Dual lattice (${\Lambda }^{\vee }$)

The dual lattice of a full-rank Euclidean lattice $\Lambda \subset {\mathbb{R}}^n$ isHere $⟨x,y⟩$ is the Euclidean inner product. If the columns of an invertible matrix $B$ form a basis of $\Lambda =B{\mathbb{Z}}^n$, thenIndeed, $y$ belongs to the dual lattice exactly when $B^{\mathsf{T}}y\in {\mathbb{Z}}^n$; $B^{-\mathsf{T}}$ denotes the transpose of the matrix inverse. The reciprocal covolume follows by taking the determinant. With the Fourier transform kernel $e^{-2\pi i⟨x,y⟩}$, these are precisely the frequencies of the characters of a real torus: translating $x$ by any $\lambda \in \Lambda$ leaves $e^{2\pi i⟨x,y⟩}$ unchanged. In one dimension, $(h\mathbb{Z}{)}^{\vee }=h^{-1}\mathbb{Z}$ for $h>0$: increasing the spacing in physical space decreases the spacing in frequency space.

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.