Sequences, series, summability General summability methods Theorem

Kojima–Schur theorem

Builds on Matrix summability method Convergent sequence

Statement

Let $A = (a_{nk})_{n, k \ge 0}$ be an infinite complex matrix. Every convergent sequence $s$ is $A$ -summable, that is, $(As)_n = \sum_k a_{nk} s_k$ is defined for all $n$ and converges as $n \to \infty$ , if and only if:

$\sup_{n} \sum_{k=0}^{\infty} \abs{a_{nk}} < \infty$ ;
for every $k$ , the limit $\alpha_k = \lim_{n \to \infty} a_{nk}$ exists;
the limit $\alpha = \lim_{n \to \infty} \sum_{k=0}^{\infty} a_{nk}$ exists.

In that case $\sum_k \abs{\alpha_k} < \infty$ , and for every convergent $s$ with limit $L$ , $\lim_{n \to \infty} (As)_n = \alpha L + \sum_{k=0}^{\infty} \alpha_k (s_k - L).$

Remarks

Such matrices are called conservative. Conditions 2 and 3 are needed because the unit sequences $e_k$ and the constant sequence $1$ are convergent; condition 1 follows from the uniform boundedness principle. The limit formula comes from writing $s = L \cdot \mathbf{1} + (s - L \cdot \mathbf{1})$ , where the second sequence tends to $0$ . Example: let $a_{nk} = 1$ if $k = 0$ , $a_{nk} = \frac{1}{n+1}$ if $1 \le k \le n$ , and $a_{nk} = 0$ otherwise. Then $\alpha_0 = 1$ , $\alpha_k = 0$ for $k \ge 1$ and $\alpha = 2$ . The formula gives $\lim (As)_n = 2L + (s_0 - L) = s_0 + L$ , which one can also check directly. With $\alpha_k = 0$ for all $k$ and $\alpha = 1$ the limit formula gives $L$ , and one recovers the Silverman–Toeplitz theorem.

Going deeper

For a student

Conservative matrices are exactly the bounded linear maps from the space of convergent sequences to itself that are given by matrices. The new limit is a continuous linear functional of $s$ , and the formula above is its general form. Regular matrices are the conservative ones for which this functional is the ordinary limit.

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Sequence definition
Real sequence definition
Limit of a sequence definition
Series definition
Convergent sequence definition
Partial sum definition
Summability method definition
Matrix summability method definition
Kojima–Schur theorem statement

Tag	`JB8GQBVW6Q`
URI	`m2://JB8GQBVW6Q`
Kind	statement, theorem
Writing	W0 stub
MSC	40C05, 40D05
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About	Matrix summability method
Uses	Matrix summability method, Convergent sequence
Changed	2 November 2026