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  "body": "\n## Definition\n\nA *locally compact abelian group* is a [[topological group|92700QAB5Y]] $G$ such that:\n\n1. the group $G$ is [[abelian|082GHNPJ53]];\n2. the topology of $G$ is Hausdorff;\n3. the topology of $G$ is [[locally compact|0FE2SB8WVJ]], that is, each point of $G$ has a compact neighbourhood.\n\n## Remarks\n\nExample: $\\R^n$ with addition and the usual topology is an LCA group.\nIt is abelian, Hausdorff, and each point $x$ has the compact neighbourhood $\\set{y : \\norm{y - x} \\le 1}$.\nNon-example: the additive group $\\Q$ of rational numbers with the topology from $\\R$ is abelian and Hausdorff, but it is not locally compact.\nEach neighbourhood of $0$ contains a set $(-\\varepsilon, \\varepsilon) \\cap \\Q$, hence a sequence of rationals that converges in $\\R$ to an irrational number.\nThis sequence has no subsequence that converges in $\\Q$, so no neighbourhood of $0$ is contained in a compact subset of $\\Q$.\nThe usual abbreviation is \"LCA group\".\nMany authors include the Hausdorff condition in the definition of a topological group.\n",
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