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

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The Parseval identity for Fourier series

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). For all $f, g \in L^{2} (T; C)$ ,

$∥ f ∥_{2}^{2} = \sum_{k \in Z} ∣ \hat{f} (k) ∣^{2}, ⟨ f, g ⟩ = \sum_{k \in Z} \hat{f} (k) \overline{\hat{g} (k)},$ where both sums are taken in the finite-subset-supremum/finite-subset-net sense of Square-summable families on an arbitrary index set and the space $ℓ^{2} (I)$ . The bilinear series is absolutely convergent, $\sum_{k \in Z} ∣ \hat{f} (k) \overline{\hat{g} (k)} ∣ \leq (\sum_{k \in Z} ∣ \hat{f} (k) ∣^{2})^{1 / 2} (\sum_{k \in Z} ∣ \hat{g} (k) ∣^{2})^{1 / 2},$

and the value is also the limit of the canonical symmetric partial sums $\sum_{∣ k ∣ \leq N} \hat{f} (k) \overline{\hat{g} (k)}$ .

Facts & Assumptions

[A1]

The characters are an orthonormal basis of the complex Hilbert space $L^{2} (T; C)$ (The trigonometric system is complete in $L^{2}$ of the torus, $L^{2}$ with the integral pairing is a Hilbert space).

[A2]

For a Hilbert space with orthonormal basis $(e_{i})_{i \in I}$ the coefficient map $x \mapsto (⟨ x, e_{i} ⟩)_{i \in I}$ is a linear bijection onto $ℓ^{2} (I, F)$ preserving norms and inner products (A Hilbert space with a given orthonormal basis is $ℓ^{2}$ of the index set).

[A3]

$\hat{f} (k) = ⟨ f, e_{k} ⟩$ (Fourier coefficients and trigonometric polynomials on the torus, The trigonometric characters are orthonormal in $L^{2}$ of the torus).

[A4]

On $ℓ^{2} (Z, C)$ the pairing is $⟨ a, b ⟩ = \sum_{k \in Z} a_{k} \overline{b_{k}}$ , the finite-dimensional Cauchy–Schwarz inequality gives $\sum_{k \in F} ∣ a_{k} \overline{b_{k}} ∣ \leq (\sum_{k \in F} ∣ a_{k} ∣^{2})^{1 / 2} (\sum_{k \in F} ∣ b_{k} ∣^{2})^{1 / 2}$ for finite $F$ , and passing to the supremum over finite $F$ bounds the total absolute sum by $∥ a ∥_{2} ∥ b ∥_{2}$ (Square-summable families on an arbitrary index set and the space $ℓ^{2} (I)$ ).

[A5]

For an absolutely summable family the finite-subset net limit equals the limit of the symmetric partial sums, because the symmetric index sets are cofinal among the finite subsets of $Z$ (Square-summable families on an arbitrary index set and the space $ℓ^{2} (I)$ , Every nonempty finite set of reals has a maximum and a minimum).

Proof

technique · direct

Given: Countable Choice and $f, g \in L^{2} (T; C)$ .

1.1

By [A1] and [A2] the coefficient map $h \mapsto (\hat{h} (k))_{k \in Z}$ is a linear bijection of $L^{2} (T; C)$ onto $ℓ^{2} (Z, C)$ preserving norms and inner products; therefore $∥ f ∥_{2}^{2} = ∥ (\hat{f} (k)) ∥_{2}^{2} = \sum_{k \in Z} ∣ \hat{f} (k) ∣^{2}$ and $⟨ f, g ⟩ = \sum_{k \in Z} \hat{f} (k) \overline{\hat{g} (k)}$ .

A1A2A3

2.1

The bilinear series converges absolutely: by the finite Cauchy–Schwarz inequality of [A4], every finite subsum of $∣ \hat{f} (k) \overline{\hat{g} (k)} ∣$ is at most $∥ (\hat{f} (k)) ∥_{2} ∥ (\hat{g} (k)) ∥_{2} = ∥ f ∥_{2} ∥ g ∥_{2}$ , and taking the supremum over finite subsets gives the displayed bound; in particular the family is summable in the sense of the finite-subset net.

step 1.1A4

3.1

The sum equals the limit of the symmetric partial sums: since the family is absolutely summable, its finite-subset net converges, and the symmetric index sets are cofinal among finite subsets by [A5], so the symmetric partial sums converge to the same value.

step 2.1A5

4.1

The norm identity and the sesquilinear identity of step 1.1, together with the absolute convergence and cofinality statements of steps 2.1 and 3.1, are exactly the assertions of the theorem.

step 1.1step 2.1step 3.1∎

Depends on

Used by

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Sources

Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §2.5, Theorem 2.17 and equation (2.48) (standard reference, not scraped)
Theo Bühler and Dietmar Salamon, Functional Analysis — Example 2.66, pp.87–88 (standard reference, not scraped)