We begin this chapter by showing that the dual of a subgroup is a quotient group and the dual of a quotient group is a subgroup.

Definition. Let H be a closed subgroup of the LCA-group G and the set of all γ in the dual group Γ of G such that (h, γ) = 0, for all h ∈ H. Then ∧ is called the annihilator of H.

For fixed h ∈ H, the continuity of (h,γ) shows that the set of all γ with (h,γ) = 0 is closed, so that ∧ is the intersection of closed sets and is therefore closed. Clearly ∧ is a group and so it is a closed subgroup of Γ.

Proposition 38. With the above notation, if Γ is the annihilator of H, then H is the annihilator of ∧.

Proof. If h ∈ H, then (h,γ) =0 for all γ ∈ A. If g ∈ G and g ∉ H then, by Corollary 1 of Theorem 21, there is a γ ∈ ∧ such that (g,γ) ≠ 0. //

Theorem 27. Let H be a closed subgroup of an LCA-group G. If Γ is the dual group of G and Γ is the annihilator of H, ∧ then ∧ and Γ/∧ are topologically isomorphic to the dual groups of G/H and H, respectively.

Proof. Let f be the canonical homomorphism of G onto G/H. Then Proposition 30 says that the mapf*: (G\H)* → G* is a continuous one–one homomorphism.