Theorems · Theorem · functional analysis

Summable.tsum_const_smul

∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} [inst : TopologicalSpace α] [inst_1 : AddCommMonoid α]
  [inst_2 : DistribSMul γ α] [ContinuousConstSMul γ α] {f : β → α} {L : SummationFilter β} [T2Space α] [L.NeBot]
  (b : γ), Summable f L → ∑'[L] (i : β), b • f i = b • ∑'[L] (i : β), f i

Infinite sums commute with scalar multiplication. Version for scalars living in a Monoid, but requiring a summability hypothesis.

Defined in

Mathlib.Topology.Algebra.InfiniteSum.Module

Cited by

4 results in Mathlib

Foundations

Depth 93 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceAddCommMonoidDistribSMulContinuousConstSMulT2SpaceSummationFilter.NeBot