Communications of the
Korean Mathematical Society CKMS

The aim of this paper is to present the classical sets of sequences and related matrix transformations with respect to the $b$-metric. Also, we introduce the relationships between these sets and their classical forms with corresponding properties including convergence and completeness. Further we determine the duals of the new spaces and characterize matrix transformations on them into the sets of $b$-bounded, $b$-convergent and $b$-null sequences.

Keywords: sequence space, $b$-metric, K\"othe-Toeplitz duals over the real field, complete metric space

MSC numbers: Primary 46A45, 54E50; Secondary 32A70