Theorems · Theorem · sequences and series

hasProdLocallyUniformlyOn_of_of_forall_exists_nhds

∀ {α : Type u_1} {β : Type u_2} {ι : Type u_3} [inst : CommMonoid α] {f : ι → β → α} {g : β → α} {s : Set β}
  [inst_1 : UniformSpace α] [inst_2 : TopologicalSpace β],
  (∀ x ∈ s, ∃ t ∈ nhdsWithin x s, HasProdUniformlyOn f g t) → HasProdLocallyUniformlyOn f g s

If every x ∈ s has a neighbourhood within s on which b ↦ ∏' i, f i b converges uniformly to g, then the product converges locally uniformly on s to g. Note that this is not a tautology, and the converse is only true if the domain is locally compact.

Defined in

Mathlib.Topology.Algebra.InfiniteSum.UniformOn

Cited by

1 results in Mathlib

Foundations

Depth 78 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommMonoidUniformSpaceTopologicalSpace