# Silverman–Toeplitz theorem

An infinite matrix $(a_{i,j})_{i,j \in \mathbb{N}}$ with complex-valued entries defines a regular summability method if and only if it satisfies all of the following properties:
$\lim_{i \to \infty} a_{i,j} = 0 \quad j \in \mathbb{N}$ (every column sequence converges to 0)
$\lim_{i \to \infty} \sum_{j=0}^{\infty} a_{i,j} = 1$ (the row sums converge to 1)
$\sup_{i} \sum_{j=0}^{\infty} \vert a_{i,j} \vert < \infty$ (the absolute row sums are bounded).