Let denote the sequence of real matrices . For a sequence , if it exists for each and . A sequence is said to be -summable to if uniformly in . This encompasses the usual summability method , ordinary convergence and among other methods almost convergence as introduce by G. G. Lorentz [Acta Math. Uppsala 80, 167-190 (1948; Zbl 0031.29501)] Let be an infinite matrix of real numbers and be two non-empty subsets of the space of all real sequences. The matrix defines a transformation from into , if for every sequence the sequence exists and is in where . By all such matrices are denoted. Let denote the spaces of all almost convergent real sequences and series respectively.
The main purpose of this paper is to determine the necessary and sufficient conditions on the matrix sequence in order that is contained in one of the classes , , and .