Theorems · Theorem · functional analysis

ContinuousAlternatingMap.hasStrictFDerivAt_compContinuousLinearMap

∀ {𝕜 : Type u_1} {ι : Type u_2} {F : Type u_4} {G : Type u_5} {H : Type u_6} [inst : NontriviallyNormedField 𝕜]
  [inst_1 : NormedAddCommGroup F] [inst_2 : NormedSpace 𝕜 F] [inst_3 : NormedAddCommGroup G] [inst_4 : NormedSpace 𝕜 G]
  [inst_5 : NormedAddCommGroup H] [inst_6 : NormedSpace 𝕜 H] [inst_7 : Fintype ι] [inst_8 : DecidableEq ι]
  (fg : G [⋀^ι]→L[𝕜] H × (F →L[𝕜] G)),
  HasStrictFDerivAt (fun fg => fg.1.compContinuousLinearMap fg.2)
    (ContinuousAlternatingMap.compContinuousLinearMapCLM fg.2 ∘SL
        ContinuousLinearMap.fst 𝕜 (G [⋀^ι]→L[𝕜] H) (F →L[𝕜] G) +
      fg.1.fderivCompContinuousLinearMap fg.2 ∘SL ContinuousLinearMap.snd 𝕜 (G [⋀^ι]→L[𝕜] H) (F →L[𝕜] G))
    fg

Defined in

Mathlib.Analysis.Calculus.FDeriv.ContinuousAlternatingMap

Cited by

3 results in Mathlib

Foundations

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

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceFintypeDecidableEq