Theorems · Theorem · functional analysis

continuousOn_tsum

∀ {α : Type u_1} {β : Type u_2} {F : Type u_3} [inst : NormedAddCommGroup F] [CompleteSpace F] {u : α → ℝ}
  [inst_2 : TopologicalSpace β] {f : α → β → F} {s : Set β},
  (∀ (i : α), ContinuousOn (f i) s) →
    Summable u → (∀ (n : α), ∀ x ∈ s, ‖f n x‖ ≤ u n) → ContinuousOn (fun x => ∑' (n : α), f n x) s

An infinite sum of functions with summable sup norm is continuous on a set if each individual function is.

Defined in

Mathlib.Analysis.Normed.Group.FunctionSeries

Cited by

3 results in Mathlib

Foundations

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

Assumes

NormedAddCommGroupCompleteSpaceTopologicalSpace