Theorems · Theorem · real analysis

HasFTaylorSeriesUpToOn.continuousLinearMap_comp

∀ {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}
  {p : E → FormalMultilinearSeries 𝕜 E F} {n : WithTop ℕ∞} (g : F →L[𝕜] G),
  HasFTaylorSeriesUpToOn n f p s →
    HasFTaylorSeriesUpToOn n (⇑g ∘ f) (fun x k => g.compContinuousMultilinearMap (p x k)) s

If f admits a Taylor series p in a set s, and g is linear, then g ∘ f admits a Taylor series whose k-th term is given by g ∘ (p k).

Defined in

Mathlib.Analysis.Calculus.ContDiff.Basic

Cited by

5 results in Mathlib

Foundations

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

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace