Silverman–Toeplitz theorem

Remarks

Necessity of conditions 2 and 3: apply $A$ to the constant sequence $1$ and to the unit sequence $e_k$. Here $e_k$ has $1$ in place $k$ and $0$ elsewhere, so it tends to $0$. Necessity of condition 1 uses the uniform boundedness principle, or a direct "gliding hump" construction of a null sequence with unbounded transform. Sufficiency: write $s_k = L + r_k$ with $r_k \to 0$; condition 3 handles $L$, and for $r$ one splits $\sum_k a_{nk} r_k$ into finitely many terms, small by condition 2, and a tail, small by condition 1. Example: the Cesàro matrix $a_{nk} = \frac{1}{n+1}$ for $k \le n$ has row sums of absolute values equal to $1$, entries tending to $0$ and row sums $1$, so Cesàro summation is regular. Non-example: the matrix with $a_{nk} = 1$ for $k \le n$ and $0$ otherwise violates condition 1, and indeed it sends the convergent sequence $s_k = \frac{1}{k+1}$ to the divergent harmonic partial sums.

Going deeper

For a student

The theorem reduces regularity, a statement about all convergent sequences, to three checks on the matrix. It is the standard tool for showing that Cesàro, Nörlund, Riesz and Euler methods are regular. The Kojima–Schur theorem characterizes the wider class of matrices that only send convergent sequences to convergent sequences, and the Silverman–Toeplitz theorem is its special case with limits $0$ and $1$. The Steinhaus theorem shows that no regular matrix sums every bounded sequence.