Theorems · Theorem · global analysis

ContinuousLinearMap.hasStrictFDerivAt_of_bilinear

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {F : Type u_3} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {G : Type u_4}
  [inst_5 : NormedAddCommGroup G] [inst_6 : NormedSpace 𝕜 G] {G' : Type u_5} [inst_7 : NormedAddCommGroup G']
  [inst_8 : NormedSpace 𝕜 G'] (B : E →L[𝕜] F →L[𝕜] G) {f : G' → E} {g : G' → F} {f' : G' →L[𝕜] E} {g' : G' →L[𝕜] F}
  {x : G'},
  HasStrictFDerivAt f f' x →
    HasStrictFDerivAt g g' x →
      HasStrictFDerivAt (fun y => (B (f y)) (g y))
        (((ContinuousLinearMap.precompR G' B) (f x)) g' + ((ContinuousLinearMap.precompL G' B) f') (g x)) x

Defined in

Mathlib.Analysis.Calculus.FDeriv.Bilinear

Cited by

1 results in Mathlib

Foundations

Depth 181 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace