THE SPACE L q OF DOUBLE SEQUENCES

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Volume51,Issue1 2009 Article10

THE SPACE L _q OF DOUBLE SEQUENCES

Feyzi Basar

^∗

Yurdal Sever

^†

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THE SPACE L _q OF DOUBLE SEQUENCES

Feyzi Basar and Yurdal Sever

Abstract

The spaces BS, BS(t), CSp, CSbp, CSrand BV of double sequences have recently been studied by Altay and Ba¸sar [J. Math. Anal. Appl. 309(1)(2005), 70–90]. In this work, following Altay and Ba¸sar [1], we introduce the Banach space Lqof double sequences corresponding to the well- known space`q of single sequences and examine some properties of the space Lq. Furthermore, we determine the β(υ)-dual of the space and establish that the α- andγ-duals of the space Lq

coincide with theβ(υ)-dual; where 1≤q<∞andυ2{p, bp, r}.

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

THE SPACE L_q OF DOUBLE SEQUENCES

Feyzi BAS¸AR and Yurdal SEVER

Abstract. The spaces BS, BS(t), CSp, CSbp, CSr and BV of double sequences have recently been studied by Altay and Ba¸sar [J. Math. Anal.

Appl. 309(1)(2005), 70–90]. In this work, following Altay and Ba¸sar [1], we introduce the Banach spaceLq of double sequences corresponding to the well-known spaceℓqof single sequences and examine some properties of the spaceLq. Furthermore, we determine theβ(υ)-dual of the space and establish that theα- and γ-duals of the space Lq coincide with the β(υ)-dual; where 1≤q <∞andυ∈ {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 scalar multiplication. Any vector subspaces ofwand Ω are called as thesingle and double sequence spaces, respectively. The space Mu of all bounded double sequences is defined by

M_u :=

(

x= (x_mn)∈Ω : kxk_∞ = sup

m,n∈N

|x_mn|<∞ )

,

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 >0 there exists n₀ =n₀(ε) ∈N and l∈R such that

|x_mn−l| < ε

for allm, n > n₀ then we call that the double sequencexisconvergent in the Pringsheim’s sense to the limit land writep−limx_mn =l; where Rdenotes the real field. By Cp we denote the space of all convergent double sequences in the Pringsheim’s sense. It is well-known that there are such sequences in the space C_p but not in the space M_u. So, we can consider the space C_bp of the double sequences which are both convergent in the Pringsheim’s sense and bounded, i.e., C_bp = C_p ∩ M_u. A sequence in the space C_p is said to be regularly convergent if it is a single convergent sequence with respect to each index and the set of all such sequences denoted byC_r. Also byC_bp0 and Cr0, we denote the spaces of all double sequences converging to 0 contained

Mathematics 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.

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150 FEYZ˙I BAS¸AR AND YURDAL SEVER

in the sequence spaces C_bp and Cr, respectively. M´oricz [7] proved that C_bp, C_bp0, C_r and C_r0 are Banach spaces with the norm k · k_∞.

Let us consider the isomorphism T which plays an essential role for the present study, defined by Zeltser [11, p. 36] as

T : Ω −→ w

x 7−→ z = (z_i) := (x_ψ⁻¹_(i)), (1.1)

where ψ :N×N→N is a bijection defined by ψ[(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 = (x_mn) and define the sequence s = (smn) via x by

s_mn :=

m,n

X

i,j=1

x_ij ; (m, n∈N), (1.2)

which will be used throughout. For the sake of brevity, here and in what follows, we abbreviate the summations P_∞

i=1

P_∞

j=1, P_m

i=1

P_n

j=1 and P_n

i=1

P_n

j=1 by P

i,j, Pm,n

i,j=1 and P_n

i,j=1, respectively. Then the pair (x, s) and the sequence s = (s_mn) are called as a double series and the sequence of partial sums of the double series, respectively. Let λ be the space of double sequences, converging with respect to some linear convergence rule υ −lim : λ → R. The sum of a double series P

i,jx_ij with respect to this rule is defined by υ−P

i,jx_ij :=υ−lims_mn.

Let us define the following sets of double sequences:

Mu(t) :=

(

(xmn)∈Ω : sup

m,n∈N

|xmn|^t^mn <∞ )

,

C_p(t) :=

(x_mn) ∈Ω :p− lim

m,n→∞|x_mn−l|^t^mn = 0 for some l ∈C

,

C_0p(t) :=

(x_mn) ∈Ω : p− lim

m,n→∞|x_mn|^t^mn = 0

,

Lu(t) :=

(

(xmn) ∈Ω :X

m,n

|xmn|^t^mn <∞ )

,

2 Mathematical Journal of Okayama University, Vol. 51 [2009], Iss. 1, Art. 10

http://escholarship.lib.okayama-u.ac.jp/mjou/vol51/iss1/10

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THE SPACE Lq OF DOUBLE SEQUENCES 151

C_bp(t) :=Cp(t)∩ Mu(t) and C_0bp(t) :=C0p(t)∩ Mu(t);

wheret= (t_mn) is the sequence of strictly positive realst_mn for allm, n ∈N. In the case t_mn = 1 for allm, n ∈N; M_u(t), C_p(t), C_0p(t), L_u(t), C_bp(t) and C_0bp(t) reduce to the sets Mu, Cp, C0p, Lu, C_bp and C_0bp, respectively. Now, we may summarize the knowledge given in some document related to the double sequence spaces. Gökhan and Ç olak [4, 5] have proved that Mu(t) and C_p(t), C_bp(t) are complete paranormed spaces of double sequences and gave the α-, β-, γ-duals of the spaces M_u(t) and C_bp(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 applied these matrices to establish a core theorem and introduced the M-core for double sequences and determined those four dimensional matrices trans- forming every bounded double sequence x = (x_jk) into one whose core is a subset of the M-core of x. More recently, Altay and Ba¸sar [1] have defined the spacesBS, BS(t),CS_p, CS_bp, CS_rand BV of double sequences consisting of all double series whose sequence of partial sums are in the spaces Mu, M_u(t), C_p, C_bp, C_r and L_u, respectively, and also examined some properties of 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 L_q L_q :=







(x_ij)∈Ω : X

i,j

|x_ij|^q <∞







, (1≤q < ∞)

of double sequences corresponding to the space ℓ_q of single sequences and examine some properties of the space.

2. The Double Sequence Space L_q

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

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Theorem 2.1. The set Lq becomes a linear space with the coordinatewise addition and scalar multiplication and L_q is a Banach space with the norm

kxk_q =



 X

i,j

|x_ij|^q





1/q

(2.1) ,

where 1 ≤q <∞.

Proof. The proof of the first part of the theorem is a routine verification and so we omit the detail.

Furthermore, the statement ”a sequence space ν is a Banach space with the norm k · k_ν if and only if the sequence space T⁻¹(ν) = λ is a Banach space with the norm k · k_λ” holds by Boos [3, Corollary 6.3.41]. Therefore, the restriction of the transformation defined by (1.1) to the space L_q which is norm preserving isomorphism yields the fact that L_q =T⁻¹(ℓ_q) is also a Banach space with the norm k · k_q defined by (2.1) because of ℓ_q is a Banach space.

This step concludes the proof.

Theorem 2.2. Let 1 ≤q < s <∞. Then, the inclusions L_q ⊂ L_s ⊂ C_r0 ⊂ M_u hold.

Proof. Let us take any x = (x_ij) ∈ L_q. Then, P

max{i,j}>n⁰|x_ij|^q < ε < 1 for sufficiently large n₀ ∈ N. Since q < s, it is obvious that |x_ij|^q ≥ |x_ij|^s for all i, j ∈N such that max{i, j} > n₀. Thus,

X

i,j

|x_ij|^s =

n0

X

i,j=1

|x_ij|^s+ X

max{i,j}>n⁰

|x_ij|^s

≤ A+ X

max{i,j}>n0

|xij|^q

≤ A+ε,

which leads us to the fact that x ∈ L_s, where A = P_n0

i,j=1|x_ij|^s. Hence, L_q ⊂ L_s.

Besides one can easily deduce, by means of the suitable restrictions of the isomorphism T defined by (1.1) and taking into account the fact that the spaceC_r0 consists of all sequencesx= (x_mn) such that lim_{max{m,n}→∞}x_mn = 0, from the known inclusions ℓ_s ⊂c₀ ⊂ℓ_∞ for 1≤s <∞ that

T⁻¹(ℓ_s) = L_s ⊂ C_r0 =T⁻¹(c₀)⊂T⁻¹(ℓ_∞) =M_u.

This step completes the proof.

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The α-dual λ^α, β(υ)-dual λ^β(υ) with respect to the υ-convergence for υ ∈ {p, bp, r}and theγ-dual λ^γ of a double sequence spaceλare respectively defined by

λ^α :=







(a_ij) ∈Ω :X

i,j

|a_ijx_ij|<∞ for all (x_ij) ∈λ





 ,

λ^β(υ) :=







(a_ij)∈Ω :υ−X

i,j

a_ijx_ij exists for all (x_ij)∈λ





 and

λ^γ :=







(a_ij)∈Ω : sup

k,l∈N

k,l

X

i,j=1

a_ijx_ij

<∞ for all (x_ij) ∈λ





 .

It is easy to see for any two spaces λ, µ of double sequences that µ^α ⊂ λ^α whenever λ ⊂µ and λ^α ⊂ λ^γ. Additionally, it is known that the inclusion λ^α ⊂ λ^β(υ) holds while the inclusion λ^β(υ) ⊂ λ^γ does not hold, since the υ-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 if λ˜ ={(u_kl) ∈Ω : ∃(x_kl)∈λ such that |u_kl| ≤ |x_kl| for all k, l ∈N} ⊂λ.

The space λ of double sequences is also said to be monotone if and only if m₀λ ⊂ λ, where m₀ is the span of the set of all sequences of zeros and ones and m₀λ = {ax = (a_ijx_ij) : a ∈ m₀, x ∈ λ}. If λ is monotone, then λ^α =λ^β(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 a solid 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

i,j=1

|x_ij| :m, n ∈N





 is a bounded set of real numbers.

Now, we may give the theorem

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Theorem 2.5. If a given double sequence space λ is solid, then the equality λ^α =λ^γ holds.

Proof. To prove the theorem, it is enough to show that the inclusionλ^γ ⊂λ^α holds. Suppose that the sequence space λ is solid and take any y = (y_kl) ∈ λ^γ. Then,

sup

m,n∈N

m,n

X

k,l=1

x_kly_kl

<∞

for any x = (xkl) ∈λ. Now, define the sequence z = (zkl) via the sequence x = (x_kl) ∈λ by z_kl := x_klsgn(x_kly_kl) for all k, l ∈N. Then, z = (z_kl) ∈λ since λ is solid and |z_kl| ≤ |x_kl| for all k, l ∈N. Therefore,

sup

m,n∈N m,n

X

k,l=1

|x_kly_kl| = sup

m,n∈N m,n

X

k,l=1

x_kly_klsgn(x_kly_kl)

= sup

m,n∈N

m,n

X

k,l=1

y_klz_kl

<∞.

This shows that the positive term double series P

k,l|x_kly_kl| is bounded which is convergent by Lemma 2.3. Therefore, one can see by Lemma 2.4 that (x_kly_kl)_k,l∈N ∈ L₁. Since x ∈ λ is arbitrary, y must be in λ^α, i.e., the inclusion λ^γ ⊂λ^α holds.

This step terminates the proof.

As an easy consequence of Theorem 2.5, we have Corollary 2.6. If λ is solid then λ^α =λ^β(υ) =λ^γ.

One can easily observe that the double sequence space L_q is solid, if q > 1. This yields to us that the double sequence space L_q is monotone which implies the fact that the α- and the β(υ)-duals of the space L_q are identical.

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 L_q is the space L_q^′, where q >1 and q⁻¹+q^′−1 = 1.

Proof. Let q > 1 and q⁻¹ +q^′−1 = 1. Let us take any x∈ L_q^′ and y ∈ Lq. Consider the inequalities

|x_mny_mn| ≤ |x_mn|^q^′

q^′ + |y_mn|^q

q ≤ |x_mn|^q^′ +|y_mn|^q

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satisfied for all m, n∈N. Therefore, we derive that X

m,n

|x_mny_mn| ≤ X

m,n

|x_mn|^q^′ +X

m,n

|y_mn|^q <∞,

which leads us to the fact that x∈ L^α_q, i.e., the inclusions L_q^′ ⊂ L^α_q ⊂ L^β(υ)_q

(2.2) hold.

Conversely, take any y = (y_mn) ∈ L^β(υ)_q . For establishing the inclusion L^β(υ)_q ⊂ L_q^′, we use the analogous idea employing by Boos [3, p. 344, Theorem 7.1.11.c] for single sequences. Let us consider the linear functional f_n and the double sequence y^[n] defined by

f_n : Lq −→ R

x= (x_kl) 7−→ f_n(x) :=P_n

k,l=1x_kly_kl and

y^[n] =







y₁₁ y₁₂ y₁₃ · · · y_1n 0 · · · y₂₁ y₂₂ y₂₃ · · · y_2n 0 · · · y₃₁ y₃₂ y₃₃ · · · y_3n 0 · · · ... ... ... . .. ... ... · · · y_n1 y_n2 y_n3 · · · y_nn 0 · · · 0 0 0 · · · 0 0 · · · ... ... ... · · · ... ... . ..







for every n ∈ N. Then, since y^[n] ∈ L_q^′, we obtain by H¨older’s inequality that

|f_n(x)| ≤

n

X

k,l=1

|x_kly_kl|=X

k,l

x_kly^[n]_kl

≤ kxk_q y^[n]

q^′

for each x= (x_kl)∈ L_q which yields the continuity of the linear functionals f_n. Therefore, we have

kf_nk ≤ y^[n]

q^′ for each n∈N. (2.3)

Let us consider the sequence x⁽ⁿ⁾ = {x⁽ⁿ⁾_kl }_k,l∈N to prove the reverse inequality, defined by

x⁽ⁿ⁾_kl :=

( |ykl|^q^′

ykl , (if y_kl 6= 0, and k, l ≤n),

0 , (otherwise).

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Then, it is clear that x⁽ⁿ⁾ ∈ Lq and one can see that

x⁽ⁿ⁾

q =





n

X

k,l=1

|y_kl|^(q^′^−1)q





1/q

=





n

X

k,l=1

|y_kl|^q^′





1/q

=

y^[n]

q^′

q^′/q

. This leads us to the consequence for all n∈N that

f_n x⁽ⁿ⁾

x⁽ⁿ⁾ q

= P_n

k,l=1|y_kl|^q^′

x⁽ⁿ⁾ q

= y^[n]

q^′. Hence,

y^[n]

q^′ ≤ kf_nk for all n ∈N. (2.4)

Therefore, we have by (2.3) and (2.4) that kf_nk =

y^[n]

q^′ for all n∈N.

By applying the Banach-Steinhauss Theorem, one can observe by our hy- pothesis that the sequence (f_n) of linear functionals converges pointwise.

Since (Lq,k · kq) and (C,| · |) are the Banach spaces, the linear functional defined by

f_y : L_q −→ R

x= (xkl) 7−→ f_y(x) := limn→∞f_n(x) = P

k,lx_kly_kl is continuous, and

kf_yk ≤ sup

n∈N

kf_nk = sup

n∈N

y^[n]

q^′ <∞ holds. Thus, we have y ∈ L_q^′, because of

kf_yk ≤ sup

n∈N

y^[n]

q^′ = sup

n∈N





n

X

k,l=1

|y_kl|^q^′





1/q^′

=



 X

k,l

|y_kl|^q^′





1/q^′

<∞.

That is to say that the inclusion

L^β(υ)_q ⊂ L_q^′ (2.5)

holds.

By combining the inclusions (2.2) and (2.5), the desired result immedi- ately 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 L_q are the space L_q^′, where q >1 and q⁻¹+q^′−1 = 1.

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Acknowledgements

We have benefited a lot from discussions with Professor Bilâl Altay, Matematik E˘gitimi Bölümü, ˙Inönü 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 and making a useful comment which improved the presentation and the read- ability of the paper.

References

[1] B. Altay, F. Ba¸sar, 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.

[4] A. G¨okhan, R. C¸ olak, The double sequence spaces c^P2(p) and c^{P B}2 (p), Appl. Math.

Comput.157(2)(2004), 491–501.

[5] A. G¨okhan, R. C¸ olak, Double sequence space ℓ^∞2 (p), ibid.160(1)(2005), 147–153.

[6] V.G. Iyer,Mathematical Analysis, Tata McGraw-Hill Publishing Company Ltd., New Delhi, 1985.

[7] F. M´oricz, 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 Meth- ods, Dissertationes Mathematicae Universitatis Tartuensis 25, Tartu University Press, Univ. of Tartu, Faculty of Mathematics and Computer Science, Tartu, 2001.

Feyzi Bas¸ar Fatih ¨Universitesi, Fen- Edebiyat Fak¨ultesi,

Matematik Bölümü, Büyükçekmece Kampüsü,

34500-˙Istanbul, T¨urkiye

e-mail address: [email protected], [email protected] Yurdal Sever

Fen Lisesi Matematik ¨O˘gretmeni, Karakavak, 44110-Malatya, T¨urkiye

e-mail address: [email protected] (Received May 22, 2007)

(Revised June 19, 2007)