Characters of a real torus (${\Lambda }^{\vee }\cong {\mathrm{Hom}}_{\mathrm{cts}}({\mathbb{R}}^n/\Lambda ,{\mathbb{C}}^{\times })$)

Every continuous group homomorphism ${\mathbb{R}}^n/\Lambda \to {\mathbb{C}}^{\times }$ is $x+\Lambda \mapsto e^{2\pi i⟨u,x⟩}$ 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.