On Some Lacunary Almost Convergent Double Sequence Spaces and Banach Limits

Abstract

The object of this paper is to introduce some new sequence spaces related with the concept of lacunary strong almost convergence for double sequences and also to characterize these spaces through sublinear functionals that both dominate and generate Banach limits and to establish some inclusion relations.

1. Introduction and Preliminaries

Let 𝑤₂ be the set of all real or complex double sequences. We mean the convergence in the Pringsheim sense, that is, a double sequence 𝑥=(𝑥_𝑖,𝑗)^∞_𝑖,𝑗=0 has a Pringsheim limit 𝜆(denoted by  𝑃−lim𝑥=𝜆) provided that given 𝜀>0 and there exists 𝑁∈ℕ such that |𝑥_𝑖,𝑗−𝜆|<𝜀 whenever 𝑖,𝑗≥𝑁 [1]. We denote by 𝑐₂, the space of 𝑃-convergent sequences. A double sequence 𝑥=(𝑥_𝑖,𝑗) is bounded if ‖𝑥‖=sup_{𝑖,𝑗≥0}|𝑥_𝑖,𝑗|<∞. Let 𝑙^∞₂ and 𝑐^∞₂ be the set of all real or complex bounded double sequences and the set of bounded and convergent double sequences, respectively. Moricz and Rhoades [2] defined the almost convergence of double sequences that 𝑥=(𝑥_𝑖,𝑗) is said to be almost convergent to a number 𝜆 if lim_{𝑝,𝑞→∞}sup_{𝑚,𝑛≥0}1𝑝+1𝑞+1^𝑚+𝑝_𝑖=𝑚^𝑛+𝑞_𝑗=𝑛𝑥_𝑖,𝑗−𝜆=0, (1.1) that is, the average value of (𝑥_𝑖,𝑗) taken over any rectangle 𝐷=𝑖,𝑗∶𝑚≤𝑖≤𝑚+𝑝,𝑛≤𝑗≤𝑛+𝑞, (1.2) tends to 𝜆 as both 𝑝 and 𝑞 tend to ∞ and this convergence is uniform in 𝑚 and 𝑛. We denote the space of almost convergent double sequences by 𝑓₂, as 𝑓₂=𝑥=𝑥_𝑖,𝑗∶lim_{𝑘,𝑙→∞}𝑡_{𝑘𝑙𝑝𝑞}(𝑥)−𝜆=0,uniformlyin𝑝,𝑞, (1.3) where 𝑡_{𝑘𝑙𝑝𝑞}(𝑥)=1(𝑘+1)(𝑙+1)^𝑝+𝑘_𝑖=𝑝^𝑞+𝑙_𝑗=𝑞𝑥_𝑖,𝑗. (1.4)

The notion of almost convergence for single sequences was introduced by Lorentz [3] and for double sequences by Moricz and Rhoades [2] and some further studies are in [4–14].

A double sequence 𝑥 is called strongly almost convergent to a number 𝜆 if lim_{𝑘,𝑙→∞}1(𝑘+1)(𝑙+1)^𝑝+𝑘_𝑖=𝑝^𝑞+𝑙_𝑗=𝑞𝑥_𝑖,𝑗−𝜆𝑒=0,uniformlyin𝑝,𝑞. (1.5)

By [𝑓₂], we denote the space of all strongly almost convergent double sequences. It is easy to see that the inclusions 𝑐^∞₂⊂[𝑓₂]⊂𝑓₂⊂𝑙^∞₂ strictly hold. As in the case of single sequences, every almost convergent double sequence is bounded. But a convergent double sequence need not be bounded. Thus, a convergent double sequence need not be almost convergent. However every bounded convergent double sequence is almost convergent.

The notion of strong almost convergence for single sequences has been introduced by Maddox [15, 16] and for double sequences by Başarir [17].

A linear functional 𝐿 on 𝑙^∞₂ is said to be Banach limit if it has the following properties [7], (1)𝐿(𝑥)≥0 if 𝑥≥0 (i.e., 𝑥_𝑖,𝑗≥0 for all 𝑖,  𝑗),(2)𝐿(𝑒)=1, where 𝑒=(𝑒_𝑖,𝑗) with 𝑒_𝑖,𝑗=1 for all 𝑖,  𝑗 and(3)𝐿(𝑥)=𝐿(𝑆₁₀𝑥)=𝐿(𝑆₀₁𝑥)=𝐿(𝑆₁₁𝑥) where the shift operators 𝑆₁₀𝑥,𝑆₀₁𝑥,𝑆₁₁𝑥 are defined by 𝑆₁₀𝑥=(𝑥_𝑖+1,𝑗),𝑆₀₁𝑥=(𝑥_𝑖,𝑗+1),𝑆₁₁𝑥=(𝑥_{𝑖+1,𝑗+1}).

Let 𝐵₂ be the set of all Banach limits on 𝑙^∞₂. A double sequence 𝑥=(𝑥_𝑖,𝑗) is said to be almost convergent to a number 𝜆 if 𝐿(𝑥)=𝜆 for all 𝐿∈𝐵₂. If 𝜑 is any sublinear functional on 𝑙^∞₂, then we write {𝑙^∞₂,𝜑} to denote the set of all linear functionals 𝐹 on 𝑙^∞₂, such that 𝐹≤𝜑, that is, 𝐹(𝑥)≤𝜑(𝑥),∀𝑥∈𝑙^∞₂. A sublinear functional 𝜑 is said to generate Banach limits if 𝐹∈{𝑙^∞₂,𝜑} implies that 𝐹 is a Banach limit; 𝜑 is said to dominate Banach limits if 𝐹 is a Banach limit implies that 𝐹∈{𝑙^∞₂,𝜑}. Then if 𝜑 both generates and dominates Banach limits, then {𝑙^∞₂,𝜑} is the set of all Banach limits.

Using the notations for single sequences, we present the notations for double-lacunary sequences that can be seen in [10]. The double sequence 𝜃_𝑟,𝑠={(𝑘_𝑟,𝑙_𝑠)} is called a double-lacunary if there exist two increasing sequences of nonnegative integers such that 𝑘₀=0,ℎ_𝑟=𝑘_𝑟−𝑘_𝑟−1→∞ as 𝑟→∞ and 𝑙₀=0,ℎ_𝑠=𝑙_𝑠−𝑙_𝑠−1→∞ as 𝑠→∞. Let 𝑘_𝑟,𝑠=𝑘_𝑟𝑙_𝑠,ℎ_𝑟,𝑠=ℎ_𝑟ℎ_𝑠,𝜃_𝑟,𝑠 is determined by 𝐼_𝑟,𝑠={(𝑖,𝑗)∶𝑘_𝑟−1<𝑖≤𝑘_𝑟and𝑙_𝑠−1<𝑗≤𝑙_𝑠}. Also ℎ_𝑟,𝑠=𝑘_𝑟𝑙_𝑠−𝑘_𝑟−1𝑙_𝑠−1 and 𝜃=𝜃_𝑟,𝑠 is determined by 𝐼_𝑟,𝑠={(𝑖,𝑗)∶𝑘_𝑟−1<𝑖≤𝑘_𝑟∪𝑙_𝑠−1<𝑗≤𝑙_𝑠}/(𝐼¹∪𝐼²), where 𝐼¹={(𝑖,𝑗)∶𝑘_𝑟−1<𝑖≤𝑘_𝑟 and 𝑙_𝑠<𝑗<∞.} and 𝐼²={(𝑖,𝑗)∶𝑙_𝑠−1<𝑗≤𝑙_𝑠 and 𝑘_𝑟<𝑖<∞.} with 𝑞_𝑟=𝑘_𝑟/𝑘_𝑟−1,𝑞_𝑠=𝑙_𝑠/𝑙_𝑠−1 and 𝑞_𝑟,𝑠=𝑞_𝑟𝑞_𝑠.

Das and Mishra [18] introduced the space of lacunary almost convergent sequences by combining the space of lacunary convergent sequences and the space of almost convergent sequences. Savaş and Patterson [10] extended the notions of lacunary almost convergence and lacunary strongly almost convergence to double-lacunary 𝑃-convergence and double-lacunary strongly almost 𝑃-convergence. They also established multidimensional analogues of Das and Patel's results.

We will use the following definition which may be called convergence in Pringsheim's sense with a bound: 𝑥_𝑖,𝑗−𝐿=𝑂(1),𝑖,𝑗⟶∞, (1.6) and also we will use the following definition which may be called convergence in Pringsheim's sense as follows: 𝑥_𝑖,𝑗−𝐿=𝑜(1),𝑖,𝑗⟶∞. (1.7)

The following sequence spaces were introduced and examined by Başarir [19]: 𝑤_𝜃=𝑥∶lim_𝑟sup_𝑖1ℎ_{𝑟𝑘∈𝐼𝑟}𝑡_𝑘𝑖(),[𝑤]𝑥−𝑠=0,forsome𝑠_𝜃=𝑥∶lim_𝑟sup_𝑖1ℎ_{𝑟𝑘∈𝐼𝑟}𝑡_𝑘𝑖(,𝑤𝑥−𝑠)=0,forsome𝑠_1𝜃=𝑥∶lim_𝑟sup_𝑖1ℎ_{𝑟𝑘∈𝐼𝑟}𝑡_𝑘𝑖(|𝑥−𝑠|)=0,forsome𝑠, (1.8) with respect to sublinear functionals on 𝑙_∞ (the set of all real or complex bounded single sequences) by 𝜙_𝜃(𝑥)=lim_𝑟sup_𝑖1ℎ_{𝑟𝑘∈𝐼𝑟}𝑡_𝑘𝑖(𝑥),𝜓_𝜃(𝑥)=lim_𝑟sup_𝑖1ℎ_{𝑟𝑘∈𝐼𝑟}𝑡_𝑘𝑖(𝑥,𝜁_𝜃)(𝑥)=lim_𝑟sup_𝑖1ℎ_{𝑟𝑘∈𝐼𝑟}𝑡_𝑘𝑖(|𝑥|), (1.9) where 𝑡_𝑘𝑖(𝑥)=(1/𝑘)∑^{𝑖+𝑘−1}_𝑗=𝑖𝑥_𝑗 and |𝑥|=(|𝑥_𝑗|)^∞_𝑗=1.

It can be easily seen that each of the above functionals are finite, well defined, and sublinear on 𝑙_∞. There is a very close connection among these sequence spaces with the sublinear functionals which were given by Başarir [19]. Recently Mursaleen and Mohiuddine [7] generalized the sequence spaces which were studied by Das and Sahoo [20] for single sequences, to the double sequences as follows: 𝑤₂=𝑥=𝑥_𝑖,𝑗∶1(𝑚+1)(𝑛+1)^𝑚𝑛_𝑘=0𝑙=0𝑡_{𝑘𝑙𝑝𝑞}(),𝑤𝑥−𝜆𝑒⟶0,as𝑚,𝑛⟶∞,uniformlyin𝑝,𝑞,forsome₂=𝑥=𝑥_𝑖,𝑗∶1(𝑚+1)(𝑛+1)^𝑚𝑛_𝑘=0𝑙=0𝑡_{𝑘𝑙𝑝𝑞}(,[𝑤]𝑥−𝜆𝑒)⟶0,as𝑚,𝑛⟶∞,uniformlyin𝑝,𝑞,forsome𝜆₂=𝑥=𝑥_𝑖,𝑗∶1(𝑚+1)(𝑛+1)^𝑚𝑛_𝑘=0𝑙=0𝑡_{𝑘𝑙𝑝𝑞}(|𝑥−𝜆𝑒|)⟶0,as𝑚,𝑛⟶∞,uniformlyin𝑝,𝑞,forsome𝜆 (1.10) by using (1.4).

The object of the present paper is to determine some new sublinear functionals involving double-lacunary sequence that both dominates and generates Banach limits. We also extend the sequence spaces which were introduced for single sequences by Başarir [19] to the double sequences with respect to these sublinear functionals. Furthermore, we present some inclusion relations with these new sequence spaces between the sequence spaces which were introduced by Mursaleen and Mohiuddine [7], earlier.

2. Sublinear Functionals and Double-Lacunary Sequence Spaces

In this section, we introduce the following sequence spaces: 𝑤²_𝜃=𝑥=𝑥_𝑖,j∶𝑃−lim_{𝑟,𝑠→∞}sup_𝑝,𝑞1ℎ_{𝑟,𝑠()∈𝑘,𝑙𝐼𝑟,𝑠}𝑡_{𝑘𝑙𝑝𝑞}(),𝑤𝑥−𝜆𝑒=0,forsome𝜆_2𝜃=𝑥=𝑥_𝑖,𝑗∶𝑃−lim_{𝑟,𝑠→∞}sup_𝑝,𝑞1ℎ_{𝑟,𝑠()∈𝑘,𝑙𝐼𝑟,𝑠}𝑡_{𝑘𝑙𝑝𝑞}(,[𝑤]𝑥−𝜆𝑒)=0,forsome𝜆_2𝜃=𝑥=𝑥_𝑖,𝑗∶𝑃−lim_{𝑟,𝑠→∞}sup_𝑝,𝑞1ℎ_{𝑟,𝑠()∈𝑘,𝑙𝐼𝑟,𝑠}𝑡_{𝑘𝑙𝑝𝑞},𝑊()=𝑥(|𝑥−𝜆𝑒|)=0,forsome𝜆𝜃,2𝑥=_𝑖,𝑗∶𝑃−lim_{𝑟,𝑠→∞}1ℎ_{𝑟,𝑠()∈𝑘,𝑙𝐼𝑟,𝑠}𝑡_𝑘𝑙00(),[𝑊(=𝑥𝑥−𝜆𝑒=0,forsome𝜆𝜃,2)]𝑥=_𝑖,𝑗∶𝑃−lim_{𝑟,𝑠→∞}sup_𝑝,𝑞1ℎ_{𝑟,𝑠(𝑘,𝑙)∈𝐼𝑟,𝑠}𝑡_𝑘𝑙00(,𝑊[]=𝑥𝑥−𝜆𝑒)=0,forsome𝜆𝜃,2𝑥=_𝑖,𝑗∶𝑃−lim_{𝑟,𝑠→∞}1ℎ_{𝑟,𝑠()∈𝑘,𝑙𝐼𝑟,𝑠}𝑡_𝑘𝑙00(|𝑥−𝜆𝑒|)=0,forsome𝜆. (2.1)

It may be noted that almost convergent double sequences are necessarily bounded but the sequence spaces 𝑤²_𝜃 and [𝑤²_𝜃] may contain unbounded sequences. Now we define the following functionals on 𝑙^∞₂ for a double-lacunary sequence 𝜃=(𝜃_𝑟,𝑠) by, 𝜙²_𝜃(𝑥)=lim_{𝑟,𝑠→∞}sup_𝑝,𝑞1ℎ_{𝑟,𝑠(𝑘,𝑙)∈𝐼𝑟,𝑠}𝑡_{𝑘𝑙𝑝𝑞}(𝑥),𝜓²_𝜃(𝑥)=lim_{𝑟,𝑠→∞}sup_𝑝,𝑞1ℎ_{𝑟,𝑠(𝑘,𝑙)∈𝐼𝑟,𝑠}𝑡_{𝑘𝑙𝑝𝑞}(𝑥,𝜑_2𝜃)(𝑥)=lim_{𝑟,𝑠→∞}sup_𝑝,𝑞1ℎ_{𝑟,𝑠(𝑘,𝑙)∈𝐼𝑟,𝑠}𝑡_{𝑘𝑙𝑝𝑞}𝑥,𝜁(||)₂(𝑥)=lim_{𝑘,𝑙→∞}sup_𝑝,𝑞𝑡_{𝑘𝑙𝑝𝑞}(𝑥),𝜂₂(𝑥)=lim_{𝑘,𝑙→∞}sup_𝑝,𝑞𝑡_{𝑘𝑙𝑝𝑞}(|𝑥|). (2.2) It is easy to see that each of the above functionals are finite, well defined, and sublinear on 𝑙^∞₂.

Throughout the paper we will write lim_𝑟,𝑠 for 𝑃−lim_{𝑟,𝑠→∞} and by this notation we shall mean the convergence in the Pringsheim sense. In the following theorem, we demonstrate that {𝑙^∞₂,𝜙²_𝜃} is the set of all Banach limits on 𝑙^∞₂ and characterize the space 𝑤²_𝜃∩𝑙^∞₂ in terms of the sublinear functional 𝜙²_𝜃.

Theorem 2.1. One has the following.
(1) The sublinear functional 𝜙²_𝜃 both dominates and generates Banach limits, that is, 𝜙²_𝜃(𝑥)=𝜁₂(𝑥), for all 𝑥=(𝑥_𝑖,𝑗)∈𝑙^∞₂.
(2) 𝑓₂=𝑥=𝑥_𝑖,𝑗∈𝑙^∞₂∶𝜙²_𝜃(𝑥)=−𝜙²_𝜃(=𝑥−𝑥)𝑥=_𝑖,𝑗∈𝑙^∞₂∶1ℎ_{𝑟,𝑠()∈𝑘,𝑙𝐼𝑟,𝑠}𝑡_{𝑘𝑙𝑝𝑞}(𝑥)⟶𝜆,as𝑟,𝑠⟶∞,𝑢𝑛𝑖𝑓𝑜𝑟𝑚𝑙𝑦𝑖𝑛𝑝,𝑞.=𝑤²_𝜃∩𝑙^∞₂. (2.3)

Proof. (1) From the definition of 𝜁₂, for given 𝜀>0 there exist 𝑘₀,𝑙₀ such that 𝑡_{𝑘𝑙𝑝𝑞}(𝑥)<𝜁₂(𝑥)+𝜀, (2.4) for 𝑘≥𝑘₀,𝑙≥𝑙₀ and for all 𝑝,𝑞. This implies that 𝜙²_𝜃(𝑥)<𝜁₂(𝑥)+𝜀, (2.5) for all 𝑥=(𝑥_𝑖,𝑗)∈𝑙^∞₂. Since 𝜀 is arbitrary, so that 𝜙²_𝜃(𝑥)≤𝜁₂(𝑥), for all 𝑥=(𝑥_𝑖,𝑗)∈𝑙^∞₂ and hence 𝑙^∞₂,𝜙²_𝜃⊂𝑙^∞₂,𝜁₂=𝐵₂, (2.6) that is, 𝜙²_𝜃 generates Banach limits.
Conversely, suppose that 𝐿∈𝐵₂. As 𝐿 is the shift invariant, that is, 𝐿(𝑆₁₁𝑥)=𝐿(𝑥)=𝐿(𝑆₁₀𝑥)=𝐿(𝑆₀₁𝑥) and using the properties of 𝐿∈𝐵₂, we obtain 𝐿(𝑥)1=𝐿(𝑘+1)(𝑙+1)^𝑝+𝑘_𝑖=𝑝^𝑞+𝑙_𝑗=𝑞𝑥_𝑖,𝑗=𝐿𝑡_{𝑘𝑙𝑝𝑞}(𝑥)=1ℎ_𝑟,𝑠𝐿_{()∈𝑘,𝑙𝐼𝑟,𝑠}𝑡_{𝑘𝑙𝑝𝑞}(𝑥)≤sup_𝑝,𝑞1ℎ_{𝑟,𝑠()∈𝑘,𝑙𝐼𝑟,𝑠}𝑡_{𝑘𝑙𝑝𝑞}(𝑥). (2.7)
It follows from the definition of 𝜙²_𝜃, that for given 𝜀>0 there exist 𝑟₀,𝑠₀ such that 1ℎ_{𝑟,𝑠(𝑘,𝑙)∈𝐼𝑟,𝑠}𝑡_{𝑘𝑙𝑝𝑞}(𝑥)<𝜙²_𝜃(𝑥)+𝜀, (2.8) for 𝑟≥𝑟₀,𝑠≥𝑠₀ and for all 𝑝,𝑞. Hence by (2.8) and properties (1) and (2) of Banach limits, we have 𝐿1ℎ_{𝑟,𝑠(𝑘,𝑙)∈𝐼𝑟,𝑠}𝑡_{𝑘𝑙𝑝𝑞}(𝑥)<𝐿𝜙²_𝜃(𝑥)+𝜀𝑒=𝜙²_𝜃(𝑥)+𝜀, (2.9) for 𝑟≥𝑟₀,𝑠≥𝑠₀ and for all 𝑝,𝑞; where 𝑒=(𝑒_𝑖,𝑗) with 𝑒_𝑖,𝑗=1 for all 𝑖,  𝑗. Since 𝜀 is arbitrary, it follows from (2.7) and (2.9) that 𝐿(𝑥)≤𝜙²_𝜃(𝑥),∀𝑥=𝑥_𝑖,𝑗∈𝑙^∞₂. (2.10) Hence 𝐵₂⊂𝑙^∞₂,𝜙²_𝜃. (2.11) That is, 𝜙²_𝜃 dominates Banach limits. Combining (2.6) and (2.11), we get 𝑙^∞₂,𝜁₂=𝑙^∞₂,𝜙²_𝜃, (2.12) this implies that 𝜙²_𝜃 dominates and generates Banach limits and 𝜙²_𝜃(𝑥)=𝜁₂(𝑥) for all 𝑥∈𝑙^∞₂.
(2) As a consequence of Hahn-Banach theorem, {𝑙^∞₂,𝜙²_𝜃} is non empty and a linear functional 𝐹∈{𝑙^∞₂,𝜙²_𝜃} is not necessarily uniquely defined at any particular value of 𝑥. This is evident in the manner the linear functionals are constructed. But in order that all the functionals {𝑙^∞₂,𝜙²_𝜃} coincide at 𝑥=(𝑥_𝑖,𝑗), it is necessary and sufficient that 𝜙²_𝜃(𝑥)=−𝜙²_𝜃(−𝑥), (2.13) we have limsup_𝑟,𝑠sup_𝑝,𝑞1ℎ_{𝑟,𝑠(𝑘,𝑙)∈𝐼𝑟,𝑠}𝑡_{𝑘𝑙𝑝𝑞}(𝑥)=liminf_𝑟,𝑠inf_𝑝,𝑞1ℎ_{𝑟,𝑠(𝑘,𝑙)∈𝐼𝑟,𝑠}𝑡_{𝑘𝑙𝑝𝑞}(𝑥). (2.14) But (2.14) holds if and only if 1ℎ_{𝑟,𝑠()∈𝑘,𝑙𝐼𝑟,𝑠}𝑡_{𝑘𝑙𝑝𝑞}(𝑥)⟶𝜆,as𝑟,𝑠⟶∞,uniformlyin𝑝,𝑞. (2.15) Hence, 𝑥=(𝑥_𝑖,𝑗)∈𝑤²_𝜃∩𝑙^∞₂. But (2.13) is equivalent to 𝜁₂(𝑥)=−𝜁₂(−𝑥), this holds if and only if 𝑥=(𝑥_𝑖,𝑗)∈𝑓₂. This completes the proof of the theorem.

If 𝐹(𝑥−𝜆𝑒)=0 for all 𝐹∈{𝑙^∞₂,𝜓²_𝜃}, then we say that 𝑥=(𝑥_𝑖,𝑗) is 𝜓²_𝜃-convergent to 𝜆. Similarly we define the 𝜑²_𝜃-convergent sequences. In the following theorem we characterize the spaces [𝑤²_𝜃]∩𝑙^∞₂ and [𝑤]²_𝜃∩𝑙^∞₂ in terms of the sublinear functionals.

Theorem 2.2. One has the following:(1)[𝑤²_𝜃]∩𝑙^∞₂  =  {𝑥=(𝑥_𝑖,𝑗)∶𝜓²_𝜃(𝑥−𝜆𝑒)=0, for  some 𝜆}  =  {𝑥=(𝑥_𝑖,𝑗)∶𝐹(𝑥−𝜆𝑒)=0, for all   𝐹∈{𝑙^∞₂,𝜓²_𝜃}, for  some   𝜆}(2)[𝑤]²_𝜃∩𝑙^∞₂  =  {𝑥=(𝑥_𝑖,𝑗)∶𝜑²_𝜃(𝑥−𝜆𝑒)=0, for  some 𝜆}  =  {𝑥=(𝑥_𝑖,𝑗)∶𝐹(𝑥−𝜆𝑒)=0, for all  𝐹∈{𝑙^∞₂,𝜑²_𝜃}, for  some 𝜆}.

Proof. (1) It can be easily verified that 𝑥=(𝑥_𝑖,𝑗)∈[𝑤²_𝜃]∩𝑙^∞₂ if and only if 𝜓²_𝜃(𝑥−𝜆𝑒)=−𝜓²_𝜃(𝜆𝑒−𝑥). (2.16) Since 𝜓²_𝜃(𝑥)=−𝜓²_𝜃(−𝑥) then (2.16) reduces to 𝜓²_𝜃(𝑥−𝜆𝑒)=0. (2.17) Now if 𝐹∈{𝑙^∞₂,𝜓²_𝜃} then from (2.17) and linearity of 𝐹, we have 𝐹(𝑥−𝜆𝑒)=0. (2.18) Conversely, suppose that 𝐹(𝑥−𝜆𝑒)=0 for all 𝐹∈{𝑙^∞₂,𝜓²_𝜃} and hence by Hahn-Banach theorem, there exists 𝐹₀∈{𝑙^∞₂,𝜓²_𝜃} such that 𝐹₀(𝑥)=𝜓²_𝜃(𝑥). Hence 0=𝐹₀(𝑥−𝜆𝑒)=𝜓²_𝜃(𝑥−𝜆𝑒). (2.19)
(2) The proof is similar to the proof of (1), above.

3. Inclusion Relations

We establish here some inclusion relations between the sequence spaces defined in Section 2.

Theorem 3.1. We have the following proper inclusions and the limit is preserved in each case.(1)[𝑓₂]⊂[𝑤]²_𝜃⊂[𝑤²_𝜃]⊂𝑤²_𝜃⊂𝑊(𝜃,2). (2)[𝑤]²_𝜃⊂[𝑤²_𝜃]⊂[𝑊(𝜃,2)]⊂𝑊(𝜃,2). (3)[𝑤]²_𝜃⊂𝑊[𝜃,2]⊂[𝑊(𝜃,2)]⊂𝑊(𝜃,2).

Proof. (1) Let 𝑥∈[𝑓₂] with [𝑓₂]−lim𝑥=𝜆. Then 𝑡_{𝑘𝑙𝑝𝑞}(|𝑥−𝜆𝑒|)⟶0,as𝑘,𝑙⟶∞,uniformlyin𝑝,𝑞. (3.1) This implies that 1ℎ_{𝑟,𝑠()∈𝑘,𝑙𝐼𝑟,𝑠}𝑡_{𝑘𝑙𝑝𝑞}(|𝑥−𝜆𝑒|)⟶0,as𝑘,𝑙⟶∞,uniformlyin𝑝,𝑞. (3.2) This proves that 𝑥∈[𝑤]²_𝜃 and [𝑓₂]−lim𝑥=[𝑤]²_𝜃−lim𝑥=𝜆. Since 1ℎ_{𝑟,𝑠(𝑘,𝑙)∈𝐼𝑟,𝑠}𝑡_{𝑘𝑙𝑝𝑞}()≤1𝑥−𝜆𝑒ℎ_{𝑟,𝑠()∈𝑘,𝑙𝐼𝑟,𝑠}𝑡_{𝑘𝑙𝑝𝑞}(≤1𝑥−𝜆𝑒)ℎ_{𝑟,𝑠(𝑘,𝑙)∈𝐼𝑟,𝑠}𝑡_{𝑘𝑙𝑝𝑞}(|𝑥−𝜆𝑒|), (3.3) this implies that [𝑤]²_𝜃⊂[𝑤²_𝜃]⊂𝑤²_𝜃 and [𝑤]²_𝜃−lim𝑥=[𝑤²_𝜃]−lim𝑥=𝑤²_𝜃−lim𝑥=𝜆. Since 1ℎ_{𝑟,𝑠(𝑘,𝑙)∈𝐼𝑟,𝑠}𝑡_{𝑘𝑙𝑝𝑞}(𝑥−𝜆𝑒) (3.4) converges uniformly in 𝑝,𝑞 as 𝑟,𝑠→∞, implies the convergence for 𝑝=0=𝑞. It follows that 𝑤²_𝜃⊂𝑊(𝜃,2) and 𝑤²_𝜃−lim𝑥=𝑊(𝜃,2)−lim𝑥=𝜆. This completes the proof of (1).
It is easy to see the proof of (2) and (3). So we omit them.

Theorem 3.2. One has the following proper inclusions; 𝑓₂⊂[𝑤]²_𝜃∩𝑙^∞₂⊂𝑤²_𝜃∩𝑙^∞₂⊂𝑓₂. (3.5)

Proof. The proof of the theorem is similar as in [7, Theorem 4.2]. So we omit it.

Prior to giving Lemmas 3.3 and 3.5, we need the following notations used in [10]: 𝐼¹_𝐶=𝑖,𝑗∶𝑞≤𝑗≤𝑞+𝑛,𝑝+𝑚<𝑖<∞,𝐼²_𝐶=,𝐶𝑖,𝑗∶𝑞+𝑛<𝑗<∞,𝑝≤𝑖≤𝑝+𝑚_{𝑚,𝑛𝑝,𝑞}=⧵𝐼𝑖,𝑗∶𝑝≤𝑖≤𝑝+𝑚or𝑞≤𝑗≤𝑞+𝑛^1𝐶∪𝐼²_𝐶,𝐼¹_𝐷=𝑖,𝑗∶𝑞+𝑦+1ℎ_𝑠−1<𝑗<∞,𝑝+𝑥ℎ_𝑟≤𝑖≤𝑝+(𝑥+1)ℎ_𝑟,𝐼−1_2𝐷=𝑖,𝑗∶𝑞+𝑦ℎ_𝑠≤𝑗≤𝑞+𝑦+1ℎ_𝑠−1,𝑝+(𝑥+1)ℎ_𝑟,𝐷−1<𝑖<∞_{𝑥,𝑦𝑝,𝑞}=𝑖,𝑗∶𝑝+𝑥ℎ_𝑟≤𝑖≤𝑝+(𝑥+1)ℎ_𝑟−1or𝑞+𝑦ℎ_𝑠≤𝑗≤𝑞+𝑦+1ℎ_𝑠⧵𝐼−1^1𝐷∪𝐼²_𝐷. (3.6)

Lemma 3.3. Suppose 𝜀>0 there exist 𝑚₀,𝑛₀,𝑝₀, and 𝑞₀ such that 1𝐶^𝑚,𝑛_{0,0()𝑘,𝑙∈𝐶}^𝑚,𝑛_0,01𝐶^{𝑝+𝑘,𝑞+𝑙}_{𝑝,𝑞(𝑖,𝑗)∈𝐶}^{𝑝+𝑘,𝑞+𝑙}_𝑝,𝑞𝑥_𝑖,𝑗−𝜆𝑒<𝜀, (3.7) for 𝑚≥𝑚₀,𝑛≥𝑛₀ and 𝑝≥𝑝₀,𝑞≥𝑞₀. Then 𝑥∈[𝑤]₂.

Proof. Let 𝜀>0 be given. Choose 𝑚¹₀,𝑛¹₀,𝑝₀ and 𝑞₀ such that 1𝐶^𝑚,𝑛_{0,0()𝑘,𝑙∈𝐶}^𝑚,𝑛_0,01𝐶^{𝑝+𝑘,𝑞+𝑙}_{𝑝,𝑞(𝑖,𝑗)∈𝐶}^{𝑝+𝑘,𝑞+𝑙}_𝑝,𝑞𝑥_𝑖,𝑗<𝜀−𝜆𝑒6 (3.8) for 𝑚≥𝑚¹₀,𝑛≥𝑛¹₀,𝑝≥𝑝₀,𝑞≥𝑞₀. We need only to show that given 𝜀>0 there exist 𝑚²₀ and 𝑛²₀ such that 1𝐶^𝑚,𝑛_{0,0()𝑘,𝑙∈𝐶}^𝑚,𝑛_0,01𝐶^{𝑝+𝑘,𝑞+𝑙}_{𝑝,𝑞(𝑖,𝑗)∈𝐶}^{𝑝+𝑘,𝑞+𝑙}_𝑝,𝑞𝑥_𝑖,𝑗−𝜆𝑒<𝜀, (3.9) for 𝑚≥𝑚²₀,𝑛≥𝑛²₀ and 0≤𝑝≤𝑝₀,0≤𝑞≤𝑞₀. If we take 𝑚₀=max{𝑚¹₀,𝑚²₀} and 𝑛₀=max{𝑛¹₀,𝑛²₀}, then (3.9) holds for 𝑚≥𝑚₀,𝑛≥𝑛₀ and for all 𝑝 and 𝑞, which gives the result. Once 𝑝₀ and 𝑞₀ have been chosen, they are fixed, so _{()𝑘,𝑙∈𝐶𝑝00,𝑞0,0}1𝐶^{𝑝+𝑘,𝑞+𝑙}_{𝑝,𝑞𝑖,𝑗∈𝐶}^𝑝00,𝑞_𝑝,𝑞𝑥_𝑖,𝑗−𝜆𝑒=𝑀, (3.10) is finite. Now taking 0≤𝑝≤𝑝₀, 0≤𝑞≤𝑞₀ and 𝑚≥𝑝₀, 𝑛≥𝑞₀ we have from (3.8) and (3.10) 1𝐶^𝑚,𝑛_{0,0()𝑘,𝑙∈𝐶}^𝑚,𝑛_0,01𝐶^{𝑝+𝑘,𝑞+𝑙}_{𝑝,𝑞𝑖,𝑗∈𝐶}^{𝑝+𝑘,𝑞+𝑙}_𝑝,𝑞𝑥_𝑖,𝑗=1−𝜆𝑒𝐶^𝑚,𝑛_{0,0()𝑘,𝑙∈𝐶}^𝑝₀₀^,𝑞_0,01𝐶^{𝑝+𝑘,𝑞+𝑙}_{𝑝,𝑞𝑖,𝑗∈𝐶}^𝑝00,𝑞_𝑝,𝑞𝑥_𝑖,𝑗+1−𝜆𝑒𝐶^𝑚,𝑛_{0,0()𝑘,𝑙∈𝐶}^𝑝₀₀^,𝑞_0,01𝐶^{𝑝+𝑘,𝑞+𝑙}_{𝑝,𝑞𝑖,𝑗∈𝐶𝑝00}^{𝑝+𝑘,𝑞+𝑙}_+1,𝑞+1𝑥_𝑖,𝑗+1−𝜆𝑒𝐶^𝑚,𝑛_{0,0()𝑘,𝑙∈𝐶𝑝00}^𝑚,𝑛_+1,𝑞+11𝐶^{𝑝+𝑘,𝑞+𝑙}_{𝑝,𝑞𝑖,𝑗∈𝐶}^𝑝00,𝑞_𝑝,𝑞𝑥_𝑖,𝑗+1−𝜆𝑒𝐶^𝑚,𝑛_{0,0𝑖,𝑗∈𝐶𝑝00}^𝑚,𝑛_+1,𝑞+11𝐶^{𝑝+𝑘,𝑞+𝑙}_{𝑝,𝑞𝑖,𝑗∈𝐶𝑝00}^{𝑝+𝑘,𝑞+𝑙}_+1,𝑞+1𝑥_𝑖,𝑗≤𝑀−𝜆𝑒𝐶^𝑚,𝑛_0,0𝜀+3⋅6=𝑀𝐶^𝑚,𝑛_0,0+𝜀2. (3.11) Therefore taking 𝑚 and 𝑛 sufficiently large, we can make 𝑀/|𝐶^𝑚,𝑛_0,0|+𝜀/2<𝜀 which gives (3.9) and hence the result.

Theorem 3.4. We have [𝑤]²_𝜃=[𝑤]₂ for every 𝜃_𝑟,𝑠.

Proof. Let 𝑥∈[𝑤]²_𝜃; then given 𝜀>0 there exist 𝑟₀,𝑠₀ and 𝜆 such that 1ℎ_{𝑟,𝑠()𝑘,𝑙∈𝐶ℎ𝑟𝑠−1,ℎ−10,0}𝑡_{𝑘𝑙𝑝𝑞}(|𝑥−𝜆𝑒|)<𝜀, (3.12) for 𝑟≥𝑟₀,𝑠≥𝑠₀, 𝑝=𝑘_𝑟−1+1+𝛼 and 𝑞=𝑙_𝑠−1+1+𝛽 where 𝛼,𝛽≥0. Let 𝑚≥ℎ_𝑟 such that 𝑚=𝛿₁ℎ_𝑟+𝜃₁−1 where 𝛿₁ is an integer. Also let 𝑛≥ℎ_𝑠 such that 𝑛=𝛿₂ℎ_𝑠+𝜃₂−1 where 𝛿₂ is an integer and 1≤𝜃₁≤ℎ_𝑟, 1≤𝜃₂≤ℎ_𝑠. Since 𝑚≥ℎ_𝑟 for 𝛿₁≥1 and 𝑛≥ℎ_𝑠 for 𝛿₂≥1 we have, 1𝐶^𝑚,𝑛_{0,0()𝑘,𝑙∈𝐶}^𝑚,𝑛_0,0𝑡_{𝑘𝑙𝑝𝑞}≤1(|𝑥−𝜆𝑒|)𝐶^𝑚,𝑛_{0,0()𝑘,𝑙∈𝐶1𝑟2𝑠(𝛿+1)ℎ−1,(𝛿+1)ℎ−10,0}𝑡_{𝑘𝑙𝑝𝑞}=1(|𝑥−𝜆𝑒|)𝐶^𝑚,𝑛_0,0^𝛿1,𝛿2_{𝑥,𝑦=0,0(𝑘,l)∈𝐷}^𝑥,𝑦_0,0𝑡_{𝑘𝑙𝑝𝑞}≤1(|𝑥−𝜆𝑒|)𝐶^𝑚,𝑛_0,0^𝛿1,𝛿2_{𝑥,𝑦=0,0}ℎ_𝑟,𝑠≤𝛿⋅𝜀¹+1𝛿₂+1𝐶^𝑚,𝑛_0,0ℎ_𝑟,𝑠⋅𝜀=𝑜(1), (3.13) which gives the result. Therefore by Lemma 3.3, [𝑤]²_𝜃⊂[𝑤]₂. It is clear that [𝑤]₂⊂[𝑤]²_𝜃 for every 𝜃_𝑟,s. This completes the proof.

Lemma 3.5. Suppose, for a given 𝜀>0, there exist 𝑚₀,𝑛₀,𝑝₀, and 𝑞₀ such that 1𝐶^𝑚,𝑛_{0,0()𝑘,𝑙∈𝐶}^𝑚,𝑛_0,01𝐶^{𝑝+𝑘,𝑞+𝑙}_{𝑝,𝑞(𝑖,𝑗)∈𝐶}^{𝑝+𝑘,𝑞+𝑙}_𝑝,𝑞𝑥_𝑖,𝑗−𝜆𝑒<𝜀, (3.14) for all 𝑚≥𝑚₀,𝑛≥𝑛₀ and 𝑝≥𝑝₀,𝑞≥𝑞₀. Then 𝑥∈[𝑤₂].

Proof. Let 𝜀>0 be given and choose 𝑚₀,𝑛₀,𝑝₀ and 𝑞₀ such that 1𝐶^𝑚,𝑛_{0,0()𝑘,𝑙∈𝐶}^𝑚,𝑛_0,01𝐶^{𝑝+𝑘,𝑞+𝑙}_{𝑝,𝑞(𝑖,𝑗)∈𝐶}^{𝑝+𝑘,𝑞+𝑙}_𝑝,𝑞𝑥_𝑖,𝑗<𝜀−𝜆𝑒4 (3.15) for all 𝑚≥𝑚₀,𝑛≥𝑛₀,𝑝≥𝑝₀, and 𝑞≥𝑞₀. As in Lemma 3.3, it is enough to show that there exist 𝑚¹₀ and 𝑛¹₀ such that for 𝑚≥𝑚¹₀,𝑛≥𝑛¹₀ implies 1𝐶^𝑚,𝑛_{0,0()𝑘,𝑙∈𝐶}^𝑚,𝑛_0,01𝐶^{𝑝+𝑘,𝑞+𝑙}_{𝑝,𝑞(𝑖,𝑗)∈𝐶}^{𝑝+𝑘,𝑞+𝑙}_𝑝,𝑞𝑥_𝑖,𝑗−𝜆𝑒<𝜀, (3.16) for all 𝑝 and 𝑞 with 0≤𝑝≤𝑝₀ and 0≤𝑞≤𝑞₀. Since 𝑝₀ and 𝑞₀ are fixed, _{()𝑘,𝑙∈𝐶𝑝00,𝑞0,0}1𝐶^{𝑝+𝑘,𝑞+𝑙}_{𝑝,𝑞(𝑖,𝑗)∈𝐶}^𝑝00,𝑞_𝑝,𝑞𝑥_𝑖,𝑗−𝜆𝑒=𝑀. (3.17) Now, let 0≤𝑝≤𝑝₀,0≤𝑞≤𝑞₀, and 𝑚≥𝑝₀,𝑛≥𝑞₀ and consider the following; 1𝐶^𝑚,𝑛_{0,0()𝑘,𝑙∈𝐶}^𝑚,𝑛_0,01𝐶^{𝑝+𝑘,𝑞+𝑙}_{𝑝,𝑞(𝑖,𝑗)∈𝐶}^{𝑝+𝑘,𝑞+𝑙}_𝑝,𝑞𝑥_𝑖,𝑗≤1−𝜆𝑒𝐶^𝑚,𝑛_{0,0()𝑘,𝑙∈𝐶}^𝑝₀₀^,𝑞_0,01𝐶^{𝑝+𝑘,𝑞+𝑙}_{𝑝,𝑞(𝑖,𝑗)∈𝐶}^𝑝00,𝑞_𝑝,𝑞𝑥_𝑖,𝑗+1−𝜆𝑒𝐶^𝑚,𝑛_{0,0()𝑘,𝑙∈𝐶}^𝑝₀₀^,𝑞_0,01𝐶^{𝑝+𝑘,𝑞+𝑙}_{𝑝,𝑞(𝑖,𝑗)∈𝐶𝑝00}^{𝑝+𝑘,𝑞+𝑙}_+1,𝑞+1𝑥_𝑖,𝑗+1−𝜆𝑒𝐶^𝑚,𝑛_{0,0()𝑘,𝑙∈𝐶𝑝00}^𝑚,𝑛_+1,𝑞+11𝐶^{𝑝+𝑘,𝑞+𝑙}_{𝑝,𝑞(𝑖,𝑗)∈𝐶}^{𝑝+𝑘,𝑞+𝑙}_𝑝,𝑞𝑥_𝑖,𝑗≤𝑀−𝜆𝑒𝐶^𝑚,𝑛_0,0+1𝐶^𝑚,𝑛_{0,0()𝑘,𝑙∈𝐶}^𝑝₀₀^,𝑞_0,01𝐶^{𝑝+𝑘,𝑞+𝑙}_{𝑝,𝑞(𝑖,𝑗)∈𝐶𝑝00}^{𝑝+𝑘,𝑞+𝑙}_+1,𝑞+1𝑥_𝑖,𝑗+1−𝜆𝑒𝐶^𝑚,𝑛_{0,0()𝑘,𝑙∈𝐶𝑝00}^𝑚,𝑛_+1,𝑞+1𝑡_{𝑘𝑙𝑝𝑞}(𝑥−𝜆𝑒). (3.18) Let 𝑘−𝑝₀≥𝑚¹₀, then 𝑘+𝑝−𝑝₀≥𝑚¹₀ for 0≤𝑝≤𝑝₀. Also if we let 𝑙−𝑞₀≥𝑛¹₀, then 𝑙+𝑞−𝑞₀≥𝑛¹₀ for 0≤𝑞≤𝑞₀. Therefore from (3.15) 1𝐶^𝑚,𝑛_{0,0()𝑘,𝑙∈𝐶}^𝑝₀₀^,𝑞_0,01𝐶^{𝑝+𝑘,𝑞+𝑙}_{𝑝,𝑞(𝑖,𝑗)∈𝐶𝑝00}^{𝑝+𝑘,𝑞+𝑙}_+1,𝑞+1𝑥_𝑖,𝑗<1−𝜆𝑒𝐶^𝑝0,𝑞0_{0,0()𝑘,𝑙∈𝐶}^𝑝₀₀^,𝑞_0,01𝐶^{𝑝0+(𝑝+𝑘−𝑝0),𝑞0+(𝑞+𝑙−𝑞0)}_{𝑝0+1,𝑞0+1(𝑖,𝑗)∈𝐶}^𝑝₀₀₀⁾_𝑝000+^{(𝑝+𝑘−𝑝),𝑞+(𝑞+𝑙−𝑞}_+1,𝑞+1𝑥_𝑖,𝑗<𝜀−𝜆𝑒4. (3.19) From (3.18) and (3.19) 1𝐶^𝑚,𝑛_{0,0()𝑘,𝑙∈𝐶}^𝑚,𝑛_0,01𝐶^{𝑝+𝑘,𝑞+𝑙}_{𝑝,𝑞(𝑖,𝑗)∈𝐶}^{𝑝+𝑘,𝑞+𝑙}_𝑝,𝑞𝑥_𝑖,𝑗≤𝑀−𝜆𝑒𝐶^𝑚,𝑛_0,0𝜀+24<𝜀, (3.20) for sufficiently large values of 𝑚 and 𝑛. Hence the result.

Theorem 3.6. For every 𝜃_𝑟,𝑠, One has [𝑤²_𝜃]∩𝑙^∞₂=[𝑤₂]∩𝑙^∞₂.

Proof. Let 𝑥∈[𝑤²_𝜃]∩𝑙^∞₂. For 𝜀>0, there exist 𝑟₀,𝑠₀,𝑝₀ and 𝑞₀ such that 1ℎ_{𝑟,𝑠()𝑘,𝑙∈𝐶ℎ𝑟𝑠−1,ℎ−10,0}𝑡_{𝑘𝑙𝑝𝑞}(<𝜀𝑥−𝜆𝑒)2 (3.21) for 𝑟≥𝑟₀,𝑠≥𝑠₀,𝑝≥𝑝₀ and 𝑞≥𝑞₀ with 𝑝=𝑘_𝑟−1+1+𝛼 where 𝛼≥0,𝑞=𝑙_𝑠−1+1+𝛽 and 𝛽≥0. Let 𝑚≥ℎ_𝑟 and 𝑛≥ℎ_𝑠 where 𝑚≥1 and 𝑛≥1. Then 1𝐶^𝑚,𝑛_{0,0()𝑘,𝑙∈𝐶}^𝑚,𝑛_0,01𝐶^{𝑝+𝑘,q+𝑙}_{𝑝,𝑞(𝑖,𝑗)∈𝐶}^{𝑝+𝑘,𝑞+𝑙}_𝑝,𝑞𝑥_𝑖,𝑗=1−𝜆𝑒𝐶^𝑚,𝑛_{0,0()𝑘,𝑙∈𝐶𝛿1ℎ𝑟2ℎ𝑠−1,𝛿−10,0}𝑡_{𝑘𝑙𝑝𝑞}(+1𝑥−𝜆𝑒)𝐶^𝑚,𝑛_{0,0()𝑘,𝑙∈𝐶𝛿1ℎ𝑟2ℎ𝑠}^𝑚,𝑛_,𝛿𝑡_{𝑘𝑙𝑝𝑞}(≤1𝑥−𝜆𝑒)𝐶^𝑚,𝑛_0,0^{𝛿1−1,𝛿2−1}_{𝑥,𝑦=0,0()𝑘,𝑙∈𝐷}^𝑥,𝑦_0,0𝑡_{𝑘𝑙𝑝𝑞}(+1𝑥−𝜆𝑒)𝐶^𝑚,𝑛_{0,0()𝑘,𝑙∈𝐶𝛿1ℎ𝑟2ℎ𝑠}^𝑚,𝑛_,𝛿1𝐶^{𝑝+𝑘,𝑞+𝑙}_{𝑝,𝑞𝑖,𝑗∈𝐶}^{𝑝+𝑘,𝑞+𝑙}_𝑝,𝑞𝑥_𝑖,𝑗−𝜆𝑒. (3.22) Since (𝑥_𝑖,𝑗)∈𝑙^∞₂ for all 𝑖 and 𝑗, there exists 𝑀 such that |𝑥_𝑖,𝑗−𝜆𝑒|≤𝑀. From (3.21) and (3.22), we have the following: 1𝐶^𝑚,𝑛_{0,0()𝑘,𝑙∈𝐶}^𝑚,𝑛_0,01𝐶^{𝑝+𝑘,𝑞+𝑙}_{𝑝,𝑞(𝑖,𝑗)∈𝐶}^{𝑝+𝑘,𝑞+𝑙}_𝑝,𝑞𝑥_𝑖,𝑗≤𝛿−𝜆𝑒¹𝛿₂𝐶^𝑚,𝑛_0,0ℎ_𝑟,𝑠𝜀2+𝑀ℎ_𝑟,𝑠𝐶^𝑚,𝑛_0,0. (3.23) Thus for 𝑚 and 𝑛 sufficiently large, we have the following: 1𝐶^𝑚,𝑛_{0,0()𝑘,𝑙∈𝐶}^𝑚,𝑛_0,01𝐶^{𝑝+𝑘,𝑞+𝑙}_{𝑝,𝑞(𝑖,𝑗)∈𝐶}^{𝑝+𝑘,𝑞+𝑙}_𝑝,𝑞𝑥_𝑖,𝑗−𝜆𝑒<𝜀, (3.24) for 𝑟≥𝑟₀,𝑠≥𝑠₀ and 𝑝≥𝑝₀,𝑞≥𝑞₀. Thus by Lemma 3.5, we have [𝑤²_𝜃]∩𝑙^∞₂⊂[𝑤₂]∩𝑙^∞₂. It is clear that [𝑤₂]∩𝑙^∞₂⊂[𝑤²_𝜃]∩𝑙^∞₂. This completes the proof of the theorem.

Corollary 3.7. [𝑓₂]⊂[𝑤]²_𝜃⊂([𝑤²_𝜃]∩𝑙^∞₂)⊂(𝑤²_𝜃∩𝑙^∞₂)=𝑓₂.

Proof. It is easy to see by combining Theorem 3.4, Theorem 3.6 with [7, Theorems 4.1 and 3.1(ii)]. So we omit it.

A paranormed space (𝑋,𝑔) is a topological linear space with the topology given by the paranorm 𝑔. It may be recalled that a paranorm 𝑔 is a real subadditive function on 𝑋 such that 𝑔(𝜃)=0,𝑔(𝑥)=𝑔(−𝑥) and scalar multiplication is continuous, that is, 𝜇_𝑛→𝜇,𝑥_𝑛→𝑥 imply that (𝜇_𝑛𝑥_𝑛)→(𝜇𝑥) where 𝜇_𝑛,𝜇 are scalars and 𝑥_𝑛,𝑥∈𝑋.

Let 𝑢=(𝑢_𝑘,𝑙) be a bounded double sequence of positive real numbers, that is, 𝑢_𝑘,𝑙>0 for all 𝑘,𝑙 with sup_𝑘,𝑙𝑢_𝑘,𝑙=𝐻<∞. Let 𝑤²_𝜃(𝑢)=𝑥=𝑥_𝑖,𝑗∶lim_𝑟,𝑠sup_𝑝,𝑞1ℎ_{𝑟,𝑠()∈𝑘,𝑙𝐼𝑟,𝑠}𝑡_{𝑘𝑙𝑝𝑞}(𝑥−𝜆𝑒)^{𝑢𝑘,𝑙}=0forsome𝜆. (3.25) If 𝑢=(𝑢_𝑘,𝑙) is constant we write [𝑤²_𝜃]_𝑢 in place of [𝑤²_𝜃(𝑢)]. If we take 𝑢=(𝑢_𝑘,𝑙) with 𝑢_𝑘,𝑙=1 for all 𝑘 and 𝑙, then [𝑤²_𝜃(𝑢)] is reduced to [𝑤²_𝜃] which is defined in Section 2.

Theorem 3.8. Let 𝑢=(𝑢_𝑘,𝑙) be a bounded sequence of positive real numbers with sup_𝑘,𝑙𝑢_𝑘,𝑙=𝐻<∞. Then [𝑤²_𝜃(𝑢)] is a complete linear topological space paranormed by 𝑔(𝑥)=sup_{𝑟,𝑠,𝑝,𝑞}1ℎ_{𝑟,𝑠(𝑘,𝑙)∈𝐼𝑟,𝑠}𝑡_{𝑘𝑙𝑝𝑞}(𝑥)^{𝑢𝑘,𝑙1/𝑀}, (3.26) where 𝑀=max(1,𝐻). In the case 𝑢 is constant, [𝑤²_𝜃]_𝑢 is a Banach space if 𝑢≥1 and is a 𝑝-normed space if  0<𝑢<1.

Proof. It is easy to see that [𝑤²_𝜃(𝑢)] is a linear space with coordinatewise addition and scalar multiplication. Clearly 𝑔(𝜃)=0,𝑔(𝑥)=𝑔(−𝑥) and 𝑔 is subadditive. To prove the continuity of multiplication, assume that 𝑥∈[𝑤²_𝜃(𝑢)]. Since 𝑢=(𝑢_𝑘,𝑙) is bounded and positive there exists a constant 𝛿>0 such that 𝑢_𝑘,𝑙≥𝛿 for all 𝑘,𝑙. Now for |𝜇|≤1,  |𝜇|^{𝑢𝑘,𝑙}≤|𝜇|^𝛿 and hence 𝑔(𝜇𝑥)≤|𝜇|^𝛿/𝑀𝑔(𝑥). This proves the fact that 𝑔 is a paranorm on [𝑤²_𝜃(𝑢)].
To prove that [𝑤²_𝜃(𝑢)] is complete, assume that (𝑥^𝑚) is a Cauchy sequence in [𝑤²_𝜃(𝑢)], that is, 𝑔(𝑥^𝑚−𝑥^𝑛)→0 as 𝑚,𝑛→∞. Since 1ℎ_{𝑟,𝑠(𝑘,𝑙)∈𝐼𝑟,𝑠}𝑡_{𝑘𝑙𝑝𝑞}𝑥^𝑚−𝑥^{𝑛𝑢𝑘,𝑙}≤𝑔𝑥^𝑚−𝑥^𝑛𝑀, (3.27) it follows that |𝑡_{𝑘𝑙𝑝𝑞}(𝑥^𝑚−𝑥^𝑛)|^{𝑢𝑘,𝑙}=0(1) as 𝑚,𝑛→∞ for each 𝑘,𝑙,𝑝, and 𝑞. In particular 𝑡_00𝑝𝑞𝑥^𝑚−𝑥^𝑛=𝑥^𝑚_𝑝,𝑞−𝑥^𝑛_𝑝,𝑞⟶0as𝑚,𝑛⟶∞,foreachfixed𝑝and𝑞. (3.28) Hence, (𝑥^𝑚) is a Cauchy sequence in ℝ(or ℂ). Since ℝ(or ℂ) is complete, there exists 𝑥∈ℝ(or ℂ) such that 𝑥^𝑚→𝑥 coordinate wise as 𝑚→∞. It follows from (3.27) that given 𝜀>0, there exists 𝑚₀∈ℕ such that 1ℎ_{𝑟,𝑠(𝑘,𝑙)∈𝐼𝑟,𝑠}𝑡_{𝑘𝑙𝑝𝑞}𝑥^𝑚−𝑥^{𝑛𝑢𝑘,𝑙1/𝑀}<𝜀, (3.29) for 𝑚,  𝑛>𝑚₀. Now making 𝑛→∞ and then taking supremum with respect to 𝑝 and 𝑞 in (3.29) we obtain 𝑔(𝑥^𝑚−𝑥)≤𝜀 for 𝑚>𝑚₀. This proves that 𝑥^𝑚→𝑥 and 𝑥∈[𝑤²_𝜃(𝑢)]. Hence [𝑤²_𝜃(𝑢)] is complete. When 𝑢 is constant, it is easy to derive the rest of the theorem.

Theorem 3.9. Let 0<𝜌_𝑘,𝑙≤𝜎_𝑘,𝑙<∞ for each 𝑘 and 𝑙. Then [𝑤²_𝜃(𝜌)]⊂[𝑤²_𝜃(𝜎)].

Proof. Let 𝑥∈[𝑤²_𝜃(𝜌)]. By the definition of [𝑤²_𝜃(𝜌)], that for given 𝜀>0 there exist 𝑟₀,𝑠₀ such that 1ℎ_{𝑟,𝑠(𝑘,𝑙)∈𝐼𝑟,𝑠}𝑡_{𝑘𝑙𝑝𝑞}(𝑥−𝜆𝑒)^{𝜌𝑘,𝑙}<𝜀, (3.30) for 𝑟>𝑟₀,𝑠>𝑠₀ and for all 𝑝,𝑞. Since 1/ℎ_𝑟,𝑠→0 as 𝑟,𝑠→∞, then _{(𝑘,𝑙)∈𝐼𝑟,𝑠}𝑡_{𝑘𝑙𝑝𝑞}(𝑥−𝜆𝑒)^{𝜌𝑘,𝑙}<∞, (3.31) for 𝑟>𝑟₀,𝑠>𝑠₀ and for all 𝑝,𝑞. This implies that 𝑡_{𝑘𝑙𝑝𝑞}(𝑥−𝜆𝑒)<1, (3.32) for sufficiently large values of 𝑘,𝑙 and for all 𝑝,𝑞. Then we get, 1ℎ_{𝑟,𝑠(𝑘,𝑙)∈𝐼𝑟,𝑠}𝑡_{𝑘𝑙𝑝𝑞}(𝑥−𝜆𝑒)^{𝜎𝑘,𝑙}≤1ℎ_{𝑟,𝑠()∈𝑘,𝑙𝐼𝑟,𝑠}𝑡_{𝑘𝑙𝑝𝑞}(𝑥−𝜆𝑒)^{𝜌𝑘,𝑙}=𝑜(1), (3.33) as 𝑟,𝑠→∞ and for all 𝑝,𝑞. Hence, lim_𝑟,𝑠sup_𝑝,𝑞1ℎ_{𝑟,𝑠(𝑘,𝑙)∈𝐼𝑟,𝑠}𝑡_{𝑘𝑙𝑝𝑞}(𝑥−𝜆𝑒)^{𝜌𝑘,𝑙}=0, (3.34) is obtained and consequently we have 𝑥∈[𝑤²_𝜃(𝜌)]. This completes the proof.

Theorem 3.10. One has the following.(1)Let 0<inf_𝑘,𝑙𝑢_𝑘,𝑙≤𝑢_𝑘,𝑙≤1 for each 𝑘 and 𝑙. Then [𝑤²_𝜃(𝑢)]⊂[𝑤²_𝜃].(2)Let 1≤𝑢_𝑘,𝑙≤sup_𝑘,𝑙𝑢_𝑘,𝑙<∞ for each 𝑘 and 𝑙. Then [𝑤²_𝜃]⊂[𝑤²_𝜃(𝑢)].

Proof. (1) It is clear from the above theorem. If we take 𝜌_𝑘,𝑙=𝑢_𝑘,𝑙 and 𝜎_𝑘,𝑙=1 for each 𝑘 and 𝑙, then we have [𝑤²_𝜃(𝑢)]⊂[𝑤²_𝜃].
(2) From the above theorem, if we take 𝜌_𝑘,𝑙=1 and 𝜎_𝑘,𝑙=𝑢_𝑘,𝑙 for each 𝑘 and 𝑙, then we have [𝑤²_𝜃]⊂[𝑤²_𝜃(𝑢)].
This completes the proof.

Acknowledgment

The authors are grateful to anonymous referees for their careful reading of the paper which improved it greatly.