On Some New Sequence Spaces and Their Duals

Abstract

In this study, we defined some new sequence spaces using regular Tribonacci matrix. We examined some properties of these spaces such as completeness, Schauder basis. We have identified , and duals of the newly created spaces.

1. Introduction and Preliminaries

Let us we denote the space of all real or complex sequence by We write the sequence spaces of all convergent, null, bounded, and absolutely summable sequences by , , and , respectively. Also we will denote the space of all bounded, convergent, and absolutely convergent series with , and , respectively. The space is Banach space with and , and are Banach spaces with

Let be a linear metric space. A function is called a paranorm, if  for   for all   for all   If is a sequence of scalars with as and is a sequence of vectors with as then as

A paranorm , where implies , is termed as a total paranorm, and the combination is referred to as a total paranormed space. It is widely recognized that the metric of any linear metric space is represented by some total paranorm (see ([1], Theorem 10.4.2, page 183)). To gain a better understanding of the theory of paranormed spaces, you can refer to these valuable articles (see Barlak [2], Zengin Alp [3], İlkhan et al. [4], and many others).

Let be a bounded sequence of real numbers such that , and For any and it has been established in [2] that

Throughout this study we will assume that provided that Maddox [5, 6] introduced the linear spaces , , and by

The linear spaces , , and are complete spaces paranormed byandrespectively.

Let be an infinite matrix of real or complex numbers and be subsets of We write and for . For a sequence space , the matrix domain of an infinite matrix is defined bywhich is also a sequence space. We denote with (X,Y) the class of all matrices A such that A : X ⟶ Y.

Recently, the literature focused on the creation of new sequence spaces through the matrix domain and the investigation of their algebraic and topological properties, and the study of matrix transformations has expanded. To enhance comprehension of the theory concerning sequence spaces, you can refer to these valuable articles (see Altay et al. [7], Gürdal [8], Şahiner and Gürdal [9], Gürdal and Şahiner [10], Et and Esi [11], Aiyub et al. [12], and many others).

The investigations into Tribonacci numbers were initially undertaken by a 14-year-old student Mark Feinberg [13] in 1963. Let be the sequence of Tribonacci numbers defined by the third-order recurrence relation for with initial values and

Hence, the initial elements of the Tribonacci sequence are 1,1,2,4,7,13,24, …. Some fundamental characteristics of the Tribonacci sequence are as follows:

Afterwards, there has appeared much research with some arguments related of Tribonacci sequence (see Bruce [14], Choi [15], Kılıç [16], Pethe [17], Scott [18], and many others).

Yaying and Hazarika [19] defined the regular matrix involving Tribonacci numbers as follows:

Equivalently,

The authors have defined the Tribonacci sequence spaces as the set of all sequences for which their transformations under , denoted as belong to the spaces and where , or

We would like to mention that the sequences and are related byfor each

In later times, Yaying and Kara [20] introduced the Tribonacci sequence spaces with the following definitions:where or

In a more recent study, Dağli and Yaying [21] have defined some new paranormed sequence spaces using regular Tribonacci matrix.

Now, we give definition of new sequence spaces.

Let be any fixed sequence of nonzero complex numbers and be the bounded sequence real numbers. We have defined the following sequence spaces:

Using (5), we may redefine these sequence spaces by , and

Remark 1. If we take and , we obtain that the sequence spaces , and reduce to the sequence spaces , and respectively. Also if and for all we obtain that the sequence space reduces to
In this paper, we examined some properties of these spaces such as completeness, Schauder basis. We establish that the novel sequence spaces , , and are linearly isomorphic to the spaces , , and , correspondingly.

2. Main Results

Now, let us give the completeness of the sequence spaces and .

Theorem 2. The sequence spaces and are complete linear metric spaces paranormed as follows:andrespectively, where It is obvious that the spaces and are paranormed spaces with when and

Proof. We will demonstrate the claim solely for with the remaining cases following similar proofs. Let and it follows from Maddox [27, page 30] thatDerived from (1) and (15), we ascertain the linearity of concerning scalar multiplication and coordinatewise addition. Additionally, it is evident that and for all in Based on (1) and (15), we establish the subadditivity of as well as for any
Let be any sequence in such that and be any sequence in such that With the help of the subadditivity of we can writefrom which one can attain the boundedness of and the fact thatThis provides the continuity of scalar multiplication. Consequently, is a paranorm on To demonstrate the completeness of let be any Cauchy sequence in such that for every For a given there exists an integer such thatfor all By utilizing the definition we havefor every and this gives that is a Cauchy sequence of real numbers for every fixed In view of the fact that is complete, we get as for each fixed Considering these infinitely numerous limits let us establish the sequence It arises from (18) thatfor all fixed and If the limit is taken for and in (20), is obtained. We consider in (20) so that Afterwards, we apply Minkowski’s inequality, and we get thatfor every fixed . Therefore, we have In view of the fact that for all we have as As a result, is complete.

Theorem 3. The sequence spaces , , and are linearly isomorphic to the spaces , , and correspondingly where

Proof. We will establish the claim exclusively for while the others can be similarly demonstrated. To achieve this, we need to establish the existence of a linear transformation between and that satisfies the properties of being injective, surjective, and preserving paranorm. Let be a transformation such that for
The linearity of is evident due to the inherent linearity found in all matrix transformations. Furthermore, the injectiveness of the transformation is established by the fact that if then it follows that If we denote the sequence for asfor any sequence then we havefrom which we get Therefore, since is surjective and preserves the paranorm, this concludes the proof.
Let us construct Schauder bases for the sequence spaces , and .
A sequence in is recognized a Schauder basis for if and only if there is a unique sequence of scalars such that as Then we writeWe are ready to provide a Schauder basis for the recently defined paranormed sequence spaces.

Theorem 4. Let us define the sequence in as follows:where is fixed. Then(i)The set is a Schauder basis for the space and any in is solely determined by where (ii)The sequence is a Schauder basis for the spaces and and any in is uniquely determined by where for each

Proof. We will establish the claim solely for with the other cases following analogous proofs.
It is obvious thatfor Let and denotefor each nonnegative integer By employing (28) and (29), we deriveandNow, for a given there exists an integer such thatfor all This provides us with the information thatfor all This results in a representation like (27). To show the uniqueness of (27), another representation of (27) is Then we writeTherefore, representation of (27) is unique.

3. The , , and Duals

In this section, we identified , and duals of the sequence spaces , , and

Now, we will provide some lemmas for our investigations. Let represent an infinite matrix of real or complex numbers and denote the family of all finite subsets of

Lemma 5 (see [22]). The subsequent statements are valid:(i)Suppose that for every Then iff there is an integer such that(ii)Suppose that for every Then iff

Lemma 6 (see [23]). The subsequent statements are valid:(i)Suppose that for every Then iff there is an integer such that(ii)Suppose that for each Then iff(iii)Suppose that for each Then iff (37) and (38) hold and for all also holds.

Theorem 7. Let and consider the sets defined byThen,(i) and (ii) and

Proof. We will establish the claim exclusively for while the others can be similarly demonstrated. In view of (22), we see the equalityholds for where is triangle defined asTherefore, whenever iff whenever This indicates that iff Hence, by employing Lemma 5, we observe thatThis indicates that

Theorem 8. Let and consider the sets defined byThen,(i) and (ii)(iii) and (iv)

Proof. We will establish the claim exclusively for while the others can be similarly demonstrated. We will only demonstrate the assertion for with the remaining cases being proven in a similar manner.
For we can write the following equation:where is a triangle defined asIn the light of (46), we see that whenever iff whenever This indicates that iff Hence, by employing Lemma 6, we observe thatandThis indicates thatOne can derive the -dual of the space using a comparable method. In order to prevent redundant repetition, we will forgo presenting the proof.

4. Conclusion

Maddox [5, 6] introduced the linear spaces , , and Recently, the literature focused on the creation of new sequence spaces through the matrix domain and the investigation of their algebraic and topological properties, and the study of matrix transformations has expanded. Yaying and Kara [20] introduced the Tribonacci sequence spaces. In this study, we defined some new sequence spaces using regular Tribonacci matrix. We examined some properties of these spaces such as completeness, Schauder basis. We have identified , and duals of the newly created spaces. In the future, new sequence spaces can be defined by taking this study into consideration.

Data Availability

No data were used to support the findings of this study.

Conflicts of Interest

The author declares that there are no conflicts of interest.