Theorems · Theorem · sequences and series

Summable.tsum_mul_tsum_eq_tsum_sum_range

∀ {α : Type u_3} [inst : TopologicalSpace α] [inst_1 : NonUnitalNonAssocSemiring α] {f g : ℕ → α} [T3Space α]
  [IsTopologicalSemiring α],
  Summable f →
    Summable g →
      (Summable fun x => f x.1 * g x.2) →
        (∑' (n : ℕ), f n) * ∑' (n : ℕ), g n = ∑' (n : ℕ), ∑ k ∈ Finset.range (n + 1), f k * g (n - k)

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

Defined in

Mathlib.Topology.Algebra.InfiniteSum.Ring

Cited by

0 results in Mathlib

Foundations

Depth 95 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceNonUnitalNonAssocSemiringT3SpaceIsTopologicalSemiring