Theorems · Theorem · functional analysis

ContinuousMultilinearMap.fderivCompContinuousLinearMap_apply

∀ {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {E₁ : ι → Type wE₁} {G : Type wG} [inst : NontriviallyNormedField 𝕜]
  [inst_1 : (i : ι) → SeminormedAddCommGroup (E i)] [inst_2 : (i : ι) → NormedSpace 𝕜 (E i)]
  [inst_3 : (i : ι) → SeminormedAddCommGroup (E₁ i)] [inst_4 : (i : ι) → NormedSpace 𝕜 (E₁ i)]
  [inst_5 : SeminormedAddCommGroup G] [inst_6 : NormedSpace 𝕜 G] [inst_7 : Fintype ι] [inst_8 : DecidableEq ι]
  (f : ContinuousMultilinearMap 𝕜 E₁ G) (g dg : (i : ι) → E i →L[𝕜] E₁ i) (v : (i : ι) → E i),
  ((f.fderivCompContinuousLinearMap g) dg) v = ∑ i, f fun j => (Function.update g i (dg i) j) (v j)

Defined in

Mathlib.Analysis.Normed.Module.Multilinear.Basic

Cited by

1 results in Mathlib

Foundations

Depth 179 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldSeminormedAddCommGroupNormedSpaceSeminormedAddCommGroupNormedSpaceSeminormedAddCommGroupNormedSpaceFintypeDecidableEq