Absolutely Convergent Generalized Sum over Union of Disjoint Index Sets

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Theorem

Let $V$ be a Banach space.

Let $I$ and $J$ be disjoint indexing sets.

Let $K = I \cup J$.

Let $\family{v_k}_{k \mathop \in K}$ be an indexed family of elements of $V$.

Then:

the generalized sum $\ds \sum_{k \mathop \in K} v_k$ converges absolutely

if and only if

the generalized sums $\ds \paren{\sum_{i \mathop \in I} v_i}$ and $\ds \paren{\sum_{j \mathop \in J} v_j}$ converge absolutely

In which case:

$\ds \sum_{k \mathop \in K} \norm{v_k} = \paren{\sum_{i \mathop \in I} \norm{v_i}} + \paren{\sum_{j \mathop \in J} \norm{v_j}}$

Proof

Necessary Condition

Let $\ds \sum_{k \mathop \in K} v_k$ converge absolutely.

By definition of absolute net convergence:

$\ds \sum_{k \mathop \in K} \norm{v_k}$ converges.

From Generalized Sum over Subset of Absolutely Convergent Generalized Sum is Absolutely Convergent:

$\ds \paren{\sum_{i \mathop \in I} \norm{v_i}}$ and $\ds \paren{\sum_{j \mathop \in J} \norm{v_j}}$ converge

By definition of absolute net convergence:

$\ds \paren{\sum_{i \mathop \in I} v_i}$ and $\ds \paren{\sum_{j \mathop \in J} v_j}$ converge absolutely.

From Generalized Sum over Union of Disjoint Index Sets:

$\ds \sum_{k \mathop \in K} \norm{v_k} = \paren{\sum_{i \mathop \in I} \norm{v_i}} + \paren{\sum_{j \mathop \in J} \norm{v_j}}$

$\Box$

Sufficient Condition

Let $\ds \paren{\sum_{i \mathop \in I} v_i}$ and $\ds \paren{\sum_{j \mathop \in J} v_j}$ converge absolutely.

By definition of absolute net convergence:

$\ds \paren{\sum_{i \mathop \in I} \norm{v_i}}$ and $\ds \paren{\sum_{j \mathop \in J} \norm{v_j}}$ converge

From Generalized Sum over Union of Disjoint Index Sets:

$\ds \sum_{k \mathop \in K} \norm{v_k} = \paren{\sum_{i \mathop \in I} \norm{v_i}} + \paren{\sum_{j \mathop \in J} \norm{v_j}}$

By definition of absolute net convergence:

$\ds \sum_{k \mathop \in K} v_k$ converges absolutely.

$\blacksquare$