Theorems · Theorem · global analysis

HasFDerivWithinAt.continuousMultilinearMap_apply

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {F : Type v} [inst_1 : NormedAddCommGroup F]
  [inst_2 : NormedSpace 𝕜 F] {ι : Type u_2} {E : ι → Type u_3} [inst_3 : (i : ι) → NormedAddCommGroup (E i)]
  [inst_4 : (i : ι) → NormedSpace 𝕜 (E i)] [inst_5 : Fintype ι] {G : Type u_4} [inst_6 : NormedAddCommGroup G]
  [inst_7 : NormedSpace 𝕜 G] [inst_8 : DecidableEq ι] {s : Set G} {x : G} {f : G → ContinuousMultilinearMap 𝕜 E F}
  {g : (i : ι) → G → E i} {f' : G →L[𝕜] ContinuousMultilinearMap 𝕜 E F} {g' : (i : ι) → G →L[𝕜] E i},
  HasFDerivWithinAt f f' s x →
    (∀ (i : ι), HasFDerivWithinAt (g i) (g' i) s x) →
      HasFDerivWithinAt (fun x => (f x) fun x_1 => g x_1 x)
        ((ContinuousMultilinearMap.apply 𝕜 E F fun x_1 => g x_1 x) ∘SL f' +
          ∑ i, (f x).toContinuousLinearMap (fun x_1 => g x_1 x) i ∘SL g' i)
        s x

Defined in

Mathlib.Analysis.Calculus.FDeriv.Analytic

Cited by

4 results in Mathlib

Foundations

Depth 189 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceFintypeNormedAddCommGroupNormedSpaceDecidableEq