LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The character topology on L^1 of an LCA group is the compact-open topology

Statement

Assume Dependent Choice. Let $G$ be a locally compact Hausdorff abelian group with Haar measure $m_{G}$ and $A = L^{1} (G, m_{G})$ . Under the bijection $γ \mapsto h_{γ}$ of Nonzero multiplicative functionals on L^1 of an LCA group are Fourier evaluations, the compact-open topology of $\hat{G}$ is the topology of pointwise evaluation on all of $A$ : a net $(h_{i})$ converges to $h_{0}$ in the topology of pointwise convergence on $A$ if and only if the corresponding characters converge to $γ_{0}$ uniformly on every compact subset of $G$ . Consequently the algebraically defined character space of $A$ carries exactly the compact-open topology of $\hat{G}$ , and the resulting identification is a homeomorphism.

Facts & Assumptions

Given: Dependent Choice, a locally compact Hausdorff abelian group $G$ with Haar measure $m_{G}$ , $A = L^{1} (G, m_{G})$ , the bijection $γ \mapsto h_{γ} (f) = \hat{f} (γ)$ onto the nonzero multiplicative linear functionals (Nonzero multiplicative functionals on L^1 of an LCA group are Fourier evaluations), a net $(γ_{i})_{i \in I}$ in $\hat{G}$ with compact-open limit $γ_{0}$ , and a net $(h_{i})$ evaluating pointwise to $h_{0}$ on $A$ (Directed preorders and nets, The topology of pointwise convergence on $Y^{X}$ , which is the product topology, and its restriction to $C (X, Y)$ , The compact-open topology on $C (X, Y)$ for arbitrary topological spaces).

[F1]

$h_{γ} (T_{x} f) = \overline{γ (x)} h_{γ} (f)$ for all $x \in G$ , $f \in A$ : substituting $y \mapsto y - x$ (a translation, so $m_{G}$ -preserving) gives $h_{γ} (T_{x} f) = \int_{G} f (y - x) \overline{γ (y)} d m_{G} (y) = \int_{G} f (z) \overline{γ (z + x)} d m_{G} (z) = \overline{γ (x)} \int_{G} f (z) \overline{γ (z)} d m_{G} (z)$ (The Pontryagin dual with the compact-open topology, The multiplicative unit circle is a compact metrizable topological abelian group).

[F2]

$∣ h_{γ} (f) ∣ \leq ∥ f ∥_{1}$ , $C_{c} (G)$ is dense in $A$ and $x \mapsto T_{x} f$ is norm-continuous with $∥ T_{x} f ∥_{1} = ∥ f ∥_{1}$ (Nonzero multiplicative functionals on L^1 of an LCA group are Fourier evaluations, C_c(X) is dense in L^p(mu) for a Radon measure, Translation continuity and normalised local approximate identities on an LCA group, The space $L^{p} (μ)$ as the quotient by null functions, Integrable real and complex functions, and their integrals, The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain).

[F3]

A continuous image of a compact set is compact (A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism, Countably compact, Lindel"of, sequentially compact, limit point compact and $σ$ -compact spaces, and relatively compact subsets); a compact subset of a metric space has a finite cover by balls of any prescribed radius, and $∥ f ∥_{\infty} m_{G} (K) < + \infty$ for compact $K$ (Left Haar integral and left Haar measure).

[F4]

For an arbitrary abelian topological group, the sets $U_{K} (γ, r) : = {η : ∣ η (x) - γ (x) ∣ < r for every x \in K}$ , with $K$ compact and $r > 0$ , form a neighbourhood basis in its compact-open dual. Thus compact-open convergence of characters is exactly uniform convergence on every compact set; no metrizability of $G$ or choice axiom is required (The compact-open character group is a Hausdorff topological abelian group, Statement and proof 1.2 and 2.1).

Proof

technique · direct

1.1F2F3F4

(Compact-open convergence gives evaluation convergence.) Assume $γ_{i} \to γ_{0}$ in the compact-open topology; by [F4] this is uniform convergence on compacta. Fix $f \in A$ , $ε > 0$ . By [F2] choose $f_{0} \in C_{c} (G)$ with $∥ f - f_{0} ∥_{1} < ε / 3$ and put $K : = supp f_{0}$ . Then for every $i$ , using [F2] and the definition of $h_{γ}$ , $∣ h_{γ_{i}} (f) - h_{γ_{0}} (f) ∣ \leq 2 ∥ f - f_{0} ∥_{1} + \int_{K} ∣ f_{0} (y) ∣ ∣ γ_{i} (y) - γ_{0} (y) ∣ d m_{G} (y) \leq \frac{2 ε}{3} + ∥ f_{0} ∥_{\infty} m_{G} (K) \sup_{y \in K} ∣ γ_{i} (y) - γ_{0} (y) ∣,$ and the supremum tends to $0$ along the net; hence $h_{γ_{i}} (f) \to h_{γ_{0}} (f)$ for every $f \in A$ .

1.2F2F3

(Evaluation convergence gives compact-open convergence.) Conversely, assume $h_{i} \to h_{0}$ pointwise on $A$ , say $h_{i} = h_{γ_{i}}$ and $h_{0} = h_{γ_{0}}$ . Choose $f \in A$ with $h_{0} (f) \neq 0$ ; then $∣ h_{i} (f) ∣ \geq ∣ h_{0} (f) ∣ / 2$ for all sufficiently large $i$ . Let $K \subseteq G$ be compact and $δ > 0$ . By [F2] the set ${T_{x} f : x \in K}$ is a continuous image of $K$ , hence compact, so finitely many of its points $T_{x_{1}} f, \dots, T_{x_{m}} f$ cover it by $δ$ -balls; put $η_{i} : = \max_{j} ∣ h_{i} (T_{x_{j}} f) - h_{0} (T_{x_{j}} f) ∣$ , which tends to $0$ by pointwise convergence. For every $x \in K$ , choosing $j$ with $∥ T_{x} f - T_{x_{j}} f ∥_{1} < δ$ and using [F2] gives $∣ h_{i} (T_{x} f) - h_{0} (T_{x} f) ∣ \leq 2 δ + η_{i}$ , so $\sup_{x \in K} ∣ h_{i} (T_{x} f) - h_{0} (T_{x} f) ∣ \leq 2 δ + η_{i}$ .

2.1F1F4step 1.2

(Uniform convergence of the characters.) By [F1], $h_{i} (T_{x} f) - h_{0} (T_{x} f) = \overline{γ_{i} (x)} h_{i} (f) - \overline{γ_{0} (x)} h_{0} (f)$ for every $x$ . For large $i$ the denominator $h_{i} (f)$ satisfies $∣ h_{i} (f) ∣ \geq ∣ h_{0} (f) ∣ / 2 > 0$ , and $\overline{γ_{i} (x)} - \overline{γ_{0} (x)} = \frac{h_{i} (T_{x} f) - h_{0} (T_{x} f) + \overline{γ_{0} (x)} (h_{0} (f) - h_{i} (f))}{h_{i} (f)} .$ Taking suprema over $x \in K$ and using step 1.2 together with $h_{i} (f) \to h_{0} (f)$ gives $\sup_{x \in K} ∣ γ_{i} (x) - γ_{0} (x) ∣ \leq 2 (2 δ + η_{i} + ∥ h_{0} (f) - h_{i} (f) ∥) / ∣ h_{0} (f) ∣$ eventually, which tends to $4 δ / ∣ h_{0} (f) ∣$ ; since $K$ and $δ > 0$ are arbitrary, $γ_{i} \to γ_{0}$ uniformly on every compact subset of $G$ , hence in the compact-open topology by [F4].

3.1step 1.1step 2.1

(The homeomorphism.) Steps 1.1 and 2.1 show that under the bijection $γ \mapsto h_{γ}$ the compact-open topology of $\hat{G}$ corresponds exactly to the topology of pointwise convergence on $A$ ; thus the algebraically defined character space of $A$ carries the compact-open topology of $\hat{G}$ and the identification is a homeomorphism.

4.1step 1.1step 2.1step 3.1∎

Together with the bijection of Nonzero multiplicative functionals on L^1 of an LCA group are Fourier evaluations, steps 1.1 and 2.1 prove both implications of the stated equivalence, and step 3.1 records the homeomorphism.

Depends on

Used by

Dependency tree · two levels

120 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Lynn H. Loomis, Introduction to Abstract Harmonic Analysis, D. Van Nostrand, 1953 (Harvard-hosted full scan) (standard reference, not scraped)
Manfred Einsiedler and Thomas Ward, Ergodic Theory with a View Towards Number Theory, Appendix C.2-C.3 (course-hosted full text) (standard reference, not scraped)