Theorems · Theorem · functional analysis

tendstoUniformly_tsum

∀ {α : Type u_1} {β : Type u_2} {F : Type u_3} [inst : NormedAddCommGroup F] [CompleteSpace F] {u : α → ℝ}
  {f : α → β → F},
  Summable u →
    (∀ (n : α) (x : β), ‖f n x‖ ≤ u n) →
      TendstoUniformly (fun t x => ∑ n ∈ t, f n x) (fun x => ∑' (n : α), f n x) Filter.atTop

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

Defined in

Mathlib.Analysis.Normed.Group.FunctionSeries

Cited by

1 results in Mathlib

Foundations

Depth 169 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupCompleteSpace