User:Leigh.Samphier/Topology/Absolutely Convergent Generalized Sum Converges to Supremum

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 $I$.

Let the generalized sum $\ds \sum_{i \mathop \in I} v_i$ converge absolutely to $c \in \R$.

Then:
 * $c = \sup \set{\ds \sum_{i \mathop \in F} \norm{v_i} : F \in \FF}$

Proof

 * $\exists E \in \FF : \ds \sum_{i \mathop \in E} \norm{v_i} > c$
 * $\exists E \in \FF : \ds \sum_{i \mathop \in E} \norm{v_i} > c$

Let:
 * $0 < \epsilon < \ds \sum_{i \mathop \in F} \norm{v_i} - c$

Let $F \in \FF$.

Let $E' = F \cup E$.

We have:

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

This contradicts the hypothesis that $\ds \sum_{i \mathop \in I} v_i$ converges absolutely to $c \in \R$.

Hence:
 * $\forall E \in \FF : \ds \sum_{i \mathop \in E} \norm{v_i} \le c$

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

From Characterizing Property of Supremum of Subset of Real Numbers:
 * $c = \sup \set{\ds \sum_{i \mathop \in F} \norm{v_i} : F \in \FF}$