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  "body": "\n## Statement\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$, and let $E$ be a topological vector space over $\\mathbb{K}$ with [[continuous dual|4A82S5FVRT]] $E'$.\nLet $U$ be a [[neighbourhood|0ZCY0Z80B6]] of $0$ in $E$.\nThen the polar\n$$U^\\circ = \\set{f \\in E' : \\abs{f(x)} \\le 1 \\text{ for all } x \\in U}$$\nis [[compact|0R6EWFKRCF]] in the weak-\\* topology $\\sigma(E', E)$, the [[weak topology|P6KHVSTPZV]] on $E'$.\nIn particular, if $E$ is a normed space, the closed unit ball $\\set{f \\in E' : \\norm{f} \\le 1}$ is weak-\\* compact.\n\n## Remarks\n\nSince $U$ is absorbing, each $x \\in E$ lies in $r_x U$ for some $r_x \u003e 0$, so $\\abs{f(x)} \\le r_x$ for $f \\in U^\\circ$.\nHence $U^\\circ$ sits inside the product $P = \\prod_{x \\in E} \\set{c \\in \\mathbb{K} : \\abs{c} \\le r_x}$, which is compact by [[Tychonoff's theorem|0JTRSJKCFB]], and the weak-\\* topology is the topology induced by the product topology.\nThe functions in $P$ that are linear and bounded by $1$ on $U$ form a closed subset of $P$.\nEach such function is continuous, since it is bounded on a neighbourhood of $0$.\nSo this closed subset is $U^\\circ$, and it is compact.\nExample: for $E = c_0$, the dual is $\\ell^1$, and the closed unit ball of $\\ell^1$ is weak-\\* compact, although it is not compact in norm.\nThe weak-\\* topology is needed: in an infinite-dimensional normed space $E$, the closed unit ball of $E'$ is never compact in the norm topology.\nThe theorem gives compactness, not sequential compactness.\nIf $E$ is a separable normed space, the closed unit ball of $E'$ is weak-\\* metrizable, and every bounded sequence in $E'$ has a weak-\\* convergent subsequence.\n\n## Explanations\n\n### L1 human\n\nIn infinite dimension, closed bounded sets are usually not compact, which makes it hard to find limits.\nThe theorem rescues compactness by weakening the topology.\nOn the dual of a normed space, the unit ball becomes compact once convergence means convergence at each vector separately.\nThis is often enough to extract a limit from a bounded family of functionals.\n\n### L2 human\n\nThe Banach–Alaoglu theorem is the standard source of compactness in functional analysis.\nIt is used to prove existence results, for instance minimizers in the calculus of variations, weak solutions of partial differential equations, and invariant means and measures.\nWith [[Goldstine's theorem|K4S9G3AQBM]], it characterizes reflexive Banach spaces by the weak compactness of their unit ball.\nWith the [[Krein–Milman theorem|RSRAY0Q68S]], it shows that the dual unit ball has many extreme points.\nIn duality theory, it says that polars of neighbourhoods of $0$ are weak-\\* compact, which is a key step of the [[Mackey–Arens theorem|RKKWS7CDNF]].\n",
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