Some New Double Sequence Spaces of Invariant Means

First, we recall some notations and definitions which will be used throughout the paper (cf. [12]). A double sequence x = (xjk) of real or complex numbers is said to be bounded if ‖x‖∞ = supj,k |xjk| < ∞. We denote the space of all bounded double sequences by L. A double sequence x = (xjk) is said to converge to the limit L in Pringsheims sense (shortly, p-convergent to L) if for every ε > 0 there exists an integer N such that |xjk −L| < ε whenever j, k > N . In this case L is called the p-limit of x. If in addition x ∈ L, then x is said to be boundedly p-convergent to L in Pringsheims sense (shortly, bp-convergent to L). In general, for any notion of convergence ν, the space of all ν-convergent double sequences will be denoted by Cν and the limit of a ν-convergent double sequence x by ν-lim j,k xjk, where ν ∈ {p, bp}. Let l∞ and c be the spaces of bounded and convergent sequences x = (xk) respectively. Let σ be a one-to-one mapping from the set N of natural numbers