\begin{definition}[Definition:Convergent Series/Normed Vector Space/Definition 1]
Let $V$ be a normed vector space.
Let $d$ be the induced metric on $V$.
Let $\ds S := \sum_{n \mathop = 1}^\infty a_n$ be a series in $V$.
$S$ is '''convergent''' {{iff}} its sequence $\sequence {s_N}$ of partial sums converges in the metric space $\struct {V, d}$.
Category:Definitions/Vector Spaces
Category:Definitions/Metric Spaces
Category:Definitions/Norm Theory
Category:Definitions/Convergent Series
\end{definition}