Correspondence between Dirichlet characters and intermediate fields

Question

I am studying from Lawrence C. Washington's book Introduction to cyclotomic field. In chapter 3 I have a group of Dirichlet characters $X$ modulo $n$ (characters of $\textrm{Gal}(\mathbb{Q}(\zeta_n)/\mathbb{Q})\cong (\mathbb{Z}_n/\mathbb{Z})^{\times}$), then I want to prove $(X^{\perp})^{\perp}=X$, where $X^{\perp}$ consists of $\sigma \in \textrm{Gal}(\mathbb{Q}(\zeta_n)/\mathbb{Q})$, for which $\chi(\sigma)=1, \forall \chi \in X$, and $ (X^{\perp})^{\perp}$ consists of all $\chi$, for which $\chi(\sigma)=1, \forall \sigma \in X^{\perp}$. Thank you.