Let $\mathbb{K}$ be either $\mathbb{R}$ or $\mathbb{C}$, and let $(X,\|\cdot\|_X)$ be a [normed vector space](/page/Normed%20Vector%20Space) over $\mathbb{K}$. Then $X$ is a [Banach space](/page/Banach%20Space) if and only if the following property holds: for every sequence $(x_n)_{n\in\mathbb{N}}$ in $X$, if the scalar series $\sum_{n=1}^{\infty}\|x_n\|_X$ converges in $\mathbb{R}$, then the vector series $\sum_{n=1}^{\infty}x_n$ converges in $X$.