Theorems · Theorem · functional analysis

HasFDerivAt.continuousAlternatingMapCompContinuousLinearMap

∀ {𝕜 : Type u_1} {ι : Type u_2} {E : Type u_3} {F : Type u_4} {G : Type u_5} {H : Type u_6}
  [inst : NontriviallyNormedField 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : NormedSpace 𝕜 E]
  [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] [inst_5 : NormedAddCommGroup G] [inst_6 : NormedSpace 𝕜 G]
  [inst_7 : NormedAddCommGroup H] [inst_8 : NormedSpace 𝕜 H] {f : E → G [⋀^ι]→L[𝕜] H} {f' : E →L[𝕜] G [⋀^ι]→L[𝕜] H}
  {g : E → F →L[𝕜] G} {g' : E →L[𝕜] F →L[𝕜] G} {x : E} [inst_9 : Fintype ι] [inst_10 : DecidableEq ι],
  HasFDerivAt f f' x →
    HasFDerivAt g g' x →
      HasFDerivAt (fun x => (f x).compContinuousLinearMap (g x))
        (ContinuousAlternatingMap.compContinuousLinearMapCLM (g x) ∘SL f' +
          (f x).fderivCompContinuousLinearMap (g x) ∘SL g')
        x

Defined in

Mathlib.Analysis.Calculus.FDeriv.ContinuousAlternatingMap

Cited by

2 results in Mathlib

Foundations

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

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceFintypeDecidableEq