Banach spaces

Proposition (Every absolutely convergent series is convergent) Let $(X,\Vert \cdot \Vert )$ be a Banach space, and ${\sum }_k{x}_k$ a series such that series of norms ${\sum }_k\Vert x_k\Vert <\infty$ conveges. Then also the series conveges.

Proof The norm series is a series of real numbers. Since it converges, it’s also a Cauchy sequence, so that $\forall ϵ>0$ there exists $N\ge 0$ such that for all $m>n\ge N$

$$\left|\sum _k^m\Vert x_k\Vert -\sum _k^n\Vert x_k\Vert \right|=\sum _{k=n+1}^m\Vert x_k\Vert <ϵ$$

using the Minkowsky’s inequality

$$\Vert \sum _k^m{x}_k-\sum _k^n{x}_k\Vert \le \sum _{k=n+1}^m\Vert x_k\Vert <ϵ$$

so that also the series is Cauchy. Since $X$ is a Banach space, the series converges. $\square$