Theorems · Theorem · several complex variables

HasFPowerSeriesWithinAt.isBigO_image_sub_norm_mul_norm_sub

∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [inst : NontriviallyNormedField 𝕜] [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {f : E → F}
  {p : FormalMultilinearSeries 𝕜 E F} {s : Set E} {x : E},
  HasFPowerSeriesWithinAt f p s x →
    (fun y => f y.1 - f y.2 - (p 1) fun x => y.1 - y.2) =O[nhdsWithin (x, x) (insert x s ×ˢ insert x s)] fun y =>
      ‖y - (x, x)‖ * ‖y.1 - y.2‖

If f has formal power series ∑ n, pₙ at x within a set s, then f y - f z - p 1 (fun _ ↦ y - z) = O(‖(y, z) - (x, x)‖ * ‖y - z‖) as (y, z) → (x, x) within s × s.

Defined in

Mathlib.Analysis.Analytic.Basic

Cited by

2 results in Mathlib

Foundations

Depth 176 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace