On Some New Sequence Spaces and Their Duals

In this study


Introduction and Preliminaries
Let us we denote the space of all real or complex sequence by w.We write the sequence spaces of all convergent, null, bounded, and absolutely p− summable sequences by c, c 0 , l ∞ , and l p , respectively.Also we will denote the space of all bounded, convergent, and absolutely convergent series with bs, cs, and l 1 , respectively.Te space l p (1 ≤ p < ∞) is Banach space with x p � ( ∞ k�0 |x k | p ) 1/p and c, c 0 , and l ∞ are Banach spaces with x ∞ � sup k |x k |.
Let X be a linear metric space.A function q: X ⟶ R is called a paranorm, if (P1) q(x) ≥ 0 for ∈ X (P2) q(−x) � q(x) for all x ∈ X (P3) q(x + y) ≤ q(x) + q(y) for all x, y ∈ X (P4) If (λ n ) is a sequence of scalars with λ n ⟶ λ as n ⟶ ∞ and (x n ) is a sequence of vectors with q(x n − x) ⟶ 0 as n ⟶ ∞, then q(λ n x n − λx) ⟶ 0 as n ⟶ ∞ A paranorm q, where q(x) � 0 implies x � θ, is termed as a total paranorm, and the combination (X, q) 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], Teorem 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 p � (p k ) be a bounded sequence of real numbers such that p k > 0, sup k∈N p k � P, and S � max 1, P { }.For any ζ ∈ R and k ∈ N, it has been established in [2] that Troughout this study we will assume that p −1 k + (p k ′ ) −1 � 1 provided that 1 < inf p k < P < ∞.Maddox [5,6] introduced the linear spaces c(p), c 0 (p), l ∞ (p), and l(p) by Te linear spaces c(p), c 0 (p), l ∞ (p), and l(p) are complete spaces paranormed by and respectively.
Let A � (a rk ) be an infnite matrix of real or complex numbers and X, Y be subsets of w.We write A r (x) �  k a rk x k and Ax � A r (x) for r, k ∈ N.For a sequence space X, the matrix domain of an infnite matrix A is defned by which 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], S ¸ahiner and Gürdal [9], Gürdal and S ¸ahiner [10], Et and Esi [11], Aiyub et al. [12], and many others).
Te investigations into Tribonacci numbers were initially undertaken by a 14-year-old student Mark Feinberg [13] in 1963.Let (t k ) k∈N be the sequence of Tribonacci numbers defned by the third-order recurrence relation 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: � 0, 54368901 . . .
Te authors have defned the Tribonacci sequence spaces X(T) as the set of all sequences z for which their transformations under T, denoted as Tz, belong to the spaces l p and l ∞ . where We would like to mention that the sequences z � (z k ) and y � (y k ) are related by for each r ∈ N.
In later times, Yaying and Kara [20] introduced the Tribonacci sequence spaces X(T) with the following defnitions: where X � c or c 0 .
In a more recent study, Dagli and Yaying [21] have defned some new paranormed sequence spaces using regular Tribonacci matrix.

2
Journal of Mathematics Now, we give defnition of new sequence spaces.
Let u � (u r ) be any fxed sequence of nonzero complex numbers and p � (p r ) be the bounded sequence real numbers.We have defned the following sequence spaces: Using ( 5), we may redefne these sequence spaces by Remark 1.If we take u � (1, 1, 1, . ..) and p � (1, 1, 1, . ..), we obtain that the sequence spaces c(T, p, u), c 0 (T, p, u), and l ∞ (T, p, u) reduce to the sequence spaces c(T), c 0 (T), and l ∞ (T), respectively.Also if u � (1, 1, 1, . ..) and p r � p for all r ∈ N, we obtain that the sequence space l(T, p, u) reduces to l p (T).
In this paper, we examined some properties of these spaces such as completeness, Schauder basis.We establish that the novel sequence spaces c(T, p, u), c 0 (T, p, u), l ∞ (T, p, u), and l(T, p, u) are linearly isomorphic to the spaces c(p), c 0 (p), l ∞ (p), and l(p), correspondingly.

Main Results
Now, let us give the completeness of the sequence spaces c 0 (T, p, u) and l(T, p, u).Theorem 2. Te sequence spaces c 0 (T, p, u) and l(T, p, u) are complete linear metric spaces paranormed as follows: and respectively, where 0 ≤ p r ≤ P < ∞.It is obvious that the spaces c(T, p, u) and l ∞ (T, p, u) are paranormed spaces with Proof.We will demonstrate the claim solely for l(T, p, u) with the remaining cases following similar proofs.
Derived from ( 1) and ( 15), we ascertain the linearity of l(T, p, u) concerning scalar multiplication and coordinatewise addition.Additionally, it is evident that q p (θ) � 0 and q p (−z) � q p (z) for all z in l(T, p, u).Based on (1) and ( 15), we establish the subadditivity of q p as well as { } be any sequence in l(T, p, u) such that q p (z r − z) ⟶ 0 and (ζ r ) be any sequence in R such that (ζ r ) ⟶ ζ.With the help of the subadditivity of q p , we can write from which one can attain the boundedness of q p (z r ) and the fact that Journal of Mathematics Tis provides the continuity of scalar multiplication.Consequently, q p is a paranorm on l(T, p, u).To demonstrate the completeness of l(T, p, u), let v i   be any Cauchy sequence in l(T, p, u) such that v i � (v i 0 , v i 1 , v i 2 , . ..) for every i ∈ N.For a given ε > 0, there exists an integer r 0 (ε) ∈ N such that for all i, j ≥ r 0 (ε).By utilizing the defnition q p , we have for every i, j ≥ r 0 (ε), and this gives that T r (v 0 ), T r (v 1 ), T r (v 2 ), . . .  is a Cauchy sequence of real numbers for every fxed r ∈ N. In view of the fact that R is complete, we get T r (v i ) ⟶ T r (v), as i ⟶ ∞ for each fxed r ∈ N. Considering these infnitely numerous limits for all fxed k ∈ N and i, j ≥ r 0 (ε).If the limit is taken for k ⟶ ∞ and j ⟶ ∞ in (20), q p (v i − v j ) < ε is obtained.We consider ε � 1 in (20) so that i ≥ r 0 (1).Afterwards, we apply Minkowski's inequality, and we get that for every fxed i ∈ N. Terefore, we have v ∈ l(T, p, u).In view of the fact that q p (v i − v) < ε for all i ≥ r 0 (ε), we have Proof.We will establish the claim exclusively for l(T, p, u) while the others can be similarly demonstrated.To achieve this, we need to establish the existence of a linear transformation between l(T, p, u) and l(p) that satisfes the properties of being injective, surjective, and preserving paranorm.Let H: l(T, p, u) ⟶ l(p) be a transformation such that Hz � ((Tz) r ) for z ∈ l(T, p, u).
Te linearity of H is evident due to the inherent linearity found in all matrix transformations.Furthermore, the injectiveness of the transformation H is established by the fact that if Hz � θ, then it follows that z � θ.If we denote the sequence z � (z r ) for r ∈ N as for any sequence y � (y r ) ∈ l(p), then we have Journal of Mathematics from which we get z ∈ l(T, p, u).Terefore, since H is surjective and preserves the paranorm, this concludes the proof.

Let us construct Schauder bases for the sequence spaces c(T, p, u), c 0 (T, p, u), and l(T, p, u).
A sequence a � (a n ) in X is recognized a Schauder basis for X if and only if there is a unique sequence of scalars We are ready to provide a Schauder basis for the recently defned paranormed sequence spaces.

□ Theorem 4. Let us defne the sequence b
where r ∈ N is fxed.Ten (ii) Te sequence b (k) is a Schauder basis for the spaces l(T, p, u) and c 0 (T, p, u) and any z in l(T, p, u) is uniquely determined by where y k � (Tz) k for each k ∈ N.
Proof.We will establish the claim solely for l(T, p, u) with the other cases following analogous proofs.It is obvious that for 0 < p k ≤ P < ∞.Let z ∈ l(T, p, u) and denote for each nonnegative integer v.By employing (28) and (29), we derive and Now, for a given ε > 0 there exists an integer v 0 such that for all v ≥ v 0 .Tis provides us with the information that for all v ≥ v 0 .Tis results in a representation like (27).To show the uniqueness of ( 27), another representation of ( 27) Terefore, representation of ( 27) is unique.

The α − , β − , and γ − Duals
In this section, we identifed α−, β−, and c−duals of the sequence spaces c(T, p, u), c 0 (T, p, u), l ∞ (T, p, u), and l (T, p, u).Now, we will provide some lemmas for our investigations.Let A � (a rk ) represent an infnite matrix of real or complex numbers and N denote the family of all fnite subsets of N. Lemma (see [22]).Te subsequent statements are valid: Lemma 6 (see [23]).Te subsequent statements are valid: 37) and ( 38) hold and for all k ∈ N, also holds.
Theorem 7. Let w k � 1/|u k |, and consider the sets H i , 1 ≤ i ≤ 5, defned by Proof.We will establish the claim exclusively for l(T, p, u) while the others can be similarly demonstrated.In view of (22), we see the equality holds for h � (h k ) ∈ w, where A(t) � (a t rk ) is triangle defned as

􏼚
Proof.We will establish the claim exclusively for l(T, p, u) while the others can be similarly demonstrated.We will only demonstrate the assertion for l(T, p, u) with the remaining cases being proven in a similar manner.For h � (h k ) ∈ w, we can write the following equation: and (49)

􏼨 (50)
One can derive the c-dual of the space l(T, u) using a comparable method.In order to prevent redundant repetition, we will forgo presenting the proof.

Conclusion
Maddox [5,6] introduced the linear spaces c(p), c 0 (p), l ∞ (p), and l(p).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 defned some new sequence spaces using regular Tribonacci matrix.We examined some properties of these spaces such as completeness, Schauder basis.We have identifed α−, β−, and c−duals of the newly created spaces.In the future, new sequence spaces can be defned by taking this study into consideration.

( i )
Te set e, b (k)   is a Schauder basis for the space c(T, p, u) and any z in c(T, p, u) is solely determined by z � ζe +  k y k − ζ b (k) , l 1 ).Let w k � 1/|u k |, and consider the sets H i , 6 ≤ i ≤ 10, defned by In the light of (46), we see that hx� (h k x k ) ∈ cs whenever x ∈ l(T, p, u) if D(t)y ∈ c whenever y ∈ l(p).Tis indicates that h � (h k ) ∈ [l(T, p, u)] β if D(t) ∈ (l (p),c).Hence, by employing Lemma 6, we observe that