusion relations concerning the space Lq , we establish thatthe α- and γ-duals of a space of double sequences are identical whenever it
is solid, and Lq is soid if q > 1 and determine the β(υ)-dual of the space Lq
for υ ∈ {p, bp, r} which coincides with the α- and γ-duals of the space Lq .

152FEYZİ BAŞAR AND YURDAL SEVERTheorem 2.1. The set Lq becomes a linear space with the coordinatewise
addition and scalar multiplication and Lq is a Banach space with the norm
1/qX
kxkq = |xij |q  ,(2.1)
i,jwhere 1 ≤ q < ∞.

Proof. The proof of the first part of the theorem is a routine verification andso we omit the detail.
Furthermore, the statement ”a sequence space ν is a Banach space withthe norm k · kν if and only if the sequence space T −1 (ν) = λ is a Banach
space with the norm k · kλ ” holds by Boos [3, Corollary 6.3.41]. Therefore,
the restriction of the transformation defined by (1.1) to the space Lq which
is norm preserving isomorphism yields the fact that Lq = T −1 (ℓq ) is also aBanach space with the norm k · kq defined by (2.1) because of ℓq is a Banach
space.
This step concludes the proof.
Theorem 2.2. Let 1 ≤ q < s < ∞. Then, the inclusions Lq ⊂ Ls ⊂ Cr0 ⊂Mu hold.
PProof. Let us take any x = (xij ) ∈ Lq . Then, max{i,j}>n0 |xij |q < ε < 1for sufficiently large n0 ∈ N. Since q < s, it is obvious that |xij |q ≥ |xij |sfor all i, j ∈ N such that max{i, j} > n0 . Thus,Xi,js|xij |=n0X|xij |s +i,j=1

X|xij |smax{i,j}>n0≤ A+X|xij |qmax{i,j}>n0≤ A + ε,P 0|xij |s . Hence,
which leads us to the fact that x ∈ Ls , where A = ni,j=1Lq ⊂ Ls .
Besides one can easily deduce, by means of the suitable restrictions of theisomorphism T defined by (1.1) and taking into account the fact that thespace Cr0 consists of all sequences x = (xmn ) such that limmax{m,n}→∞ xmn =0, from the known inclusions ℓs ⊂ c0 ⊂ ℓ∞ for 1 ≤ s < ∞ thatT −1 (ℓs ) = Ls ⊂ Cr0 = T −1 (c0 ) ⊂ T −1 (ℓ∞ ) = Mu .
This step completes the proof.

THE SPACE Lq OF DOUBLE SEQUENCES153

The α-dual λα , β(υ)-dual λβ(υ) with respect to the υ-convergence forυ ∈ {p, bp, r} and the γ-dual λγ of a double sequence space λ are respectively
defined byXαλ := (aij ) ∈ Ω :|aij xij | < ∞ for all (xij ) ∈ λ ,i,jλβ(υ) :=andλγ :=






(aij ) ∈ Ω : υ −

Xi,j(aij ) ∈ Ω : supk,lXk,l∈N i,j=1


aij xij exists for all (xij ) ∈ λ
aij xij < ∞ for all (xij ) ∈ λ .
It is easy to see for any two spaces λ, µ of double sequences that µα ⊂ λα
whenever λ ⊂ µ and λα ⊂ λγ . Additionally, it is known that the inclusion
λα ⊂ λβ(υ) holds while the inclusion λβ(υ) ⊂ λγ does not hold, since theυ-convergence of the sequence of partial sums of a double series does not
imply its boundedness.
The space λ of double sequences is said to be solid if and only ifλ̃ = {(ukl ) ∈ Ω : ∃(xkl ) ∈ λ such that |ukl | ≤ |xkl | for all k, l ∈ N} ⊂ λ.
The space λ of double sequences is also said to be monotone if and only
if m0 λ ⊂ λ, where m0 is the span of the set of all sequences of zeros andones and m0 λ = {ax = (aij xij ) : a ∈ m0 , x ∈ λ}. If λ is monotone, thenλα = λβ(υ) (cf. Zeltser [11, p. 36]) and λ is monotone whenever λ is solid.
Prior to giving the theorem which asserts that the α- and γ-duals of asolid space of double sequences are identical, we quote two lemmas which
are needed in proving the theorem.
Lemma 2.3. [6, Theorem 2, p. 279] A positive term double series converges
to its l.u.b. (that is the l.u.b. of its partial sums) if it is bounded above.
Otherwise it diverges to +∞.
Lemma 2.4. [2, p. 382] A double series is absolutely convergent if and only
if the setm,nX|xij | : m, n ∈ N

i,j=1

is a bounded set of real numbers.

Now, we may give the theorem

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FEYZİ BAŞAR AND YURDAL SEVER

Theorem 2.5. If a given double sequence space λ is solid, then the equality
λα = λγ holds.
Proof. To prove the theorem, it is enough to show that the inclusion λγ ⊂ λα
holds. Suppose that the sequence space λ is solid and take any y = (ykl ) ∈λγ . Then,m,nX

supm,n∈N k,l=1

xkl ykl < ∞for any x = (xkl ) ∈ λ. Now, define the sequence z = (zkl ) via the sequencex = (xkl ) ∈ λ by zkl := xkl sgn(xkl ykl ) for all k, l ∈ N. Then, z = (zkl ) ∈ λsince λ is solid and |zkl | ≤ |xkl | for all k, l ∈ N. Therefore,
supm,nXm,n∈N k,l=1|xkl ykl | ==supm,nX

xkl ykl sgn(xkl ykl )m,nXykl zkl < ∞.

m,n∈N k,l=1

supm,n∈N k,l=1

PThis shows that the positive term double seriesk,l |xkl ykl | is bounded
which is convergent by Lemma 2.3. Therefore, one can see by Lemma 2.4that (xkl ykl )k,l∈N ∈ L1 . Since x ∈ λ is arbitrary, y must be in λα , i.e., theinclusion λγ ⊂ λα holds.
This step terminates the proof.
As an easy consequence of Theorem 2.5, we have
Corollary 2.6. If λ is solid then λα = λβ(υ) = λγ .
One can easily observe that the double sequence space Lq is solid, ifq > 1. This yields to us that the double sequence space Lq is monotone
which implies the fact that the α- and the β(υ)-duals of the space Lq areidentical.
Now, we may give the theorem on the β(υ)-dual of the space Lq .
Theorem 2.7. The β(υ)-dual of the space Lq is the space Lq′ , where q > 1
and q −1 + q ′−1 = 1.
Proof. Let q > 1 and q −1 + q ′−1 = 1. Let us take any x ∈ Lq′ and y ∈ Lq .
Consider the inequalities′|xmn |q|ymn |q′|xmn ymn | ≤+≤ |xmn |q + |ymn |q′qq

THE SPACE Lq OF DOUBLE SEQUENCES

155

satisfied for all m, n ∈ N. Therefore, we derive thatXXX′|xmn ymn | ≤|xmn |q +|ymn |q < ∞,m,nm,nm,nwhich leads us to the fact that x ∈ Lαq , i.e., the inclusions
Lq′ ⊂ Lαq ⊂ Lβ(υ)
q(2.2)hold.
β(υ)
Conversely, take any y = (ymn ) ∈ Lq . For establishing the inclusionβ(υ)⊂ Lq′ , we use the analogous idea employing by Boos [3, p. 344,
Lq
Theorem 7.1.11.c] for single sequences. Let us consider the linear functional
fn and the double sequence y [n] defined by
fn :andLq−→ RP
x = (xkl ) 7−→ fn (x) := nk,l=1 xkl ykl

y11
 y21

 y31
 .
[n]y =  ..
 yn1 0..
.
y12y22y32..
.
yn20..
.

y13 · · ·y23 · · ·y33 · · ·..
..
.
.
yn3 · · ·0 ···
..
. ···

y1ny2ny3n..
.
ynn0..
.

000..
.

······ ··· ··· 
0 ··· 0 ··· .. . .
.
.

for every n ∈ N. Then, since y [n] ∈ Lq′ , we obtain by Hölder’s inequalitythat|fn (x)| ≤n
X|xkl ykl | =k,l=1

X[n]xkl ykl ≤ kxkq y [n]k,lq′

for each x = (xkl ) ∈ Lq which yields the continuity of the linear functionals
fn . Therefore, we have
(2.3)

kfn k ≤ y [n]q′for each n ∈ N.
(n)

Let us consider the sequence x(n) = {xkl }k,l∈N to prove the reverse inequality, defined by(′|ykl |q(n), (if ykl 6= 0, and k, l ≤ n),ykl
xkl :=0,(otherwise).

156

FEYZİ BAŞAR AND YURDAL SEVERThen, it is clear that x(n) ∈ Lq and one can see that1/q 
1/q
nnX
X(n)
(q ′ −1)q q′ x=|ykl |==y [n]|ykl |qk,l=1k,l=1q′q′ /q.

This leads us to the consequence for all n ∈ N thatPnq′
fn x(n)k,l=1 |ykl |== y [n] ′ .
(n)(n)
qxx
qqHence,y [n](2.4)

q′≤ kfn k for all n ∈ N.

Therefore, we have by (2.3) and (2.4) that
kfn k = y [n]q′for all n ∈ N.

By applying the Banach-Steinhauss Theorem, one can observe by our hypothesis that the sequence (fn ) of linear functionals converges pointwise.
Since (Lq , k · kq ) and (C, | · |) are the Banach spaces, the linear functional
defined byfy :Lq−→ RPx = (xkl ) 7−→ fy (x) := limn→∞ fn (x) = k,l xkl ykl

is continuous, andkfy k ≤ sup kfn k = sup y [n]n∈Nn∈Nq′<∞holds. Thus, we have y ∈ Lq′ , because of
1/q′1/q′ nXX′′
kfy k ≤ sup y [n] ′ = sup < ∞.
=|ykl |q |ykl |q n∈N

qn∈Nk,l=1k,lThat is to say that the inclusion
(2.5)

Lβ(υ)
⊂ Lq ′qholds.
By combining the inclusions (2.2) and (2.5), the desired result immediately follows.
This completes the proof.
As a direct consequence of Theorem 2.7, we have
Corollary 2.8. The α-, β(υ)- and γ-duals of the space Lq are the space
Lq′ , where q > 1 and q −1 + q ′−1 = 1.

THE SPACE Lq OF DOUBLE SEQUENCES

157

Acknowledgements
We have benefited a lot from discussions with Professor Bilâl Altay,
Matematik Eğitimi Bölümü, İnönü Üniversitesi, Malatya-44280/Türkiye,about this work. We would like to express our gratitude for his valuable
helps. Finally, we thank to the reviewer for his/her careful reading andmaking a useful comment which improved the presentation and the readability of the paper.
References[1] B. Altay, F. Başar, Some new spaces of double sequences, J. Math. Anal. Appl.
309(1)(2005), 70–90.
[2] R.G. Bartle, The Elements of Real Analysis, John Wiley & Sons Inc., New York, 1964.
[3] J. Boos, Classical and Modern Methods in Summability, Oxford University Press Inc.,New York, 2000.
PB[4] A. Gökhan, R. Çolak, The double sequence spaces cP
2 (p) and c2 (p), Appl. Math.
Comput. 157(2)(2004), 491–501.
[5] A. Gökhan, R. Çolak, Double sequence space ℓ∞
2 (p), ibid. 160(1)(2005), 147–153.
[6] V.G. Iyer, Mathematical Analysis, Tata McGraw-Hill Publishing Company Ltd., NewDelhi, 1985.
[7] F. Móricz, Extensions of the spaces c and c0 from single to double sequences, Acta
Math. Hung. 57(1-2)(1991), 129–136.
[8] Mursaleen, O.H.H. Edely, Statistical convergence of double sequences, J. Math. Anal.
Appl. 288(1)(2003), 223–231.
[9] Mursaleen, Almost strongly regular matrices and a core theorem for double sequences,J. Math. Anal. Appl. 293(2)(2004), 523–531.
[10] Mursaleen, O.H.H. Edely, Almost convergence and a core theorem for double sequences, J. Math. Anal. Appl. 293(2)(2004), 532–540.
[11] M. Zeltser, Investigation of Double Sequence Spaces by Soft and Hard Analitical Methods, Dissertationes Mathematicae Universitatis Tartuensis 25, Tartu University Press,
Univ. of Tartu, Faculty of Mathematics and Computer Science, Tartu, 2001.
Feyzi BaşarFatih Üniversitesi,Fen- Edebiyat Fakültesi,
Matematik Bölümü,Büyükçekmece Kampüsü,34500-İstanbul, Türkiye
e-mail address: fbasar@fatih.edu.tr, feyzibasar@gmail.com
Yurdal Sever
Fen Lisesi Matematik Öğretmeni,Karakavak, 44110-Malatya, Türkiye
e-mail address: ysever@hotmail.com(Received May 22, 2007 )(Revised June 19, 2007 )