Theorems · Theorem · global analysis

fderiv_inner_apply

∀ (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}
  {x : G},
  DifferentiableAt ℝ f x →
    DifferentiableAt ℝ g x →
      ∀ (y : G),
        (fderiv ℝ (fun t => inner 𝕜 (f t) (g t)) x) y =
          inner 𝕜 (f x) ((fderiv ℝ g x) y) + inner 𝕜 ((fderiv ℝ f x) y) (g x)

Defined in

Mathlib.Analysis.InnerProductSpace.Calculus

Cited by

0 results in Mathlib

Foundations

Depth 183 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceNormedSpaceNormedAddCommGroupNormedSpace