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  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$.\nA *dual pair* over $\\mathbb{K}$ is a triple $(X, Y, \\langle \\cdot, \\cdot \\rangle)$ such that:\n\n1. $X$ and $Y$ are [[vector spaces|05EGHEQZAW]] over $\\mathbb{K}$, and $\\langle \\cdot, \\cdot \\rangle \\colon X \\times Y \\to \\mathbb{K}$ is a [[bilinear form|0SQZMXB6GP]].\n2. For every $x \\in X$ with $x \\ne 0$ there is $y \\in Y$ with $\\langle x, y \\rangle \\ne 0$.\n3. For every $y \\in Y$ with $y \\ne 0$ there is $x \\in X$ with $\\langle x, y \\rangle \\ne 0$.\n\n## Remarks\n\nSome authors call any triple satisfying condition 1 a dual pair, and call it *separated* when 2 and 3 hold.\nExample: $X = Y = \\R^n$ with $\\langle x, y \\rangle = \\sum_i x_i y_i$; if $x \\ne 0$ then $\\langle x, x \\rangle \u003e 0$, which gives 2, and 3 holds in the same way.\nExample: for a Hausdorff locally convex space $E$, the triple $(E, E', \\langle x, f \\rangle = f(x))$ is a dual pair; condition 2 follows from the Hahn–Banach theorem, and condition 3 is trivial.\nNon-example: $X = C([0, 1])$, $Y = \\R$, $\\langle f, t \\rangle = t f(0)$ satisfies 1 and 3 but not 2, since the function $f(s) = s$ is nonzero and $\\langle f, t \\rangle = 0$ for all $t$.\n",
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