Let $K$ be a nondiscrete locally compact field. Then fixing a character $\chi$ on $K$, any character on $K$ can be written as $t \mapsto \chi(xt)$ for some $x \in K$. For $E \leq K$ a closed subgroup of the additive group and $\hat{K}$ the group of characters of $K$, let $E^{\perp} = \{\rho \in \hat{K} \mid \rho(E) = 1\}.$ The claim is that
$\mathcal{O}_{K}^{\perp} = \mathfrak{p}^{-\nu}$ for some integer $\nu$
($\mathfrak{p}$ is the maximal ideal), but I can't see where this comes from.
(This is out of Weil's Basic Number Theory, end of section 2.5)
EDIT: I have sort of a back way to show this. On the next page, Weil states that the basic character $\chi$ may be taken as $$\chi(t) = \lambda(a_{-1})$$ where $t = \sum a_{i}\pi^{i}$ ($\mathfrak{p} = (\pi)$) and $\lambda$ a character of the additive group of $\mathbb{F}_{q}$ ($q = \lvert \mathcal{O}_{K}/\mathfrak{p}\rvert$). Then any other character $\rho$ may be taken as $\rho: t \mapsto \chi(xt)$, so that taking the dual of $\mathcal{O}_{K}$ with respect to $\rho$, $$\mathcal{O}_{K}^{\perp} = \{y \in K \mid \rho(yt) = 1 \forall t \in \mathcal{O}_{K}\} = \{y \in K \mid \chi(xyt) = 1 \forall t \in \mathcal{O}_{K}\}\\= \{y \in K \mid xy \in \mathcal{O}_{K}\} = \mathfrak{p}^{-\text{ord}_{\mathfrak{p}}(x)}.$$
However, this is not mentioned until after the above claim, which Weil says is clear., so I would still like to understand why it is true from a more abstract point of view.
Because the dual of a $K$-lattice (a compact and open $\mathcal{O}_{K}$-module in $K$) is a $K$-lattice and the (nonzero) $K$-lattices of $K$ are the sets $\mathfrak{p}^{-\nu}$ and $K$, we know that $\mathcal{O}_{K}^{\perp}$ is either $\mathfrak{p}^{-\nu}$ or $K$. But if $\mathcal{O}_{K}^{\perp} = K$, then $$K = \mathcal{O}_{K}^{\perp} = \{x \in K \mid \chi(xt) = 1 \,\,\forall t \in \mathcal{O}_{K}\}.$$ That is, $\chi(xt) = 1$ for all $x \in K, t \in \mathcal{O}_{K}$. In particular, taking $t = 1$, $\chi(x) = 1$ for all $x \in K$, in contradiction to the assumption that $\chi$ is not the trivial character. Thus, $\mathcal{O}_{K}^{\perp} = \mathfrak{p}^{-\nu}$ for some finite $\nu$.