Theorems · Theorem · functional analysis

Asymptotics.IsLittleOTVS.exists_eventuallyLE_mul_ennreal

∀ {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] {f : α → E} {g : α → F} {l : Filter α},
  f =o[𝕜; l] g →
    ∀ {U : Set E},
      U ∈ nhds 0 →
        ∃ V ∈ nhds 0, ∀ (ε : ENNReal), ε ≠ 0 → (fun x => egauge 𝕜 U (f x)) ≤ᶠ[l] fun x => ε * egauge 𝕜 V (g x)

A version of IsLittleOTVS.exists_eventuallyLE_mul where ε is quantified over ℝ≥0∞ instead of ℝ≥0.

Defined in

Mathlib.Analysis.Asymptotics.TVS

Cited by

0 results in Mathlib

Foundations

Depth 159 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldAddCommGroupTopologicalSpaceModuleAddCommGroupTopologicalSpaceModule