Products of one plus a complex number #
We gather some results about the uniform convergence of the product of 1 + f n x for a
sequence f n x or complex numbers.
theorem
tendstoUniformlyOn_tprod'
{α : Type u_1}
[TopologicalSpace α]
{f : ℕ → α → ℂ}
{K : Set α}
(hK : IsCompact K)
{u : ℕ → ℝ}
(hu : Summable u)
(h : ∀ (n : ℕ), ∀ x ∈ K, ‖f n x‖ ≤ u n)
(hcts : ∀ (n : ℕ), ContinuousOn (fun (x : α) => f n x) K)
:
TendstoUniformlyOn (fun (n : ℕ) (a : α) => ∏ i ∈ Finset.range n, (1 + f i a)) (fun (a : α) => ∏' (i : ℕ), (1 + f i a))
Filter.atTop K
This is the version for infinite products of with terms of the from 1 + f n x.