Theorems · Theorem · sequences and series

summable_mul_prod_iff_summable_mul_sigma_antidiagonal

∀ {α : Type u_3} {A : Type u_4} [inst : AddCommMonoid A] [inst_1 : Finset.HasAntidiagonal A]
  [inst_2 : TopologicalSpace α] [inst_3 : NonUnitalNonAssocSemiring α] {f g : A → α},
  (Summable fun x => f x.1 * g x.2) ↔ Summable fun x => f (↑x.snd).1 * g (↑x.snd).2

The family (k, l) : ℕ × ℕ ↦ f k * g l is summable if and only if the family (n, k, l) : Σ (n : ℕ), antidiagonal n ↦ f k * g l is summable.

Defined in

Mathlib.Topology.Algebra.InfiniteSum.Ring

Cited by

2 results in Mathlib

Foundations

Depth 85 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

AddCommMonoidFinset.HasAntidiagonalTopologicalSpaceNonUnitalNonAssocSemiring