Theorems · Theorem · global analysis

ContDiffWithinAt.restrictScalars_iteratedFDerivWithin_eventuallyEq

∀ {𝕜 : Type u_1} {𝕜' : Type u_2} [inst : NontriviallyNormedField 𝕜] [inst_1 : NontriviallyNormedField 𝕜']
  [inst_2 : NormedAlgebra 𝕜 𝕜'] {E : Type u_3} [inst_3 : NormedAddCommGroup E] [inst_4 : NormedSpace 𝕜 E]
  [inst_5 : NormedSpace 𝕜' E] [inst_6 : IsScalarTower 𝕜 𝕜' E] {F : Type u_4} [inst_7 : NormedAddCommGroup F]
  [inst_8 : NormedSpace 𝕜 F] [inst_9 : NormedSpace 𝕜' F] [inst_10 : IsScalarTower 𝕜 𝕜' F] {x : E} {f : E → F} {n : ℕ}
  {s : Set E},
  ContDiffWithinAt 𝕜' (↑n) f s x →
    UniqueDiffOn 𝕜 s →
      x ∈ s →
        ContinuousMultilinearMap.restrictScalars 𝕜 ∘ iteratedFDerivWithin 𝕜' n f s =ᶠ[nhdsWithin x s]
          iteratedFDerivWithin 𝕜 n f s

If f is n times continuously differentiable at x within s, then the nth iterated Fréchet derivative within s with respect to 𝕜 equals scalar restriction of the nth iterated Fréchet derivative within s with respect to 𝕜'.

Defined in

Mathlib.Analysis.Calculus.ContDiff.RestrictScalars

Cited by

1 results in Mathlib

Foundations

Depth 187 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNontriviallyNormedFieldNormedAlgebraNormedAddCommGroupNormedSpaceNormedSpaceIsScalarTowerNormedAddCommGroupNormedSpaceNormedSpaceIsScalarTower