DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generated

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.
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Orthogonal complement of a nuisance tangent space

Definition

Assume Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )), so that $L_{0}^{2} (P)$ is the real Hilbert space of The mean-zero L² space is a Hilbert space. The orthogonal complement of the nuisance tangent space within $L_{0}^{2} (P)$ is

$T_{η} (P)^{⊥} : = {h \in L_{0}^{2} (P) : ⟨ h, g ⟩_{P} = 0 for every g \in T_{η} (P)}, ⟨ h, g ⟩_{P} : = P (h g) .$

This is the ordinary orthogonal complement from Orthogonality and the orthogonal complement applied in the mean-zero Hilbert space, not in all of $L^{2} (P)$ . It is a closed linear subspace by Orthogonal complements are closed.

Depends on

Used by

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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)