Theorems · Theorem · functional analysis

fderivWithin_continuousMultilinearMapCompContinuousLinearMap

∀ {𝕜 : Type u_1} {ι : Type u_2} {E : Type u_3} {F : ι → Type u_4} {G : ι → Type u_5} {H : Type u_6}
  [inst : NontriviallyNormedField 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : NormedSpace 𝕜 E]
  [inst_3 : (i : ι) → NormedAddCommGroup (F i)] [inst_4 : (i : ι) → NormedSpace 𝕜 (F i)]
  [inst_5 : (i : ι) → NormedAddCommGroup (G i)] [inst_6 : (i : ι) → NormedSpace 𝕜 (G i)] [inst_7 : NormedAddCommGroup H]
  [inst_8 : NormedSpace 𝕜 H] {f : E → ContinuousMultilinearMap 𝕜 G H} {g : (i : ι) → E → F i →L[𝕜] G i} {s : Set E}
  {x : E} [inst_9 : Fintype ι] [inst_10 : DecidableEq ι],
  DifferentiableWithinAt 𝕜 f s x →
    (∀ (i : ι), DifferentiableWithinAt 𝕜 (g i) s x) →
      UniqueDiffWithinAt 𝕜 s x →
        fderivWithin 𝕜 (fun x => (f x).compContinuousLinearMap fun x_1 => g x_1 x) s x =
          (ContinuousMultilinearMap.compContinuousLinearMapL fun x_1 => g x_1 x) ∘SL fderivWithin 𝕜 f s x +
            ((f x).fderivCompContinuousLinearMap fun x_1 => g x_1 x) ∘SL
              ContinuousLinearMap.pi fun i => fderivWithin 𝕜 (g i) s x

Defined in

Mathlib.Analysis.Calculus.FDeriv.ContinuousMultilinearMap

Cited by

0 results in Mathlib

Foundations

Depth 190 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceFintypeDecidableEq