Generalized Sum with Countable Non-zero Summands

Theorem
Let $V$ be a Banach space.

Let $\norm {\, \cdot \,}$ denote the norm on $V$.

Let $\family{v }_{i \in I}$ be an indexed subset of $V$.

Let $\sequence{i_n}_{n \in \N}$ be a sequence of distinct terms in $I$:
 * $\set{i_0, i_1, i_2, \ldots} = \set{i \in I : v_i \ne 0}$

Then:
 * the generalized sum $\ds \sum_{i \in I} g_i$ converges absolutely


 * the series $\ds \sum_{n = 1}^\infty g_{i_n}$ converges absolutely
 * the series $\ds \sum_{n = 1}^\infty g_{i_n}$ converges absolutely

In which case:
 * $\ds \sum_{i \in I} \norm{g_i} = \sum_{n = 1}^\infty \norm{g_{i_n}}$