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  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$, and let $(X, Y, \\langle \\cdot, \\cdot \\rangle)$ be a [[dual pair|R68HNB5SVR]] over $\\mathbb{K}$.\nLet $\\mathcal{K}$ be the family of subsets $K \\subseteq Y$ that are [[convex|02AS45T3H9]], [[balanced|2TDSHJJYS4]] and [[compact|0R6EWFKRCF]] for the [[weak topology|P6KHVSTPZV]] $\\sigma(Y, X)$.\nFor $K \\in \\mathcal{K}$, put $p_K(x) = \\sup_{y \\in K} \\abs{\\langle x, y \\rangle}$ for $x \\in X$.\nThe *Mackey topology* $\\tau(X, Y)$ is the [[topology generated by the seminorms|ZVFXQ7S6FY]] $p_K$, $K \\in \\mathcal{K}$.\n\n## Remarks\n\nEach $p_K$ is finite, because $y \\mapsto \\langle x, y \\rangle$ is $\\sigma(Y, X)$-continuous and $K$ is $\\sigma(Y, X)$-compact.\nFor a locally convex space $E$, the Mackey topology of $E$ is $\\tau(E, E')$, and $E$ is called a *Mackey space* if its topology equals $\\tau(E, E')$.\nExample: every metrizable locally convex space, in particular every normed space, is a Mackey space.\nCheck for a Banach space $E$: the closed unit ball $K$ of $E'$ is convex, balanced and weak-\\* compact by the [[Banach–Alaoglu theorem|1BXEJXQ2YT]], and $p_K(x) = \\norm{x}$ by the Hahn–Banach theorem.\nSo $\\tau(E, E')$ is at least as fine as the norm topology.\nConversely, each weak-\\* compact $K \\subseteq E'$ is bounded at each point of $E$, hence norm bounded by the uniform boundedness principle, say by $r$.\nThen $p_K(x) \\le r \\norm{x}$, so $\\tau(E, E')$ is no finer than the norm topology.\nNon-example: on an infinite-dimensional normed space $E$, the weak topology $\\sigma(E, E')$ has the same continuous dual $E'$ but is strictly coarser than $\\tau(E, E')$, so it is not the Mackey topology.\n",
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