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 $\ds \sum \set {v_i: i \in I}$ converge absolutely to $c \in \R$.

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

Proof

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

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

Let $F \in \FF$.

Let $E' = F \cup E$.

We have:

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