Abel summability in topological spaces

Abstract

The classical summability theory can not be used in the topological spaces as it needs addition operator. Recently some authors have studied the summability theory in the topological spaces by assuming the topological space to have a group structure or a linear structure or introducing some summability methods those do not need a linear structure in the topological space as statistical convergence and distributional convergence. In the present paper we introduce a new concept of density and we study the summability theory in an arbitrary Hausdorff space by introducing a new type of statistical convergence and distirbutional convergence via Abel method that is a sequence-to-function transformation. Moreover we give a Bochner integral representation of \(Abel\)-summability in the Banach spaces.