Remark 2

The definition of a summable family does not involve any ordering of the index set $I$, and we may therefore say that the notion of a sum thus defined is commutative. More precisely, we have the following property: let $(x_i)_{i \in I}$ be a summable family and let $\varphi$ be a bijective mapping of an index set $K$ onto the set $I$; then if we put $y_x = x_{\varphi(x)}$, the family $(y_x)_{x \in K}$ is summable and has the same sum as $(x_i)$. For if $s = \sum_{i \in I} x_i$, and if $\sum_{i \in J} x_i \in s + V$ for every finite subset $J$ containing the finite subset $J_0$, then we shall have $\sum_{x \in L} y_x \in s + V$ for every finite subset $L$ of $K$ containing $\overline{\varphi}(J_0)$.