Classically, a lattice in the Cartesian space$\mathbb{R}^n$ is a discretesubgroup (of the underlying topologicalabelian group) that spans$\mathbb{R}^n$ as a vector space over $\mathbb{R}$. This may be generalized, from $\mathbb{R}^n$ to a general locally compact abelian group.

Applying Pontryagin duality, the dual of the quotient map $q: A \to A/L$ is in that case a discrete subgroup $\widehat{A/L} \hookrightarrow \widehat{A}$ that is also cocompact (its cokernel being the compact group $\widehat{L}$). This is called the dual lattice of $L$.

Examples

Notable examples of classical lattices (in $\mathbb{R}^n$) include

The standard diagonal inclusion of a global field (such as a number field) $k$ into its ring of adeles$A_k$ is a lattice in the more general sense. Recalling that $A_k$ is Pontryagin dual to itself, the lattice $k$ is identified with its dual lattice.