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  "body": "\n## Summary\n\nThe summability field of an infinite matrix $A$ is the set of all sequences $x$ that $A$ transforms into a convergent sequence $Ax$.\nIt is the domain of the [[matrix summability method|M6311DWJ1E]] given by $A$, and it is a [[sequence space|XYTN19EPF2]].\nMany questions of summability theory are questions about summability fields.\nFor instance, one asks whether a summability field contains the space $c$ of convergent sequences or whether two of them are equal.\nWith a natural topology, the summability field is an [[FK-space|71DMQHW6JZ]], which brings functional analysis into summability.\n\n## Explanations\n\n### L1 human\n\nA matrix method takes a sequence, forms weighted averages of its terms, and looks at whether these averages converge.\nThe summability field is the collection of all sequences for which this works.\nFor the identity matrix it is just the convergent sequences.\nFor the Cesàro averages it is bigger: it also contains $1, 0, 1, 0, \\dots$, whose averages tend to $1/2$.\n",
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