Theorems · Theorem · functional analysis

Filter.Tendsto.zero_smul_isBoundedUnder_le

∀ {𝕜 : Type u_1} {E : Type u_3} {α : Type u_5} [inst : NormedField 𝕜] [inst_1 : SeminormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {f : α → 𝕜} {g : α → E} {l : Filter α},
  Filter.Tendsto f l (nhds 0) →
    Filter.IsBoundedUnder (fun x1 x2 => x1 ≤ x2) l (norm ∘ g) → Filter.Tendsto (fun x => f x • g x) l (nhds 0)

Defined in

Mathlib.Analysis.Normed.Module.Basic

Cited by

1 results in Mathlib

Foundations

Depth 121 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedFieldSeminormedAddCommGroupNormedSpace