CorollaryStatement: AI-adaptedProof: AI-adaptedPipeline-generatedaudited 2026-09-22

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.

Trigonometric polynomials are uniformly dense in continuous functions on the torus

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). The trigonometric polynomials of Fourier coefficients and trigonometric polynomials on the torus are uniformly dense in the complex Banach space $C (T, C)$ of continuous complex functions on the torus: for every continuous $f : T \to C$ and every real $ε > 0$ there is a trigonometric polynomial $p$ with $∥ f - p ∥_{\infty} < ε$ .

The same holds on every finite torus $T^{n}$ , $n \geq 1$ , for the trigonometric polynomials in the $n$ coordinate characters.

Facts & Assumptions

[A1]

$T$ and each $T^{n}$ are compact Hausdorff spaces (Finite tori are compact Hausdorff spaces separated by characters, The one-dimensional torus and its normalized Haar integral).

[A2]

Every unital point-separating self-adjoint complex function algebra on a compact Hausdorff space is uniformly dense in the complex continuous functions (Complex Stone–Weierstrass dichotomy for separating self-adjoint algebras; the unital case is dense).

[A3]

The trigonometric polynomials form a complex vector subspace of $C (T, C)$ closed under multiplication, with $e_{0} = 1$ and $\overline{e_{k}} = e_{- k}$ ; on $T^{n}$ the same holds for the coordinate-character polynomials (Fourier coefficients and trigonometric polynomials on the torus, $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ ).

[A4]

The characters separate points: for distinct $x, y \in T^{n}$ there is $j < n$ with $e^{(j)} (x) = \exp (2 π i x_{j}) \neq \exp (2 π i y_{j}) = e^{(j)} (y)$ (Finite tori are compact Hausdorff spaces separated by characters).

Proof

technique · direct

Given: Countable Choice and a natural $n \geq 1$ ; write $T^{1} = T$ .

1.1

Let $A$ be the set of trigonometric polynomials on $T^{n}$ . Then $A$ is a complex vector subspace of $C (T^{n}, C)$ , contains the constant function $e_{0} = 1$ , is closed under multiplication because $e_{k} e_{l} = e_{k + l}$ , and is closed under complex conjugation because $\overline{e_{k}} = e_{- k}$ ; hence $A$ is a unital self-adjoint complex function algebra.

1.2

The algebra $A$ separates points of $T^{n}$ : if $x \neq y$ then some coordinate character gives different values at $x$ and $y$ , and that character lies in $A$ .

A4A3

2.1

Since $T^{n}$ is compact Hausdorff and $A$ is a unital point-separating self-adjoint complex function algebra, the unital case of complex Stone–Weierstrass gives that $A$ is uniformly dense in $C (T^{n}, C)$ .

step 1.1step 1.2A1A2

3.1

Thus for every continuous $f : T^{n} \to C$ and every $ε > 0$ there is a trigonometric polynomial $p$ with $∥ f - p ∥_{\infty} < ε$ , which for $n = 1$ is the first claim and for general $n$ the second.

step 2.1∎

Depends on

Used by

The trigonometric system is complete in L² of the torus Theorem

Dependency tree · two levels

77 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

Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §2.5, p.68, Problem 2.19 (standard reference, not scraped)
Theo Bühler and Dietmar Salamon, Functional Analysis — Example 2.66, pp.87–88 (standard reference, not scraped)