A double sequence of real numbers is called almost convergent to a limit s if
uniformly in m and n. The definition is an extension of Lorentz’s definition of almost convergence of single sequences. The sequence x is said to be A-summable to limit t if , where is doubly infinite matrix of real numbers for all . The matrix A is said to be bounded- regular if every bounded and convergent sequence x is A-summable to the same limit and the A-means are also bounded. The matrix A is strongly regular if every almost convergent sequence x is A-summable to the same limit and the A-means are also bounded. It is shown that the necessary and sufficient conditions for a matrix A to be strongly regular are that A is bounded-regular and
where and . Then the authors define as a hump matrix if (i) for each m, n, k there exists a positive integer such that if and if ; (ii) for each m, n, j there exists a positive integer such that if and if . Let be the set of all hump matrices which are bounded regular and for which and Let be the set of all double sequences x which are almost convergent and be the set of all bounded double sequences whose A-means converge, then it is shown that .