Generalized Sum over Subset of Absolutely Convergent 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 the generalized sum $\ds \sum_{i \mathop \in I} v_i$ be absolutely net convergent.
Let $J \subseteq I$.
Then:
- the generalized sum $\ds \sum_{j \mathop \in J} v_j$ is absolutely net convergent.
Proof
By definition of absolute net convergence, let:
- $\ds \sum_{i \mathop \in I} \norm{v_i} = M$
Let $F \subseteq J$ be finite.
From Subset Relation is Transitive:
- $F \subseteq I$
From Absolutely Convergent Generalized Sum Converges to Supremum:
- $\ds \sum_{j \mathop \in F} \norm{v_j} \le M$
Since $F \subseteq J$ was arbitrary, it follows that:
- $\forall F \subseteq J : F$ is finite $: \ds \sum_{j \mathop \in F} \norm{v_j} \le M$
From Bounded Generalized Sum is Absolutely Convergent:
- $\ds \sum_{j \mathop \in J} \norm{v_j}$ is absolutely net convergent
$\blacksquare$