Theorems · Theorem · functional analysis

Asymptotics.isBigOTVS_iff

∀ (𝕜 : Type u_1) {α : Type u_2} {E : Type u_3} {F : Type u_4} [inst : ENorm 𝕜] [inst_1 : TopologicalSpace E]
  [inst_2 : TopologicalSpace F] [inst_3 : Zero E] [inst_4 : Zero F] [inst_5 : SMul 𝕜 E] [inst_6 : SMul 𝕜 F]
  (l : Filter α) (f : α → E) (g : α → F),
  f =O[𝕜; l] g ↔ ∀ U ∈ nhds 0, ∃ V ∈ nhds 0, (fun x => egauge 𝕜 U (f x)) ≤ᶠ[l] fun x => egauge 𝕜 V (g x)

Defined in

Mathlib.Analysis.Asymptotics.TVS

Cited by

3 results in Mathlib

Foundations

Depth 130 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

ENormTopologicalSpaceTopologicalSpaceZeroZeroSMulSMul