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  "body": "\n## Definition\n\nA *locally compact group* is a [[topological group|92700QAB5Y]] $G$ such that:\n\n1. the topology of $G$ is [[Hausdorff|0MXB4CKHFW]];\n2. the topology of $G$ is [[locally compact|0EQMWQ4WMJ]], that is, every point of $G$ has a compact neighbourhood.\n\n## Remarks\n\nExample: $(\\R^n, +)$ is a locally compact group, since the closed ball of radius $1$ around $x$ is a compact neighbourhood of $x$ by the Heine–Borel theorem.\nEvery discrete group is locally compact, since $\\set{x}$ is a compact neighbourhood of $x$.\n\nNon-example: the additive group of an infinite-dimensional Hilbert space $H$ is a Hausdorff topological group but is not locally compact.\nSuppose $0$ had a compact neighbourhood $K$.\nThen $K$ contains a closed ball of some radius $r \u003e 0$, and this ball is compact as a closed subset of $K$.\nFor an orthonormal sequence $(e_k)$, the points $r e_k$ lie in this ball.\nThey have no convergent subsequence, a contradiction, since $\\norm{r e_j - r e_k} = r\\sqrt{2}$ for $j \\ne k$.\n\nBy homogeneity it is enough that the identity has a compact neighbourhood.\nSome authors omit the Hausdorff condition; a locally compact group in that wider sense becomes Hausdorff after dividing by the closure of the identity.\n",
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