Bounded Generalized Sum is Absolutely Convergent
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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 $I$.
Then:
- the generalized sum $\ds \sum_{i \mathop \in I} v_i$ is absolutely net convergent
- there exists $M \in \R_{\mathop \ge 0}$ such that for all $F \in \FF$:
- the summation $\ds \sum_{i \mathop \in F} \norm{v_i} \le M$
Proof
Necessary Condition
Let the generalized sum $\ds \sum_{i \mathop \in I} v_i$ be absolutely net convergent.
Let:
- $M = \ds \sum_{i \mathop \in I} \norm{v_i}$
Aiming for a contradiction, suppose
- $\exists F_0 \in \FF : \sum_{i \mathop \in F_0} \norm{v_i} > M$
Let:
- $(1) \quad \epsilon \in \R_{\mathop > 0} : \epsilon < \paren{\ds \sum_{i \mathop \in F_0} \norm{v_i} } - M$
By definition of absolutely net convergence:
- $(2) \quad \exists F \in \FF : \forall E \in \FF : E \supseteq F \implies \ds \size{\sum_{i \mathop \in E} \norm{v_i} - M} < \epsilon$
Let:
- $E = F \cup F_0$
From Set is Subset of Union:
- $E \supseteq F$
We have:
\(\ds \sum_{i \mathop \in E} \norm{v_i}\) | \(<\) | \(\ds M + \epsilon\) | from $(2)$ | |||||||||||
\(\ds \) | \(<\) | \(\ds \sum_{i \mathop \in F_0} \norm{v_i}\) | from $(1)$ | |||||||||||
\(\ds \) | \(\le\) | \(\ds \sum_{i \mathop \in F_0} \norm{v_i} + \sum_{i \mathop \in E \setminus F_0} \norm{v_i}\) | Norm Axiom $\text N 1$: Positive Definiteness | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{i \mathop \in E} \norm{v_i}\) | Summation over Union of Disjoint Finite Index Sets |
We have a contradiction, so it follows that:
- $\forall F \in \FF : \ds \sum_{i \mathop \in F} \norm{v_i} \le M$
$\Box$
Sufficient Condition
Let there exist $M \in \R_{\mathop \ge 0}$ 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:
- $(3)\quad \forall F \in \FF : \ds \sum_{i \mathop \in F} \norm{v_i} \le c$
and
- $(4)\quad \forall \epsilon \in \R_{> 0}: \exists F \in \FF : \ds \sum_{i \mathop \in F} \norm{v_i} > c - \epsilon$
Let $\epsilon \in \R_{> 0}$.
From $(4)$:
- $\exists F \in \FF : \ds \sum_{i \mathop \in F} \norm{v_i} > c - \epsilon$
Let $E \in \FF : E \supseteq F$.
We have:
\(\ds c\) | \(\ge\) | \(\ds \sum_{i \mathop \in E} \norm{v_i}\) | from $(3)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{i \mathop \in F} \norm{v_i} + \sum_{i \mathop \in E \setminus F} \norm{v_i}\) | Summation over Union of Disjoint Finite Index Sets | |||||||||||
\(\ds \) | \(\ge\) | \(\ds \sum_{i \mathop \in F} \norm{v_i}\) | Norm Axiom $\text N 1$: Positive Definiteness | |||||||||||
\(\ds \) | \(>\) | \(\ds c - \epsilon\) | from $(4)$ |
Since $E$ was arbitrary, it follows:
- $\forall E \in \FF : E \supseteq F \leadsto \ds \sum_{i \mathop \in E} \norm{v_i} \in \openint {c - \epsilon} {c + \epsilon}$
Since $\epsilon$ was arbitrary, it follows:
- $\forall \epsilon \in \R_{> 0} : \exists F \in \FF : \forall E \in \FF : E \supseteq F \leadsto \ds \sum_{i \mathop \in E} \norm{v_i} \in \openint {c - \epsilon} {c + \epsilon}$
It follows that the generalized sum $\ds \sum_{i \mathop I} \norm{v_i}$ is convergent to $c$.
The result follows.
$\blacksquare$