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  "body": "\n## Summary\n\nThe Pontryagin dual of a locally compact abelian group $G$ is the group $\\widehat{G}$ of continuous characters of $G$, with the compact-open topology.\nIt is again a locally compact abelian group.\nThe dual is the domain of the Fourier transform on $G$, and the Pontryagin duality theorem says that the dual of $\\widehat{G}$ is $G$ again.\n\n## Explanations\n\n### L1 human\n\nThe dual group lists all the ways the group can \"vibrate\": each element of the dual is a wave on $G$.\nFor the circle, the waves are $e^{in\\theta}$ with $n$ an integer, so the dual of the circle is $\\Z$.\nFor the integers, the waves are $n \\mapsto z^n$ with $\\abs{z} = 1$, so the dual of $\\Z$ is the circle.\nFor the real line, the waves are $x \\mapsto e^{i\\xi x}$ with $\\xi$ real, so $\\R$ is its own dual.\nFourier series and the Fourier transform are the cases $G = $ circle and $G = \\R$ of one theory built on this dual.\n",
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