Theorems · Theorem · functional analysis

Asymptotics.isBigOTVS_prodMk_left

∀ {α : Type u_1} {𝕜 : Type u_3} {E : Type u_4} {F : Type u_5} {G : Type u_6} [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] [inst_7 : AddCommGroup G] [inst_8 : TopologicalSpace G]
  [inst_9 : Module 𝕜 G] {l : Filter α} {f : α → E} {g : α → F} [ContinuousSMul 𝕜 E] [ContinuousSMul 𝕜 F] {k : α → G},
  (fun x => (f x, g x)) =O[𝕜; l] k ↔ f =O[𝕜; l] k ∧ g =O[𝕜; l] k

Defined in

Mathlib.Analysis.Asymptotics.TVS

Cited by

0 results in Mathlib

Foundations

Depth 164 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldAddCommGroupTopologicalSpaceModuleAddCommGroupTopologicalSpaceModuleAddCommGroupTopologicalSpaceModuleContinuousSMulContinuousSMul