LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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The Pontryagin dual of the integer lattice is the torus

Statement

Give $Z^{d}$ the discrete topology and let $T : = {z \in C : ∣ z ∣ = 1}$ be the multiplicative unit circle. The map $Ψ : T^{d} ⟶ \hat{Z^{d}}, Ψ (z_{1}, \dots, z_{d}) (k_{1}, \dots, k_{d}) : = \prod_{r = 1}^{d} z_{r}^{k_{r}}$ is a topological group isomorphism, where the dual has the compact-open topology and $T^{d}$ has the product topology.

Facts & Assumptions

Given: $d \geq 1$ , the discrete additive group $Z^{d}$ , the multiplicative unit circle $T$ , and the map $Ψ$ in the Statement.

[F1]

The Pontryagin dual with the compact-open topology: a character is a continuous group homomorphism to $T$ , and the dual consists of these characters with pointwise multiplication and the compact-open topology.

[F2]

On a discrete domain the compact-open topology is the topology of pointwise convergence: on a discrete domain, the compact-open topology on the character set is the subspace topology inherited from $T^{Z^{d}}$ ; in particular, evaluation at any fixed lattice point is continuous.

[F3]

The integers as equivalence classes of pairs of naturals, The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies, Monoid homomorphism and group homomorphism, Integer powers in the complex field, and Exponent laws in a group: $g^{m + n} = g^{m} g^{n}$ and $(g^{m})^{n} = g^{m n}$ for all $m, n \in Z$ , and $(g h)^{n} = g^{n} h^{n}$ when $g$ and $h$ commute: $Z^{d}$ has the discrete topology, each $k \in Z^{d}$ has integer coordinates, a homomorphism preserves addition, and integer powers satisfy $z^{a + b} = z^{a} z^{b}$ for $z \in T$ .

[F4]

The multiplicative unit circle is a compact metrizable topological abelian group and Topological group: multiplication and inversion are continuous and A map into a product is continuous iff each of its components is; the projections are continuous and open; and each projection is surjective when every factor is nonempty, which for an infinite index set uses the Axiom of Choice: $T$ is a topological abelian group, integer-power maps on it are continuous by repeated multiplication and inversion, and a map into a product is continuous when each coordinate map is continuous.

[F5]

Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological: a homeomorphism is a bijection that is continuous with continuous inverse.

[F6]

The integer lattice is a discrete locally compact abelian group: $Z^{d}$ with coordinatewise addition and the discrete topology is a Hausdorff locally compact abelian topological group; maps out of it are continuous.

Verification

1.1F1F3F6

For $z = (z_{1}, \dots, z_{d}) \in T^{d}$ , the formula for $Ψ (z)$ is a homomorphism because for $k, ℓ \in Z^{d}$ , $Ψ (z) (k + ℓ) = \prod_{r} z_{r}^{k_{r} + ℓ_{r}} = (\prod_{r} z_{r}^{k_{r}}) (\prod_{r} z_{r}^{ℓ_{r}}) .$ It is continuous because its domain $Z^{d}$ is discrete, so every preimage of an open set is open. Therefore $Ψ (z) \in \hat{Z^{d}}$ .

1.2F1F3F4F6

The map $Ψ$ is a group homomorphism because $Ψ (z z^{'}) (k) = \prod_{r} (z_{r} z_{r}^{'})^{k_{r}} = (\prod_{r} z_{r}^{k_{r}}) (\prod_{r} (z_{r}^{'})^{k_{r}})$ by [F3] and commutativity of $T$ [F4]. If $χ \in \hat{Z^{d}}$ , put $z_{r} : = χ (e_{r})$ , where $e_{r}$ is the $r$ th standard basis vector. Then $z_{r} \in T$ by [F1], and the decomposition $k = \sum_{r} k_{r} e_{r}$ together with homomorphism and power laws gives $χ (k) = \prod_{r} z_{r}^{k_{r}}$ . Thus $χ = Ψ (z)$ ; also $z_{r} = χ (e_{r})$ recovers each coordinate uniquely, so $Ψ$ is bijective.

1.3F2F4

For each fixed $k$ , the coordinate map $z \mapsto Ψ (z) (k)$ is a finite product of continuous integer-power maps on $T$ , hence continuous by [F4]. The product-topology criterion and [F2] then imply that $Ψ$ is continuous into $\hat{Z^{d}}$ .

2.1F1F2F5step 1.2step 1.3∎

The inverse map is $χ \mapsto (χ (e_{1}), \dots, χ (e_{d}))$ . Each coordinate is continuous by [F2], so the inverse is continuous into the finite product $T^{d}$ . Thus $Ψ$ is a homeomorphism by [F5] and a group isomorphism by step 1.2 and the pointwise group law in [F1].

Depends on

Used by

Dependency tree · two levels

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Sources

Oxford Mathematical Institute, Harmonic Analysis lecture notes, characters of Z^d, pp. 17–18 (standard reference, not scraped)