Definition:Generalized Sum/Net Convergence

Definition

Let $\struct {G, +}$ be a commutative topological semigroup.

Let $\sequence {g_n}_{n \mathop \in \N}$ be a sequence in $G$.

The series $\ds \sum_{n \mathop = 1}^\infty g_n$ converges as a net or has net convergence if and only if the generalized sum $\ds \sum \set {g_n: n \in \N}$ converges.

Also see

• Net Convergence Equivalent to Absolute Convergence: when $G$ is a Banach space, net convergence is equivalent to absolute convergence.

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next) $\text I.4.11$