Definition:Convergent Series/Normed Vector Space/Definition 1

Definition
Let $V$ be a normed vector space.

Let $d$ be the induced metric on $V$.

Let $\displaystyle S := \sum_{n \mathop = 1}^\infty a_n$ be a series in $V$.

$S$ is convergent its sequence $\left \langle {s_N} \right \rangle$ of partial sums converges in the metric space $\left({V, d}\right)$.