Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6.1-sol)
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The dual of a closed subgroup is a quotient of the dual

Statement

Assume the Axiom of Choice (The Axiom of Choice) and Dependent Choice (The axiom of dependent choice: a relation in which every element is related to something admits an N-indexed chain). Let G be a locally compact Hausdorff abelian group with dual G^ and let H≤G be a closed subgroup. Then restriction R:G^→H^,R(γ):=γ∣H, is an open continuous surjection with kernel H⊥ (The annihilator of a subgroup), and it induces an isomorphism of topological groups G^/H⊥  ≅  H^.

Facts & Assumptions

Given: A locally compact Hausdorff abelian group G, a closed subgroup H≤G, the restriction map R, and the quotient map q:G^→G^/H⊥.

[F1]

R is a continuous group homomorphism with kernel H⊥, a closed subgroup of G^; the quotient G^/H⊥ is a locally compact Hausdorff abelian group, and q is a continuous surjection. (The annihilator of a subgroup, Dual homomorphisms: continuity, and the annihilator of a closed subgroup, The quotient of an LCA group by a closed subgroup is LCA, The quotient group G/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, Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological)

[F2]

R is surjective: every continuous character of H extends to a continuous character of G. (Characters of a closed subgroup extend to the ambient LCA group)

[F3]

The quotient-dual theorem applied to the group G^ and its closed subgroup H⊥ gives a topological isomorphism Ξ:(G^/H⊥) ^→(H⊥)⊥, Ξ(ξ)=ξ∘q, and by the double-annihilator identity in G one has (H⊥)⊥=ΦG(H), the image of H under the biduality identification. (The dual of a quotient is the annihilator, Annihilators reverse inclusions and the double annihilator closes the subgroup, Pontryagin biduality: the evaluation map is a topological isomorphism)

[F4]

A closed subgroup H of an LCA group is LCA (A locally compact subgroup of a Hausdorff topological group is closed), and its dual is LCA under AC (The dual of a locally compact abelian group is locally compact abelian). The evaluation maps are topological isomorphisms and natural: for a continuous homomorphism ψ:A→B of locally compact Hausdorff abelian groups one has ΦB∘ψ=ψ^^∘ΦA, and the dual of a topological isomorphism is a topological isomorphism. (Pontryagin biduality: the evaluation map is a topological isomorphism, Dual homomorphisms: continuity, and the annihilator of a closed subgroup, The Pontryagin dual with the compact-open topology)

[F5]

An open continuous surjection is a quotient map, and a map out of a quotient is continuous exactly when its composite with the quotient map is; the quotient map of a topological group by a subgroup is open. (For a quotient map q:X→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, A continuous open surjection, a continuous closed surjection, and a continuous surjection admitting a continuous section are all quotient maps, The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection, Subgroup)

Proof

1.1F1F5

The map ψ:G^/H⊥→H^ given by ψ(γH⊥):=γ∣H is well defined, because R has kernel H⊥; it is continuous and injective by the quotient universal property, and R=ψ∘q.

2.1F2step 1.1

The map R is surjective by [F2], hence ψ is bijective and its image is all of H^.

2.2F3F4step 1.1

The transpose ψ^:H^^→(G^/H⊥) ^ is a topological isomorphism. Indeed for h∈H and γ∈G^ one computes ψ^(ΦH(h))(γH⊥)=ΦH(h)(ψ(γH⊥))=γ(h)=ΦG(h)(γ), so ψ^=Ξ−1∘ΦG∣H∘ΦH−1 is a composite of topological isomorphisms by [F3] and [F4].

3.1F4step 2.2

Naturality of evaluation, ΦH^∘ψ=ψ^^∘ΦG^/H⊥ (a direct computation from ΦA(a)(λ)=λ(a)), writes ψ=ΦH^−1∘ψ^^∘ΦG^/H⊥ as a composite of topological isomorphisms, so ψ is a topological isomorphism of G^/H⊥ onto H^.

4.1F5step 1.1step 2.1step 3.1∎

Finally R=ψ∘q is continuous, open (a composite of the open quotient map q of [F5] with the homeomorphism ψ) and surjective, its kernel is ker⁡q=H⊥, and the induced map G^/H⊥→H^ is the topological isomorphism ψ; this is the statement.

Depends on

Used by

Dependency tree · two levels

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Sources