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  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$, let $E$ be a topological vector space over $\\mathbb{K}$, and let $E'$ be its [[continuous dual space|4A82S5FVRT]].\nFor each [[von Neumann bounded set|3BAJ8ZTVPX]] $B \\subseteq E$, put $p_B(f) = \\sup_{x \\in B} \\abs{f(x)}$ for $f \\in E'$.\nThe *strong dual space* $E'_b$ of $E$ is $E'$ with the [[topology generated by the seminorms|ZVFXQ7S6FY]] $p_B$, for all bounded subsets $B$ of $E$.\n\n## Remarks\n\nEach $p_B$ is finite, because a continuous linear form maps bounded sets to bounded subsets of $\\mathbb{K}$.\nA net $(f_i)$ converges to $f$ in $E'_b$ if and only if $f_i \\to f$ uniformly on every bounded subset of $E$.\nThe topology is also written $\\beta(E', E)$.\nExample: let $E$ be a normed space with closed unit ball $B_1$.\nEvery bounded set lies in some $r B_1$, and $p_{r B_1}(f) = r \\norm{f}$, so the strong dual topology is the topology of the dual norm.\nNon-example: for an infinite-dimensional Banach space $E$, the weak-\\* topology $\\sigma(E', E)$ uses only the finite sets $B$.\nIt is not the strong topology, since its neighbourhoods of $0$ are not norm bounded.\n",
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