= Silverman-Toeplitz theorem
{c}
{title2=$\sup_i\sum_j|a_{ij}|<\infty,\quad a_{ij}\to0,\quad\sum_j a_{ij}\to1$}
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.
Back to article page
