Matrix domain of a sequence space β-and γ -duals and matrix transformations a b s t r a c t Let f denotes the space of almost convergent sequences introduced by Lorentz [G.G.Lorentz, A contribution to the theory of divergent sequences, Acta Math. 80 (1948), 167-190] and f also be the domain of the generalized difference matrix B(r, s) in the sequence space f . In this paper, the β-and γ -duals of the spaces f , fs and f are determined. Furthermore, two basic results on the space f are proved and the classes ( f : µ) and (µ: f ) of infinite matrices are characterized, and the characterizations of some other classes are also given as an application of those main results, where µ is any given sequence space. Preliminaries, background and notationBy a sequence space, we understand a linear subspace of the space ω = C N of all complex sequences which contains φ, the set of all finitely non-zero sequences, where N = {0, 1, 2, . . .}. We write ℓ ∞ , c, c 0 and ℓ p for the classical sequence spaces of all bounded, convergent, null and absolutely p-summable sequences which are Banach spaces with the sup-norm ‖x‖ ∞ = sup k∈N |x k | and ‖x‖ p = ( ∑ ∞ k=0 |x k | p ) 1/p , while φ is not a Banach space with respect to any norm, respectively, where 1 ≤ p < ∞. Also by bs and cs, we denote the spaces of all bounded and convergent series, respectively. bv is the space consisting of all sequences (x k ) such that (x k − x k+1 ) in ℓ 1 and bv 0 is the intersection of the spaces bv and c 0 . We assume throughout unless stated otherwise that q is the conjugate number of p for 1 ≤ p ≤ ∞, that is, q = ∞ for p = 1, q = p/(p−1) for 1 < p < ∞, and q = 1 for p = ∞, and use the convention that any term with negative subscript is equal to zero.Let A = (a nk ) be an infinite matrix of complex numbers a nk , where n, k ∈ N and write (Ax) n := − k a nk x k ; (n ∈ N, x ∈ D 00 (A)),( 1.1) where D 00 (A) denotes the subspace of ω consisting of x = (x k ) ∈ ω for which the sum exists as a finite sum. For simplicity in notation, here and in what follows, the summation without limits runs from 0 to ∞. More generally if µ is a normed sequence space, we can write D µ (A) for x ∈ ω for which the sum in (1.1) converges in the norm of µ. We writefor the space of those matrices which send the whole of the sequence space λ into the sequence space µ in this sense. A matrix A = (a nk ) is called a triangle if a nk = 0 for k > n and a nn ̸ = 0 for all n ∈ N. It is trivial that A(Bx) = (AB)x holds for the triangle matrices A, B and a sequence x. Further, a triangle matrix U uniquely has an inverse U −1 = V that is also a triangle matrix. Then, x = U(Vx) = V (Ux) holds for all x ∈ ω.✩ The main results of this paper were partially presented in an invited talk at the conference International Workshop on Mathematical Analysis 1 (IWOMA 1) to be held June 22-24,
The most straightforward approaches to checking the degrees of similarity and differentiation between two sets are to use distance and cosine similarity metrics. The cosine of the angle between two n-dimensional vectors in n-dimensional space is called cosine similarity. Even though the two sides are dissimilar in size, cosine similarity may readily find commonalities since it deals with the angle in between. Cosine similarity is widely used because it is simple, ideal for usage with sparse data, and deals with the angle between two vectors rather than their magnitude. The distance function is an elegant and canonical quantitative tool to measure the similarity or difference between two sets. This work presents new metrics of distance and cosine similarity amongst Fermatean fuzzy sets. Initially, the definitions of the new measures based on Fermatean fuzzy sets were presented, and their properties were explored. Considering that the cosine measure does not satisfy the axiom of similarity measure, then we propose a method to construct other similarity measures between Fermatean fuzzy sets based on the proposed cosine similarity and Euclidean distance measures and it satisfies the axiom of the similarity measure. Furthermore, we obtain a cosine distance measure between Fermatean fuzzy sets by using the relationship between the similarity and distance measures, then we extend the technique for order of preference by similarity to the ideal solution method to the proposed cosine distance measure, which can deal with the related decision-making problems not only from the point of view of geometry but also from the point of view of algebra. Finally, we give a practical example to illustrate the reasonableness and effectiveness of the proposed method, which is also compared with other existing methods.
In this paper, we investigate integrated and differentiated sequence spaces which emerge from the concept of the sequence space ℓ 1 . The integrated and differentiated sequence spaces were initiated by Goes and Goes [4]. The main propose of the present paper, we study matrix domains and some properties of the integrated and differentiated sequence spaces. In Section 3, we compute the alpha-, beta-and gamma duals of these spaces. Afterward, we characterize the matrix classes of these spaces with well-known sequence spaces.
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