Theorems · Theorem · sequences and series

tprod_bot

∀ {α : Type u_1} {β : Type u_2} [inst : CommMonoid α] [inst_1 : TopologicalSpace α] {L : SummationFilter β},
  ¬L.NeBot → ∀ (f : β → α), ∏'[L] (b : β), f b = ∏ᶠ (b : β), f b

If the summation filter is the trivial filter ⊥, then the topological product is equal to the finite product (which is taken to be 1 if the multiplicative support of f is infinite). Note that in this case HasProd f a is satisfied for every element a of the target, so the value assigned to the tprod is a question of conventions.

Defined in

Mathlib.Topology.Algebra.InfiniteSum.Defs

Cited by

6 results in Mathlib

Foundations

Depth 83 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommMonoidTopologicalSpace