Net Convergence Equivalent to Absolute Convergence

Theorem
Let $V$ be a Banach space.

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

Then the following two statements are equivalent:


 * $(1): \quad \ds \sum_{n \mathop = 1}^\infty \norm {v_n}$ converges (absolute convergence)
 * $(2): \quad \ds \sum \set {v_n: n \in \N}$ converges (generalised or net convergence)