Theorems · Theorem · general topology

UniformContinuousOn.comp_tendstoUniformly

∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {ι : Type u_4} [inst : UniformSpace β] [inst_1 : UniformSpace γ]
  {p : Filter ι} {s : Set β} {F : ι → α → β} {f : α → β} {g : β → γ},
  (∀ (i : ι) (x : α), F i x ∈ s) →
    (∀ (x : α), f x ∈ s) →
      UniformContinuousOn g s → TendstoUniformly F f p → TendstoUniformly (fun i x => g (F i x)) (fun x => g (f x)) p

Composing on the left by a uniformly continuous function preserves uniform convergence

Defined in

Mathlib.Topology.UniformSpace.UniformConvergenceTopology

Cited by

1 results in Mathlib

Foundations

Depth 75 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

UniformSpaceUniformSpace