Measure, Integration & Real Analysis · Problem 8C.4
Real and Complex Measures — Sheldon Axler
Mathematics — Measure Theory · unordered-sums-linearity
Problem
Suppose {fk}k∈Γ and {gk}k∈Γ are families in a normed vector space such that ∑k∈Γfk and ∑k∈Γgk converge. Prove that ∑k∈Γ(fk+gk) converges and
k∈Γ∑(fk+gk)=k∈Γ∑fk+k∈Γ∑gk.