Theorems · Theorem · real analysis

norm_iteratedFDerivWithin_prod_le

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type uE} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {s : Set E} {ι : Type u_2} {A' : Type u_4} [inst_3 : NormedCommRing A']
  [inst_4 : NormedAlgebra 𝕜 A'] [inst_5 : DecidableEq ι] [NormOneClass A'] {u : Finset ι} {f : ι → E → A'}
  {N : WithTop ℕ∞},
  (∀ i ∈ u, ContDiffOn 𝕜 N (f i) s) →
    UniqueDiffOn 𝕜 s →
      ∀ {x : E},
        x ∈ s →
          ∀ {n : ℕ},
            ↑n ≤ N →
              ‖iteratedFDerivWithin 𝕜 n (fun x => ∏ j ∈ u, f j x) s x‖ ≤
                ∑ p ∈ u.sym n, ↑(↑p).countPerms * ∏ j ∈ u, ‖iteratedFDerivWithin 𝕜 (Multiset.count j ↑p) (f j) s x‖

Defined in

Mathlib.Analysis.Calculus.ContDiff.Bounds

Cited by

1 results in Mathlib

Foundations

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

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedCommRingNormedAlgebraDecidableEqNormOneClass