Generalized Sum Restricted to Non-zero Summands

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Theorem

Let $G$ be a commutative topological semigroup with identity $0_G$.

Let $\family{g_i}_{i \in I}$ be an indexed family of elements of $G$.

Let $J = \set{i \in I : g_i \ne 0_G}$

Let $h \in G$.

Then:

the generalized sum $\ds \sum_{i \mathop \in I} g_i$ converges to $h$

if and only if:

the generalized sum $\ds \sum_{j \mathop \in J} g_j$ converges to $h$

Corollary

Let $K \subseteq I : \set{i \in I : g_i \ne 0_G} \subseteq K$

Let $h \in G$.

Then:

the generalized sum $\ds \sum_{i \mathop \in I} g_i$ converges to $h$

if and only if:

the generalized sum $\ds \sum_{k \mathop \in K} g_k$ converges to $h$

Proof

Necessary Condition

Let the generalized sum $\ds \sum_{i \mathop \in I} g_i$ converge to $h$.

It will be shown that $\ds \sum_{j \mathop \in J} g_j$ converges to $h$.

Let $U \subseteq G$ be an open subset of $G$ such that $h \in U$.

By definition of convergent net:

$(1) \quad \exists F \subseteq I : F \ne \O : \forall E \subseteq I : E \supseteq F \implies \ds \sum_{i \mathop \in E} g_i \in U$

where $\ds \sum_{i \mathop \in E} g_i$ is the summation over $E$.

Let:

$F'= F \cap J$

From Set Difference and Intersection form Partition:

$F = F' \cup F \setminus J$

Let $E' \subseteq J$:

$E' \supseteq F'$

We have:

$E' \cap F \setminus J = \O$

Let:

$E = E' \cup F \setminus J$

From Set Union Preserves Subsets:

$E \supseteq F$

From $(1)$:

$\ds \sum_{i \mathop \in E} g_i \in U$

Case : $F \setminus J = \O$

Let:

$F \setminus J = \O$

From Union with Empty Set:

$E = E'$

Hence:

$\ds \sum_{i \mathop \in E'} g_i \in U$

$\Box$

Case : $F \setminus J \ne \O$

Let:

$F \setminus J \ne \O$

We have:

\(\ds \sum_{i \mathop \in E} g_i\)

\(=\)

\(\ds \sum_{i \mathop \in E'} g_i + \sum_{i \mathop \in F \setminus J} g_i\)

Summation over Union of Disjoint Finite Index Sets

\(\ds \)

\(=\)

\(\ds \sum_{i \mathop \in E'} g_i + \sum_{i \mathop \in F \setminus J} 0_G\)

definitions of $F$ and $J$

\(\ds \)

\(=\)

\(\ds \sum_{i \mathop \in E'} g_i\)

Definition of identity

Hence:

$\ds \sum_{i \mathop \in E'} g_i \in U$

$\Box$

In either case:

$\ds \sum_{i \mathop \in E'} g_i \in U$

Since $U$ was arbitrary, it follows that $\ds \sum_{j \mathop \in J} g_j$ converges to $h$ by definition.

$\Box$

Sufficient Condition

Let the generalized sum $\ds \sum_{j \mathop \in J} g_j$ converge to $h \in G$.

It will be shown that $\ds \sum_{i \mathop \in I} g_j$ converges to $h$.

Let $U \subseteq G$ be an open subset of $G$ such that $h \in U$.

By definition of convergent net:

$(2) \quad \exists F' \subseteq J : F' \ne \O : \forall E' \subseteq J : E' \supseteq F' \implies \ds \sum_{j \mathop \in E'} g_j \in U$

where $\ds \sum_{j \mathop \in E'} g_j$ is the summation over $E$.

We have:

$F' \subseteq J \subseteq I$.

Let $E \subseteq I$:

$E \supseteq F'$

Let:

$E' = E \cap J$

From Set Intersection Preserves Subsets and Intersection with Subset is Subset:

$E' \supseteq F'$

From $(2)$:

$\ds \sum_{j \mathop \in E'} g_j \in U$

From Set Difference Union Intersection:

$E = E' \cup E \setminus J$

From Set Difference and Intersection are Disjoint:

$E' \cap E \setminus J = \O$

Case : $E \setminus J = \O$

Let:

$E \setminus J = \O$

From Union with Empty Set:

$E = E'$

Hence:

$\ds \sum_{i \mathop \in E} g_i \in U$

$\Box$

Case : $E \setminus J \ne \O$

Let:

$E \setminus J \ne \O$

We have:

\(\ds \sum_{i \mathop \in E} g_i\)

\(=\)

\(\ds \sum_{i \mathop \in E'} g_i + \sum_{i \mathop \in E \setminus J} g_i\)

Summation over Union of Disjoint Finite Index Sets

\(\ds \)

\(=\)

\(\ds \sum_{i \mathop \in E'} g_i + \sum_{i \mathop \in E \setminus J} 0_G\)

definitions of $E$ and $J$

\(\ds \)

\(=\)

\(\ds \sum_{i \mathop \in E'} g_i\)

Definition of identity

Hence:

$\ds \sum_{i \mathop \in E} g_i \in U$

$\Box$

In either case:

$\ds \sum_{i \mathop \in E} g_i \in U$

Since $U$ was arbitrary, it follows that $\ds \sum_{i \mathop \in I} g_i$ converges to $h$ by definition.

$\blacksquare$