User:Leigh.Samphier/Topology/Bounded Generalized Sum is Absolutely Convergent

Theorem
Let $V$ be a Banach space.

Let $\family {v_i}_{i \mathop \in I}$ be an indexed family of elements of $V$.

Let $\FF$ denote the set of finite subsets of $I$.

Then:
 * the generalized sum $\ds \sum \set {v_i: i \in I}$ is absolutely net convergent


 * there exists $M \in \R_{\mathop \ge 0}$ such that for all $F \in \FF$: the summation $\ds \sum_{i \mathop \in F} \norm{v_i} \le M$

Necessary Condition
Let the generalized sum $\ds \sum \set {v_i: i \in I}$ be absolutely net convergent to $c \in \R$.

Be definition of absolutely net convergent:
 * $\forall \epsilon \in \R_{\mathop > 0}: \exists F \in \FF : \forall E \in \FF : E \supseteq F : \ds \sum_{i \in E} \norm{v_i} \in \openint {c - \epsilon} {c + \epsilon}$

Sufficient Condition
Let $M \in \R_{\mathop \ge 0}$ such that for all $F \in \FF$:
 * the summation $\ds \sum_{i \mathop \in F} \norm{v_i} \le M$

Let $S = \set{\ds \sum_{i \mathop \in F} \norm{v_i} : F \in \FF}$

From Least Upper Bound Property, let:
 * $c = \sup S$

From Characterizing Property of Supremum of Subset of Real Numbers:
 * $(1)\quad \forall F \in \FF : \ds \sum_{i \mathop \in F} \norm{v_i} \le c$

and
 * $(2)\quad \forall \epsilon \in \R_{> 0}: \exists F \in \FF : \ds \sum_{i \mathop \in F} \norm{v_i} > c - \epsilon$

Let $\epsilon \in \R_{> 0}$.

From $(2)$:
 * $\exists F \in \FF : \ds \sum_{i \mathop \in F} \norm{v_i} > c - \epsilon$

Let $E \in \FF : E \supseteq F$.

We have:

Since $E$ was arbitrary, it follows:
 * $\forall E \in \FF : E \supseteq F : \ds \sum_{i \mathop \in E} \norm{v_i} \in \openint {c - \epsilon} {c + \epsilon}$

Since $\epsilon$ was arbitrary, it follows:
 * $\forall \epsilon \in \R_{> 0} : \exists F \in \FF : \forall E \in \FF : E \supseteq F : \ds \sum_{i \mathop \in E} \norm{v_i} \in \openint {c - \epsilon} {c + \epsilon}$

It follows that the generalized sum $\ds \sum_{i \mathop I} \norm{v_i}$ is convergent to $c$.

The result follows.