Theorems · Theorem · real analysis

HasFTaylorSeriesUpToOn.prodMk

∀ {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∞},
  HasFTaylorSeriesUpToOn n f p s →
    ∀ {g : E → G} {q : E → FormalMultilinearSeries 𝕜 E G},
      HasFTaylorSeriesUpToOn n g q s →
        HasFTaylorSeriesUpToOn n (fun y => (f y, g y)) (fun y k => (p y k).prod (q y k)) s

If two functions f and g admit Taylor series p and q in a set s, then the Cartesian product of f and g admits the Cartesian product of p and q as a Taylor series.

Defined in

Mathlib.Analysis.Calculus.ContDiff.Basic

Cited by

2 results in Mathlib

Foundations

Depth 175 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace