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.

Hadamard differentiability tangentially to a subspace

Definition

Let $E, F$ be normed spaces, let $D \subseteq E$ , let $θ \in D$ , and let $E_{0} \subseteq E$ be a linear subspace. A map $Ψ : D \to F$ is Hadamard differentiable at $θ$ , tangentially to $E_{0}$ if there is a continuous linear map $Ψ_{θ}^{'} : E_{0} \to F$ such that, for every sequence $t_{m} > 0$ with $t_{m} \to 0$ , and every sequence $h_{m} \to h \in E_{0}$ for which $θ + t_{m} h_{m} \in D$ ,

$\frac{Ψ (θ + t_{m} h_{m}) - Ψ (θ)}{t_{m}} ⟶ Ψ_{θ}^{'} (h) in F .$

The subspace is part of the hypothesis: directions outside $E_{0}$ are not covered by this definition.

Used by

Conditional bootstrap delta method for Hadamard differentiable maps Theorem

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

D. Pilavakis, E. Paparoditis, and T. Sapatinas, Moving block and tapered block bootstrap for functional time series with an application to the K-sample mean problem, Bernoulli 25 (2019), Remark 2.1 (standard reference, not scraped)