Measure, Integration & Real Analysis · Problem 8C.5
Real and Complex Measures — Sheldon Axler
Mathematics — Measure Theory · unordered-sums-scalar-multiplication
Problem
Suppose {fk}k∈Γ is a family in a normed vector space such that ∑k∈Γfk converges. Prove that if c∈F, then ∑k∈Γ(cfk) converges and
k∈Γ∑(cfk)=ck∈Γ∑fk.