Theorems · Theorem · real analysis

HasFTaylorSeriesUpToOn.comp_continuousAffineMap

∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [inst : NontriviallyNormedField 𝕜]
  [inst_1 : NormedAddCommGroup E] [inst_2 : NormedSpace 𝕜 E] [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F]
  [inst_5 : NormedAddCommGroup G] [inst_6 : NormedSpace 𝕜 G] {s : Set E} {f : E → F} {n : WithTop ℕ∞}
  {p : E → FormalMultilinearSeries 𝕜 E F},
  HasFTaylorSeriesUpToOn n f p s →
    ∀ (g : G →ᴬ[𝕜] E),
      HasFTaylorSeriesUpToOn n (f ∘ ⇑g) (fun x k => (p (g x) k).compContinuousLinearMap fun x => g.contLinear)
        (⇑g ⁻¹' s)

If f admits a Taylor series p in a set s, and g is affine, then f ∘ g admits a Taylor series in g ⁻¹' s, whose k-th term at x is given by p (g x) k (g.contLinear v₁, ..., g.contLinear vₖ) .

Defined in

Mathlib.Analysis.Calculus.ContDiff.Basic

Cited by

1 results in Mathlib

Foundations

Depth 174 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace