Definition:Convergent Series/Normed Vector Space

Definition

Definition 1

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

Definition 2

Let $\struct {V, \norm {\, \cdot \,}}$ be a normed vector space.

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

$S$ is convergent if and only if its sequence $\sequence {s_N}$ of partial sums converges in the normed vector space $\struct {V, \norm {\, \cdot \,}}$.