Theorems · Theorem · functional analysis

Asymptotics.IsBigOTVS.of_egauge_le_mul

∀ {α : Type u_1} {𝕜 : Type u_3} {E : Type u_4} {F : Type u_5} [inst : NontriviallyNormedField 𝕜]
  [inst_1 : AddCommGroup E] [inst_2 : TopologicalSpace E] [inst_3 : Module 𝕜 E] [inst_4 : AddCommGroup F]
  [inst_5 : TopologicalSpace F] [inst_6 : Module 𝕜 F] {l : Filter α} {f : α → E} {g : α → F} [ContinuousConstSMul 𝕜 F]
  {ι : Sort u_7} {p : ι → Prop} {U : ι → Set E},
  (nhds 0).HasBasis p U →
    (∀ (i : ι), p i → ∃ C, ∃ V ∈ nhds 0, (fun x => egauge 𝕜 (U i) (f x)) ≤ᶠ[l] fun x => ↑C * egauge 𝕜 V (g x)) →
      f =O[𝕜; l] g

The definition of IsBigOTVS says that for each neighborhood U of the origin in the codomain of f, there exists a neighborhood V of the origin in the codomain of g such that egauge 𝕜 U (f x) ≤ egauge 𝕜 V (g x) eventually along l. This lemma shows that it suffices to make this inequality work up to a constant multiplier.

Defined in

Mathlib.Analysis.Asymptotics.TVS

Cited by

0 results in Mathlib

Foundations

Depth 162 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldAddCommGroupTopologicalSpaceModuleAddCommGroupTopologicalSpaceModuleContinuousConstSMul