Silverman-Toeplitz theorem (${\sup }_i{\sum }_j|a_{ij}|<\infty ,a_{ij}\to 0,{\sum }_j{a}_{ij}\to 1$)

For a real or complex matrix, the three displayed conditions are equivalent to being a regular matrix summability method. For necessity, first apply the Uniform boundedness principle to each row's finite partial sums on the convergent sequence space, obtaining absolute summability of that row. Apply the principle again to the full row functionals, obtaining a uniform bound. Coordinate vectors and the constant-one sequence give the remaining conditions. For sufficiency, write $x_j=L+(x_j-L)$ and split the error into a finite head, controlled by columnwise convergence, and a uniformly small tail, controlled by the absolute row bound.