I am currently working through the exercises in 'Principles of Harmonic Analysis'
For a closed subgroup B of the LCA-group A and a closed subgroup L of $\widehat{A}$.
Let $B^{\bot}=\{\chi\in\widehat{A}:\chi(B)=1\}$
$L^{\bot}=\{x\in{A}:\delta_{x}(L)=1\}$
where $\delta$ is the Pontryagin map.
How do you show that $B^{\bot}$ is canonically isomorphic to $\widehat{A/B}$, $(B^{\bot})^{\bot}=B$ and $\widehat{A}/B^{\bot}$ is canonically isomorphic to $\widehat{B}$