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  "body": "\n## Summary\n\nA dual pair, or dual system, consists of two vector spaces and a [[bilinear form|0SQZMXB6GP]] between them that separates points on both sides.\nIt is the abstract framework of duality theory for [[locally convex spaces|B6GY2KJMWW]].\nThe basic example is a Hausdorff locally convex space $E$ paired with its [[continuous dual|4A82S5FVRT]] $E'$ by $\\langle x, f \\rangle = f(x)$.\nTopologies such as the [[weak topology|P6KHVSTPZV]] and the [[Mackey topology|W18EY4E5TH]] are defined from a dual pair alone.\n\n## Explanations\n\n### L1 human\n\nIn a dual pair, each side can be used to test the other.\nA vector $x$ is known once one knows all the numbers $\\langle x, y \\rangle$, and likewise for $y$.\nThe simplest case is $\\R^n$ paired with itself by the dot product.\nIn functional analysis one pairs a space of functions with a space of measures or of other functions, for example $\\langle f, g \\rangle = \\int f g$.\n",
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