DefinitionDefinition: AI-adaptedProof: 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.

Weighted trajectory space for a local center-manifold construction

Definition

Let $E$ be a finite-dimensional real normed space and let $η > 0$ . Define

$C_{η} (E) : = {u \in C (R, E) : ∥ u ∥_{η} : = \sup_{t \in R} e^{- η ∣ t ∣} ∥ u (t) ∥ < \infty} .$

The norm is $∥ u ∥_{η}$ . It permits at most the fixed exponential growth rate $η$ in either time direction. It does not say that a trajectory is bounded.

Depends on

A norm on a real vector space, the induced metric, and the dictionary with the metric axioms

Used by

Dependency tree · two levels

13 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

J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Springer-Verlag, 1976 (standard reference, not scraped)
Yuri A. Kuznetsov, Elements of Applied Bifurcation Theory, 2nd ed., Springer (standard reference, not scraped)