For any vector space X, we call a function g:X↦ℝ+ a paranorm on X if g satisfies the following conditions:

• 1. g(θ)=0, where 𝜃 is the zero element in X,

• 2. g(x)=g(−x) for all x∈X,

• 3. g(x+y)≤g(x)+g(y) for all x,y∈X

• 4. If {αn} is a sequence of scalars with |αn−α|→0 and {xn} is a sequence of vectors with g(xn−x)→0, then g(αnxn−αx)→0.

A paranormed space is a pair (X,g) of a vector space X and a paranorm g on X. If (X,g) is a paranormed space, then the function d:X×X→ℝ defined by d(x,y)=g(x-y) is a pseudometric on X, and hence it becomes a metric on the set X/ of all equivalence classes of elements of X under the equivalence relation ∼ on X defined by x y⇔d(x,y)=0. With this consideration, every paranormed space can be regarded as a metric space.

Let p={pk} be a bounded sequence of real numbers such that pk≥1 for all k∈ℕ, and let M=max{1,suppk}. It is well-known that the following sequence spaces, defined by Maddox [11] and known as the sequence spaces of Maddox (see further in [20] and [14]):

c0(p)=xk:|xk|pk→0 as k→∞,c(p)=xk:|xk−l|pk→0 as k→∞ for some l∈ℝ,l∞(p)=xk:supk|xk|pk<∞,l(p)=xk:∑k|xk|pk<∞,

are complete paranormed spaces, where the first three spaces are equipped with the paranorm g₁ defined by

g1{xk}=supk|xk|pkM,

and the last one is equipped with the paranorm g₂ defined by

g2{xk}=∑ k|xk|pk1M.

We see that when p_k=p for all k, the above sequence spaces of Maddox become the classical Banach sequence spaces

c0={xk}:|xk|→0 as k→∞,c={xk}:|xk| is convergent,l∞={xk}:supk|xk|<∞,andlp={xk}:∑ k |xk|p<∞.

Let E be a subspace of the vector space of all X -valued sequences, called an X -valued sequence space. The α -dual and β -dual of E introduced by Maddox [13] are defined as follows:

Eα=Ak⊆L(X,Y):∑kAkxk converges for all xk∈E,Eβ=Ak⊆L(X,Y):∑k‖Akxk‖<∞ for all xk∈E,

where L(X,Y) is the set of all linear operator from a normed space X into a normed space Y.

There have been several works on the notions of α -dual and β -dual defined by Maddox mentioned above (see [6], [7], [8], and [10] for some references). Grosse-Erdmann [6] investigated some topological and sequence structure of scalar-valued sequence spaces of Maddox. In 2002, Suantai and Sanhan [21] provided some general properties of β -dual of the vector-valued sequence spaces of Maddox, and gave characterizations of β -dual of the sequence spaces l(p) when pk>1 for all k∈ℕ. In 2012, Rakbud and Suantai [18] gave a general theorem on duality for a class of Banach-valued function spaces which is a generalization of the classical sequence space lp for 1≤p<∞. The β -dual of Banach-space-valued difference sequence spaces E(Δ)={xk}:{Δxk}∈E where E={l∞,c,c0} and Δxk=xk−xk+1 for all k∈ℕ, was studied by Bhardwaj and Gupta [2].

For any Banach space X, an X -valued sequence space E is called a BK -space if it is a Banach space and the k -th coordinate mapping pk:E→X, pk(x)=xk, is continuous for all k∈ℕ. In 2013, Faroughi, Osgooei and Rahimi provided some properties of α -dual and β -dual of a BK -space. Furthermore, the concepts of α -dual and β -dual of a BK -space were used by the same authors in their other works to define some new spaces (see [4], [5], and [15]).

Let 1≤p,q<∞. An infinite scalar matrix A=[ajk] is said to define a linear operator from l_p into l_q if for every {xk} in l_p, the ∑kajkxk converges for all j and the sequence Ax=∑kajkxk is in l_q. If a matrix A defines a linear operator from l_p into l_q, we call the operator x↦Ax the linear operator defined by A. By the uniform boundedness principle, the linear operator defined by A is bounded. Let B(lp,lq) be the set of all bounded linear operator from l_p into l_q. Then B(lp,lq) is the Banach space, and it is isometrically isomorphic to the space of matrices defining a linear operator from l_p into l_q endowed with the operator norm on B(lp,lq).

For any two matrices A=[ajk] and C=[cjk] of the same size, the Schur product of A and C is the matrix A•C given by A•C=[ajkcjk]. Schur [19] showed that the Banach space B(l2) is a commutative Banach algebra under the operator norm and the Schur product multiplication. The Banach space B(lp,lq) under the Schur product operation is a Banach algebra proven by Bennett [1]. Let 1≤r<∞, for a fixed Banach algebra B with identity, Chaisuriya and Ong [3] considered the space of matrices

Sp,qr(B)=A:A[r]∈B(lp,lq)

where A[r]=‖ajk‖r. They obtained that it is a Banach algebra under the absolute Schur r -norm defined by ‖A‖p,q,r=‖A[r]‖p,q1r, and also proved that Sp,q2(ℂ) contains B(lp,lq) as an ideal. In 2001, Livshits, Ong and Wang [9] studied the duality in the absolute Schur algebra S2,2r(ℂ) by a way analogous to Dixmier’s theorem and Schatten’s theorem. After that, Rakbud and Chaisuriya [17] generalized the results of Livshits, Ong and Wang [9] to the absolute Schur algebra SΛ,Σ2(B), where Λ,Σ∈{lp:1≤p<∞}∪{c0}, which was examined by the same authors in 2005 (see [16]).

In this work, we extend the definition of the set Sp,qr(B) defined by Chaisuriya and Ong [3] from the fixed real number r, which is greater than or equal to 1, to a fixed bounded matrix R=[rjk] of scalars, which is greater than or equal to 1. Hence our setting becomes

Sp,qR(B)=A:A[R]∈B(lp,lq)

where A[R]=‖ajk‖rjk. Our goal is to define a paranorm on the vector space Sp,qr(B) and investigate the properties of this paranormed space, including its existence, completeness, and duality.

In this section, the two versions of the Minkowski's inequality that Maddox demonstrated in [12] are mentioned to achieve our results.

Theorem 2.1. Let p≥1, a1,a2,⋯,an≥0 and b1,b2,⋯,bn≥0. Then

∑ k=1n(ak+bk)p1p≤∑ k=1nakp1p+∑ k=1nbkp1p.

Theorem 2.2. Let 0<p≤1, a1,a2,⋯,an≥0 and b1,b2,⋯,bn≥0. Then

∑ k=1n(ak+bk)p≤∑ k=1nakp+∑ k=1nbkp.

In the following theorems, we investigate some elementary properties of Sp,qR(B).

Theorem 2.3. Sp,qR(B) is a linear space.

Proof. Let A=[ajk] and B=[bjk] be matrices in Sp,q[R](B). Then A[R] and B[R]∈B(lp,lq). Let M=max(1,supj,krjk). For any fixed positive integers J and K, and any fixed unit vector x={xk}∈lp, by Theorem 2.1 and Theorem 2.2,

∑j=1J ∑k=1K ‖a jk+b jk‖ r jk|xk|q=∑j=1J ∑k=1K ‖a jk+b jk ‖ r jk |xk| 1MMq=∑j=1J ∑k=1K ‖a jk+b jk ‖ r jkM|xk | 1MMq≤∑j=1J ∑k=1K ‖a jk ‖ r jkM|xk | 1M+‖b jk ‖ r jkM|xk | 1MMq≤∑j=1J ∑k=1K ‖a jk ‖ rjk|xk| 1M+ ∑k=1K ‖b jk ‖ rjk|xk| 1M qM≤ ∑ j=1J ∑ k=1 K ‖ a jk ‖ r jk | x k | q 1 qM + ∑ j=1J ∑ k=1 K ‖ b jk ‖ r jk | x k | q 1 qM qM≤ ∑ j=1J ∑ k=1 K ‖ a jk ‖ r jk | x k | q 1 qM + ∑ j=1J ∑ k=1 K ‖ b jk ‖ r jk | x k | q 1 qM qM≤ ∑ j=1J ∑ k=1 K ‖ a jk ‖ r jk | x k | q 1 q + ∑ j=1J ∑ k=1 K ‖ b jk ‖ r jk | x k | q 1 q qM≤‖(A[R] )(|x|)‖ q+‖(B[R] )(|x|)‖ q qM≤‖A[R] ‖p,q‖x‖p+‖B[R] ‖p,q‖x‖p qM≤‖A[R] ‖p,q+‖B[R] ‖p,q qM.

Since J and K are arbitrary, we have

‖(A+B)[R]x‖qq=∑ j=1∞∑k=1∞‖a jk+b jk‖ r jk xkq      ≤∑ j=1∞∑k=1∞‖a jk+b jk‖ r jk |xk|q      ≤ ‖A [R] ‖ p,q+‖B [R] ‖ p,q qM.

This implies that ‖(A+B)[R]x‖q≤‖A[R] ‖p,q+‖B[R] ‖p,qM. Since A^[R] and B[R]∈B(lp,lq), ‖(A+B)[R]‖p,q<∞. So A+B∈Sp,qR(B). Next, we let α∈B. For any fixed unit vector x={xk}∈lp, we obtain

(αA)[R]xq≤∑ j=1∞∑k=1∞ ‖αa jk ‖ r jk |xk| q1q    ≤∑ j=1∞∑k=1∞ ‖α‖M ‖a jk‖ r jk |xk| q1q    ≤‖α‖M‖(A[R])|x|‖q    ≤‖α‖M‖A[R]‖p,q

Since A[R]∈B(lp,lq), ‖(αA)[R]‖p,q<∞. Then αA∈Sp,qR(B). By the termwise sum and any scalar multiple of any matrices in Sp,qR(B), for any matrices A,B,C∈Sp,qR(B) and any scalars α,β∈B,

• 1. (A+B)[R]=(B+A)[R],

• 2. (A+(B+C))[R]=((A+B)+C)[R],

• 3. there exists 0_∈Sp,q[R](B) such that (A+0_)[R]=A[R],

• 4. there is −A∈Sp,qR(B) such that (A+(−A))[R]=0_,

• 5. (α(βA))[R]=((αβ)A)[R],

• 6. ((α+β)A)[R]=(αA+βA)[R],

• 7. (α(A+B))[R]=(αA+αB)[R],

• 8. since the identity 1∈B, (1A)[R]=A[R].

This completes the proof.

We define G:Sp,qR(B)→ℝ+ by

G(A):=‖A[R]‖p,q1M

where M=max(1,supj,krjk).

Theorem 2.4. Sp,qR(B) equipped with G(⋅) is a paranormed space.

Proof. It is obvious that G(0_)=0. Since ‖A[R]‖p,q=‖(−A)[R]‖p,q, G(A)=G(−A) for all A∈Sp,qR(B). Next, let A,B∈Sp,qR(B). To show G(A+B)≤G(A)+G(B), we use the same technique of the proof in Theorem 2.3. For any fixed positive integers J and K, and any unit vector x={xk}∈lp,

∑ j=1J ∑k=1K‖ajk+bjk ‖rjk|xk|q1M=∑ j=1J ∑k=1K ‖ajk+bjk ‖ rjk M |xk | 1 M Mq1M≤∑ j=1J ∑k=1K ‖ajk ‖ rjk M |xk | 1 M +‖bjk ‖ rjk M |xk | 1 M Mq1M≤∑ j=1J ∑ k=1K ‖ajk ‖rjk|xk| 1 M+ ∑ k=1K ‖bjk ‖rjk|xk| 1 MqM1M≤ ∑ j=1J ∑ k=1K ‖ajk ‖rjk|xk|q 1qM +∑ j=1J ∑ k=1K ‖bjk ‖rjk|xk|q 1qM q≤‖(A[R] )|x|‖ q 1 M +‖(B[R] )|x|‖ q 1 M q≤‖A[R] ‖p,q 1 M +‖B[R] ‖p,q 1 M q.

Therefore

‖(A+B)[R]x‖qqM= ∑j=1∞ ∑k=1∞ ‖a jk+b jk‖ rjk xk q 1M      ≤ ∑j=1∞ ∑k=1∞ ‖a jk+b jk‖ rjk |xk| q 1M      ≤ ‖A [R] ‖ p,q 1 M +‖B [R] ‖ p,q 1 M q

which implies that ‖(A+B)[R]‖p,q1M≤‖A[R]‖p,q1M+‖B[R]‖p,q1M. Thus G(A+B)≤G(A)+G(B). Finally, we assume that {αn} is a sequence in B such that ‖αn−α‖→0 and {An=[ajk(n)]} is a sequence in Sp,qR(B) with G(An−A)→0 as n→∞. We claim that ‖(An)[R]‖p,q1M is bounded. Since G(An−A)→0 as n→∞, there are T∈ℕ and K>0 such that for all n≥T,

(2.1)G(An−A)<K.

Because of finiteness of G(A), there is L>0 such that G(A)<L. By (2.1), we see that for all n≥T,

‖(An)[R]‖p,q1M=G(An)    =G(An−A+A)    ≤G(An−A)+G(A)    ≤K+L.

So we get the claim. Next, we want to show that G(αnAn−αA)→0 as n→∞. Let ε>0. Since {An} is a sequence in Sp,qR(B), by the boundedness of ‖(An)[R]‖p,q1M, there exists J>0 such that

‖(An)[R]‖p,q1M<J for  all  n∈ℕ.

Because ‖αn−α‖→0 as n→∞, there is Q∈ℕ such that for all n≥Q,

‖αn−α‖<ε2J1q.

By the assumption that G(An−A)→0 as n→∞, there is P∈ℕ such that for all n≥P,

G(An−A)<ε2‖α‖q.

That is

‖(An−A)[R]‖p,q1M<ε2‖α‖q.

Choose N=max(Q,P) and let n≥N,

‖(αnAn−αA)[R]‖p,q1M   =‖(αnAn−αAn+αAn−αA)[R]‖p,q1M   ≤‖(αnAn−αAn)[R]‖p,q1M+‖(αAn−αA)[R]‖p,q1M   ≤ ‖αn−α ‖ Mq 1M‖(An)[R]‖p,q1M+ ‖α ‖ Mq 1M‖(An−A)[R]‖p,q1M   <‖αn−α‖qJ+‖α‖qε2‖α‖q   = ε 2J 1 q qJ+ε2   =ε.

Hence we have the theorem.

Theorem 2.5. Sp,qR(B) is a complete paranormed space.

Proof. Let {An=[ajk(n)]} be a Cauchy sequence in Sp,q[R](B). We will show that there is A∈Sp,q[R](B) such that G(An−A)→0 as n→∞. Since {An} is a Cauchy sequence in Sp,q[R](B),

(2.2)G(An−Am)→0 as m,n→∞.

For a fixed j and k, we have

(2.3)‖ajk(n)−ajk(m)‖rjk|xk|≤ ∑j=1∞ ∑k=1∞ ‖ a jk (n) − a jk (m) ‖ r jk | x k|q 1qM        =‖(An −Am )[R]‖p,q1M.

From (2.2) and (2.3), the sequence {ajk(n)} is a Cauchy sequence in B. Since B is a Banach algebra, there is ajk∈B such that

‖ajk(n)−ajk‖→0 as n→∞

for any j and k. Let A=[ajk]. To show that A∈Sp,q[R](B). Let x={xk}∈lp with ‖x‖p≤1. Since {An} is a Cauchy sequence in Sp,q[R](B), there exists N1∈ℕ such that for all m,n≥N1,

‖(An−Am)[R]x‖q1M=G(An−Am)<1.

For a fixed J and K, we have

∑ j=1J ∑k=1K ‖a jk (n)−a jk (m)‖ r jk|xk|q<1 for all m,n≥N1.

Thus by taking the limit on m→∞, we get

∑ j=1J ∑k=1K ‖a jk (n)−a jk‖ r jk|xk|q<1 for all n≥N1.

Consider AN1=[ajk(N1)]∈Sp,q[R](B). Since (AN1)[R]∈B(lp,lq), there is T>0 such that

‖(AN1)[R]x‖q1M<T.

Therefore

∑ j=1J ∑k=1K‖ajk ‖rjkxkq1 qM≤‖(A−AN 1+AN 1)[R] x‖q 1M          ≤‖(A−AN 1 )[R]x‖q1M+‖(AN 1 )[R]x‖q1M          ≤1+T.

Then

∑ j=1J ∑k=1K ‖a jk‖ r jkxkq≤(1+T)qM.

This implies that

‖A[R]x‖q=∑ j=1∞∑ k=1∞‖ajk‖rjk xk q1q≤(1+T)M.

So A[R]∈B(lp,lq) and then A∈Sp,q[R](B). Next, we will prove that G(An−A)→0 as n→∞. Let ε>0 and x={xk}∈lp with ‖x‖p≤1. Since {An} is a Cauchy sequence in Sp,q[R](B), there exists N2∈ℕ such that

‖(An−Am)[R]x‖q1M=G(An−Am)<ε2 for all m,n≥N2.

For a fixed J and K, we have

∑ j=1J∑ k=1K‖ajk(n)−ajk(m) ‖rjkxkq1qM<ε2 for all m,n≥N2.

Thus for all J and K, and n≥N2, by taking the limit as m→∞, we get

∑ j=1J∑ k=1K‖ajk(n)−ajk ‖rjkxkq1qM<ε2.

By taking the limits as K→∞ and then J→∞,

‖(An−A)[R]x‖q1M=∑ j=1∞∑ k=1∞‖ajk(n)−ajk‖rjk xk q1qM<ε2,

as asserted.

Let E be a vector subspace of the vector space of all infinite matrices over a Banach algebra B. We call E a matrix space.

For any A=[ajk]∈E, we define a partial sum of ∑ j=1∞∑k=1∞a jk by the finite sum

smn=∑ j=1m∑ k=1najk

for all m,n∈ℕ.

Definition 3.1. The double series ∑ j=1∞∑k=1∞a jk is said to converge if ∑ k=1∞ajk converges for all j∈ℕ, and ∑ j=1∞∑k=1∞a jk converges.

Definition 3.2. A matrix space E is said to be normal if A=[ajk]∈E whenever ‖ajk‖≤‖bjk‖ for all j,k∈ℕ and B=[bjk]∈E.

Define

Eα=A=[ajk]:∑ j=1∞∑ k=1∞‖ajkbjk‖<∞ for all  B=[bjk]∈E,

and

Eβ=A=[ajk]:∑ j=1∞∑ k=1∞ajkbjk converges for all  B=[bjk]∈E.

Theorem 3.1. Let E,E1,E2 be matrix spaces.

• 1. Eα⊆Eβ.

• 2. If E1⊆E2, then E2β⊆E1β.

• 3. If E=E1+E2, then Eβ=E1β∩E2β.

• 4. If E is normal, then Eα=Eβ.

Proof. By using the properties of absolutely convergent double series, the proof is completed.

Subsequently, we present characterizations of the β -dual of the paranormed space Sp,qR(B).

Theorem 3.2. Let R=[rjk] be a bounded matrix of scalars with rjk>1 for all j,k∈ℕ. Then

Sp,qR(B)β=SQ(B)

where SQ(B)={A=[ajk]:∑ j=1∞∑ k=1∞‖ajk‖qjk L−qjk<∞ for some L∈ℕ } and Q=[qjk] is a bounded matrix of scalars such that 1rjk+1qjk=1 for all j,k∈ℕ.

Proof. Suppose that A∈SQ(B). Then ∑ j=1∞∑k=1∞‖a jk‖ q jkL−qjk<∞ for some L∈ℕ. We will show that A∈Sp,qR(B). Let B=[bjk]∈Sp,qR(B). Then ∑ j=1∞ ∑k=1∞ ‖b jk‖ r jkxkq<∞ for any x=(xk)∈lp. This implies that ‖B[R]‖p,q<∞. For any positive integer j, and any unit vector x={xk}∈lp, by using Hölder's inequality, we obtain

(3.1)∑k=1∞‖a jkb jk‖≤∑k=1∞‖a jk‖L−1 r jkL1 r jk‖bjk‖|xk|−1 r jk|xk|1 r jk    ≤ ∑ k=1 ∞ ‖ a jk ‖ q jk (L| x k |) − q jk r jk 1 q jk ∑ k=1 ∞ ‖ b jk ‖ r jk L| x k |1 r jk    = ∑ k=1 ∞ ‖ a jk ‖ q jk (L| x k |)−( q jk −1) 1 q jk ∑ k=1 ∞ ‖ b jk ‖ r jk L| x k |1 r jk    ≤L ∑ k=1 ∞ ‖ a jk ‖ q jk L− q jk 1 q jkL ∑ k=1 ∞ ‖ b jk ‖ r jk | x k |1 r jk    ≤L∑k=1∞ ‖a jk‖ q jkL −q jk∑k=1∞ ‖b jk‖ r jk|xk|.

For each j∈ℕ, ∑ k=1∞‖ajk‖q jkL−qjk≤∑ j=1∞∑k=1∞‖a jk‖ q jkL−qjk<∞ and ∑ k=1∞‖bjk‖r jk|xk|≤∑ j=1∞∑k=1∞‖b jk‖ r jk |xk|≤‖B[R]‖p,q<∞. Thus ∑ k=1∞ajkbjk converges for all j∈ℕ. From (3.1), we get

∑j=1∞∑k=1∞ ‖a jkb jk‖≤∑j=1∞L∑k=1∞ ‖a jk‖ q jkL −q jk∑k=1∞ ‖b jk‖ r jk|xk|      ≤L∑j=1∞ ∑k=1∞ ‖a jk‖ q jkL −q jk∑j=1∞ ∑k=1∞ ‖b jk‖ r jk|xk|      <∞.

Therefore ∑ j=1∞∑k=1∞a jkbjk converges. Consequently, A∈Sp,qR(B)β.

Next, we assume that A∈Sp,qR(B)β. For each B=[bjk]∈Sp,qR(B), we choose a scalar matrix T=[tjk] such that ‖tjk‖=1 and ‖ajktjkbjk‖=ajktjkbjk for all j,k∈ℕ. Note that ‖tjkbjk‖=‖bjk‖ for all j,k∈ℕ. By normality of Sp,qR(B) and B∈Sp,qR(B), [tjkbjk]∈Sp,qR(B). Since A∈Sp,qR(B)β, ∑ j=1∞∑k=1∞a jktjkbjk converges. Then

∑j=∞∑k=1∞ajktjkbjk<∞.

Because ‖ajktjkbjk‖=ajktjkbjk for all j,k∈ℕ,

(3.2)∑ j=1∞∑k=1∞‖a jkb jk‖=∑ j=1∞∑k=1∞‖a jkt jkb jk‖=∑ j=1∞∑k=1∞a jktjkbjk<∞.

To prove that A∈SQ(B). Suppose that A∉SQ(B). Then

∑ j=1∞∑k=1∞‖a jk‖ q jkL−qjk=∞ for all L∈ℕ.

It follows that for all j,k∈ℕ,

(3.3)∑ j>J∑k>K‖a jk‖ q jkL−qjk=∞ for all L∈ℕ.

By (3.3), we let L₁=1. Then there are j1,k1∈ℕ such that

∑ j≤j1∑k≤k1‖a jk ‖ q jk L1−qjk>1.

By (3.3), we can choose L2>L1, j2>j1 and k2>k1 with L₂>2² such that

∑ j1<j≤j2∑k1<k≤k2‖a jk ‖ q jk L2−qjk>1.

Continuing this process, we can choose sequences of positive integers {Li}, {ji} and {ki} with 1=k0<k1<k2<…, 1=j0<j1<j2<… and L1<L2<… such that Li>2i and

∑ ji−1<j≤ji∑ki−1<k≤ki‖a jk ‖ q jk Li−qjk>1 for all i∈ℕ.

For each i∈ℕ, we let ai=∑ ji−1 <j≤j i ∑ ki−1<k≤k i‖a jk‖ q jk Li−qjk where k0=j0=1. Define C=[cjk] where cjk=ai−1Li−qjk‖ajk‖qjk−1 if ji−1<j≤ji and ki−1<k≤ki. For any fixed positive integers i and any unit vector x={xk}∈lp, by the fact that rjkqjk=rjk+qjk and rjk(qjk−1)=qjk for all j,k∈ℕ,

∑ji−1<j≤ji ∑ ki−1 <k≤ki ‖c jk‖ rjk |xk |q=∑ji−1<j≤ji ∑ ki−1 <k≤ki a i −rjk L i −(r jk+q jk)‖a jk ‖ q jk |xk |q≤∑ji−1<j≤ji ∑ k i−1<k≤k i ai−rjk Li−(rjk+qjk) ‖ajk ‖ qjk q 1q ∑ k i−1<k≤k i |x k |p 1p q≤∑ji−1<j≤ji ∑ ki−1 <k≤ki a i −1 L i −1L i −q jk ‖a jk ‖ q jk q=ai−qLi−q∑ji−1<j≤ji ∑ ki−1 <k≤ki L i −qjk ‖a jk ‖ q jk q≤ai−1Li−1ai<12i.

By taking the limit as i→∞, we have

∑ j=1∞ ∑k=1 ∞ ‖c jk‖ r jk|xk|q≤∑ i=1∞12i<∞.

Thus C=[cjk]∈Sp,qR(B). For each i∈ℕ,

∑ji−1<j≤ji∑ki−1<k≤ki ‖a jk c jk ‖=∑ji−1<j≤ji∑ki−1<k≤ki a jk a i −1L i −q jk ‖a jk ‖ q jk−1         =∑ji−1<j≤ji∑ki−1<k≤ki a i −1 Li−q jk‖ajk‖q jk        =ai−1∑ji−1<j≤ji∑ki−1<k≤ki L i −q jk ‖ajk‖q jk        =ai−1ai'        =1.

By taking the limit as i→∞, we get

∑j=∞∑k=1∞‖ajkcjk‖=∑i=1∞1=∞

which contradicts (3.2). Hence A∈SQ(B) and the proof is finished.