Theorems · Theorem · real analysis

norm_iteratedFDeriv_smul_le

∀ {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] {𝕜' : Type u_2}
  [inst_5 : NormedField 𝕜'] [inst_6 : NormedAlgebra 𝕜 𝕜'] [inst_7 : NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F]
  {f : E → 𝕜'} {g : E → F} {N : WithTop ℕ∞},
  ContDiff 𝕜 N f →
    ContDiff 𝕜 N g →
      ∀ (x : E) {n : ℕ},
        ↑n ≤ N →
          ‖iteratedFDeriv 𝕜 n (fun y => f y • g y) x‖ ≤
            ∑ i ∈ Finset.range (n + 1), ↑(n.choose i) * ‖iteratedFDeriv 𝕜 i f x‖ * ‖iteratedFDeriv 𝕜 (n - i) g x‖

Defined in

Mathlib.Analysis.Calculus.ContDiff.Bounds

Cited by

0 results in Mathlib

Foundations

Depth 204 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedFieldNormedAlgebraNormedSpaceIsScalarTower