DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Closed 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 established by The mean-zero L² space is a Hilbert space. For a tangent set $\dot{M} (P) \subseteq L_{0}^{2} (P)$ , its linear span is the set of all finite real linear combinations of elements of $\dot{M} (P)$ ; the span of the empty set is ${0}$ . Its tangent space $T (P)$ is the closure of that span in the $L^{2} (P)$ norm. The closure consists of limits of sequences from the span, equivalently of points every positive-radius norm ball around which meets the span.

The tangent set itself is retained separately and may be a cone rather than a vector space.

Depends on

Used by

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

Dependency tree · two levels

20 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)