Inequality Rule for Absolutely Convergent Generalized Sums

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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 $\ds \sum \set {v_i: i \in I}$ be absolutely net convergent.

Let $\family {w_i}_{i \mathop \in I}$ be an indexed family of elements of $V$:

$\forall i \in I : \norm{w_i} \le \norm{v_i}$

Then:

the generalized sum $\ds \sum_{i \mathop \in I} \norm{w_i}$ is absolutely net convergent

and:

$\ds \sum_{i \mathop \in I} \norm{w_i} \le \sum_{i \mathop \in I} \norm{v_i}$

Proof

By definition of absolutely net convergence, let:

$\ds \sum_{i \mathop \in I} \norm{v_i} = M$

Let $F \subseteq I$ be finite.

From Absolutely Convergent Generalized Sum Converges to Supremum:

$\ds \sum_{i \mathop \in F} \norm{v_i} \le M$

So by hypothesis:

$\ds \sum_{i \mathop \in F} \norm{w_i} \le \ds \sum_{i \mathop \in F} \norm{v_i} \le M$

Since $F \subseteq J$ was arbitrary, it follows that:

$\forall F \subseteq F : F$ is finite $: \ds \sum_{i \mathop \in F} \norm{w_i} \le M$

From Bounded Generalized Sum is Absolutely Convergent:

$\ds \sum_{i \mathop \in I} \norm{w_i}$ is absolutely net convergent

From Inequality Rule for Real Convergent Nets:

$\ds \sum_{i \mathop \in I} \norm{w_i} \le \sum_{i \mathop \in I} \norm{v_i}$

$\blacksquare$