Theorems · Theorem · real analysis

norm_iteratedFDerivWithin_clm_apply

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type uE} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {F : Type uF} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {G : Type uG}
  [inst_5 : NormedAddCommGroup G] [inst_6 : NormedSpace 𝕜 G] {f : E → F →L[𝕜] G} {g : E → F} {s : Set E} {x : E}
  {N : WithTop ℕ∞} {n : ℕ},
  ContDiffOn 𝕜 N f s →
    ContDiffOn 𝕜 N g s →
      UniqueDiffOn 𝕜 s →
        x ∈ s →
          ↑n ≤ N →
            ‖iteratedFDerivWithin 𝕜 n (fun y => (f y) (g y)) s x‖ ≤
              ∑ i ∈ Finset.range (n + 1),
                ↑(n.choose i) * ‖iteratedFDerivWithin 𝕜 i f s x‖ * ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖

Defined in

Mathlib.Analysis.Calculus.ContDiff.Bounds

Cited by

1 results in Mathlib

Foundations

Depth 203 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace