Theorems · Theorem · Lie groups

summable_iff_summable_compl_and_tsum_mem

∀ {α : Type u_1} {β : Type u_2} [inst : AddCommGroup α] [inst_1 : UniformSpace α] [inst_2 : IsUniformAddGroup α]
  {L : SummationFilter β} (f : β → α),
  Summable f L ↔
    Summable (⇑UniformSpace.Completion.toCompl ∘ f) L ∧
      ∑'[L] (i : β), UniformSpace.Completion.toCompl (f i) ∈ Set.range ⇑UniformSpace.Completion.toCompl

A function f is summable in a uniform additive group α if and only if it is summable in Completion α and its sum in Completion α lies in the range of toCompl : α →+ Completion α.

Defined in

Mathlib.Topology.Algebra.InfiniteSum.GroupCompletion

Cited by

1 results in Mathlib

Foundations

Depth 114 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

AddCommGroupUniformSpaceIsUniformAddGroup