TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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The mean-zero L² space is a Hilbert space

Statement

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). For any probability measure $P$ , the space $L_{0}^{2} (P)$ of Mean-zero square-integrable functions is a closed real linear subspace of real $L^{2} (P)$ with pairing $⟨ f, g ⟩_{P} : = \int f g d P$ ; consequently it is a real Hilbert space (Hilbert space). The only choice use is the Countable Choice assumption already required by the general $L^{2} (P)$ completeness supplier.

Facts & Assumptions

Given: Countable Choice and a probability measure $P$ on $(X, A)$ .

[A1]

Countable Choice is the stated selection principle (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

Under Countable Choice, real $L^{2} (P)$ with pairing $⟨ f, g ⟩ = \int f g d P$ is a complete real inner-product space ( $L^{2}$ with the integral pairing is a Hilbert space).

[F2]

Cauchy–Schwarz in an inner-product space gives $∣ ⟨ f, g ⟩ ∣ \leq ∥ f ∥ ∥ g ∥$ (Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs).

[F3]

$L_{0}^{2} (P) = {f \in L^{2} (P) : P f = 0}$ , with $P f$ well-defined on equivalence classes (Mean-zero square-integrable functions).

Proof

technique · Identify the mean-zero space as the kernel of the continuous expectation functional

1.1F1F2F3algebra

The constant function $1$ belongs to $L^{2} (P)$ and has norm $1$ , since $P (X) = 1$ . For each $f \in L^{2} (P)$ , the pairing in [F1] satisfies $P f = ⟨ f, 1 ⟩_{P}$ , so [F2] gives $∣ P f ∣ \leq ∥ f ∥_{2}$ . Thus $f \mapsto P f$ is a bounded linear functional, and its kernel is exactly $L_{0}^{2} (P)$ by [F3].

2.1step 1.1F1F3

The kernel is linear. If $f_{n} \in L_{0}^{2} (P)$ and $f_{n} \to f$ in $L^{2} (P)$ , then $∣ P f ∣ = ∣ P (f - f_{n}) ∣ \leq ∥ f - f_{n} ∥_{2} \to 0$ by step 1.1 and linearity of the pairing [F1], so $f \in L_{0}^{2} (P)$ . Hence the kernel is closed.

3.1A1F1step 2.1∎

Every Cauchy sequence in $L_{0}^{2} (P)$ is Cauchy in $L^{2} (P)$ , which is complete by [F1] under the Countable Choice assumption [A1]; its $L^{2} (P)$ limit lies in $L_{0}^{2} (P)$ by step 2.1. The restricted inner product is therefore complete, proving the asserted Hilbert-space property.

Depends on

Used by

Closed tangent space Definition
Orthogonal complement of a nuisance tangent space Definition
Pathwise differentiability and parameter influence curves Definition
A boundary score cone differs from its tangent space Example
Bounded mean-zero scores generate the unrestricted tangent space Example
A tangent space is a closed linear subspace Theorem
Efficient score gives the canonical gradient Theorem

Dependency tree · two levels

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Sources

A. W. van der Vaart, Semiparametric Statistics, Lecture 1 §1.2 (standard reference, not scraped)
Edward H. Kennedy, Semiparametric theory, §4 (standard reference, not scraped)