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A tangent space is a closed linear subspace

Statement

Assume Countable Choice. Let $\dot{M} (P)$ be any tangent set in the real Hilbert space $L_{0}^{2} (P)$ . Its tangent space $T (P)$ , defined as the norm closure of its finite real linear span (Closed tangent space), is a closed linear subspace of $L_{0}^{2} (P)$ (Linear subspace of a vector space).

Facts & Assumptions

Given: Countable Choice, a probability law $P$ , and a tangent set $\dot{M} (P) \subseteq L_{0}^{2} (P)$ .

[A1]

Countable Choice is the selection principle required by the general $L^{2} (P)$ completeness supplier (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

The tangent space is the norm closure of all finite real linear combinations of tangent-set scores (Closed tangent space).

[F2]

Under Countable Choice, $L_{0}^{2} (P)$ is a complete real inner-product space (The mean-zero L² space is a Hilbert space).

[F3]

A linear subspace is a subset containing the zero vector and closed under addition and real scalar multiplication (Linear subspace of a vector space).

Proof

technique · Prove the closure is linear, closed, and complete in the inherited norm

1.1F1F3algebra

Let $V$ be the finite real span in [F1]. By collecting coefficients of each score, the sum of two finite linear combinations is another finite linear combination; scalar multiples have the same form, and $0$ is the empty sum. Thus $V$ is a linear subspace by [F3].

1.2F1algebra

If $x \notin \overline{V}$ , some $r > 0$ has $B (x, r) \cap V = \emptyset$ . For any $y$ with $∥ y - x ∥_{2} < r / 2$ , the triangle inequality gives $B (y, r / 2) \subset B (x, r)$ , so $y \notin \overline{V}$ . Hence the complement of $\overline{V}$ is open and $\overline{V}$ is closed.

2.1step 1.1F1algebra

Write $\overline{V} = T (P)$ . If $x, y \in \overline{V}$ and $ε > 0$ , choose $u, v \in V$ with $∥ x - u ∥_{2} < ε / 2$ and $∥ y - v ∥_{2} < ε / 2$ . Then $u + v \in V$ and $∥ (x + y) - (u + v) ∥_{2} < ε$ , so $x + y \in \overline{V}$ . For $a \in R$ , if $a = 0$ then $a x = 0 \in V$ ; if $a \neq 0$ , choose $u \in V$ with $∥ x - u ∥_{2} < ε / ∣ a ∣$ , giving $a u \in V$ and $∥ a x - a u ∥_{2} < ε$ . Thus $\overline{V}$ is linear.

3.1F2A1step 1.2step 2.1∎

A Cauchy sequence in $\overline{V}$ converges in $L_{0}^{2} (P)$ by [F2] under [A1]. Since $\overline{V}$ is closed by step 1.2, that limit lies in $\overline{V}$ . Therefore the tangent space is complete as well as a closed linear subspace, as asserted.

Depends on

Used by

Efficient score and efficient information Definition
Nuisance tangent space Definition
Pathwise differentiability and parameter influence curves Definition
Efficient score gives the canonical gradient Theorem

Dependency tree · two levels

18 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

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)