Theorems · Theorem · functional analysis

tendstoUniformly_tsum_nat

∀ {β : Type u_2} {F : Type u_3} [inst : NormedAddCommGroup F] [CompleteSpace F] {f : ℕ → β → F} {u : ℕ → ℝ},
  Summable u →
    (∀ (n : ℕ) (x : β), ‖f n x‖ ≤ u n) →
      TendstoUniformly (fun N x => ∑ n ∈ Finset.range N, f n x) (fun x => ∑' (n : ℕ), f n x) Filter.atTop

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

Defined in

Mathlib.Analysis.Normed.Group.FunctionSeries

Cited by

0 results in Mathlib

Foundations

Depth 170 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupCompleteSpace