Absolutely Convergent Generalized Sum Converges

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Theorem

Let $V$ be a Banach space.

Let $\norm {\, \cdot \,}$ denote the norm on $V$.

Let $d$ denote the corresponding induced metric.

Let $\family {v_i}_{i \mathop \in I}$ be an indexed subset of $V$ such that the generalized sum $\displaystyle \sum_{i \mathop \in I} \set {v_i}$ converges absolutely.


Then the generalized sum $\displaystyle \sum \set {v_i: i \in I}$ converges.


Proof

The proof proceeds in two stages:

$(1): \quad$ Finding a candidate $v \in V$ where the sum might converge to
$(2): \quad$ Showing that the candidate is indeed sought limit.


That $\displaystyle \sum \set {v_i: i \mathop \in I}$ converges absolutely means that $\displaystyle \sum \set {\norm {v_i}: i \mathop \in I}$ converges.

Now, for all $n \in \N$, let $F_n \subseteq I$ be finite such that:

$\displaystyle \sum_{i \mathop \in G} \norm {v_i} > \sum \set {\norm {v_i}: i \mathop \in I} - 2^{-n}$

for all finite $G$ with $F_n \subseteq G \subseteq I$

It may be arranged that $n \ge m \implies F_m \subseteq F_n$ by passing over to $\displaystyle F'_n = \bigcup_{m \mathop = 1}^n F_m$ if necessary.

Define:

$\displaystyle v_n = \sum_{i \mathop \in F_n} v_i$

Next, it is to be shown that the sequence $\sequence {v_n}_{n \mathop \in \N}$ is Cauchy.


So let $\epsilon > 0$, and let $N \in \N$ be such that $2^{-N} < \epsilon$.

Then for $m \ge n \ge N$, have:

\(\displaystyle \map d {v_m, v_n}\) \(=\) \(\displaystyle \norm {\paren {\sum_{i \mathop \in F_m} v_i} - \paren {\sum_{i \mathop \in F_n} v_i} }\) Definition of Metric Induced by Norm
\(\displaystyle \) \(=\) \(\displaystyle \norm {\sum_{i \mathop \in F_m \setminus F_n} v_i}\) $F_n \subseteq F_m$
\(\displaystyle \) \(\le\) \(\displaystyle \sum_{i \mathop \in F_m \setminus F_n} \norm {v_i}\) Triangle Inequality for $\norm {\, \cdot \,}$


Now to estimate this last quantity, observe:

\(\displaystyle \sum \set {\norm {v_i}: i \mathop \in I} - 2^{-n} + \sum_{i \mathop \in F_m \setminus F_n} \norm {v_i}\) \(<\) \(\displaystyle \sum_{i \mathop \in F_n} \norm {v_i} + \sum_{i \mathop \in F_m \setminus F_n} \norm {v_i}\) Defining property of $F_n$
\(\displaystyle \) \(=\) \(\displaystyle \sum_{i \mathop \in F_m} \norm {v_i}\) Union with Relative Complement, $F_n \subseteq F_m$
\(\displaystyle \) \(\le\) \(\displaystyle \sum \set {\norm {v_i}: i \mathop \in I}\) Generalized Sum is Monotone
\(\displaystyle \leadsto \ \ \) \(\displaystyle \sum_{i \mathop \in F_m \setminus F_n} \norm {v_i}\) \(<\) \(\displaystyle 2^{-n}\)


Finally, by the defining property of $N$, as $n \ge N: :2^{-n} < 2^{-N} < \epsilon$

Combining all of these estimates leads to the conclusion that:

$\map d {v_m, v_n} < \epsilon$

It follows that $\sequence {v_n}_{n \mathop \in \N}$ is a Cauchy sequence.

As $V$ is a Banach space:

$\displaystyle \exists v \in V: \lim_{n \mathop \to \infty} v_n = v$


Having identified a candidate $v$ for the sum $\displaystyle \sum \set {v_i: i \in I}$ to converge to, it remains to verify that this is indeed the case.


According to the definition of considered sum, the convergence is convergence of a net.

Next, Metric Induces Topology ensures that we can limit the choice of opens $U$ containing $v$ to open balls centered at $v$.

Now let $\epsilon > 0$.

We want to find a finite $F \subseteq I$ such that:

$\map d {\displaystyle \sum_{i \mathop \in G} v_i, v} < \epsilon$

for all finite $G$ with $F \subseteq G \subseteq I$.


Now let $N \in \N$ such that:

$\forall n \ge N: \map d {v_n, v} < \dfrac \epsilon 2$

with the $v_n$ as above.

By taking a larger $N$ if necessary, ensure that $2^{-N} < \dfrac \epsilon 2$ holds as well.

Let us verify that the set $F_N$ defined above has sought properties.

So let $G$ be finite with $F_N \subseteq G \subseteq I$.

Then:

\(\displaystyle \map d {\sum_{i \mathop \in G} v_i, v}\) \(=\) \(\displaystyle \norm {\paren {\sum_{i \mathop \in G} v_i} - v}\) Definition of Metric Induced by Norm
\(\displaystyle \) \(\le\) \(\displaystyle \norm {\paren {\sum_{i \mathop \in G} v_i} - \paren {\sum_{i \mathop \in F_N} v_i} } + \norm {\paren {\sum_{i \mathop \in F_N} v_i} - v}\) Triangle inequality for $\norm {\, \cdot \,}$
\(\displaystyle \) \(<\) \(\displaystyle \norm {\sum_{i \mathop \in G \setminus F_N} v_i} + \frac \epsilon 2\) $F_N \subseteq G$, defining property of $N$
\(\displaystyle \) \(\le\) \(\displaystyle \sum_{i \mathop \in G \setminus F_N} \norm {v_i} + \frac \epsilon 2\) Triangle inequality for $\norm {\, \cdot \,}$


For the first of these terms, observe:

\(\displaystyle \sum \set {\norm {v_i}: i \in I} - 2^{-N} + \sum_{i \mathop \in G \setminus F_N} \norm {v_i}\) \(<\) \(\displaystyle \sum_{i \mathop \in F_N} \norm {v_i} + \sum_{i \mathop \in G \setminus F_N} \norm {v_i}\) defining property of $F_N$
\(\displaystyle \) \(=\) \(\displaystyle \sum_{i \mathop \in G} \norm {v_i}\) Union with Relative Complement, $F_N \subseteq G$
\(\displaystyle \) \(\le\) \(\displaystyle \sum \set {\norm {v_i}: i \in I}\) Generalized Sum is Monotone
\(\displaystyle \leadsto \ \ \) \(\displaystyle \sum_{i \mathop \in G \setminus F_N} \norm {v_i}\) \(<\) \(\displaystyle 2^{-N}\)


Using that $2^{-N} < \dfrac \epsilon 2$, combine these inequalities to obtain:

$\displaystyle \map d {\sum_{i \mathop \in G} v_i, v} < \frac \epsilon 2 + \frac \epsilon 2 = \epsilon$


By definition of convergence of a net, it follows that:

$\displaystyle \sum \set {v_i: i \in I} = v$

$\blacksquare$


Also see