Crossposted from [math.stackexchange](https://math.stackexchange.com/questions/64016/periodicity-of-the-nullspace-of-a-module) since I'm not getting any answer.


Let $W$ be the finite $\mathbb{Z}$-module obtained from $\mathbb{Z}\_q^n$ with addition componentwise where $\mathbb{Z}\_q$ is the integers mod $q$. Let $V$ be a submodule of $W$. Let $V^{\perp} = \{w \in W \, : \, \forall v \in V,\, v \cdot w = 0 \}$ where "$\cdot$" is the dot product. Is it true that ${(V^{\perp})}^{\perp} = V$ for all $q \geq 2$? If not, when is it the case?



Double orthogonal complement of a finite module
The answer is yes. The easiest way for me is to appeal to character theory. If $\zeta$ is
a complex $q$-th root of 1 then the map from $\mathbb{Z}\_q^n$ to $\mathbb{C}$ given by
$$
 x \mapsto \zeta^{a^Tx},\qquad (a\in\mathbb{Z}\_q^n)
$$
is a character of the abelian group $W=\mathbb{Z}\_q^n$. The set of characters obtained as
$a$ varies over the elements of $W$ is the character group $W^\*$ of $W$. 
If $V$ is a subgroup of $W$, then $V^\perp=V^\*$ is isomorphic to the subgroup $(W/V)^\*$ 
of $W^\*$.


A convenient source for the relevant character theory is from our own KConrad: `http://www.math.uconn.edu/~kconrad/blurbs/` (under characters of finite abelian groups).


