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  "body": "\n## Statement\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$, let $(X, Y, \\langle \\cdot, \\cdot \\rangle)$ be a [[dual pair|R68HNB5SVR]] over $\\mathbb{K}$, and let $\\mathcal{T}$ be a [[locally convex|B6GY2KJMWW]] topology on $X$.\nSay that $\\mathcal{T}$ is *compatible* with the pair if the [[continuous dual|4A82S5FVRT]] of $(X, \\mathcal{T})$ is exactly the set of linear forms $x \\mapsto \\langle x, y \\rangle$ with $y \\in Y$.\nThen $\\mathcal{T}$ is compatible with the pair if and only if\n$$\\sigma(X, Y) \\subseteq \\mathcal{T} \\subseteq \\tau(X, Y),$$\nwhere $\\sigma(X, Y)$ is the [[weak topology|P6KHVSTPZV]] and $\\tau(X, Y)$ is the [[Mackey topology|W18EY4E5TH]].\n\n## Remarks\n\nThe weak topology is compatible by construction, and it is the coarsest topology for which all the forms $\\langle \\cdot, y \\rangle$ are continuous.\nIf $\\mathcal{T}$ is compatible, each $\\mathcal{T}$-neighbourhood of $0$ contains a closed convex balanced neighbourhood $U$ of $0$.\nIts polar $U^\\circ$ in $Y$ is convex, balanced and $\\sigma(Y, X)$-compact by the [[Banach–Alaoglu theorem|1BXEJXQ2YT]], and $U = U^{\\circ\\circ}$ by the [[bipolar theorem|FX5PWKCMBQ]].\nSo $U$ is a $\\tau(X, Y)$-neighbourhood of $0$, which gives $\\mathcal{T} \\subseteq \\tau(X, Y)$.\nThat $\\tau(X, Y)$ itself is compatible is the harder half; it uses the [[bipolar theorem|FX5PWKCMBQ]] and a compactness argument in the algebraic dual of $Y$.\nExample: for a normed space $E$, the weak topology $\\sigma(E, E')$ and the norm topology are both compatible with the pair $(E, E')$, so they have the same continuous linear forms.\nA consequence: all compatible topologies have the same closed convex sets.\nIndeed, by the Hahn–Banach separation theorem, a closed convex set is an intersection of closed half-spaces defined by continuous forms.\n\n## Explanations\n\n### L1 human\n\nDifferent topologies on a space can give the same continuous linear forms.\nThe theorem describes all of them at once: they are exactly the locally convex topologies between the weak topology and the Mackey topology.\nSo the set of continuous linear forms pins down a whole interval of topologies, with a smallest and a largest element.\n\n### L2 human\n\nIn duality theory one often starts from a dual pair and asks which topologies on one side give the other side as dual.\nThe Mackey–Arens theorem answers this completely.\nIt explains why closed convex sets and bounded sets depend only on the dual pair: they are the same for all compatible topologies.\nFor bounded sets this is Mackey's theorem.\nIt also shows that the [[Mackey topology|W18EY4E5TH]] is natural: metrizable and barrelled locally convex spaces carry their Mackey topology.\nThe proof rests on the [[bipolar theorem|FX5PWKCMBQ]] and on [[Banach–Alaoglu|1BXEJXQ2YT]] compactness.\n",
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