Editing Dual lattice (section)

==Definition==

Let <math display = "inline"> L \subseteq \mathbb{R}^n </math> be a lattice. That is, <math display = "inline"> L = B \mathbb{Z}^n </math> for some matrix <math display = "inline"> B </math>.

The dual lattice is the set of [[linear form|linear]] [[Functional (mathematics)|functionals]] on <math display = "inline"> L </math> which take integer values on each point of <math display = "inline"> L </math>:

:<math> L^* = \{ f \in (\text{span}(L))^* : \forall x \in L, f(x) \in \mathbb{Z} \}. </math>

If <math display = "inline"> (\mathbb{R}^n)^* </math> is identified with <math display = "inline"> \mathbb{R}^n </math> using the [[dot-product]], we can write <math display="inline"> L^* = \{ v \in \text{span}(L) : \forall x \in L, v \cdot x \in \mathbb{Z} \}. </math> It is important to restrict to [[Vector (mathematics and physics)|vector]]s in the [[linear span|span]] of <math display="inline"> L </math>, otherwise the resulting object is not a [[Lattice (group)|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:

:<math> L^* = \{ f : L \to \mathbb{Z} : \text{f is a linear function} \} = \text{Hom}_{\text{Ab}}(L, \mathbb{Z}). </math>

However, we note that the dual is not considered just as an abstract [[Abelian group]] of functionals, but comes with a natural inner product: <math display="inline"> f \cdot g = \sum_i f(e_i) g(e_i) </math>, where <math display="inline"> e_i </math> is an [[orthonormality|orthonormal]] basis of <math display="inline"> \text{span}(L)</math>. (Equivalently, one can declare that, for an orthonormal basis  <math display="inline"> e_i </math> of  <math display="inline"> \text{span}(L) </math>, the dual vectors <math display="inline"> e^*_i </math>, defined by  <math display="inline"> e_i^*(e_j) = \delta_{ij} </math> 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.