In the theory of lattices, the dual lattice is a construction analogous to that of a dual vector space. In certain respects, the geometry of the dual lattice of a lattice ${\textstyle L}$ is the reciprocal of the geometry of ${\textstyle L}$, a perspective which underlies many of its uses.

Dual lattices have many applications inside of lattice theory, theoretical computer science, cryptography and mathematics more broadly. For instance, it is used in the statement of the Poisson summation formula, transference theorems provide connections between the geometry of a lattice and that of its dual, and many lattice algorithms exploit the dual lattice.

For an article with emphasis on the physics / chemistry applications, see Reciprocal lattice. This article focuses on the mathematical notion of a dual lattice.

Let ${\textstyle L\subseteq \mathbb {R} ^{n}}$ be a lattice. That is, ${\textstyle L=B\mathbb {Z} ^{n}}$ for some matrix ${\textstyle B}$.

The dual lattice is the set of linear functionals on ${\textstyle L}$ which take integer values on each point of ${\textstyle L}$:

If ${\textstyle (\mathbb {R} ^{n})^{*}}$ is identified with ${\textstyle \mathbb {R} ^{n}}$ using the dot-product, we can write ${\textstyle L^{*}=\{v\in {\text{span}}(L):\forall x\in L,v\cdot x\in \mathbb {Z} \}.}$ It is important to restrict to vectors in the span of ${\textstyle L}$, otherwise the resulting object is not a lattice.

Despite this identification of ambient Euclidean spaces, it should be emphasized that a lattice and its dual are fundamentally different kinds of objects; one consists of vectors in Euclidean space, and the other consists of a set of linear functionals on that space. Along these lines, one can also give a more abstract definition as follows:

However, we note that the dual is not considered just as an abstract Abelian group of functionals, but comes with a natural inner product: ${\textstyle f\cdot g=\sum _{i}f(e_{i})g(e_{i})}$, where ${\textstyle e_{i}}$ is an orthonormal basis of ${\textstyle {\text{span}}(L)}$. (Equivalently, one can declare that, for an orthonormal basis ${\textstyle e_{i}}$ of ${\textstyle {\text{span}}(L)}$, the dual vectors ${\textstyle e_{i}^{*}}$, defined by ${\textstyle e_{i}^{*}(e_{j})=\delta _{ij}}$ are an orthonormal basis.) One of the key uses of duality in lattice theory is the relationship of the geometry of the primal lattice with the geometry of its dual, for which we need this inner product. In the concrete description given above, the inner product on the dual is generally implicit.