Theorems · Theorem · sequences and series

summableLocallyUniformlyOn_of_of_forall_exists_nhds

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

If every x ∈ s has a neighbourhood within s on which b ↦ ∑' i, f i b converges uniformly, then the sum converges locally uniformly. 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

0 results in Mathlib

Foundations

Depth 85 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

AddCommMonoidUniformSpaceTopologicalSpace