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  "body": "\n## Statement\n\nLet $A = (a_{nk})_{n, k \\ge 0}$ be an infinite complex matrix.\nEvery [[convergent sequence|0AZ3R52593]] $s$ is [[$A$-summable|RKFB6AHPBE]], 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:\n\n1. $\\sup_{n} \\sum_{k=0}^{\\infty} \\abs{a_{nk}} \u003c \\infty$;\n2. for every $k$, the limit $\\alpha_k = \\lim_{n \\to \\infty} a_{nk}$ exists;\n3. the limit $\\alpha = \\lim_{n \\to \\infty} \\sum_{k=0}^{\\infty} a_{nk}$ exists.\n\nIn that case $\\sum_k \\abs{\\alpha_k} \u003c \\infty$, and for every convergent $s$ with limit $L$,\n$$\\lim_{n \\to \\infty} (As)_n = \\alpha L + \\sum_{k=0}^{\\infty} \\alpha_k (s_k - L).$$\n\n## Remarks\n\nSuch matrices are called *conservative*.\nConditions 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.\nThe limit formula comes from writing $s = L \\cdot \\mathbf{1} + (s - L \\cdot \\mathbf{1})$, where the second sequence tends to $0$.\nExample: let $a_{nk} = 1$ if $k = 0$, $a_{nk} = \\frac{1}{n+1}$ if $1 \\le k \\le n$, and $a_{nk} = 0$ otherwise.\nThen $\\alpha_0 = 1$, $\\alpha_k = 0$ for $k \\ge 1$ and $\\alpha = 2$.\nThe formula gives $\\lim (As)_n = 2L + (s_0 - L) = s_0 + L$, which one can also check directly.\nWith $\\alpha_k = 0$ for all $k$ and $\\alpha = 1$ the limit formula gives $L$, and one recovers the [[Silverman–Toeplitz theorem|S4J7SRPXMZ]].\n\n## Explanations\n\n### L1 human\n\nThe theorem tells exactly which infinite matrices turn every convergent sequence into a convergent sequence.\nIt also gives the new limit in terms of the old limit and of the limits of the columns.\n\n### L2 human\n\nConservative matrices are exactly the bounded linear maps from the space of convergent sequences to itself that are given by matrices.\nThe new limit is a continuous linear functional of $s$, and the formula above is its general form.\nRegular matrices are the conservative ones for which this functional is the ordinary limit.\n",
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