Domain of the double sequential band matrix in some sequence spaces

The main purpose of this study, which is prepared as a thesis topic, is to define the sequence space λ_{B(r,s)} and to specify the β- and γ- duals of this space. Here, the sequence space λ is any of the ℓ_{∞}, c, c₀ or ℓ_{p} spaces. In addition, in our study, Schauder basis for c, c₀ and ℓ_{p} spaces is given and some topological properties of c, ℓ₁ and ℓ_{p} spaces are examined. Finally, some classes of matrix transformations on the space λ_{B(r,s)} are characterised. This study is organized as follows. In the first chapter, after mentioning the method of constructing a new sequence space via matrix domain and the different infinite band matrices used for this technique available in the literature, a brief summary of the chapters of the thesis is presented. In the second chapter; basic definitions and theorems are given. In the third chapter; The studies on difference sequence spaces are summarized. In the fourth chapter; We have defined the domain of λ_{B(r,s)} of the sequential double band matrix B(r,s) in λ sequence space, λ∈{ℓ_{∞},c,c₀,ℓ_{p}} and λ_{B(r,s)} We have determined the β- and γ- duals of . After proving under which conditions it is λ⊂λ_{B(r,s)} and the equality of λ=λ_{B(r,s)} is satisfied, we have given the Schauder basis of (c₀)_{B(r,s)}, (ℓ₁)_{B(r,s)} and (ℓ_{p})_{B(r,s)} spaces. Finally, we examined some topological properties of spaces c₀, ℓ₁ and ℓ_{p} when p>1. In the fifth chapter; A general theorem characterizing matrix transformations from the domain of a triangular matrix to any sequence space is stated and proved. As an application of this simple theorem, λ∈{ℓ_{∞},c,c₀,ℓ_{p}} and μ∈{ℓ_{∞},c,c₀,ℓ₁}.We have made a table giving necessary and sufficient conditions for matrix transformations from λ_{B(r,s)} to μ. In the sixth chapter, the results we have obtained in the thesis are mentioned.