Theorems · Theorem · functional analysis

tendstoUniformlyOn_tsum_of_cofinite_eventually

∀ {β : Type u_2} {F : Type u_3} [inst : NormedAddCommGroup F] [CompleteSpace F] {ι : Type u_4} {f : ι → β → F}
  {u : ι → ℝ},
  Summable u →
    ∀ {s : Set β},
      (∀ᶠ (n : ι) in Filter.cofinite, ∀ x ∈ s, ‖f n x‖ ≤ u n) →
        TendstoUniformlyOn (fun t x => ∑ n ∈ t, f n x) (fun x => ∑' (n : ι), f n x) Filter.atTop s

An infinite sum of functions with eventually summable sup norm is the uniform limit of its partial sums. Version relative to a set, with general index set.

Defined in

Mathlib.Analysis.Normed.Group.FunctionSeries

Cited by

6 results in Mathlib

Foundations

Depth 168 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupCompleteSpace