Category:Generalized Sum with Countable Non-zero Summands
Jump to navigation
Jump to search
This category contains pages concerning Generalized Sum with Countable Non-zero Summands:
Let $V$ be a Banach space.
Let $\norm {\, \cdot \,}$ denote the norm on $V$.
Let $\family{v }_{i \in I}$ be an indexed family of elements of $V$.
Let $J$ be a countably infinite subset of $I$ such that $\set{i \in I : v_i \ne 0} \subseteq J$.
Let $\set{j_0, j_1, j_2, \ldots}$ be a countably infinite enumeration of $J$.
Let $r \in \R_{\mathop > 0}$.
Then:
- the generalized sum $\ds \sum_{i \mathop \in I} v_i$ converges absolutely to $r$
- the series $\ds \sum_{n \mathop = 1}^\infty v_{j_n}$ converges absolutely to $r$
Pages in category "Generalized Sum with Countable Non-zero Summands"
The following 2 pages are in this category, out of 2 total.