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  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$, let $\\omega$ be the space of all sequences in $\\mathbb{K}$, and let $X \\subseteq \\omega$ be a [[sequence space|XYTN19EPF2]].\nThe *$\\alpha$-dual* of $X$ is\n$$X^\\alpha = \\set{a \\in \\omega : \\textstyle\\sum_{n} \\abs{a_n x_n} \u003c \\infty \\text{ for every } x \\in X}.$$\nThe *$\\beta$-dual* of $X$ is\n$$X^\\beta = \\set{a \\in \\omega : \\textstyle\\sum_{n} a_n x_n \\text{ converges for every } x \\in X}.$$\nThe $\\alpha$-dual is called the Köthe–Toeplitz dual of $X$, and the $\\beta$-dual is also called the generalized Köthe–Toeplitz dual.\n\n## Remarks\n\nBoth duals are sequence spaces, and $X^\\alpha \\subseteq X^\\beta$ because absolutely [[convergent series|0CVVKY7GY2]] converge.\nIf $X \\subseteq Y$, then $Y^\\alpha \\subseteq X^\\alpha$ and $Y^\\beta \\subseteq X^\\beta$.\n\nExample: $(\\ell^1)^\\alpha = (\\ell^1)^\\beta = \\ell^\\infty$, with $\\ell^1$ and $\\ell^\\infty$ as in the [[spaces ℓ^p|08P6T2WH6R]].\nIf $a$ is bounded, then $\\sum \\abs{a_n x_n} \\le \\sup_n \\abs{a_n} \\sum \\abs{x_n}$ for every $x \\in \\ell^1$.\nIf $a$ is unbounded, choose indices $n_1 \u003c n_2 \u003c \\cdots$ with $\\abs{a_{n_k}} \\ge k^2$, and let $x_{n_k} = \\overline{a_{n_k}} / \\abs{a_{n_k}} \\cdot k^{-2}$ and $x_n = 0$ otherwise.\nThen $x \\in \\ell^1$, but the terms $a_{n_k} x_{n_k} = \\abs{a_{n_k}} k^{-2} \\ge 1$ do not tend to $0$, so $a \\notin (\\ell^1)^\\beta$.\n\nNon-example: $a = (n)_{n \\ge 1}$ is not in $(\\ell^1)^\\beta$; for $x_n = 1/n^2$ the series $\\sum a_n x_n = \\sum 1/n$ diverges.\n\nOther standard values: $\\omega^\\beta$ is the space $\\varphi$ of sequences with finitely many nonzero terms, $\\varphi^\\beta = \\omega$, and $(c_0)^\\beta = (\\ell^\\infty)^\\beta = \\ell^1$.\n",
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