Can any one explain to me,what is meant by convergence of a series defined over an arbitary indexing set?
Context:I was checking definition of holomorphic map of several variables where the power series is defined without mention of any absolute convergence(so rearrangement theorem is not applicable).
Thanks in advance.
If $(a_i)_{i \in \mathcal I}$ is an indexed set of real numbers, split $\mathcal I$ into two parts: $$ \mathcal I_+ = \{i : a_i \ge 0\}, \quad \mathcal I_- = \{i : a_i < 0\} .$$ Then define $$ S_+ = \sup\left\{\sum_{a \in \mathcal J} a_i : \text{$\mathcal J$ is a finite subset of $\mathcal I_+$} \right\} ,$$ $$ S_- = \sup\left\{-\sum_{a \in \mathcal J} a_i : \text{$\mathcal J$ is a finite subset of $\mathcal I_-$} \right\} .$$ Then $$ \sum_{i \in \mathcal I} a_i = S_+ - S_- .$$ Note if both $S_+$ and $S_-$ are infinite, then the sum is undefined. So the sum is only defined and finite if the indexed set is absolutely summable.