Remark

Given a subgroup H of $\mathbf{R}^n$, there corresponds to it its orthogonal $H^{\bot}$, a subgroup of $\widehat{\mathbf{R}}^n=\mathbf{R}^n$, which is none other than the subgroup associated with H defined in TG, VII, p. 6, n$^o3$.

In the sequel, one will identify the dual of $\mathbf{R}^n$ (resp. of $\mathbf{T}^n$) with $\mathbf{R}^n$ (resp. with $\mathbf{Z}^n$) by the duality of the corollary. In particular, for $f\in L^1(\mathbf{R}^n)$, its Fourier transform is identified with the function from $\mathbf{R}^n$ into $\mathbf{C}$ which to $y\in \mathbf{R}^n$ associates

$\mathscr{F}(f)(y) =\int_{\mathbf{R}^n}f(x)$ exp($-2i\pi x\cdot y$)$dx$.