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 $\set{i \in I : v_i \ne 0}$ be countably infinite.

Let $\set{i_0, i_1, i_2, \ldots}$ be a countably infinite enumeration of $\set{i \in I : v_i \ne 0}$.

Let $r \in \R_{\mathop > 0}$.

Then:
 * the generalized sum $\ds \sum_{i \mathop \in I} v_i$ converges absolutely to $r$


 * the series $\ds \sum_{n \mathop = 1}^\infty v_{i_n}$ converges absolutely to $r$
 * the series $\ds \sum_{n \mathop = 1}^\infty v_{i_n}$ converges absolutely to $r$

Proof
Let $J = \set{i \in I : v_i \ne 0}$.

From User:Leigh.Samphier/Topology/Generalized Sum Restricted to Non-zero Summands:
 * the generalized sum $\ds \sum_{i \mathop \in I} v_i$ converges absolutely to $r$


 * the generalized sum $\ds \sum_{i \mathop \in J} v_i$ converges absolutely to $r$

From User:Leigh.Samphier/Topology/Absolute Net Convergence Equivalent to Absolute Convergence:
 * the generalized sum $\ds \sum_{i \mathop \in J} v_i$ converges absolutely to $r$


 * the series $\ds \sum_{n \mathop = 1}^\infty v_{i_n}$ converges absolutely to $r$
 * the series $\ds \sum_{n \mathop = 1}^\infty v_{i_n}$ converges absolutely to $r$