Measure, Integration & Real Analysis · Problem 8C.7

Real and Complex Measures — Sheldon Axler

Mathematics — Measure Theory · unordered-sums-reordering

Problem

Suppose

\{f_k\}_{k\in\mathbf{Z}^+}

is a family in a normed vector space

V

and

f \in V

. Prove that the unordered sum

\sum_{k\in\mathbf{Z}^+} f_k

equals

f

if and only if the usual ordered sum

\sum_{k=1}^{\infty} f_{p(k)}

equals

f

for every injective and surjective function

p: \mathbf{Z}^+ \to \mathbf{Z}^+