Theorems · Theorem · global analysis

contMDiffWithinAt_vectorSpace_iff_contDiffWithinAt

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {n : WithTop ℕ∞} {E : Type u_2} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {V : (x : E) → TangentSpace (modelWithCornersSelf 𝕜 E) x} {s : Set E} {x : E},
  ContMDiffWithinAt (modelWithCornersSelf 𝕜 E) (modelWithCornersSelf 𝕜 E).tangent n (fun x => ⟨x, V x⟩) s x ↔
    ContDiffWithinAt 𝕜 n V s x

A vector field on a vector space is C^n in the manifold sense iff it is C^n in the vector space sense.

Defined in

Mathlib.Geometry.Manifold.VectorBundle.Tangent

Cited by

2 results in Mathlib

Foundations

Depth 221 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpace