Theorems · Theorem · sequences and series

Summable.tsum_mul_tsum_eq_tsum_sum_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} [T3Space α]
  [IsTopologicalSemiring α],
  Summable f →
    Summable g →
      (Summable fun x => f x.1 * g x.2) →
        (∑' (n : A), f n) * ∑' (n : A), g n = ∑' (n : A), ∑ kl ∈ Finset.HasAntidiagonal.antidiagonal n, f kl.1 * g kl.2

The Cauchy product formula for the product of two infinite sums indexed by ℕ, expressed by summing on Finset.HasAntidiagonal.antidiagonal. See also tsum_mul_tsum_eq_tsum_sum_antidiagonal_of_summable_norm if f and g are absolutely summable.

Defined in

Mathlib.Topology.Algebra.InfiniteSum.Ring

Cited by

3 results in Mathlib

Foundations

Depth 94 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

AddCommMonoidFinset.HasAntidiagonalTopologicalSpaceNonUnitalNonAssocSemiringT3SpaceIsTopologicalSemiring