A normed space is complete exactly when every absolutely convergent series converges in the space.Suggest a slogan
Let \((V,\norm{\cdot})\) be a normed vector space. Then \(V\) is complete if and only if every absolutely convergent series in \(V\) converges: whenever \(x_n\in V\) and \(\sum_{n=1}^\infty\norm{x_n}<\infty\), the series \(\sum_{n=1}^\infty x_n\) converges in \(V\).
This is the equivalence proved in FD-WN55Z.
Characterize completeness of a normed space in terms of absolutely convergent series.