= Characters of a real torus
{title2=$\Lambda^\vee\cong\operatorname{Hom}_{\mathrm{cts}}(\mathbb R^n/\Lambda,\mathbb C^\times)$}
= Torus character
{synonym}
Every continuous <group homomorphism> $\mathbb R^n/\Lambda\to\mathbb C^\times$ is $x+\Lambda\mapsto e^{2\pi i\langle u,x\rangle}$ for a unique $u$ in the <dual lattice>. Compactness forces unit modulus, and lifting the character to the simply connected cover makes its phase a real linear functional. These characters are the frequencies of torus <Fourier series>.
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