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  "body": "\n## Definition\n\nLet $G$ be a [[locally compact abelian group|ZR09S9R3AW]], and let $\\mathbb{T}$ be the [[circle group|04EAFVFTF7]].\nThe *Pontryagin dual* of $G$ is the set $\\widehat{G}$ of all continuous [[characters|H1WNVN5V7F]] $\\chi \\colon G \\to \\mathbb{T}$, with:\n\n1. the group operation $(\\chi\\psi)(x) = \\chi(x)\\psi(x)$;\n2. the [[compact-open topology|RG15F0TWPW]], that is, the topology of uniform convergence on compact subsets of $G$.\n\n## Remarks\n\nExample: for $G = \\Z$ with the discrete topology, a character $\\chi$ is fixed by $z = \\chi(1) \\in \\mathbb{T}$, since $\\chi(n) = z^n$, and each $z \\in \\mathbb{T}$ gives a character.\nUniform convergence on finite sets is pointwise convergence, so $\\widehat{\\Z} \\cong \\mathbb{T}$ as topological groups.\nNon-example: the group of all characters of $\\R$, continuous or not, is not the Pontryagin dual of $\\R$.\nThere are discontinuous homomorphisms $\\R \\to \\mathbb{T}$, built from a Hamel basis, and the dual excludes them.\nWith the compact-open topology, $\\widehat{G}$ is again a locally compact abelian group.\nIf $G$ is compact, then $\\widehat{G}$ is discrete, and if $G$ is discrete, then $\\widehat{G}$ is compact.\n",
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