Theorems · Theorem · functional analysis

Summable.hasProdUniformlyOn_one_add

∀ {α : Type u_1} {ι : Type u_2} {K : Set α} {u : ι → ℝ} {R : Type u_3} [inst : NormedCommRing R] [NormOneClass R]
  [CompleteSpace R] [inst_3 : TopologicalSpace α] {f : ι → α → R},
  IsCompact K →
    Summable u →
      (∀ᶠ (i : ι) in Filter.cofinite, ∀ x ∈ K, ‖f i x‖ ≤ u i) →
        (∀ (i : ι), ContinuousOn (f i) K) →
          HasProdUniformlyOn (fun i x => 1 + f i x) (fun x => ∏' (i : ι), (1 + f i x)) K

If a sequence of continuous functions f i x on an open compact K have norms eventually bounded by a summable function, then ∏' i, (1 + f i x) is uniformly convergent on K.

Defined in

Mathlib.Analysis.Normed.Module.MultipliableUniformlyOn

Cited by

3 results in Mathlib

Foundations

Depth 281 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedCommRingNormOneClassCompleteSpaceTopologicalSpace