Generalized Sum is Linear
Theorem
Let $\family {z_i}_{i \mathop \in I}$ and $\family {w_i}_{i \mathop \in I}$ be $I$-indexed families of complex numbers.
That is, let $z_i, w_i \in \C$ for all $i \in I$.
Let $\ds \sum \set {z_i: i \in I}$ and $\sum \set {w_i: i \mathop \in I}$ converge to $z, w \in \C$, respectively.
Then:
- $(1): \quad \ds \sum \set {z_i + w_i: i \in I}$ converges to $z + w$
- $(2): \quad \forall \lambda \in \C: \ds \sum \set {\lambda z_i: i \in I}$ converges to $\lambda z$
Proof
Proof of $(1)$
Let $\epsilon > 0$.
To verify the convergence, it is necessary to find a finite $F \subseteq I$ such that:
- $\ds \map d {\sum_{i \mathop\in G} z_i + w_i, z + w} < \epsilon$ for all finite $G$ with $F \subseteq G \subseteq I$
Now let $F_z, F_w \subseteq I$ be finite subsets of $I$ such that:
- $\ds \map d {\sum_{i \mathop \in G} z_i, z} < \frac \epsilon 2$ for all finite $G$ with $F_z \subseteq G \subseteq I$
- $\ds \map d {\sum_{i \mathop \in G} w_i, w} < \frac \epsilon 2$ for all finite $G$ with $F_w \subseteq G \subseteq I$
The set $F_z \cup F_w$ will be the sought $F$. Let $G$ be finite such that $F_z \cup F_w \subseteq G \subseteq I$.
It follows that:
| \(\ds \map d {\sum_{i \mathop \in G} z_i + w_i, z + w}\) | \(=\) | \(\ds \size {\paren {\sum_{i \mathop \in G} z_i + w_i} - \paren {z + w} }\) | Definition of Metric Induced by Norm | |||||||||||
| \(\ds \) | \(=\) | \(\ds \size {\paren {\sum_{i \mathop \in G} z_i - z} + \paren {\sum_{i \mathop \in G} w_i - w} }\) | ||||||||||||
| \(\ds \) | \(\le\) | \(\ds \size {\paren {\sum_{i \mathop \in G} z_i - z} } + \size {\paren {\sum_{i \mathop \in G} w_i, w} }\) | Triangle Inequality for $\size {\, \cdot \,}$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map d {\sum_{i \mathop\in G} z_i, z} + \map d {\sum_{i \mathop\in G} w_i, w}\) | Definition of Metric Induced by Norm | |||||||||||
| \(\ds \) | \(<\) | \(\ds \epsilon\) | $F_z, F_w \subseteq G$ |
From the definition of convergence, it is concluded that:
- $\ds \sum \set {z_i + w_i: i \in I} = z + w$
$\Box$
Proof of $(2)$
Let $\epsilon > 0$.
To verify the convergence, it is necessary to find a finite $F \subseteq I$ such that:
- $\ds \map d {\sum_{i \mathop\in G} \lambda z_i, \lambda z} < \epsilon$ for all finite $G$ with $F \subseteq G \subseteq I$
Now let $F_z \subseteq I$ be a finite subset of $I$ such that:
- $\ds \map d {\sum_{i \mathop \in G} z_i, z} < \frac \epsilon {\size \lambda}$
for all finite $G$ with $F_z \subseteq G \subseteq I$
The set $F_z$ will be the sought $F$.
Let $G$ be finite such that $F_z \subseteq G \subseteq I$.
It follows that:
| \(\ds \map d {\sum_{i \mathop \in G} \lambda z_i, \lambda z}\) | \(=\) | \(\ds \size {\paren {\sum_{i \mathop \in G} \lambda z_i} - \lambda z}\) | Definition of Metric Induced by Norm | |||||||||||
| \(\ds \) | \(=\) | \(\ds \size {\lambda \paren {\sum_{i \mathop \in G} z_i - z} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \size \lambda \size {\sum_{i \mathop \in G} z_i - z}\) | Multiplicativity for $\size {\, \cdot \,}$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \size \lambda \map d {\sum_{i \mathop \in G} z_i, z}\) | Definition of Metric Induced by Norm | |||||||||||
| \(\ds \) | \(<\) | \(\ds \epsilon\) | $F_z \subseteq G$ |
From the definition of convergence, conclude that:
- $\ds \sum \set {\lambda z_i: i \in I} = \lambda z$
$\blacksquare$
Also see
- Convergence of Generalized Sum of Complex Numbers establishes a partial converse.