Abstract

We introduce a new class of ideal convergent (shortly I-convergent) sequence spaces using an Orlicz function and difference operator of order 𝑠β‰₯1 defined over the n-normed spaces. We investigate these spaces for some linear topological structures. These investigations will enhance the acceptability of the notion of n-norm by giving a way to construct different sequence spaces with elements in n-normed spaces. We also give some relations related to these sequence spaces.

1. Introduction

Throughout the paper w, β„“βˆž, c, 𝑐0, and ℓ𝑝  denote the classes of all, bounded, convergent, null, and p-absolutely summable sequences of complex numbers.

The notion of statistical convergence is a very useful functional tool for studying the convergence problems of numerical sequences/matrices (double sequences) through the concept of density. It was first introduced by Fast [1], and Schoenberg [2], independently for the real sequences. Later on, it was further investigated from sequence point of view and linked with the summability theory by Fridy [3] and many others. The idea is based on the notion of natural density of subsets of β„•, the set of positive integers, which is defined as follows: the natural density of a subset E of β„• is denoted by Ξ΄(E) and is defined by𝛿(𝐸)=limπ‘›β†’βˆž1𝑛||||,{π‘˜βˆˆπΈβˆΆπ‘˜β‰€π‘›}(1.1) where the vertical bar denotes the cardinality of the respective set.

The notion of I-convergence (I denotes the ideal of subsets of β„•), which is a generalization of statistical convergence, was introduced by Kastyrko et al. [4] and further studied by many other authors.

The notion of difference sequence space was introduced by Kizmaz [5], who studied the difference sequence spaces β„“βˆž(Ξ”), 𝑐(Ξ”), and 𝑐0(Ξ”). The notion was further generalized by Et and Γ‡olak [6] by introducing the sequence spaces β„“βˆž(Δ𝑠), 𝑐(Δ𝑠), and 𝑐0(Δ𝑠). Another type of generalization of the difference sequence spaces is due to Tripathy and Esi [7], who studied the spaces β„“βˆž(Ξ”π‘š), 𝑐(Ξ”π‘š), and 𝑐0(Ξ”π‘š). Tripathy et al. [8] generalized the above notion and unified these as follows.

For nonnegative integers m, s, the generalized difference sequence spaces are defined as follows: for a given sequence space Z, we have π‘ξ€·Ξ”π‘ π‘šξ€Έ=ξ€½ξ€·π‘₯π‘₯=π‘˜ξ€Έξ€·Ξ”βˆˆπ‘€βˆΆπ‘ π‘šπ‘₯π‘˜ξ€Έξ€Ύβˆˆπ‘,(1.2) where Ξ”π‘ π‘šπ‘₯= (Ξ”π‘ π‘šπ‘₯π‘˜)=(Ξ”π‘šπ‘ βˆ’1π‘₯π‘˜βˆ’Ξ”π‘šπ‘ βˆ’1π‘₯π‘˜+π‘š), Ξ”0π‘šπ‘₯π‘˜=π‘₯π‘˜,for all π‘˜βˆˆβ„•, the difference operator is equivalent to the following binomial representation: Ξ”π‘ π‘šπ‘₯π‘˜=π‘ ξ“πœˆ=0(βˆ’1)πœˆξƒ©π‘ πœˆξƒͺπ‘₯π‘˜+π‘šπœˆ.(1.3) Taking 𝑠=1, we get the spaces β„“βˆž(Ξ”π‘š), 𝑐(Ξ”π‘š), and 𝑐0(Ξ”π‘š), introduced and studied by Tripathy and Et [7]. Taking π‘š=1, we get the spaces β„“βˆž(Δ𝑠), 𝑐(Δ𝑠), and 𝑐0(Δ𝑠) studied by Et and Γ‡olak [6]. Taking 𝑠=π‘š=1, we get the spaces β„“βˆž(Ξ”), 𝑐(Ξ”), and 𝑐0(Ξ”) introduced and studied by Kizmaz [5].

Let m, s be nonnegative integers, then for a given sequence space Z, we introducedπ‘ξ‚€Ξ”π‘š(𝑠)=π‘₯π‘₯=π‘˜ξ€Έξ‚€Ξ”βˆˆπ‘€βˆΆπ‘š(𝑠)π‘₯π‘˜ξ‚ξ‚‡βˆˆπ‘,(1.4) where Ξ”(s)mπ‘₯=(Ξ”(s)mπ‘₯π‘˜)=(Ξ”(sβˆ’1)mπ‘₯π‘˜βˆ’Ξ”(sβˆ’1)mπ‘₯π‘˜βˆ’π‘š),Ξ”(0)mπ‘₯π‘˜=π‘₯π‘˜, for allβ€‰β€‰π‘˜βˆˆβ„•, the difference operator is equivalent to the following binomial representation: Ξ”π‘š(𝑠)π‘₯π‘˜=π‘ ξ“πœˆ=0(βˆ’1)πœˆξƒ©π‘ πœˆξƒͺπ‘₯π‘˜βˆ’π‘šπœˆ,(1.5) where π‘₯π‘˜=0, for π‘˜<0.

The concept of 2-normed space was initially introduced by GΓ€hler [9], in the mid of 1960s, while that of 𝑛-normed spaces can be found in Misiak [10]. Since then, many others authors have used this concept and obtained various results, see, for instance, Gunawan [11] and Gunawan and Mashadi [12, 13]. Recently, a lot of activities have started to study summability, sequence spaces, and related topics in these spaces (see [14, 15]).

The notion of ideal-convergence in 2-normed spaces was initially introduced by GΓΌrdal [16]. Later on, it was extended to n-normed spaces by GΓΌrdal and Sahnier [17].

An Orlicz function is a function π‘€βˆΆ[0,∞)β†’[0,∞), which is continuous, nondecreasing and convex with 𝑀(0)=0, 𝑀(0)>0, as π‘₯>0 and 𝑀(π‘₯)β†’βˆž, as π‘₯β†’βˆž.

An Orlicz function 𝑀 can always be represented in the following integral form:ξ€œπ‘€(π‘₯)=π‘₯0𝑝(𝑑)𝑑𝑑,(1.6) where p is the known kernel of M, right differentiable for 𝑑β‰₯0, 𝑝(0)=0, 𝑝(𝑑)>0, for 𝑑>0 and 𝑝(𝑑)β†’βˆž, as π‘‘β†’βˆž.

If convexity of Orlicz function M is replaced by 𝑀(π‘₯+𝑦)≀𝑀(π‘₯)+𝑀(𝑦), then this function is called Modulus function, which was presented and discussed by Maddox [18].

Lindenstrauss and Tzafriri [19] used the idea of Orlicz function to construct the sequence spaceℓ𝑀=ξƒ―ξ€·π‘₯π‘˜ξ€Έβˆˆπ‘€βˆΆβˆžξ“π‘˜=1𝑀||π‘₯π‘˜||πœŒξ‚Άξƒ°<∞,forsome𝜌>0.(1.7) The space ℓ𝑀 with the normβ€–ξƒ―π‘₯β€–=inf𝜌>0βˆΆβˆžξ“π‘˜=1𝑀||π‘₯π‘˜||πœŒξ‚Άξƒ°β‰€1(1.8) becomes a Banach space which is called an Orlicz sequence space.

The space ℓ𝑀  is closely related to the space ℓ𝑝 which is an Orlicz sequence space with 𝑀(𝑑)=|𝑑|𝑝, for 1≀𝑝<∞.

Subsequently, Orlicz function was used to define sequence spaces by Parashar and Choudhary [20] and others (see [10, 15, 21–23]).

Remark 1.1. It is well known if M is a convex function and M(0) = 0, then 𝑀(πœ†π‘₯)β‰€πœ†π‘€(π‘₯),βˆ€πœ†with0<πœ†<1.(1.9) In this paper, we introduce some new ideal convergent sequences over n-normed spaces by using Orlicz functions and generalized difference sequence spaces.

2. Definitions and Preliminaries

Let n be a nonnegative integer and X a real vector space of dimension 𝑑β‰₯𝑛 (d may be infinite). A real-valued function β€–β‹…,…,β‹…β€– on 𝑋𝑛 satisfying the following conditions:(1)β€–π‘₯1,π‘₯2,…,π‘₯𝑛‖=0 if and only if π‘₯1,π‘₯2,…,π‘₯𝑛 are linearly dependent,(2)β€–π‘₯1,π‘₯2,…,π‘₯𝑛‖ is invariant under permutation,(3)‖𝛼π‘₯1,π‘₯2,…,π‘₯𝑛‖=|𝛼|β€–π‘₯1,π‘₯2,…,π‘₯𝑛‖, for any α∈ ℝ,(4)β€–π‘₯+π‘₯ξ…ž,π‘₯2,…,π‘₯𝑛‖≀‖π‘₯,π‘₯2,…,π‘₯𝑛‖+β€–π‘₯ξ…ž,π‘₯2,…,π‘₯𝑛‖, is called an n-norm on 𝑋 and the pair (𝑋,β€–.,…,.β€–) is called an 𝑛-normed space.

A trivial example of an n-normed space is 𝑋=ℝ𝑛, equipped with the Euclidean n-norm β€–π‘₯1,π‘₯2,…,π‘₯𝑛‖𝐸= the volume of the 𝑛-dimensional parallelopiped spanned by the vectors π‘₯1,π‘₯2,…,π‘₯𝑛, which may be given explicitly by the formulaβ€–β€–π‘₯1,π‘₯2,…,π‘₯𝑛‖‖𝐸=||ξ€·π‘₯det𝑖𝑗||βŽ›βŽœβŽœβŽœβŽœβŽœβŽ|||||||||=abs⟨π‘₯1,π‘₯1βŸ©β‹―βŸ¨π‘₯1,π‘₯π‘›βŸ©β‹…β‹―β‹…β‹…β‹―β‹…βŸ¨π‘₯𝑛,π‘₯1βŸ©β‹―βŸ¨π‘₯𝑛,π‘₯π‘›βŸ©|||||||||⎞⎟⎟⎟⎟⎟⎠,(2.1) where π‘₯𝑖=(π‘₯𝑖1,π‘₯𝑖2,…,π‘₯𝑖𝑛)βˆˆβ„π‘›, for each 𝑖=1,2,…,𝑛.

Let (𝑋,β€–.,…,.β€–) be an n-normed space of dimension 𝑑β‰₯𝑛β‰₯2 and {π‘Ž1,π‘Ž2,…,π‘Žπ‘›} a linearly independent set in 𝑋. Then, the function β€–β‹…,…,β‹…β€–βˆžonπ‘‹π‘›βˆ’1 is defined by β€–β€–π‘₯1,π‘₯2,…,π‘₯π‘›β€–β€–βˆž=max1≀𝑖≀𝑛‖‖π‘₯1,π‘₯2,…,π‘₯π‘›βˆ’1,π‘Žπ‘–β€–β€–ξ€Ύ(2.2) defines as (π‘›βˆ’1)-norm on 𝑋 with respect to {π‘Ž1,π‘Ž2,…,π‘Žπ‘›}, and this is known as the derived (π‘›βˆ’1)-norm (see [12] for details).

The standard n-norm on 𝑋, a real inner product space of dimension 𝑑β‰₯𝑛, is as follows:β€–β€–π‘₯1,π‘₯2,…,π‘₯𝑛‖‖𝑆=|||||||||⟨π‘₯1,π‘₯1βŸ©β‹―βŸ¨π‘₯1,π‘₯π‘›βŸ©β‹…β‹―β‹…β‹…β‹―β‹…βŸ¨π‘₯𝑛,π‘₯1βŸ©β‹―βŸ¨π‘₯𝑛,π‘₯π‘›βŸ©|||||||||1/2,(2.3) where βŸ¨β‹…,β‹…βŸ© denotes the inner product on 𝑋. If we take 𝑋=ℝ𝑛, then this n-norm is exactly the same as the Euclidean n-norm β€–π‘₯1,π‘₯2,…,π‘₯𝑛‖𝐸  mentioned earlier.

For 𝑛=1, this n-norm is the usual norm β€–π‘₯1βˆšβ€–=⟨π‘₯1,π‘₯1⟩.

Definition 2.1. A sequence (π‘₯π‘˜) in an n-normed space (𝑋,β€–β‹…,…,β‹…β€–) is said to converge to some π‘™βˆˆπ‘‹ with respect the n-norm if for each πœ€>0, there exists an positive integer 𝑛0 such that β€–β€–π‘₯π‘˜βˆ’π‘™,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–<πœ€,βˆ€π‘˜β‰₯𝑛0forevery𝑧1,𝑧2,…,π‘§π‘›βˆ’1βˆˆπ‘‹.(2.4)

Definition 2.2. A sequence (π‘₯π‘˜) in an n-normed space (𝑋,β€–.,…,.β€–) is said to be Cauchy with respect the n-norm if for each πœ€>0, there exists an positive integer 𝑛0=𝑛0(πœ€) such that β€–β€–π‘₯π‘˜βˆ’π‘₯π‘š,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–<πœ€,βˆ€π‘˜,π‘šβ‰₯𝑛0forevery𝑧1,𝑧2,…,π‘§π‘›βˆ’1βˆˆπ‘‹.(2.5) If every Cauchy sequence in 𝑋 converges to some π‘™βˆˆπ‘‹, then 𝑋 is said to be complete with respect to the n-norm. Any complete n-normed space is said to be n-Banach space.

Definition 2.3. Let S be a nonempty set. A nonempty family of sets πΌβŠ†π‘ƒ(𝑆) (power set of S) is called an ideal in S if (i) for each 𝐴,𝐡∈𝐼, we have 𝐴βˆͺ𝐡∈𝐼; (ii) for each 𝐴∈𝐼 and π΅βŠ†π΄, we have 𝐡∈𝐼.

Definition 2.4. Let S be a nonempty set. A family πΉβŠ†π‘ƒ(𝑆) (power set of S) is called a filter on S if (i) βˆ…βˆ‰πΉ; (ii) for each A, 𝐡∈𝐹, we have 𝐴∩𝐡∈𝐹; (iii) for each 𝐴∈𝐹 and π΅βŠ‡π΄, we have 𝐡∈𝐹.

An ideal 𝐼 is called nontrivial if πΌβ‰ βˆ… and π‘†βˆ‰πΌ. It is clear that πΌβŠ†π‘ƒ(𝑆) is a nontrivial ideal if and only if the class 𝐹=𝐹(𝐼)={π‘†βˆ’π΄βˆΆπ΄βˆˆπΌ} is a filter on S. The filter 𝐹(𝐼) is called the filter associated with the ideal 𝐼.

A nontrivial ideal πΌβŠ†π‘ƒ(𝑆) is called an admissible ideal in 𝑆 if and only if it contains all singletons, that is, if it contains {{π‘₯}∢π‘₯βˆˆπ‘†}.

Definition 2.5. A sequence (π‘₯π‘˜) in a normed space (𝑋,||β‹…||) is said to be statistically convergent to π‘₯0βˆˆπ‘‹ if for each πœ€>0, the set 𝐸(πœ€)={π‘˜βˆˆβ„•βˆΆβ€–π‘₯π‘˜βˆ’π‘₯0β€–β‰₯πœ€} has natural density zero.

Definition 2.6. A sequence (π‘₯π‘˜) in a normed space (𝑋,||β‹…||) is said to be I-convergent to π‘₯0βˆˆπ‘‹ if for each πœ€>0, the set 𝐸‖‖π‘₯(πœ€)=π‘˜βˆˆβ„•βˆΆπ‘˜βˆ’π‘₯0β€–β€–ξ€Ύβ‰₯πœ€belongsto𝐼.(2.6)

Definition 2.7. A sequence (π‘₯π‘˜) in a normed space (𝑋,||β‹…||) is said to be I-bounded if there exists 𝑀>0 such that the set {π‘˜βˆˆβ„•βˆΆβ€–π‘₯π‘˜β€–>𝑀} belongs to 𝐼.

Definition 2.8. A sequence (π‘₯π‘˜) in a normed space (𝑋,β€–β‹…β€–) is said to be I-Cauchy if for each πœ€>0, there exists an positive integer π‘š=π‘š(πœ€) such that the set 𝐸‖‖π‘₯(πœ€)=π‘˜βˆˆβ„•βˆΆπ‘˜βˆ’π‘₯π‘šβ€–β€–ξ€Ύβ‰₯πœ€belongsto𝐼.(2.7)

Definition 2.9. A sequence (π‘₯π‘˜) in an n-normed space (𝑋,β€–β‹…,…,β‹…β€–) is said to be I-convergent to π‘₯0βˆˆπ‘‹ with respect the n-norm, if for each πœ€>0, the set 𝐸‖‖π‘₯(πœ€)=π‘˜βˆˆβ„•βˆΆπ‘˜βˆ’π‘₯0,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–β‰₯πœ€,forevery𝑧1,𝑧2,…,π‘§π‘›βˆ’1ξ€Ύβˆˆπ‘‹belongsto𝐼.(2.8)

Definition 2.10. A sequence (π‘₯π‘˜) in an n-normed space (𝑋,β€–β‹…,…,β‹…β€–) is said to be I-Cauchy with respect to the n-norm if for each πœ€>0, there exists a positive integer π‘š=π‘š(πœ€) such that the set 𝐸‖‖π‘₯(πœ€)=π‘˜βˆˆβ„•βˆΆπ‘˜βˆ’π‘₯π‘š,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–β‰₯πœ€,forevery𝑧1,𝑧2,…,π‘§π‘›βˆ’1ξ€Ύβˆˆπ‘‹belongsto𝐼.(2.9)

Definition 2.11. A sequence space 𝐸 is said to be normal (or solid) if (π›Όπ‘˜π‘₯π‘˜)∈𝐸, whenever (π‘₯π‘˜)∈𝐸 and for all sequence (π›Όπ‘˜) of scalars with |π›Όπ‘˜|≀1, for all π‘˜βˆˆβ„•.

Let 𝐾={π‘˜1<π‘˜2<β‹―}βŠ†β„• and 𝐸 a sequence space.A K-step space of 𝐸 is a sequence space πœ†πΈπΎ={(π‘₯π‘˜π‘›)βˆˆπ‘€βˆΆ(π‘˜π‘›)∈𝐸}.

A canonical preimage of a sequence {π‘₯π‘˜π‘›}βˆˆπœ†πΈπΎ is a sequence {π‘¦π‘˜}βˆˆπ‘€ defined as π‘¦π‘˜=ξƒ―π‘₯π‘˜,ifπ‘˜βˆˆπΎ,0,otherwise.(2.10) A canonical preimage of a step space πœ†πΈπΎ is a set of canonical preimages of all elements inπœ†πΈπΎ; that is, y is in canonical preimage of πœ†πΈπΎ if and only if y is canonical preimage of some π‘₯βˆˆπœ†πΈπΎ.

Definition 2.12. A sequence space 𝐸 is said to be monotone if it contains the canonical preimages of all its step spaces.

Example 2.13. If we take 𝐼=𝐼𝑓={π΄βŠ†β„•βˆΆπ΄isafinitesubset}. Then, 𝐼𝑓 is a nontrivial admissible ideal of  ℕ  and the corresponding convergence coincide with usual convergence of sequences in n-normed spaces.

Example 2.14. If we take 𝐼=𝐼𝛿={π΄βŠ†β„•βˆΆπ›Ώ(𝐴)=0}, where 𝛿(𝐴) denote the asymptotic density of the set A then 𝐼𝛿 is a nontrivial admissible ideal of β„• and the corresponding convergence coincide with statistical convergence in n-normed spaces.

Now, we state the following lemmas (see [12] for details).

Lemma 2.15. Every n-normed space is an (π‘›βˆ’π‘Ÿ)-normed space for all π‘Ÿ=1,2,…,π‘›βˆ’1. In particular, every n-normed space is a normed space.

Lemma 2.16. A standard n-normed space is complete if and only if it is complete with respect to the usual norm βˆšβ€–β‹…β€–=⟨,⟩.

Lemma 2.17. On a standard n-normed space 𝑋, the derived (π‘›βˆ’1)-norm β€–β‹…,…,β‹…β€–βˆž defined with respect to the orthogonal set {𝑒1,𝑒2,…,𝑒𝑛} is equivalent to the standard (π‘›βˆ’1)-norm β€–β‹…,…,⋅‖𝑆. To be precise, one has β€–β€–π‘₯1,π‘₯2,…,π‘₯π‘›βˆ’1β€–β€–βˆžβ‰€β€–β€–π‘₯1,π‘₯2,…,π‘₯π‘›βˆ’1β€–β€–π‘†β‰€βˆšπ‘›β€–β€–π‘₯1,π‘₯2,…,π‘₯π‘›βˆ’1β€–β€–βˆž,(2.11) for all π‘₯1,π‘₯2,…,π‘₯π‘›βˆ’1βˆˆπ‘‹, whereβ€–π‘₯1,π‘₯2,…,π‘₯π‘›βˆ’1β€–βˆž=max1≀𝑖≀𝑛{β€–π‘₯1,π‘₯2,…,π‘₯π‘›βˆ’1,𝑒𝑖‖𝑆}.

Lemma 2.18 (Kamthan and Gupta [24]). Every normal space is monotone.

3. Main Results

In this section, we define some new ideal convergent sequence spaces and investigate their linear topological structures. Also, we find out some relations related to these sequence spaces.

Let 𝐼 be an admissible ideal of β„•, 𝑀 an Orlicz function, and (𝑋,β€–β‹…,…,β‹…β€–) an n-normed space. Further, let 𝑝=(π‘π‘˜) be any bounded sequence of positive real numbers, and let 𝑀(π‘›βˆ’π‘‹), denote the spaces of all X-valued sequences. For each πœ€>0, we define the following sequence spaces:π‘ŠπΌξ€·Ξ”π‘ π‘šξ€Έ=ξƒ―ξ€·π‘₯,𝑀,𝑝,β€–β‹…,…,β‹…β€–π‘˜ξ€Έξƒ―1βˆˆπ‘€(π‘›βˆ’π‘‹)βˆΆπ‘›βˆˆβ„•βˆΆπ‘›π‘›ξ“π‘˜=1ξ‚Έπ‘€ξ‚΅β€–β€–β€–Ξ”π‘ π‘šπ‘₯π‘˜βˆ’l𝜌,𝑧1𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–ξ‚Άξ‚Ήπ‘π‘˜ξƒ°β‰₯πœ€βˆˆπΌ,forsome𝜌>0,π‘™βˆˆπ‘‹andβˆ€π‘§1,𝑧2β€¦π‘§π‘›βˆ’1ξƒ°,π‘Šβˆˆπ‘‹(3.1)πΌξ€·Ξ”π‘ π‘šξ€Έ=ξƒ―ξ€·π‘₯,𝑀,𝑝,β€–β‹…,…,β‹…β€–π‘˜ξ€Έξƒ―1βˆˆπ‘€(π‘›βˆ’π‘‹)βˆΆπ‘›βˆˆβ„•βˆΆπ‘›π‘›ξ“π‘˜=1ξ‚Έπ‘€ξ‚΅β€–β€–β€–Ξ”π‘ π‘šπ‘₯π‘˜πœŒ,𝑧1𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–ξ‚Άξ‚Ήπ‘π‘˜ξƒ°β‰₯πœ€βˆˆπΌ,forsome𝜌>0andβˆ€π‘§1,𝑧2β€¦π‘§π‘›βˆ’1ξƒ°,π‘Šβˆˆπ‘‹(3.2)πΌξ€·Ξ”π‘ π‘šξ€Έ=ξƒ―ξ€·π‘₯,𝑀,𝑝,β€–β‹…,…,β‹…β€–π‘˜ξ€Έβˆˆπ‘€(π‘›βˆ’π‘‹)∢supπ‘›βˆˆβ„•1π‘›π‘›ξ“π‘˜=1ξ‚Έπ‘€ξ‚΅β€–β€–β€–Ξ”π‘ π‘šπ‘₯π‘˜πœŒ,𝑧1𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–ξ‚Άξ‚Ήπ‘π‘˜<∞,forsome𝜌>0andβˆ€π‘§1,𝑧2β€¦π‘§π‘›βˆ’1ξƒ°,π‘Šβˆˆπ‘‹(3.3)πΌξ€·Ξ”π‘ π‘šξ€Έ=ξƒ―ξ€·π‘₯,𝑀,𝑝,β€–β‹…,…,β‹…β€–π‘˜ξ€Έξƒ―1βˆˆπ‘€(π‘›βˆ’π‘‹)βˆΆβˆƒπΎ>0s.t.π‘›βˆˆβ„•βˆΆπ‘›π‘›ξ“π‘˜=1ξ‚Έπ‘€ξ‚΅β€–β€–β€–Ξ”π‘ π‘šπ‘₯π‘˜πœŒ,𝑧1𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–ξ‚Άξ‚Ήπ‘π‘˜ξƒ°β‰₯𝐾∈𝐼,forsome𝜌>0andβˆ€π‘§1,𝑧2β€¦π‘§π‘›βˆ’1ξƒ°.βˆˆπ‘‹(3.4)

The following well-known inequality will be used throughout the article. Let 𝑝=(π‘π‘˜) be any sequence of positive real numbers with 0β‰€π‘π‘˜β‰€supπ‘˜π‘π‘˜=𝐺,𝐷=max{1,2πΊβˆ’1}, then  |π‘Žπ‘˜+π‘π‘˜|π‘π‘˜β‰€π·(|π‘Žπ‘˜|π‘π‘˜+|π‘π‘˜|π‘π‘˜), for all π‘˜βˆˆβ„• and π‘Žπ‘˜,π‘π‘˜βˆˆβ„‚. Also, |π‘Ž|π‘π‘˜β‰€max{1,|π‘Ž|𝐺},  for all π‘Žβˆˆβ„‚.

Theorem 3.1. The spaces π‘ŠπΌ(Ξ”π‘ π‘š,𝑀,𝑝,β€–β‹…,…,β‹…β€–), π‘ŠπΌ0(Ξ”π‘ π‘š,𝑀,𝑝,||β‹…,…,β‹…||)  and π‘ŠπΌβˆž(Ξ”π‘ π‘š,𝑀,𝑝, β€–β‹…,…,β‹…β€–)   are linear.

Proof. We will prove the result for the space π‘ŠπΌ0(Ξ”π‘ π‘š,𝑀,𝑝,β€–β‹…,…,β‹…β€–) only and the others can be proved similarly.
Let π‘₯=(π‘₯π‘˜) and 𝑦=(π‘¦π‘˜) be any two elements of the space π‘ŠπΌ0(Ξ”π‘ π‘š,𝑀,𝑝,β€–β‹…,…,β‹…β€–). Then, there exist 𝜌1>0 and 𝜌2>0 such that the sets π΄πœ€/2ξƒ―1(π‘₯)=π‘›βˆˆβ„•βˆΆπ‘›π‘›ξ“π‘˜=1ξ‚Έπ‘€ξ‚΅β€–β€–β€–Ξ”π‘ π‘šπ‘₯π‘˜πœŒ1,𝑧1,𝑧2,....,π‘§π‘›βˆ’1β€–β€–β€–ξ‚Άξ‚Ήπ‘π‘˜β‰₯πœ€2,forevery𝑧1,𝑧2,…,π‘§π‘›βˆ’1ξƒ°,π΅βˆˆπ‘‹πœ€/2ξƒ―1(𝑦)=π‘›βˆˆβ„•βˆΆπ‘›π‘›ξ“π‘˜=1ξ‚Έπ‘€ξ‚΅β€–β€–β€–Ξ”π‘ π‘šπ‘¦π‘˜πœŒ2,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–ξ‚Άξ‚Ήπ‘π‘˜β‰₯πœ€2,forevery𝑧1,𝑧2,…,π‘§π‘›βˆ’1ξƒ°βˆˆπ‘‹(3.5) belong to 𝐼.
Let 𝛼,π›½βˆˆβ„ be two scalars. Since β€–β‹…,…,β‹…β€– is an n-norm, the operator Δ𝑠 is linear and the Orlicz function M is continuous, the following inequality holds: 1π‘›π‘›ξ“π‘˜=1ξƒ¬π‘€ξƒ©β€–β€–β€–β€–Ξ”π‘ π‘šξ€·π›Όπ‘₯π‘˜+π›½π‘¦π‘˜ξ€Έξ€·|𝛼|𝜌1+||𝛽||𝜌2ξ€Έ,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–β€–ξƒͺξƒ­π‘π‘˜1β‰€π·π‘›π‘›ξ“π‘˜=1|𝛼|ξ€·|𝛼|𝜌1+||𝛽||𝜌2ξ€Έπ‘€ξ‚΅β€–β€–β€–Ξ”π‘ π‘šπ‘₯π‘˜πœŒ1,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–ξ‚Άξƒ­π‘π‘˜1+π·π‘›π‘›ξ“π‘˜=1||𝛽||ξ€·|𝛼|𝜌1+||𝛽||𝜌2ξ€Έπ‘€ξ‚΅β€–β€–β€–Ξ”π‘ π‘šπ‘¦π‘˜πœŒ2,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–ξ‚Άξƒ­π‘π‘˜1β‰€π·πΎπ‘›π‘›ξ“π‘˜=1ξ‚Έπ‘€ξ‚΅β€–β€–β€–Ξ”π‘ π‘šπ‘₯π‘˜πœŒ1,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–ξ‚Άξ‚Ήπ‘π‘˜1+π·πΎπ‘›π‘›ξ“π‘˜=1ξ‚Έπ‘€ξ‚΅β€–β€–β€–Ξ”π‘ π‘šπ‘¦π‘˜πœŒ2,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–ξ‚Άξ‚Ήπ‘π‘˜,(3.6) where 𝐾=max{1,(|𝛼|/(|𝛼|𝜌1+|𝛽|𝜌2)),(|𝛽|/(|𝛼|𝜌1+|𝛽|𝜌2))}.
From the above relation, we get ξƒ―1π‘›βˆˆβ„•βˆΆπ‘›π‘›ξ“π‘˜=1ξƒ¬π‘€ξƒ©β€–β€–β€–β€–Ξ”π‘ π‘šξ€·π›Όπ‘₯π‘˜+π›½π‘¦π‘˜ξ€Έξ€·|𝛼|𝜌1+||𝛽||𝜌2ξ€Έ,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–β€–ξƒͺξƒ­π‘π‘˜ξƒ°βŠ†ξƒ―1β‰₯πœ€π‘›βˆˆβ„•βˆΆπ·πΎπ‘›π‘›ξ“π‘˜=1ξ‚Έπ‘€ξ‚΅β€–β€–β€–Ξ”π‘ π‘šπ‘₯π‘˜πœŒ1,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–ξ‚Άξ‚Ήπ‘π‘˜β‰₯πœ€2ξƒ°βˆͺξƒ―1π‘›βˆˆβ„•βˆΆπ·πΎπ‘›π‘›ξ“π‘˜=1ξ‚Έπ‘€ξ‚΅β€–β€–β€–Ξ”π‘ π‘šπ‘¦π‘˜πœŒ2,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–ξ‚Άξ‚Ήπ‘π‘˜β‰₯πœ€2ξƒ°.(3.7) Since both the sets on the right hand side of the relation (3.7) belong to 𝐼, this completes the proof.

Note 1. It is easy to verify that the space π‘Šβˆž(Ξ”π‘ π‘š,𝑀,𝑝,||β‹…,…,β‹…||) is a linear space.

Theorem 3.2. The space π‘ŠπΌ(Ξ”π‘ π‘š,𝑀,𝑝,||β‹…,…,β‹…||), π‘ŠπΌ0(Ξ”π‘ π‘š,𝑀,𝑝,||β‹…,…,β‹…||), π‘ŠπΌβˆž(Ξ”π‘ π‘š,𝑀,𝑝, ||β‹…,…,β‹…||),   and π‘Šβˆž(Ξ”π‘ π‘š,𝑀,𝑝, ||β‹…,…,β‹…||)are paranormed spaces (not totally paranormed ) with respect to the paranorm 𝑔Δ defined by 𝑔Δ(π‘₯)=π‘šπ‘ ξ“π‘˜=1β€–β€–π‘₯π‘˜,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–ξ‚»πœŒ+infπ‘π‘˜/𝐻∢supπ‘˜π‘€ξ‚΅β€–β€–β€–Ξ”π‘ π‘šπ‘₯π‘˜πœŒ,𝑧1,𝑧2,…,π‘§π‘›βˆ’1‖‖‖≀1,forsome𝜌>0andβˆ€π‘§1,𝑧2,…,π‘§π‘›βˆ’1ξƒ°βˆˆπ‘‹(3.8) where 𝐻=max{1,supπ‘π‘˜}.

Proof. Clearly, 𝑔Δ(βˆ’π‘₯)=𝑔Δ(π‘₯) and 𝑔Δ(πœƒ)=0. Let π‘₯=(π‘₯π‘˜) and 𝑦=(π‘¦π‘˜)be any two elements of the space 𝑍(Ξ”π‘ π‘š,𝑀,𝑝,||β‹…,…,β‹…||), where 𝑍=π‘ŠπΌ,π‘ŠπΌ0,π‘ŠπΌβˆžπ‘Žπ‘›π‘‘π‘Šβˆž.
Then,for 𝜌>0, we set 𝐴1=ξ‚»πœŒ>0∢supπ‘˜π‘€ξ‚΅β€–β€–β€–Ξ”π‘ π‘šπ‘₯π‘˜πœŒ,𝑧1,𝑧2,…,π‘§π‘›βˆ’1‖‖‖≀1,βˆ€π‘§1,𝑧2,…,π‘§π‘›βˆ’1ξ‚Ό,π΄βˆˆπ‘‹2=ξ‚»πœŒ>0∢supπ‘˜π‘€ξ‚΅β€–β€–β€–Ξ”π‘ π‘šπ‘¦π‘˜πœŒ,𝑧1,𝑧2,…,π‘§π‘›βˆ’1‖‖‖≀1,βˆ€π‘§1,𝑧2,…,π‘§π‘›βˆ’1ξ‚Ό.βˆˆπ‘‹(3.9) Let 𝜌1∈𝐴1 and 𝜌2∈𝐴2. If 𝜌=𝜌1+𝜌2, then we have π‘€ξƒ©β€–β€–β€–β€–Ξ”π‘ π‘šξ€·π‘₯π‘˜+π‘¦π‘˜ξ€ΈπœŒ,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–β€–ξƒͺβ‰€πœŒ1𝜌1+𝜌2π‘€ξ‚΅β€–β€–β€–Ξ”π‘ π‘šπ‘₯π‘˜πœŒ1,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–ξ‚Ά+𝜌2𝜌1+𝜌2π‘€ξ‚΅β€–β€–β€–Ξ”π‘ π‘šπ‘¦π‘˜πœŒ2,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–ξ‚Ά.(3.10) Thus, we have supπ‘˜ξƒ¬π‘€ξƒ©β€–β€–β€–β€–Ξ”π‘ π‘šξ€·π‘₯π‘˜+π‘¦π‘˜ξ€ΈπœŒ,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–β€–ξƒͺξƒ­π‘π‘˜π‘”β‰€1,Ξ”(π‘₯+𝑦)=π‘šπ‘ ξ“π‘˜=1β€–β€–π‘₯π‘˜+π‘¦π‘˜,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–ξ‚†ξ€·πœŒ+inf1+𝜌2ξ€Έπ‘π‘˜/𝐻∢𝜌1∈𝐴1,𝜌2∈𝐴2ξ‚‡β‰€π‘šπ‘ ξ“π‘˜=1β€–β€–π‘₯π‘˜,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–ξ‚†ξ€·πœŒ+inf1ξ€Έπ‘π‘˜/𝐻∢𝜌1∈𝐴1+π‘šπ‘ ξ“π‘˜=1β€–β€–π‘¦π‘˜,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–ξ‚†ξ€·πœŒinf2ξ€Έπ‘π‘˜/𝐻∢𝜌2∈𝐴2=𝑔Δ(π‘₯)+𝑔Δ(𝑦).(3.11) Let πœ†π‘ β†’πœ†, where πœ†π‘ ,πœ†βˆˆβ„‚, and let 𝑔Δ(π‘₯π‘ βˆ’π‘₯)β†’0 as π‘ β†’βˆž. We have to show that 𝑔Δ(πœ†π‘ π‘₯π‘ βˆ’πœ†π‘₯)β†’0 as π‘ β†’βˆž.
We set 𝐴3=ξ‚»πœŒπ‘ >0∢supπ‘˜ξ‚Έπ‘€ξ‚΅β€–β€–β€–Ξ”π‘ π‘šπ‘₯π‘ π‘˜πœŒπ‘ ,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–ξ‚Άξ‚Ήπ‘π‘˜β‰€1,βˆ€π‘§1,𝑧2,…,π‘§π‘›βˆ’1ξ‚Ό,π΄βˆˆπ‘‹4=ξƒ―πœŒπ‘ ξ…ž>0∢supπ‘˜ξƒ¬π‘€ξƒ©β€–β€–β€–β€–Ξ”π‘ π‘šξ€·π‘₯π‘ π‘˜βˆ’π‘₯π‘˜ξ€ΈπœŒπ‘ ξ…ž,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–β€–ξƒͺξƒ­π‘π‘˜β‰€1,βˆ€π‘§1,𝑧2,…,π‘§π‘›βˆ’1ξƒ°.βˆˆπ‘‹(3.12) If πœŒπ‘ βˆˆπ΄3 andβ€‰β€‰πœŒπ‘ ξ…žβˆˆπ΄4, we observe by the continuity of the Orlicz function M that π‘€ξƒ©β€–β€–β€–β€–Ξ”π‘ π‘šξ€·πœ†π‘ π‘₯π‘ π‘˜βˆ’πœ†π‘₯π‘˜ξ€Έξ€·||πœ†π‘ ||πœŒβˆ’πœ†π‘ +||πœ†||πœŒπ‘ ξ…žξ€Έ,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–β€–ξƒͺ‖‖‖‖Δ(3.13)β‰€π‘€π‘ π‘šξ€·πœ†π‘ π‘₯π‘ π‘˜βˆ’πœ†π‘₯π‘ π‘˜ξ€Έξ€·||πœ†π‘ ||πœŒβˆ’πœ†π‘ +||πœ†||πœŒπ‘ ξ…žξ€Έ,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–β€–ξƒͺ‖‖‖‖Δ+π‘€π‘ π‘šξ€·πœ†π‘₯π‘ π‘˜βˆ’πœ†π‘₯π‘˜ξ€Έξ€·||πœ†π‘ ||πœŒβˆ’πœ†π‘ +||πœ†||πœŒπ‘ ξ…žξ€Έ,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–β€–ξƒͺ≀||πœ†(3.14)𝑠||πœŒβˆ’πœ†π‘ ||πœ†π‘ ||πœŒβˆ’πœ†π‘ +||πœ†||πœŒπ‘ ξ…žπ‘€ξƒ©β€–β€–β€–β€–Ξ”π‘ π‘šξ€·π‘₯π‘ π‘˜ξ€ΈπœŒπ‘ ,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–β€–ξƒͺ+||πœ†||πœŒπ‘ ξ…ž||πœ†π‘ ||πœŒβˆ’πœ†π‘ +||πœ†||πœŒπ‘ ξ…žπ‘€ξƒ©β€–β€–β€–β€–Ξ”π‘ π‘šξ€·π‘₯π‘ π‘˜βˆ’π‘₯π‘˜ξ€ΈπœŒπ‘ ξ…ž,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–β€–ξƒͺ.(3.15) From the above inequality, it follows that supπ‘˜ξƒ¬π‘€ξƒ©β€–β€–β€–β€–Ξ”π‘ π‘šξ€·πœ†π‘ π‘₯π‘ π‘˜βˆ’πœ†π‘₯π‘˜ξ€Έξ€·||πœ†π‘ ||πœŒβˆ’πœ†π‘ +||πœ†||πœŒπ‘ ξ…žξ€Έ,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–β€–ξƒͺξƒ­π‘π‘˜β‰€1,(3.16) and consequently 𝑔Δ(πœ†π‘ π‘₯𝑠=βˆ’πœ†π‘₯)π‘šπ‘ ξ“π‘˜=1β€–β€–πœ†π‘ π‘₯π‘ π‘˜βˆ’πœ†π‘₯π‘˜,𝑧1,𝑧2,…,π‘§π‘›βˆ’1‖‖||πœ†+inf𝑠||πœŒβˆ’πœ†π‘ +||πœ†||πœŒπ‘ ξ…žξ€Έπ‘π‘˜/π»βˆΆπœŒπ‘ βˆˆπ΄3,πœŒπ‘ ξ…žβˆˆπ΄4≀||πœ†π‘ ||βˆ’πœ†π‘šπ‘ ξ“π‘˜=1β€–β€–π‘₯π‘ π‘˜,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–+||πœ†π‘ ||βˆ’πœ†π‘π‘˜/π»ξ‚†ξ€·πœŒinfπ‘ ξ€Έπ‘π‘˜/π»βˆΆπœŒπ‘ βˆˆπ΄3+||πœ†||π‘šπ‘ ξ“π‘˜=1β€–β€–π‘₯π‘ π‘˜βˆ’π‘₯π‘˜,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–+||πœ†||π‘π‘˜/π»ξ‚†ξ€·πœŒinfπ‘ ξ…žξ€Έπ‘π‘˜/π»βˆΆπœŒπ‘ ξ…žβˆˆπ΄4||πœ†β‰€max𝑠||,||πœ†βˆ’πœ†π‘ ||βˆ’πœ†π‘π‘˜/𝐻𝑔Δ(π‘₯𝑠||πœ†||,||πœ†||)+maxπ‘π‘˜/𝐻𝑔Δ(π‘₯π‘ βˆ’π‘₯).(3.17) Note that 𝑔Δ(π‘₯𝑠)≀𝑔Δ(π‘₯)+𝑔Δ(π‘₯π‘ βˆ’π‘₯), for all π‘ βˆˆβ„•.
Hence, by our assumption, the right hand side (3.17) tends to 0 as π‘ β†’βˆž, and the result follows. This completes the proof of the theorem.

Theorem 3.3. Let 𝑀,𝑀1, and 𝑀2  be Orlicz functions. Then, the following hold: (a)π‘ŠπΌ0(Ξ”π‘ π‘š,𝑀,𝑝,β€–β‹…,…,β‹…β€–)βŠ†π‘ŠπΌ0(Ξ”π‘ π‘š,𝑀.𝑀1,𝑝,β€–β‹…,…,β‹…β€–), provided (π‘π‘˜) be such that 𝐺0=infπ‘π‘˜>0,(b)π‘ŠπΌ0(Ξ”π‘ π‘š,𝑀1,𝑝,β€–β‹…,…,β‹…β€–)βˆ©π‘ŠπΌ0(Ξ”π‘ π‘š,𝑀2,𝑝,β€–β‹…,…,β‹…β€–)βŠ†π‘ŠπΌ0(Ξ”π‘ π‘š,𝑀1+𝑀2,𝑝,β€–β‹…,…,β‹…β€–).

Proof. (a) Let πœ€>0 be given. Choose πœ€1>0 such that max{πœ€πΊ1,πœ€πΊ01}<πœ€.
Using the continuity of the Orlicz function M, choose 0<𝛿<1 such that 0<𝑑<𝛿 implies that 𝑀(𝑑)<πœ€1.
Let π‘₯=(π‘₯π‘˜) be any element in π‘ŠπΌ0(Ξ”π‘ π‘š,𝑀1,𝑝,β€–β‹…,…,β‹…β€–). Put 𝐴𝛿=ξƒ―1π‘›βˆˆβ„•βˆΆπ‘›π‘›ξ“π‘˜=1𝑀1ξ‚΅β€–β€–β€–Ξ”π‘ π‘šπ‘₯π‘˜πœŒ,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–ξ‚Άξ‚Ήπ‘π‘˜β‰₯𝛿𝐺.(3.18) Then, by definition of ideal, we have the set π΄π›ΏβˆˆπΌ.
If π‘›βˆ‰π΄π›Ώ, then we have 1π‘›π‘›ξ“π‘˜=1𝑀1ξ‚΅β€–β€–β€–Ξ”π‘ π‘šπ‘₯π‘˜πœŒ,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–ξ‚Άξ‚Ήπ‘π‘˜<π›ΏπΊβŸΉπ‘›ξ“π‘˜=1𝑀1ξ‚΅β€–β€–β€–Ξ”π‘ π‘šπ‘₯π‘˜πœŒ,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–ξ‚Άξ‚Ήπ‘π‘˜<π‘›π›ΏπΊβŸΉξ‚Έπ‘€1ξ‚΅β€–β€–β€–Ξ”π‘ π‘šπ‘₯π‘˜πœŒ,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–ξ‚Άξ‚Ήπ‘π‘˜<𝛿𝐺,βˆ€π‘˜=1,2,…,π‘›βŸΉπ‘€1ξ‚΅β€–β€–β€–Ξ”π‘ π‘šπ‘₯π‘˜πœŒ,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–ξ‚Ά<𝛿,βˆ€π‘˜=1,2,…,𝑛.(3.19) Using the continuity of the Orlicz function M, then from the relation (3.19), we have π‘›ξ“π‘˜=1𝑀𝑀1ξ‚΅β€–β€–β€–Ξ”π‘ π‘šπ‘₯π‘˜πœŒ,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–ξ‚Άξ‚Ά<πœ€1,βˆ€π‘˜=1,2,…,𝑛.(3.20) Consequently, we get π‘›ξ“π‘˜=1𝑀𝑀1ξ‚΅β€–β€–β€–Ξ”π‘ π‘šπ‘₯π‘˜πœŒ,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–ξ‚Άξ‚Άξ‚Ήπ‘π‘˜ξ‚†πœ€<𝑛max𝐺1,πœ€πΊ01ξ‚‡βŸΉ1<π‘›πœ€π‘›π‘›ξ“π‘˜=1𝑀𝑀1ξ‚΅β€–β€–β€–Ξ”π‘ π‘šπ‘₯π‘˜πœŒ,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–ξ‚Άξ‚Άξ‚Ήπ‘π‘˜<πœ€.(3.21) This shows that ξƒ―1π‘›βˆˆβ„•βˆΆπ‘›π‘›ξ“π‘˜=1𝑀𝑀1ξ‚΅β€–β€–β€–Ξ”π‘ π‘šπ‘₯π‘˜πœŒ,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–ξ‚Άξ‚Άξ‚Ήπ‘π‘˜ξƒ°β‰₯πœ€,βŠ†π΄π›ΏβˆˆπΌ.(3.22) This proves the result.
(b) Let π‘₯=(π‘₯π‘˜)βˆˆπ‘ŠπΌ0(Ξ”π‘ π‘š,𝑀1,𝑝,β€–β‹…,…,β‹…β€–)βˆ©π‘ŠπΌ0(Ξ”π‘ π‘š,𝑀2,𝑝,β€–β‹…,…,β‹…β€–). Then, by the following inequality, the result follows: 1π‘›π‘›ξ“π‘˜=1𝑀1+𝑀2ξ€Έξ‚΅β€–β€–β€–Ξ”π‘ π‘šπ‘₯π‘˜πœŒ,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–ξ‚Άξ‚Ήπ‘π‘˜1β‰€π·π‘›π‘›ξ“π‘˜=1𝑀1ξ‚΅β€–β€–β€–Ξ”π‘ π‘šπ‘₯π‘˜πœŒ,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–ξ‚Άξ‚Ήπ‘π‘˜1+π·π‘›π‘›ξ“π‘˜=1𝑀2ξ‚΅β€–β€–β€–Ξ”π‘ π‘šπ‘₯π‘˜πœŒ,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–ξ‚Άξ‚Ήπ‘π‘˜.(3.23)

Theorem 3.4. The inclusions 𝑍(Ξ”π‘šπ‘ βˆ’1,𝑀,𝑝,β€–β‹…,…,β‹…β€–)βŠ†π‘(Ξ”π‘ π‘š,𝑀,𝑝,β€–β‹…,…,β‹…β€–)   are strict for 𝑠β‰₯1. In general, 𝑍(Ξ”π‘—π‘š,𝑀,𝑝,β€–β‹…,…,β‹…β€–)βŠ†π‘(Ξ”π‘ π‘š,𝑀,𝑝,β€–β‹…,…,β‹…β€–), for 𝑗=0,1,2,…,π‘ βˆ’1,  and the inclusion is strict, where 𝑍=π‘ŠπΌ,π‘ŠπΌ0,andπ‘ŠπΌβˆž.

Proof. We will give the proof for π‘ŠπΌ0(Ξ”π‘ π‘š,𝑀,𝑝,β€–β‹…,…,β‹…β€–) only. The others can be proved by similar arguments.
Let π‘₯=(π‘₯π‘˜) be any element in the space π‘ŠπΌ0(Ξ”π‘ π‘š,𝑀,𝑝,β€–β‹…,…,β‹…β€–). Let πœ€>0 be given. Then, there exists ρ > 0 such that the set ξƒ―1π‘›βˆˆβ„•βˆΆπ‘›π‘›ξ“π‘˜=1ξ‚Έπ‘€ξ‚΅β€–β€–β€–Ξ”π‘ π‘šπ‘₯π‘˜πœŒ,𝑧1𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–ξ‚Άξ‚Ήπ‘π‘˜ξƒ°β‰₯πœ€βˆˆπΌ.(3.24) Since M is nondecreasing and convex, it follows that 1π‘›π‘›ξ“π‘˜=1ξ‚Έπ‘€ξ‚΅β€–β€–β€–Ξ”π‘ π‘šπ‘₯π‘˜2𝜌,𝑧1𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–ξ‚Άξ‚Ήπ‘π‘˜=1π‘›π‘›ξ“π‘˜=1ξƒ¬π‘€ξƒ©β€–β€–β€–Ξ”π‘šπ‘ βˆ’1π‘₯π‘˜+1βˆ’Ξ”π‘šπ‘ βˆ’1π‘₯π‘˜2𝜌,𝑧1𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–ξƒͺξƒ­π‘π‘˜β‰€π·π‘›π‘›ξ“π‘˜=112π‘€ξƒ©β€–β€–β€–Ξ”π‘šπ‘ βˆ’1π‘₯π‘˜+1𝜌,𝑧1𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–ξƒͺξƒ­π‘π‘˜+π·π‘›π‘›ξ“π‘˜=112π‘€ξƒ©β€–β€–β€–Ξ”π‘šπ‘ βˆ’1π‘₯π‘˜πœŒ,𝑧1𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–ξƒͺξƒ­π‘π‘˜β‰€π·π»π‘›π‘›ξ“π‘˜=1π‘€ξƒ©β€–β€–β€–Ξ”ξƒ©ξƒ¬π‘šπ‘ βˆ’1π‘₯π‘˜+1𝜌,𝑧1𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–ξƒͺξƒ­π‘π‘˜+ξƒ¬π‘€ξƒ©β€–β€–β€–Ξ”π‘šπ‘ βˆ’1π‘₯π‘˜πœŒ,𝑧1𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–ξƒͺξƒ­π‘π‘˜ξƒͺ,(3.25) where 𝐻=max{1,(1/2)𝐺}.
Thus, we have ξƒ―1π‘›βˆˆβ„•βˆΆπ‘›π‘›ξ“π‘˜=1ξ‚Έπ‘€ξ‚΅β€–β€–β€–Ξ”π‘ π‘šπ‘₯π‘˜2𝜌,𝑧1𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–ξ‚Άξ‚Ήπ‘π‘˜ξƒ°βŠ†ξƒ―β‰₯πœ€π‘›βˆˆβ„•βˆΆπ·π»π‘›π‘›ξ“π‘˜=1ξƒ¬π‘€ξƒ©β€–β€–β€–Ξ”π‘šπ‘ βˆ’1π‘₯π‘˜+1𝜌,𝑧1𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–ξƒͺξƒ­π‘π‘˜β‰₯πœ€2ξƒ°βˆͺξƒ―π‘›βˆˆβ„•βˆΆπ·π»π‘›π‘›ξ“π‘˜=1ξƒ¬π‘€ξƒ©β€–β€–β€–Ξ”π‘šπ‘ βˆ’1π‘₯π‘˜πœŒ,𝑧1𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–ξƒͺξƒ­π‘π‘˜β‰₯πœ€2ξƒ°.(3.26) Since both the sets in the right side of the relation (3.26) belongs to I, we get the set ξƒ―1π‘›βˆˆβ„•βˆΆπ‘›π‘›ξ“π‘˜=1ξ‚Έπ‘€ξ‚΅β€–β€–β€–Ξ”π‘ π‘šπ‘₯π‘˜2𝜌,𝑧1𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–ξ‚Άξ‚Ήπ‘π‘˜ξƒ°β‰₯πœ€βˆˆπΌ.(3.27) Let 𝑀(π‘₯)=π‘₯2, for all π‘₯∈[0,∞), π‘π‘˜=1, for all π‘˜βˆˆβ„•. Consider a sequence π‘₯=(π‘₯π‘˜)=((π‘˜+1)π‘Ÿ). Then, x belongs to π‘ŠπΌ0(Ξ”π‘šπ‘ βˆ’1,𝑀,𝑝,β€–β‹…,…,β‹…β€–) but does not belong to π‘ŠπΌ0(Ξ”π‘ π‘š,𝑀,𝑝,β€–β‹…,…,β‹…β€–). This shows that the inclusion π‘ŠπΌ0(Ξ”π‘šπ‘ βˆ’1,𝑀,𝑝,β€–β‹…,…,β‹…β€–)βŠ‚π‘ŠπΌ0(Ξ”π‘ π‘š,𝑀,𝑝,β€–β‹…,…,β‹…β€–) is strict.

Theorem 3.5. Let 0<π‘π‘˜β‰€π‘žπ‘˜β€‰β€‰and (π‘žπ‘˜/π‘π‘˜)  be bounded, then π‘ŠπΌξ€·Ξ”π‘ π‘šξ€Έ,𝑀,π‘ž,β€–β‹…,…,β‹…β€–βŠ†π‘ŠπΌξ€·Ξ”π‘ π‘šξ€Έ,𝑀,𝑝,β€–β‹…,…,β‹…β€–.(3.28)

Proof. Let π‘₯=(π‘₯π‘˜)βˆˆπ‘ŠπΌ(Ξ”π‘ π‘š,𝑀,π‘ž,β€–β‹…,…,β‹…β€–). We put π‘¦π‘˜=ξ‚΅π‘€ξ‚΅β€–β€–β€–Ξ”π‘ π‘šπ‘₯π‘˜βˆ’π‘™πœŒ,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–ξ‚Άξ‚Άπ‘žπ‘˜andπ›½π‘˜=π‘π‘˜π‘žπ‘˜,βˆ€π‘˜βˆˆβ„•.Then,0<π›½π‘˜β‰€1.(3.29) For all π‘˜βˆˆβ„•. Let 𝛽 be such that 0<𝛽<π›½π‘˜, for all π‘˜βˆˆβ„•.
Define the sequences (π‘Žπ‘˜) and (π‘π‘˜) as follows:  for π‘¦π‘˜β‰₯1, let π‘Žπ‘˜=π‘¦π‘˜ and π‘π‘˜=0; for π‘¦π‘˜<1, let π‘Žπ‘˜=0 and π‘π‘˜=π‘¦π‘˜.
Then, clearly, for all π‘˜βˆˆβ„•, we have π‘¦π‘˜=π‘Žπ‘˜+π‘π‘˜,π‘¦π›½π‘˜π‘˜=π‘Žπ›½π‘˜π‘˜+π‘π›½π‘˜π‘˜;π‘Žπ›½π‘˜π‘˜β‰€π‘Žπ‘˜β‰€π‘¦π‘˜ and π‘π›½π‘˜π‘˜β‰€π‘π›½π‘˜. Therefore, we have 1π‘›π‘›ξ“π‘˜=1π‘¦π›½π‘˜π‘˜β‰€1π‘›π‘›ξ“π‘˜=1π‘¦π‘˜β‰€ξƒ¬1π‘›π‘›ξ“π‘˜=1π‘π‘˜ξƒ­π›½.(3.30) Hence, (π‘₯π‘˜)βˆˆπ‘ŠπΌ(Ξ”π‘ π‘š,𝑀,𝑝,β€–β‹…,…,β‹…β€–).

Theorem 3.6. For any two sequences 𝑝=(π‘π‘˜) and π‘ž=(π‘žπ‘˜)   of positive real numbers and for any two n-norms β€–β‹…,…,β‹…β€–1  and   ‖⋅,…,β‹…β€–2  on X, the following holds: π‘ξ€·Ξ”π‘ π‘š,𝑀,𝑝,β€–β‹…,…,β‹…β€–1ξ€Έξ€·Ξ”βˆ©π‘π‘ π‘š,𝑀,𝑝,β€–β‹…,…,β‹…β€–2ξ€Έβ‰ βˆ…,(3.31) where 𝑍=π‘ŠπΌ,π‘ŠπΌ0,π‘ŠπΌβˆž,andπ‘Šβˆž.

Proof. Proof of the theorem is obvious, because the zero element belongs to each of the sequence spaces involved in the intersection.

Theorem 3.7. The sequence spaces 𝑍(Ξ”π‘ π‘š,𝑀,𝑝,β€–β‹…,…,β‹…β€–) are normal as well as monotone, where 𝑍=π‘ŠπΌ0,π‘ŠπΌβˆž.

Proof. We will give the proof for π‘ŠπΌ0(Ξ”π‘ π‘š,𝑀,𝑝,β€–β‹…,…,β‹…β€–) only. Let π‘₯=(π‘₯π‘˜) be any element in π‘ŠπΌ0(Ξ”π‘ π‘š,𝑀,𝑝,||β‹…,…,β‹…||), and let (π›Όπ‘˜)  be a sequence of scalars such that |π›Όπ‘˜|≀1, for all π‘˜βˆˆβ„•. Then, we have ξƒ―1π‘›βˆˆβ„•βˆΆπ‘›π‘›ξ“π‘˜=1ξƒ¬π‘€ξƒ©β€–β€–β€–β€–Ξ”π‘ ξ€·π›Όπ‘˜π‘₯π‘˜ξ€ΈπœŒ,𝑧1𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–β€–ξƒͺξƒ­π‘π‘˜ξƒ°βŠ†ξƒ―πΈβ‰₯πœ€π‘›βˆˆβ„•βˆΆπ‘›π‘›ξ“π‘˜=1𝑀‖‖‖Δ𝑠π‘₯π‘˜πœŒ,𝑧1𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–ξ‚Άξ‚Ήπ‘π‘˜ξƒ°,β‰₯πœ€(3.32) where 𝐸=max{1,|π›Όπ‘˜|𝐺}.
Hence, (π›Όπ‘˜π‘₯π‘˜)βˆˆπ‘ŠπΌ0(Ξ”π‘ π‘š,𝑀,𝑝,||β‹…,…,β‹…||), for all sequence (π›Όπ‘˜) of scalars with |π›Όπ‘˜|≀1, for all π‘˜βˆˆβ„•, whenever (π‘₯π‘˜)βˆˆπ‘ŠπΌ0(Ξ”π‘ π‘š,𝑀,𝑝,||β‹…,…,β‹…||).
By Lemma 2.18, we have the space π‘ŠπΌ0(Ξ”π‘ π‘š,𝑀,𝑝,||β‹…,…,β‹…||) is monotone.

Remark 3.8. If we replace the difference operator Ξ”π‘ π‘š by Ξ”π‘š(𝑠), then for each πœ€>0, we get the following sequence spaces: π‘ŠπΌξ‚€Ξ”π‘š(𝑠)=ξƒ―ξ€·π‘₯,𝑀,𝑝,β€–β‹…,…,β‹…β€–π‘˜ξ€Έξƒ―1βˆˆπ‘€(π‘›βˆ’π‘‹)βˆΆπ‘›βˆˆβ„•βˆΆπ‘›π‘›ξ“π‘˜=1ξƒ¬π‘€ξƒ©β€–β€–β€–β€–Ξ”π‘š(𝑠)π‘₯π‘˜βˆ’π‘™πœŒ,𝑧1,𝑧2β€¦π‘§π‘›βˆ’1β€–β€–β€–β€–ξƒͺξƒ­π‘π‘˜ξƒ°β‰₯πœ€βˆˆπΌ,forsome𝜌>0,π‘™βˆˆπ‘‹π‘Žπ‘›π‘‘βˆ€π‘§1,𝑧2β€¦π‘§π‘›βˆ’1ξƒ°,π‘Šβˆˆπ‘‹(3.33)𝐼0ξ‚€Ξ”π‘š(𝑠)=ξƒ―ξ€·π‘₯,𝑀,𝑝,β€–β‹…,…,β‹…β€–π‘˜ξ€Έξƒ―1βˆˆπ‘€(π‘›βˆ’π‘‹)βˆΆπ‘›βˆˆβ„•βˆΆπ‘›π‘›ξ“π‘˜=1ξƒ¬π‘€ξƒ©β€–β€–β€–β€–Ξ”π‘š(𝑠)π‘₯π‘˜πœŒ,𝑧1,𝑧2β€¦π‘§π‘›βˆ’1β€–β€–β€–β€–ξƒͺξƒ­π‘π‘˜ξƒ°β‰₯πœ€βˆˆπΌ,forsome𝜌>0,π‘™βˆˆπ‘‹π‘Žπ‘›π‘‘βˆ€π‘§1,𝑧2β€¦π‘§π‘›βˆ’1ξƒ°,π‘Šβˆˆπ‘‹(3.34)βˆžξ‚€Ξ”π‘š(𝑠)=ξƒ―ξ€·π‘₯,𝑀,𝑝,β€–β‹…,…,β‹…β€–π‘˜ξ€Έβˆˆπ‘€(π‘›βˆ’π‘‹)∢supπ‘›βˆˆβ„•1π‘›π‘›ξ“π‘˜=1ξƒ¬π‘€ξƒ©β€–β€–β€–β€–Ξ”π‘š(𝑠)π‘₯π‘˜πœŒ,𝑧1𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–β€–ξƒͺξƒ­π‘π‘˜<∞,forsome𝜌>0,π‘™βˆˆπ‘‹π‘Žπ‘›π‘‘βˆ€π‘§1,𝑧2β€¦π‘§π‘›βˆ’1ξƒ°,π‘Šβˆˆπ‘‹(3.35)πΌβˆžξ‚€Ξ”π‘š(𝑠)=ξƒ―ξ€·π‘₯,𝑀,𝑝,β€–β‹…,…,β‹…β€–π‘˜ξ€Έξƒ―1βˆˆπ‘€(π‘›βˆ’π‘‹)βˆΆβˆƒπΎ>0s.t.π‘›βˆˆβ„•βˆΆπ‘›π‘›ξ“π‘˜=1ξƒ¬π‘€ξƒ©β€–β€–β€–β€–Ξ”π‘š(𝑠)π‘₯π‘˜πœŒ,𝑧1𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–β€–ξƒͺξƒ­π‘π‘˜ξƒ°β‰₯𝐾∈𝐼,forsome𝜌>0,π‘™βˆˆπ‘‹π‘Žπ‘›π‘‘βˆ€π‘§1,𝑧2β€¦π‘§π‘›βˆ’1ξƒ°,βˆˆπ‘‹(3.36)

Note 2. For 𝑠=0, we write the above spaces as 𝑍(𝑀,𝑝,β€–β‹…,…,β‹…β€–), where 𝑍=π‘ŠπΌ, π‘ŠπΌ0, π‘Šβˆž,  and π‘ŠπΌβˆž.
It is clear from definitions that π‘ŠπΌ0ξ‚€Ξ”π‘š(𝑠),𝑀,𝑝,β€–β‹…,…,β‹…β€–βŠ‚π‘ŠπΌξ‚€Ξ”π‘š(𝑠),𝑀,𝑝,β€–β‹…,…,β‹…β€–βŠ‚π‘Šβˆžξ‚€Ξ”π‘š(𝑠),𝑀,𝑝,β€–β‹…,…,β‹…β€–.(3.37)

Corollary 3.9. The sequence spaces 𝑍(Ξ”π‘š(𝑠),𝑀,𝑝,β€–β‹…,…,β‹…β€–) are paranormed spaces (not totally paranormed ) with respect to the paranormed β„ŽΞ”β€‰β€‰defined by β„ŽΞ”ξƒ―πœŒ(π‘₯)=infπ‘π‘˜/𝐻∢supπ‘˜π‘€ξƒ©β€–β€–β€–β€–Ξ”π‘š(𝑠)π‘₯π‘˜πœŒ,𝑧1,𝑧2,…,π‘§π‘›βˆ’1β€–β€–β€–β€–ξƒͺ≀1,forsome𝜌>0andβˆ€π‘§1,𝑧2,…,π‘§π‘›βˆ’1ξƒ°,βˆˆπ‘‹(3.38) where 𝐻=max{1,supπ‘π‘˜}and 𝑍=π‘ŠπΌ, π‘ŠπΌ0, π‘Šβˆž, and π‘ŠπΌβˆž.

Remark 3.10. It is obvious that (π‘₯π‘˜)βˆˆπ‘(Ξ”π‘ π‘š,𝑀,𝑝,β€–β‹…,…,β‹…β€–) if and only if 𝑍(Ξ”π‘š(𝑠),𝑀,𝑝,β€–β‹…,…,β‹…β€–), for 𝑍=π‘ŠπΌ, π‘ŠπΌ0, π‘ŠπΌβˆžβ€‰β€‰and π‘Šβˆž.
Also it is clear that the paranorm 𝑔Δ and β„ŽΞ” are equivalent.

We state the following Theorem in view of the Lemma 2.17.

Theorem 3.11. Let X be a standard n-normed space and {𝑒1,𝑒2,…,𝑒𝑛} an orthogonal set in X. Then, the following hold: (a)π‘ŠπΌ(Ξ”π‘ π‘š,𝑀,𝑝,β€–β‹…,…,β‹…β€–βˆž)=π‘ŠπΌ(Ξ”π‘ π‘š,𝑀,𝑝,β€–β‹…,…,β‹…β€–(π‘›βˆ’1)), (b)π‘ŠπΌ0(Ξ”π‘ π‘š,𝑀,𝑝,β€–β‹…,…,β‹…β€–βˆž)=π‘ŠπΌ0(Ξ”π‘ π‘š,𝑀,𝑝,β€–β‹…,…,β‹…β€–(π‘›βˆ’1)), (c)π‘Šβˆž(Ξ”π‘ π‘š,𝑀,𝑝,β€–β‹…,…,β‹…β€–βˆž)=π‘Šβˆž(Ξ”π‘ π‘š,𝑀,𝑝,β€–β‹…,…,β‹…β€–(π‘›βˆ’1)), (d)π‘ŠπΌβˆž(Ξ”π‘ π‘š,𝑀,𝑝,β€–β‹…,…,β‹…β€–βˆž)=π‘ŠπΌβˆž(Ξ”π‘ π‘š,𝑀,𝑝,β€–β‹…,…,β‹…β€–(π‘›βˆ’1)),   where β€–β‹…,…,β‹…β€–βˆž is the derived (π‘›βˆ’1)-norm defined with respect to the set {𝑒1,𝑒2,…,𝑒𝑛} and β€–β‹…,…,β‹…β€–(π‘›βˆ’1) is the standard (π‘›βˆ’1)-norm on X.

Remark 3.12. Theorem 3.11 also holds if we replace the difference operator Ξ”π‘ π‘šβ€‰β€‰by the difference operator Ξ”π‘š(𝑠).

Theorem 3.13. The spaces 𝑍(Ξ”π‘š(𝑠),𝑀,𝑝,β€–β‹…,…,β‹…β€–) and 𝑍(𝑀,𝑝,β€–β‹…,…,β‹…β€–) are equivalent as topological spaces, where 𝑍=π‘ŠπΌ,π‘ŠπΌ0,π‘ŠπΌβˆž, and π‘Šβˆž.

Proof. Consider the mapping π‘‡βˆΆπ‘(Ξ”π‘š(𝑠),𝑀,𝑝,β€–β‹…,…,β‹…β€–)→𝑍(𝑀,𝑝,β€–β‹…,…,β‹…β€–) defined by Δ𝑇π‘₯=(𝑠)π‘₯π‘˜ξ€Έξ€·π‘₯,foreachπ‘₯=π‘˜ξ€Έξ‚€Ξ”βˆˆπ‘π‘š(𝑠),𝑀,𝑝,β€–β‹…,…,β‹…β€–.(3.39) Then, clearly, T is a linear homeomorphism and the proof follows.