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  "body": "\n## Summary\n\nThe Köthe–Toeplitz duals of a [[sequence space|XYTN19EPF2]] $X$ are the sequence spaces of multipliers $a$ for which $\\sum a_n x_n$ behaves well for every $x$ in $X$.\nThe $\\alpha$-dual asks for absolute convergence of $\\sum a_n x_n$, and the $\\beta$-dual asks for convergence.\nThey play the role of a dual space that is again a space of sequences, without any topology.\nFor example, the $\\beta$-dual of $\\ell^1$ is $\\ell^\\infty$, which matches the dual space of $\\ell^1$ in functional analysis.\n\n## Explanations\n\n### L1 human\n\nGiven a family of sequences $X$, ask which sequences $a$ can be multiplied term by term with every $x$ in $X$ so that the series $\\sum a_n x_n$ still converges.\nThese $a$ form a new space of sequences, the dual of $X$.\nIf $X$ consists of the sequences with $\\sum \\abs{x_n} \u003c \\infty$, then any bounded $a$ works, and no unbounded $a$ does.\nThe bigger $X$ is, the smaller its dual.\n",
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