The dual of a quotient is the annihilator

Given: A locally compact Hausdorff abelian group $G$, a closed subgroup $H\le G$, and the quotient homomorphism $q:G\to G\mathrm{/}H$.

$G\mathrm{/}H$ is the abelian quotient group of cosets with the quotient topology, and $q:G\to G\mathrm{/}H$ is a continuous surjective homomorphism; continuity of a map out of $G\mathrm{/}H$ is equivalent to continuity after composition with $q$. (The quotient group $G\mathrm{/}N$ and coset product $(gN)(hN)=ghN$, The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection, Every quotient group of an abelian group is abelian, For a quotient map $q:X\to Y$, a map out of $Y$ is continuous iff its composite with $q$ is; a continuous map on $X$ constant on the fibres of $q$ factors uniquely through $q$; and a composite of quotient maps is a quotient map)

The annihilator is $H^{\perp }=\{\gamma \in \widehat{G}:\gamma (h)=1\text{ for every }h\in H\}$; it is a subgroup of $\widehat{G}$ and the kernel of the restriction homomorphism $\widehat{G}\to \widehat{H}$, hence closed in $\widehat{G}$. (The annihilator of a subgroup)

For a continuous homomorphism $\phi$ of abelian topological groups the pullback $\widehat{\phi }(\gamma )=\gamma \circ \phi$ is a continuous group homomorphism; and for a closed subgroup $H$ of a locally compact Hausdorff abelian group $G$, the pullback $\widehat{q}:\widehat{G\mathrm{/}H}\to \widehat{G}$ of the quotient map is a topological group isomorphism onto $H^{\perp }=\{\gamma \in \widehat{G}:\gamma (h)=1\text{ for all }h\in H\}$, a closed subgroup of $\widehat{G}$. (Dual homomorphisms: continuity, and the annihilator of a closed subgroup, The Pontryagin dual with the compact-open topology)