On Generalized Convergence in Sequence Spaces

Abstract

In the present thesis, we have made efforts to develop some classical methods of summability and introduce some summability theories in differentspaces like 2-normed Spaces, probabilistic normed spaces and double sequence spacesfollowed by characterization of certain properties of these generalized convergence methods in the above-mentioned sequence spaces.The whole work in the thesis has been divided into five chapters. Chapter and#8722; I is an introductory one in which, the basic terminology of some generalized summability methods and abstract spaces under consideration in present thesis has been presented. We start with the definition of statistical convergence, which was initially introduced by H. Fast [54] and see how it has been extended in various abstract spaces to resolve many problems in the area of mathematical analysis. We also record some facts related to statistical convergence and its generalizations from the history, which will be needed in the sequel and form the base for the present thesis.Chapter and#8722; II is devoted to the development of new sequence spaces V fand#955; [and#916;mp, I]0,V fand#955; [and#916;mp, I]L and V fand#955; [and#916;mp, I]and#8734;. These sequence spaces have been defined by the use ofde la vallee poussin mean, modulus function f and the difference operator and#916;. After making some early remarks on these spaces, some of their important properties and inclusion relations have been established in this chapter. Some concrete examples in support of our results have also been provided. Subsequently, the concept of Sand#916;mand#955; (I)-convergence has also been introduced and a condition is obtained under which Sand#916;m and#955; (I)-convergence coincides with above-mentioned sequence spaces. In Chapter and#8722; III, we consider one of the generalized form of abstract spaces i.e.2-norm space.