Category:Absolute Net Convergence Equivalent to Absolute Convergence

This category contains pages concerning Absolute Net Convergence Equivalent to Absolute Convergence:

Let $V$ be a Banach space.

Let $\sequence {v_n}_{n \mathop \in \N}$ be a sequence of elements in $V$.

Let $r \in \R_{\mathop \ge 0}$

Then the following two statements are equivalent:

$(1): \quad$ the generalized sum $\ds \sum \set {v_n: n \in \N}$ is absolutely net convergent to $r$

$(2): \quad$ the series $\ds \sum_{n \mathop = 1}^\infty v_n$ is absolutely convergent to $r$