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  "body": "\n## Definition\n\nLet $G$ be a [[topological group|92700QAB5Y]].\nA subgroup $K$ of $G$ is a *maximal compact subgroup* if two conditions hold.\n\n1. The subgroup $K$ is [[compact|0TVJZYB6SH]] in the subspace topology.\n2. If $L$ is a compact subgroup of $G$ with $K \\subseteq L$, then $L = K$.\n\n## Remarks\n\nExample: $K = \\mathrm{O}(n)$ in $G = \\mathrm{GL}_n(\\R)$.\nIt is closed and bounded in the space of matrices, so it is compact.\nLet $L \\supseteq \\mathrm{O}(n)$ be a compact subgroup.\nAveraging the standard inner product over $L$ with the Haar measure gives an inner product $\\ip{\\cdot}{\\cdot}'$ that $L$ preserves.\nThe group $\\mathrm{O}(n)$ preserves it too, and the only $\\mathrm{O}(n)$-invariant inner products are the positive multiples of the standard one.\nSo $L$ preserves the standard inner product, and $L \\subseteq \\mathrm{O}(n)$.\n\nNon-example: $\\mathrm{SO}(n)$ in $\\mathrm{GL}_n(\\R)$ is compact, but it fails clause 2, since $\\mathrm{SO}(n) \\subsetneq \\mathrm{O}(n)$.\n\nA maximal compact subgroup need not exist.\nIn the countable direct sum $\\bigoplus_{i \\in \\N} \\Z/2$ with the discrete topology, the compact subgroups are the finite ones, and each finite subgroup lies in a larger finite subgroup.\n",
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