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  "body": "\n## Definition\n\nLet $G$ be an abelian [[topological group|92700QAB5Y]], written additively.\nLet $C_b(G)$ be the space of bounded [[continuous functions|0X3WGX95FE]] $G \\to \\C$ with the norm $\\norm{f}_\\infty = \\sup_{x \\in G} \\abs{f(x)}$.\nFor $f \\in C_b(G)$ and $a \\in G$, let $f_a(x) = f(x + a)$ be the translate of $f$ by $a$.\nA function $f \\in C_b(G)$ is *almost periodic* if the set of its translates $\\set{f_a : a \\in G}$ is relatively [[compact|0R6EWFKRCF]] in $C_b(G)$.\nThis means that its closure in the norm $\\norm{\\cdot}_\\infty$ is compact.\n\n## Remarks\n\nEquivalently, every sequence of translates $f_{a_1}, f_{a_2}, \\dots$ has a subsequence that converges uniformly on $G$.\nThis is Bochner's form of the definition.\nFor $G = \\R$ it is equivalent to Bohr's definition through relatively dense sets of almost periods, see [[Bohr almost periodic function|ED69H4C3PK]].\nFor a locally compact abelian group $G$, the almost periodic functions are exactly the functions $F \\circ b$, where $b \\colon G \\to bG$ is the [[Bohr compactification|G852FHZ3W1]] and $F$ is continuous on $bG$.\n\nExample: on $\\R$, let $f(x) = \\cos x + \\cos(\\sqrt{2} x)$.\nThe translates of $\\cos$ are $\\cos(x + a) = \\cos a \\cos x - \\sin a \\sin x$, the image of the circle $\\set{(\\cos a, \\sin a)}$ under a continuous map into $C_b(\\R)$, so they form a compact set.\nThe same holds for $\\cos(\\sqrt{2} x)$, and the translates of $f$ lie in the sum of these two compact sets, which is compact.\nSo $f$ is almost periodic.\n\nNon-example: $f(x) = \\arctan x$ on $\\R$ is bounded and continuous, but not almost periodic.\nThe translates $f_n$, $n = 1, 2, \\dots$, converge pointwise to the constant $\\pi/2$.\nA uniformly convergent subsequence would have the same limit, but $\\abs{f_n(-n) - \\pi/2} = \\pi/2$ for every $n$.\n",
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