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  "body": "\n## Statement\n\nLet $G$ be a [[locally compact abelian group|ZR09S9R3AW]], let $\\widehat{G}$ be its [[Pontryagin dual|FMW57WNPDB]], and let $\\widehat{\\widehat{G}}$ be the Pontryagin dual of $\\widehat{G}$.\nFor $x \\in G$, define $\\alpha_G(x) \\in \\widehat{\\widehat{G}}$ by $\\alpha_G(x)(\\chi) = \\chi(x)$ for all [[characters|H1WNVN5V7F]] $\\chi \\in \\widehat{G}$.\nThen the map $\\alpha_G \\colon G \\to \\widehat{\\widehat{G}}$ is an isomorphism of topological groups.\n\n## Remarks\n\nIt is easy to check that $\\alpha_G$ is a continuous homomorphism.\nInjectivity means that continuous characters separate the points of $G$.\nThis is a hard step; one proof uses the Gelfand–Raikov theorem, or the Peter–Weyl theorem in the compact case.\nSurjectivity and openness are usually proved with the Plancherel theorem, or by reducing to compactly generated groups of the form $\\R^n \\times \\Z^m \\times K$ with $K$ compact.\nExample: $\\widehat{\\Z} \\cong \\mathbb{T}$ and $\\widehat{\\mathbb{T}} \\cong \\Z$, where $n \\in \\Z$ gives the character $z \\mapsto z^n$ of $\\mathbb{T}$, and the theorem says that $n \\mapsto (z \\mapsto z^n)$ is the natural isomorphism.\nThe map $\\alpha_G$ is natural in $G$, so duality is a contravariant equivalence of the category of locally compact abelian groups with itself.\nThe theorem needs local compactness: for general abelian topological groups the dual may fail to be locally compact, and $\\alpha_G$ may fail to be an isomorphism.\n\n## Explanations\n\n### L1 human\n\nEvery locally compact abelian group has a dual group made of its waves, its continuous characters.\nThe theorem says that taking the dual twice brings back the original group, in a natural way.\nSo the group and its dual carry the same information, seen from two sides.\nThe circle and the integers are dual to each other, and the real line is dual to itself.\n\n### L2 human\n\nDuality turns properties of $G$ into properties of $\\widehat{G}$: compact groups have discrete duals, and discrete groups have compact duals.\nA compact group is connected exactly when its discrete dual is torsion-free.\nClosed subgroups $H \\subseteq G$ correspond to closed subgroups of $\\widehat{G}$ through annihilators, with $\\widehat{G/H}$ equal to the annihilator of $H$.\nThe theorem is the background of the [[Fourier transform on LCA groups|VHN5VW5Q92]].\nThe transform turns functions on $G$ into functions on $\\widehat{G}$, and duality lets the inverse transform go back from $\\widehat{G}$ to $G$.\nIt also underlies the [[Plancherel theorem for abelian groups|V9K9TG0K3J]] and [[Bochner's theorem|ZRHAG4NXHC]].\n",
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