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  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$, and let $E$ be a topological vector space over $\\mathbb{K}$.\nThe *continuous dual space* of $E$ is the set $E'$ of all [[continuous linear maps|03TM135G0N]] $f \\colon E \\to \\mathbb{K}$, that is, of all continuous [[linear forms|07YQ1C78G0]] on $E$.\nWith pointwise addition and scalar multiplication, $E'$ is a linear subspace of the algebraic [[dual space|07JBSZE0V4]] of $E$.\n\n## Remarks\n\nThe notations $E^*$ and $E'$ are both common; some texts reserve $E^*$ for the algebraic dual.\nA linear form $f$ is continuous if and only if it is bounded on some neighbourhood of $0$.\nFor a normed space $E$, $f$ is continuous if and only if $\\norm{f} = \\sup_{\\norm{x} \\le 1} \\abs{f(x)} \u003c \\infty$, and $E'$ with this norm is a Banach space.\nExample: on $\\ell^2$, for fixed $y \\in \\ell^2$, the form $f(x) = \\sum_n x_n y_n$ is in $(\\ell^2)'$, since $\\abs{f(x)} \\le \\norm{y} \\norm{x}$ by the Cauchy–Schwarz inequality.\nNon-example: on the space $c_{00}$ of finitely supported sequences with the sup norm, $f(x) = \\sum_n x_n$ is linear but not continuous, since $x = e_1 + \\cdots + e_k$ has norm $1$ and $f(x) = k$.\nFor a Hausdorff locally convex space $E$, the Hahn–Banach theorem gives that $E'$ separates the points of $E$; for $L^p([0, 1])$ with $0 \u003c p \u003c 1$, by contrast, $E' = \\set{0}$.\n",
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