DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

Fredholm maps and regular values on countable-base Banach manifolds

Definition

A bounded linear map $T : E \to F$ of Banach spaces is Fredholm when its range is closed and both $\ker T$ and $F / im T$ are finite-dimensional; its index is $\dim \ker T - \dim (F / im T)$ . A $C^{h}$ map $P : X \to Y$ between countable-base Banach manifolds is Fredholm of index $m$ if every $d P_{x}$ is Fredholm of index $m$ . A value $y$ is regular if each $x \in P^{- 1} (y)$ has surjective derivative (vacuously if the fibre is empty).

Used by

Dependency tree · 0 levels

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

Sources

Stephen Smale, An Infinite Dimensional Version of Sard's Theorem, §1 (standard reference, not scraped)