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  "body": "\n## Statement\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$, and let $(X, Y, \\langle \\cdot, \\cdot \\rangle)$ be a [[dual pair|R68HNB5SVR]] over $\\mathbb{K}$.\nFor $A \\subseteq X$, its *polar* is $A^\\circ = \\set{y \\in Y : \\abs{\\langle x, y \\rangle} \\le 1 \\text{ for all } x \\in A}$, and for $B \\subseteq Y$, its polar is $B^\\circ = \\set{x \\in X : \\abs{\\langle x, y \\rangle} \\le 1 \\text{ for all } y \\in B}$.\nLet $A \\subseteq X$, and let $C$ be the [[convex hull|0FQSDC2EQ2]] of $\\set{\\lambda a : a \\in A, \\lambda \\in \\mathbb{K}, \\abs{\\lambda} \\le 1}$.\nThen the bipolar $A^{\\circ\\circ} = (A^\\circ)^\\circ$ is the [[closure|0QEAJB91GZ]] of $C$ in the [[weak topology|P6KHVSTPZV]] $\\sigma(X, Y)$.\n\n## Remarks\n\nThe set described is the smallest $\\sigma(X, Y)$-closed, convex and [[balanced|2TDSHJJYS4]] set containing $A$, the closed absolutely convex hull of $A$.\nThe bipolar contains $A$ and is convex, balanced and $\\sigma(X, Y)$-closed, so it contains this hull.\nFor the reverse inclusion, take $x_0$ outside the hull.\nThe continuous dual of $(X, \\sigma(X, Y))$ is $Y$, so the Hahn–Banach separation theorem gives $y \\in Y$ with $\\abs{\\langle a, y \\rangle} \\le 1$ on the hull and $\\abs{\\langle x_0, y \\rangle} \u003e 1$.\nHence $x_0 \\notin A^{\\circ\\circ}$.\nExample: take a normed space $E$ with the pair $(E, E')$, and let $A$ be the open unit ball of $E$.\nThen $A^\\circ$ is the closed unit ball of $E'$, and $A^{\\circ\\circ}$ is the closed unit ball of $E$, the weak closure of the open ball.\nOver $\\R$ with one-sided polars $\\set{y : \\langle x, y \\rangle \\le 1}$, the bipolar is instead the closed convex hull of $A \\cup \\set{0}$.\n\n## Explanations\n\n### L1 human\n\nThe polar of a set collects all test vectors that keep the set inside the \"unit band\" $\\abs{\\langle x, y \\rangle} \\le 1$.\nTaking the polar twice gives back the original set, filled in to be convex, symmetric and closed.\nSo a closed convex symmetric set is completely described by its polar.\n\n### L2 human\n\nThe bipolar theorem is the convex-geometric form of the Hahn–Banach theorem for a dual pair.\nIt shows that every closed absolutely convex set is an intersection of \"slabs\" $\\set{x : \\abs{\\langle x, y \\rangle} \\le 1}$.\nIt is the main tool in the proof of the [[Mackey–Arens theorem|RKKWS7CDNF]] and in describing equicontinuous sets and bounded sets through polars.\nCombined with the [[Banach–Alaoglu theorem|1BXEJXQ2YT]], it describes the closed absolutely convex neighbourhoods of $0$.\nEach of them is the polar of its own polar, which is a weak-\\* compact absolutely convex set of the dual.\n",
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