Generalizations of Köthe-Toeplitz duals and Null duals of new difference sequence spaces

Abstract

The main purpose of the paper is to generalize the notions of the Köthe-Toeplitz duals and Null duals of sequence spaces by introducing the concepts of αEF-, βEF-, γEF-duals and NEF-duals, where E = (E_n) and F = (F_n) are two partitions of finite subsets of the positive integers. These duals are computed for the classical sequence spaces l_{∞, c} and c₀. The other purpose of the paper is to introduce the sequence spaces

\(X\left( {E,\;\Delta } \right) = \left\{ {x = \left( {{x_k}} \right):\left( {\sum\limits_{i \in {E_k}} {{x_i} - \sum\limits_{i \in {E_{k - 1}}} {{x_i}} } } \right) \in X} \right\}\)

. where \(X \in \left\{ {{l_\infty },\;c,\;{c_0}} \right\}\). We investigate the topological properties of these spaces, establish some inclusion relations between them, and compute the NEF-(or Null) duals for these spaces.