TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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Fourier transform is a topological automorphism of tempered distributions

Statement

Assume Countable Choice. Fourier transformation is a topological automorphism of $S^{'} (R^{n})$ for both the weak and strong dual topologies. If

$⟨ R u, φ ⟩ = ⟨ u, φ (- \cdot) ⟩,$

then $F^{2} u = R u$ and $F^{- 1} = R F = F R$ .

Facts & Assumptions

Given: Countable Choice and $u \in S^{'} (R^{n})$ .

[F1]

Fourier transformation on $S^{'}$ is weakly and strongly continuous (Fourier transform on tempered distributions is well defined and continuous).

[F2]

On Schwartz space, $F^{2} = R$ , $R^{2} = I$ , and $F^{- 1} = R F$ (Fourier transform is a topological automorphism of Schwartz space).

Proof

technique · transpose the Schwartz identities

1.1

Evaluate the second transform on an arbitrary Schwartz test $φ$ .

$⟨ F^{2} u, φ ⟩ = ⟨ u, F^{2} φ ⟩ = ⟨ u, R φ ⟩ = ⟨ R u, φ ⟩ .$

Thus $F^{2} u = R u$ . This is a direct test calculation and uses no density assertion about $S$ inside its dual. [F2]

2.1

Reflection on the dual satisfies $R^{2} = I$ . Since step 1.1 gives $R = F^{2}$ as operators on $S^{'}$ , associativity gives the following two-sided inverse calculation.

step 1.1algebra

$(R F) F = R^{2} = I, F (R F) = F^{4} = R^{2} = I .$

Hence $F^{- 1} = R F$ ; also $R F = F^{3} = F R$ . [step 1.1, algebra]

3.1

The inverse $R F = F^{3}$ is a composition of weakly continuous maps and also of strongly continuous maps. Therefore $F$ is a topological automorphism for both topologies. Countable Choice is used only through [F1]–[F2].

F1step 2.1∎

Depends on

Used by

Dependency tree · two levels

13 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

Semyon Dyatlov, Lecture notes for 18.155 (2022) (standard reference, not scraped)
Radu Gelca, Functional Analysis (standard reference, not scraped)