Abstract

We determine the necessary and sufficient conditions to characterize the matrices which transform convex sequences and Maddox sequences into 𝑉𝜎(𝜃) and 𝑉𝜎(𝜃).

1. Introduction and Preliminaries

By 𝑤, we denote the space of all real-valued sequences 𝑥=(𝑥𝑘)𝑘=1. Any vector subspace of 𝑤 is called a sequence space. We write that , 𝑐, and 𝑐0 denote the sets of all bounded, convergent, and null sequences, respectively, and note that 𝑐; also 𝑐𝑠 and 𝑝 are the set of all convergent and 𝑝-absolutely convergent series, respectively, where 𝑝={𝑥𝑤𝑘=0|𝑥𝑘|𝑝<} for 1𝑝<. In the theory of sequence spaces, a beautiful application of the well-known Hahn-Banach extension theorem gave rise to the concept of the Banach limit. That is, the lim functional defined on 𝑐 can be extended to the whole , and this extended functional is known as the Banach limit [1]. In 1948, Lorentz [2] used this notion of a weak limit to define a new type of convergence, known as the almost convergence. Later on, Raimi [3] gave a slight generalization of almost convergence and named it the 𝜎-convergence.

A sequence space 𝑋 with a linear topology is called a 𝐾-𝑠𝑝𝑎𝑐𝑒 if each of the maps 𝑝𝑖𝑋 defined by 𝑝𝑖(𝑥)=𝑥𝑖 is continuous for all 𝑖. A 𝐾-space 𝑋 is called an 𝐹𝐾-𝑠𝑝𝑎𝑐𝑒 if 𝑋 is a complete linear metric space. An 𝐹𝐾-space whose topology is normable is called a 𝐵𝐾-space. An 𝐹𝐾-space 𝑋 is said to have 𝐴𝐾 property if 𝑋𝜙 and (𝑒(𝑘)) is a basis for 𝑋, where (𝑒(𝑘)) is a sequence whose only nonzero term is a 1 in 𝑘th place for each 𝑘 and 𝜙=span{𝑒(𝑘)}, the set of all finitely nonzero sequences. If 𝜙 is dense in 𝑋, then 𝑋 is called an 𝐴𝐷-𝑠𝑝𝑎𝑐𝑒, thus 𝐴𝐾 implies 𝐴𝐷. For example, the spaces 𝑐0, 𝑐𝑠, and 𝑝 are 𝐴𝐾-spaces.

Let 𝑋 and 𝑌 be two sequence spaces, and let 𝐴=(𝑎𝑛𝑘)𝑛;𝑘=1 be an infinite matrix of real or complex numbers. We write 𝐴𝑥=(𝐴𝑛(𝑥)), 𝐴𝑛(𝑥)=𝑘𝑎𝑛𝑘𝑥𝑘 provided that the series on the right converges for each 𝑛. If 𝑥=(𝑥𝑘)𝑋 implies that 𝐴𝑥𝑌, then we say that 𝐴 defines a matrix transformation from 𝑋 into 𝑌, and by (𝑋,𝑌), we denote the class of such matrices.

Let 𝜎 be a one-to-one mapping from the set of natural numbers into itself. A continuous linear functional 𝜑 on the space is said to be an 𝑖𝑛𝑣𝑎𝑟𝑖𝑎𝑛𝑡𝑚𝑒𝑎𝑛 or a 𝜎-𝑚𝑒𝑎𝑛 if and only if (i) 𝜑(𝑥)0 if 𝑥0 (i.e., 𝑥𝑘0 for all 𝑘), (ii) 𝜑(𝑒)=1, where 𝑒=(1,1,1,), and (iii) 𝜑(𝑥)=𝜑((𝑥𝜎(𝑘))) for all 𝑥.

Throughout this paper, we consider the mapping 𝜎 which has no finite orbits, that is, 𝜎𝑝(𝑘)𝑘 for all integer 𝑘0 and 𝑝1, where 𝜎𝑝(𝑘) denotes the 𝑝th iterate of 𝜎 at 𝑘. Note that a 𝜎-mean extends the limit functional on the space 𝑐 in the sense that 𝜑(𝑥)=lim

𝑥 for all 𝑥𝑐, (cf. [4]). Consequently, 𝑐𝑉𝜎, the set of bounded sequences all of whose 𝜎-means are equal. We say that a sequence 𝑥=(𝑥𝑘) is 𝜎-𝑐𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑡 if and only if 𝑥𝑉𝜎. Using this concept, Schaefer [5] defined and characterized 𝜎-conservative, 𝜎-regular, and 𝜎-coercive matrices. If 𝜎 is translation, then 𝑉𝜎 is reduced to the set 𝑓 of almost convergent sequences [2].

The notion of 𝜎-convergence for double sequences has been introduced in [6] and further studied in [79].

Recently, the sequence spaces 𝑉𝜎(𝜃) and 𝑉𝜎(𝜃) have been introduced in [10] which are related to the concept of 𝜎-mean and the lacunary sequence 𝜃=(𝑘𝑟).

In this section, we establish the necessary and sufficient conditions on the matrix 𝐴=(𝑎𝑛𝑘)𝑛,𝑘=1 which transforms 𝑟-convex sequences in to the spaces 𝑉𝜎(𝜃) and 𝑉𝜎(𝜃).

By a lacunary sequence, we mean an increasing integer sequence 𝜃=(𝑘𝑟) such that 𝑘0=0 and 𝑟=𝑘𝑟𝑘𝑟1 as 𝑟. Throughout this paper, the intervals determined by 𝜃 will be denoted by 𝐼𝑟=(𝑘𝑟1,𝑘𝑟], and the ratio 𝑘𝑟/𝑘𝑟1 will be abbreviated by 𝑞𝑟.

A bounded sequence 𝑥=(𝑥𝑘) of real numbers is said to be 𝜎-lacunary convergent to a number 𝐿 if and only if lim𝑟(1/𝑟)𝑗𝐼𝑟𝑥𝜎𝑗(𝑛)=𝐿, uniformly in 𝑛, and let 𝑉𝜎(𝜃) denote the set of all such sequences, that is,𝑉𝜎(𝜃)=𝑥lim𝑚1𝑟𝑗𝐼𝑟𝑥𝜎𝑗(𝑛)=𝐿uniformlyin𝑛.(1.1) In this case, 𝐿 is called the (𝜎,𝜃)-limit of 𝑥. We remark that(i) if 𝜎(𝑛)=𝑛+1, then 𝑉𝜎(𝜃) is reduced to the space 𝑓(𝜃) (cf. [11]),(ii)𝑐𝑉𝜎(𝜃).

A bounded sequence 𝑥=(𝑥𝑘) of real numbers is said to be 𝜎-lacunary bounded if and only if sup𝑟,𝑛|(1/𝑟)𝑗𝐼𝑟𝑥𝜎𝑗(𝑛)|<, and let 𝑉𝜎(𝜃) denote the set of all such sequences, that is,𝑉𝜎(𝜃)=𝑥sup𝑟,𝑛|||||1𝑟𝑗𝐼𝑟𝑥𝜎𝑗(𝑛)|||||<.(1.2) We remark that 𝑐𝑉𝜎(𝜃)𝑉𝜎(𝜃) and the spaces 𝑉𝜎(𝜃) and 𝑉𝜎(𝜃) are BK spaces with the norm𝑥=sup𝑟,𝑛|||||1𝑟𝑗𝐼𝑟𝑥𝜎𝑗(𝑛)|||||.(1.3)

2. Convex Sequence Spaces

Pati and Sinha [12] defined 𝑟-convex sequences as follows: a real sequence 𝑥=(𝑥𝑘)𝑘=0 is said to be 𝑟-𝑐𝑜𝑛𝑣𝑒𝑥,𝑟, if Δ𝑟𝑥𝑘0 for all 𝑘, where Δ𝑟𝑥𝑘 is defined byΔ0𝑥𝑘=𝑥𝑘,Δ1𝑥𝑘=𝑥𝑘𝑥𝑘+1,Δ𝑟𝑥𝑘Δ=Δ𝑟1𝑥𝑘,𝑟.(2.1) The space of all bounded 𝑟-convex sequences with 𝑟2 is denoted by SC𝑟, that is,SC𝑟=𝑥𝑙Δ𝑟𝑥𝑘,0𝑘SC1=𝑥𝑙𝑥𝑘𝑥𝑘+1.0(2.2) It is clear that SC1𝑐.

It is well known that (Zygmund [13]) a bounded convex sequence (𝑥𝑘) is nonincreasing. It is easy to prove the identity Δ(𝑟+𝑠)𝑥𝑘=Δ𝑟(Δ𝑠𝑥𝑘),𝑟,𝑠0,which shows that SC𝑟SC𝑟1, when 𝑟2. Properties of bounded 𝑟-convex sequences have been investigated by Rath [14]. Note that SC𝑟𝑣𝑐. Recently, using the generalized difference operator Δ𝑟, Çolak and Et [15], and Et and Çolak [16] defined and studied the sequence spaces 𝑐0(Δ𝑟),𝑐(Δ𝑟), and (Δ𝑟). In this section, we establish the necessary and sufficient conditions on the matrix 𝐴=(𝑎𝑛𝑘)𝑛,𝑘=1 which transforms 𝑟-convex sequences into the spaces 𝑉𝜎(𝜃) and 𝑉𝜎(𝜃).

Write1𝑡(𝑛,𝑘,𝑚)=𝑚𝑗𝐼𝑚𝑎𝜎𝑗(𝑛),𝑘,𝑔(𝑟)(𝑛,𝑘,𝑚)=𝑘𝑗=1𝑟1𝑘𝑗𝑡(𝑛,𝑗,𝑚)𝑘𝑟1,𝜆𝑚𝑛(𝑥)=𝑘=1𝑔(𝑛,𝑘,𝑚)𝑥𝑘,(2.3) where for our convenience, we use 𝑔(𝑛,𝑘,𝑚) instead of 𝑔(𝑟)(𝑛,𝑘,𝑚) for 𝑟2 throughout the paper.

Theorem 2.1. 𝐴(𝑆𝐶𝑟,𝑉𝜎(𝜃)) if and only if(i)sup𝑖,𝑝|𝑘=𝑝𝑎𝑖𝑘|<, (ii) there exists a constant 𝑀 such that for 𝑠,𝑛=1,2,,sup𝑚𝑘=𝑠||||𝑔(𝑛,𝑘,𝑚)𝑀,(2.4)(iii)lim𝑚𝑔(𝑛,𝑘,𝑚)=𝛼𝑘, uniformly in 𝑛, for each 𝑘,(iv)lim𝑚𝑘𝑔(𝑛,𝑘,𝑚)=𝛼, uniformly in 𝑛.

Proof. In [17], a characterization of 𝐴(SC𝑟,𝐹) was given, where 𝐹, in the sense of [18], is the bounded domain of a sequence =(𝐵(𝑖)) of matrices 𝐵(𝑖)=(𝑏(𝑖)𝑟𝑘). Now, by the setting 𝑏(𝑖)𝑟𝑘=1𝑟0if𝑘=𝜎𝑗(𝑖),𝑗𝐼𝑟,otherwise.(𝑟,𝑘,𝑖),(2.5) then 𝑉𝜎(𝜃)=𝐹, and the proof follows from Theorem 2.1 of [17].

Theorem 2.2. 𝐴(𝑆𝐶𝑟,𝑉𝜎(𝜃)) if and only if the condition (i) of Theorem 2.1 holds and sup𝑛,𝑚𝑘||||𝑡(𝑛,𝑘,𝑚)<.(2.6)

Proof. Sufficiency
Suppose that the conditions (i) and (2.6) hold and 𝑥=(𝑥𝑘)SC𝑟. Therefore, 𝐴𝑥 is bounded, and we have ||𝜆𝑚𝑛||(𝑥)𝑘||𝑔(𝑛,𝑘,𝑚)𝑥𝑘||𝑘||||𝑔(𝑛,𝑘,𝑚)sup𝑘||𝑥𝑘||.(2.7) Taking the supremum over 𝑛,𝑚 on both sides and using (2.6), we get 𝐴𝑥𝑉𝜎(𝜃) for 𝑥SC𝑟.

Necessity
Let 𝐴(SC𝑟,𝑉𝜎(𝜃)). Condition (i) follows as in the proof of Theorem 2.1. Write 𝑞𝑛(𝑥)=sup𝑚|𝜆𝑚𝑛(𝐴𝑥)|. It is easy to see that 𝑞𝑛 is a continuous seminorm on SC𝑟, since for 𝑥SC𝑟, ||𝑞𝑛||(𝑥)𝑀𝑥,𝑀>0.(2.8) Suppose that (2.6) is not true, then there exists 𝑥SC𝑟 with sup𝑛𝑞𝑛(𝑥)=. By the principle of condensation of singularities (cf. [19]), the set 𝑥SC𝑟sup𝑛𝑞𝑛(𝑥)=(2.9) is of second category in SC𝑟 and hence nonempty, that is, there is 𝑥SC𝑟 with sup𝑛𝑞𝑛(𝑥)=. But this contradicts the fact that 𝑞𝑛 is pointwise bounded on SC𝑟. Now, by the Banach-Steinhaus theorem, there is a constant 𝑀 such that 𝑞𝑛(𝑥)𝑀𝑥.(2.10) Now, we define a sequence 𝑥=(𝑥𝑘) by 𝑥𝑘=sgn𝑔(𝑛,𝑘,𝑚)𝑘foreach𝑚,𝑛1𝑘𝑘0,0for𝑘>𝑘0.(2.11) Then, 𝑥SC𝑟. Applying this sequence to (2.10), we get (2.6).
This completes the proof of the theorem.

3. Maddox Sequence Spaces

A linear topological space 𝑋 over the real field is said to be a paranormed space if there is a subadditive function 𝑔𝑋 such that 𝑔(𝜃)=0,𝑔(𝑥)=𝑔(𝑥), and scalar multiplication is continuous, that is, |𝛼𝑛𝛼|0 and 𝑔(𝑥𝑛𝑥)0 imply 𝑔(𝛼𝑛𝑥𝑛𝛼𝑥)0 for all 𝑥,𝑥𝑛 in 𝑋 and 𝛼,𝛼𝑛 in , where 𝜃 is the zero vector in the linear space 𝑋. Assume here and after that 𝑥=(𝑥𝑘) is a sequence such that 𝑥𝑘0 for all 𝑘. Let 𝑝=(𝑝𝑘)𝑘=0 be a bounded sequence of positive real numbers with sup𝑘𝑝𝑘=𝐻 and 𝑀=max{1,𝐻}. The sequence spaces𝑐0(𝑝)=𝑥𝜔lim𝑘||𝑥𝑘||𝑝𝑘,=0𝑐(𝑝)=𝑥𝜔𝑥𝑙𝑒𝑐0,𝑙(𝑝)forsome𝑙(𝑝)=𝑥𝜔sup𝑘||𝑥𝑘||𝑝𝑘,<𝑙(𝑝)=𝑥𝜔𝑘=0||𝑥𝑘||𝑝𝑘,<(3.1) were defined and studied by Et and Çolak [16] and Pati and Sinha [12]. If 𝑝𝑘=𝑝(𝑘=0,1,) for some constant 𝑝>0, then these sets reduce to 𝑐0,𝑐,𝑙, and 𝑙𝑝, respectively. Note that 𝑐0(𝑝) is a linear metric space paranormed by 𝑔(𝑥)=sup𝑘||𝑥𝑘||𝑝𝑘/𝑀(3.2) where 𝑀=max(1,sup𝑝𝑘). 𝑙(𝑝) and 𝑐(𝑝) fail to be linear metric spaces because the continuity of scalar multiplication does not hold for them, but these two turn out to be linear metric spaces if and only if inf𝑘𝑝𝑘>0. 𝑙(𝑝) is linear metric space paranormed by 1(𝑥)=(𝑘|𝑥𝑘|𝑝𝑘)1/𝑀. All these spaces are complete in their respective topologies; however, these are not normed spaces in general (cf. [20]).

In this section, we characterize the matrix classes (𝑙(𝑝),𝑉𝜎(𝜃)) and (𝑙(𝑝),𝑉𝜎(𝜃)).

Let 𝐴𝑥 be defined, then, for all 𝑟,𝑛, we write𝜏𝑟(𝐴𝑥)=𝑘=1𝑡(𝑛,𝑘,𝑟)𝑥𝑘,(3.3) where1𝑡(𝑛,𝑘,𝑟)=𝑟𝑗𝐼𝑟𝑎𝜎𝑗(𝑛),𝑘,(3.4) and 𝑎(𝑛,𝑘) denotes the element 𝑎𝑛𝑘 of the matrix 𝐴.

Theorem 3.1. 𝐴((𝑝),𝑉𝜎(𝜃)) if and only if there exists 𝐵>1 such that for every 𝑛,(i)sup𝑟𝑘||||𝑡(𝑛,𝑘,𝑟)𝑞𝑘𝐵𝑞𝑘<,1<𝑝𝑘,𝑝<𝑘1+𝑞𝑘1;=1sup𝑟,𝑘||||𝑡(𝑛,𝑘,𝑟)𝑝𝑘<,0<𝑝𝑘,1(3.5)(ii)𝑎(𝑘)={𝑎𝑛𝑘}𝑛=1𝑉𝜎(𝜃) for each 𝑘, that is, lim𝑟𝑡(𝑛,𝑘,𝑟)=𝑢𝑘 uniformly in 𝑛.In this case, the (𝜎,𝜃)-limit of 𝐴𝑥 is 𝑘𝑢𝑘𝑥𝑘.

Proof. Necessity
We consider the case 1<𝑝𝑘<. Let 𝐴((𝑝),𝑉𝜎(𝜃)). Since 𝑒𝑘(𝑝), the condition (ii) holds. Put 𝑓𝑟𝑛(𝑥)=𝜏𝑟𝑛(𝐴𝑥), since 𝜏𝑟𝑛(𝐴𝑥) exists for each 𝑟 and 𝑥𝑙(𝑝), therefore {𝑓𝑟𝑛(𝑥)}𝑟 is a sequence of continuous real functionals on 𝑙(𝑝) and further sup𝑟|𝑓𝑟𝑛(𝑥)|< on 𝑙(𝑝). Now condition (i) follows by arguing with uniform boundedness principle. The case 0<𝑝𝑘1 can be proved similarly.

Sufficiency
Suppose that the conditions (i) and (ii) hold and 𝑥(𝑝). Now for every 𝑚1, we have 𝑚𝑘=1||||𝑡(𝑛,𝑘,𝑟)𝑞𝑘𝐵𝑞𝑘sup𝑟𝑘||||𝑡(𝑛,𝑘,𝑟)𝑞𝑘𝐵𝑞𝑘.(3.6) Therefore, 𝑘||𝑢𝑘||𝑞𝑘𝐵𝑞𝑘=lim𝑚lim𝑟𝑚𝑘=1||||𝑡(𝑛,𝑘,𝑟)𝑞𝑘𝐵𝑞𝑘sup𝑟𝑘||||𝑡(𝑛,𝑘,𝑟)𝑞𝑘𝐵𝑞𝑘<.(3.7) Thus, the series 𝑘𝑡(𝑛,𝑘,𝑟)𝑥𝑘 and 𝑘𝑢𝑘𝑥𝑘 converge for each 𝑟 and 𝑥(𝑝). For a given 𝜀>0 and 𝑥(𝑝), choose 𝑘0 such that 𝑘=𝑘0+1||𝑥𝑘||𝑝𝑘1/𝐻<𝜀,(3.8) where 𝐻=sup𝑘𝑝𝑘. Since (ii) holds, therefore there exists 𝑟0 such that |||||𝑘0𝑘=1𝑡(𝑛,𝑘,𝑟)𝑢𝑘|||||<𝜀,(3.9) for every 𝑟>𝑟0. Hence, by the condition (ii), it follows that |||||𝑘=𝑘0+1𝑡(𝑛,𝑘,𝑟)𝑢𝑘|||||(3.10) is arbitrary small, and we have lim𝑟𝑘𝑡(𝑛,𝑘,𝑟)𝑥𝑘=𝑘𝑢𝑘𝑥𝑘,(3.11) uniformly in 𝑛.
This completes the proof of the theorem.

Theorem 3.2. 𝐴((𝑝),𝑉𝜎(𝜃)) if and only if there exists 𝑁>1 such that(i)𝑀𝑛=sup𝑟𝑘|𝑡(𝑛,𝑘,𝑟)|𝑁1/𝑝𝑘< for every 𝑛,(ii)𝑎(𝑘)={𝑎𝑛𝑘}𝑛=1𝑉𝜎(𝜃) for each 𝑘, that is, lim𝑟𝑡(𝑛,𝑘,𝑟)=𝑢𝑘 uniformly in 𝑛,(iii)lim𝑟𝑘|𝑡(𝑛,𝑘,𝑟)𝑢𝑘|=0 uniformly in 𝑛.In this case, the (𝜎,𝜃)-limit of 𝐴𝑥 is 𝑘𝑢𝑘𝑥𝑘.

Proof. Necessity
Let 𝐴((𝑝),𝑉𝜎(𝜃)), then 𝐴(,𝑉𝜎(𝜃)), and the conditions (ii) and (iii) follow from Theorem 3 of Schaefer [5]. Now on the contrary, suppose that (i) does not hold, then there exists 𝑁>1 such that 𝑀𝑛=. Therefore, by Theorem 3 of Schaefer [5], the matrix 𝑏𝐵=𝑛𝑘=𝑎𝑛𝑘𝑁1/𝑝𝑘,𝑉𝜎(𝜃),(3.12) that is, there exists 𝑥 such that 𝐵𝑥𝑉𝜎(𝜃). Now, let 𝑦=(𝑁1/𝑝𝑘𝑥𝑘), then 𝑦(𝑝) and 𝐵𝑥=𝐴𝑦𝑉𝜎(𝜃), which contradicts that 𝐴((𝑝),𝑉𝜎(𝜃)). Therefore, (i) must hold.

Sufficiency
Suppose that the conditions hold and 𝑥(𝑝), then for every 𝑛, |||||𝑘𝑡(𝑛,𝑘,𝑟)𝑥𝑘|||||sup𝑘||𝑥𝑘||𝑝𝑘sup𝑟𝑘||||𝑁𝑡(𝑛,𝑘,𝑟)1/𝑝𝑘<.(3.13) Therefore 𝐴𝑥 is defined. Now arguing as in Theorem 3.1, we get 𝐴𝑥𝑉𝜎(𝜃), and the series 𝑘𝑡(𝑛,𝑘,𝑟)𝑥𝑘 and 𝑘𝑢𝑘𝑥𝑘 converge for 𝑥(𝑝). Hence, by the condition (iii), we get lim𝑟𝑘𝑡(𝑛,𝑘,𝑟)𝑥𝑘=𝑘𝑢𝑘𝑥𝑘,(3.14) uniformly in 𝑛.
This completes the proof of the theorem.

Theorem 3.3. Let 1<𝑝𝑘<sup𝑘𝑝𝑘=𝐻< for every 𝑘, then 𝐴((𝑝),𝑉𝜎(𝜃)) if and only if there exists an integer 𝑁>1 such thatsup𝑟,𝑛𝑘||||𝑡(𝑛,𝑘,𝑟)𝑞𝑘𝑁𝑞𝑘<.(3.15)

Proof. Sufficiency
Let (3.15) hold and that 𝑥(𝑝) using the following inequality (see [21]): ||||𝑎𝑏𝐶|𝑎|𝑞𝐶𝑞+||𝑏||𝑝,(3.16) for 𝐶>0 and 𝑎,𝑏, are two complex numbers (𝑞1+𝑝1=1), we have ||𝜏𝑟(||=𝐴𝑥)𝑘||𝑡(𝑛,𝑘,𝑟)𝑥𝑘||𝑘𝑁||||𝑡(𝑛,𝑘,𝑟)𝑞𝑘𝑁𝑞𝑘+||𝑥𝑘||𝑝𝑘,(3.17) where 𝑞𝑘1+𝑝𝑘1=1. Taking the supremum over 𝑟,𝑛 on both sides and using (3.15), we get 𝐴𝑥𝑉𝜎(𝜃)) for 𝑥(𝑝), that is, 𝐴((𝑝),𝑉𝜎(𝜃)).

Necessity
Let 𝐴((𝑝),𝑉𝜎(𝜃)). Write 𝑞𝑛(𝑥)=sup𝑟|𝜏𝑟(𝐴𝑥)|. It is easy to see that for 𝑛0, 𝑞𝑛 is a continuous seminorm on (𝑝), and (𝑞𝑛) is pointwise bounded on (𝑝). Suppose that (3.15) is not true, then there exists 𝑥(𝑝) with sup𝑛𝑞𝑛(𝑥)=. By the principle of condensation of singularities [19], the set 𝑥(𝑝)sup𝑛𝑞𝑛(𝑥)=(3.18) is of second category in (𝑝) and hence nonempty, that is, there is 𝑥(𝑝) with sup𝑛𝑞𝑛(𝑥)=. But this contradicts the fact that (𝑞𝑛) is pointwise bounded on (𝑝). Now, by the Banach-Steinhaus theorem, there is constant 𝑀 such that 𝑞𝑛(𝑥)𝑀𝑔(𝑥).(3.19) Now, we define a sequence 𝑥=(𝑥𝑘) by 𝑥𝑘=𝛿𝑀/𝑝𝑘||||(sgn𝑡(𝑛,𝑘,𝑟))𝑡(𝑛,𝑘,𝑟)𝑞𝑘1𝑆1𝑁𝑞𝑘/𝑝𝑘,1𝑘𝑘0,0,for𝑘>𝑘0,(3.20) where 0<𝛿<1 and 𝑆=𝑘0𝑘=1||||𝑡(𝑛,𝑘,𝑟)𝑞𝑘𝑁𝑞𝑘.(3.21) Then it is easy to see that 𝑥(𝑝) and 𝑔(𝑥)𝛿. Applying this sequence to (3.19), we get the condition (3.15).
This completes the proof of the theorem.