Net Convergence Equivalent to Absolute Convergence

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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)


Proof




Sources