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The trigonometric system is complete in $L^{2}$ of the torus

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). The characters $(e_{k})_{k \in Z}$ form an orthonormal basis of the complex Hilbert space $L^{2} (T; C)$ (Orthonormal families, complete orthonormal systems and Hilbert bases, $L^{2}$ with the integral pairing is a Hilbert space): they are orthonormal and their closed linear span is all of $L^{2} (T; C)$ .

Facts & Assumptions

[A1]

The characters are orthonormal in $L^{2} (T; C)$ (The trigonometric characters are orthonormal in $L^{2}$ of the torus).

[A2]

The complex continuous functions on $T$ are dense in $L^{2} (T; C)$ for the $L^{2}$ norm (Continuous functions are dense in $L^{p}$ of finite tori and of bounded intervals).

[A3]

The trigonometric polynomials are uniformly dense in $C (T, C)$ , and on the probability space $T$ one has $∥ g ∥_{2} \leq m_{T} (T)^{1 / 2} ∥ g ∥_{\infty} = ∥ g ∥_{\infty}$ for continuous $g$ (Trigonometric polynomials are uniformly dense in continuous functions on the torus, Finite-measure $L^{r}$ includes into $L^{p}$ for $p < r$ ).

[A4]

The span of the characters is exactly the set of trigonometric polynomials, and an orthonormal family is complete exactly when its closed linear span is the whole space (Fourier coefficients and trigonometric polynomials on the torus, Orthonormal families, complete orthonormal systems and Hilbert bases).

Proof

technique · direct

Given: Countable Choice, the Hilbert space $L^{2} (T; C)$ and the characters $e_{k}$ .

1.1

The characters are orthonormal by [A1], and their linear span is the set of trigonometric polynomials by [A4].

A1A4

1.2

The closure of the span contains every continuous function: given continuous $g$ and $ε > 0$ , uniform density of trigonometric polynomials gives a polynomial $p$ with $∥ g - p ∥_{\infty} < ε$ , and then $∥ g - p ∥_{2} \leq ∥ g - p ∥_{\infty} < ε$ .

2.1

Hence the closed span contains the closure of $C (T, C)$ , which is all of $L^{2} (T; C)$ by the density of continuous functions.

step 1.2A2

3.1

Therefore the characters are an orthonormal family whose closed linear span is $L^{2} (T; C)$ , that is, an orthonormal basis of that Hilbert space.

step 1.1step 2.1A4∎

Depends on

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Sources

Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §2.5, pp.64–65, Theorem 2.17 (standard reference, not scraped)
Theo Bühler and Dietmar Salamon, Functional Analysis — Example 2.66, pp.87–88 (standard reference, not scraped)

The trigonometric system is complete in L2L2 of the torus