Theorems · Theorem · real analysis

Convex.isLittleO_pow_succ

∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {𝕜 : Type u_3} {G : Type u_4}
  [inst_2 : NontriviallyNormedField 𝕜] [IsRCLikeNormedField 𝕜] [inst_4 : NormedSpace 𝕜 E]
  [inst_5 : NormedAddCommGroup G] [inst_6 : NormedSpace 𝕜 G] {f : E → G} {s : Set E} {f' : E → E →L[𝕜] G} {x₀ : E}
  {n : ℕ},
  Convex ℝ s →
    x₀ ∈ s →
      (∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) →
        (f' =o[nhdsWithin x₀ s] fun x => ‖x - x₀‖ ^ n) →
          (fun x => f x - f x₀) =o[nhdsWithin x₀ s] fun x => ‖x - x₀‖ ^ (n + 1)

Defined in

Mathlib.Analysis.Calculus.MeanValue

Cited by

1 results in Mathlib

Foundations

Depth 192 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNontriviallyNormedFieldIsRCLikeNormedFieldNormedSpaceNormedAddCommGroupNormedSpace