Theorems · Theorem · functional analysis

ContinuousMultilinearMap.hasStrictFDerivAt_compContinuousLinearMap

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

Defined in

Mathlib.Analysis.Calculus.FDeriv.ContinuousMultilinearMap

Cited by

4 results in Mathlib

Foundations

Depth 188 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceFintypeDecidableEq