= Dual lattice
{title2=$\Lambda^\vee$}
{wiki}
The <dual lattice> of a full-rank <Euclidean lattice> $\Lambda\subset\mathbb R^n$ is
$$
\Lambda^\vee=\{y\in\mathbb R^n:\langle x,y\rangle\in\mathbb Z\text{ for every }x\in\Lambda\}.
$$
Here $\langle x,y\rangle$ is the Euclidean <inner product>. If the columns of an invertible <matrix> $B$ form a <basis> of $\Lambda=B\mathbb Z^n$, then
$$
\Lambda^\vee=B^{-\mathsf T}\mathbb Z^n,\qquad m(\Lambda^\vee)=|\det B|^{-1}=m(\Lambda)^{-1}.
$$
Indeed, $y$ 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 $e^{-2\pi i\langle x,y\rangle}$, these are precisely the <frequencies> of the <characters of a real torus>: translating $x$ by any $\lambda\in\Lambda$ leaves $e^{2\pi i\langle x,y\rangle}$ 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.
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