On some new paranormed difference sequence spaces and their geometric properties

In this study, we define some new paranormed difference sequencespaces by using generalized weighted mean and difference matricesand investigate some topological and geometrical properties of them.This study consists of four chapters.The first chapter reviews some basic definitions and theorems whichwill be used in the next chapters.In the second chapter the sequence spaces$c_{0}(u,v;p,\Delta),c(u,v;p,\Delta),\ell_{\infty}(u,v;p,\Delta)$and $\ell(u,v;p,\Delta)$ are defined and the spaces to which thesespaces are isomorphic are determined and it is shown that thesespaces are complete linear metric spaces. In addition,$\alpha-,\beta-$ and $\gamma-$ duals of these sequence spaces areestablished and their Schauder bases are calculated.The third chapter starts with discussing a new kind of method pairs,which enables us to characterize the matrix classes $(X:Y)$ where$X\in \{c_{0}(u,v;p,\Delta),\\ c(u,v;p,\Delta),\ell_{\infty}(u,v;p,\Delta)\}$ and$Y\in\{c_{0}(q),c(q),\ell_{\infty}(q),\ell(q)\}$. Also; the theoremsthat characterize the matrix classes from the sequence space$\ell(u,v;p,\Delta)$ to the sequence spaces $\ell_{\infty},c$ and$c_{0}$ are stated and proved.In the fourth chapter some geometric properties of the modularsequence space $\ell_{\sigma}(u,v;p,\Delta)$ are investigated whichis based on defining a modular on $\ell(u,v;p,\Delta)$.