Theorems · Theorem · global analysis

HasStrictFDerivAt.inner

∀ (𝕜 : Type u_1) {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
  [inst_3 : NormedSpace ℝ E] {G : Type u_4} [inst_4 : NormedAddCommGroup G] [inst_5 : NormedSpace ℝ G] {f g : G → E}
  {f' g' : G →L[ℝ] E} {x : G},
  HasStrictFDerivAt f f' x →
    HasStrictFDerivAt g g' x →
      HasStrictFDerivAt (fun t => inner 𝕜 (f t) (g t)) (fderivInnerCLM 𝕜 (f x, g x) ∘SL f'.prod g') x

Defined in

Mathlib.Analysis.InnerProductSpace.Calculus

Cited by

2 results in Mathlib

Foundations

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

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceNormedSpaceNormedAddCommGroupNormedSpace