Absolutely Convergent Generalized Sum Converges to Supremum
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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
Aiming for a contradiction, suppose:
- $\exists E \in \FF : \ds \sum_{i \mathop \in E} \norm{v_i} > c$
Let:
- $0 < \epsilon < \ds \sum_{i \mathop \in F} \norm{v_i} - c$
Let $F \in \FF$.
Let $E' = F \cup E$.
We have:
\(\ds \sum_{i \mathop \in E'} \norm{v_i}\) | \(=\) | \(\ds \sum_{i \mathop \in F} \norm{v_i} + \sum_{i \mathop \in E' \setminus F} \norm{v_i}\) | Summation over Union of Disjoint Finite Index Sets | |||||||||||
\(\ds \) | \(\ge\) | \(\ds \sum_{i \mathop \in F} \norm{v_i}\) | ||||||||||||
\(\ds \) | \(>\) | \(\ds c + \epsilon\) | Summation over Union of Disjoint Finite Index Sets | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \epsilon\) | \(<\) | \(\ds \size{\sum_{i \mathop \in E'} \norm{v_i} - c}\) |
Since $F$ was arbitrary, it follows:
- $\forall F \in \FF : \exists E' \in \FF : E' \supseteq F$ and $\size{\ds \sum_{i \mathop \in E'} \norm{v_i} - c} > \epsilon$
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:
- $\forall \epsilon \in \R_{\mathop > 0}: \exists F \in \FF : \forall E \in \FF : E \supseteq F \leadsto \ds \sum_{i \mathop \in E} \norm{v_i} \in \hointl {c - \epsilon} c$
From Characterizing Property of Supremum of Subset of Real Numbers:
- $c = \sup \set{\ds \sum_{i \mathop \in F} \norm{v_i} : F \in \FF}$
$\blacksquare$