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  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$, and let $(X, Y, \\langle \\cdot, \\cdot \\rangle)$ be a [[dual pair|R68HNB5SVR]] over $\\mathbb{K}$.\nThe *weak topology* $\\sigma(X, Y)$ on $X$ is the [[topology generated by the family of seminorms|ZVFXQ7S6FY]] $p_y(x) = \\abs{\\langle x, y \\rangle}$, $y \\in Y$.\nLet $E$ be a Hausdorff locally convex space with [[continuous dual|4A82S5FVRT]] $E'$, paired by $\\langle x, f \\rangle = f(x)$.\nThe *weak topology* of $E$ is $\\sigma(E, E')$, and the *weak-\\* topology* of $E'$ is $\\sigma(E', E)$.\n\n## Remarks\n\n$\\sigma(X, Y)$ is the coarsest topology on $X$ for which every map $x \\mapsto \\langle x, y \\rangle$, $y \\in Y$, is continuous; it is a Hausdorff locally convex topology.\nA net $(x_i)$ converges to $x$ in $\\sigma(X, Y)$ if and only if $\\langle x_i, y \\rangle \\to \\langle x, y \\rangle$ for every $y \\in Y$.\nThe continuous linear forms on $(X, \\sigma(X, Y))$ are exactly the maps $x \\mapsto \\langle x, y \\rangle$ with $y \\in Y$.\nExample: in $\\ell^2$, the unit vectors $e_n$ converge to $0$ in $\\sigma(\\ell^2, \\ell^2)$, since $\\langle e_n, y \\rangle = y_n \\to 0$ for each $y \\in \\ell^2$.\nNon-example: on $\\ell^2$, the norm topology makes every map $x \\mapsto \\langle x, y \\rangle$ continuous.\nBut it is not the coarsest such topology, so it is not the weak topology: the sequence $(e_n)$ converges weakly but not in norm.\n",
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