Let \((V,\norm{\cdot})\) be a normed vector space. The following are equivalent:
\(V\) is complete: every Cauchy sequence in \(V\) converges.
Every absolutely convergent series in \(V\) converges: if \(x_n\in V\) and \(\sum_{n=1}^\infty\norm{x_n}<\infty\), then \(\sum_{n=1}^\infty x_n\) converges in \(V\).
Assume \(V\) is complete and \(\sum_n\norm{x_n}<\infty\). For \(M>N\) the partial sums \(s_N\coloneqq\sum_{n=1}^N x_n\) satisfy \(\norm{s_M-s_N}\leq\sum_{n=N+1}^M\norm{x_n}\), so \((s_N)\) is Cauchy and converges.
Conversely, assume every absolutely convergent series converges, and let \((y_m)\) be a Cauchy sequence. Choose \(m_1<m_2<\cdots\) with \(\norm{y_p-y_q}<2^{-k}\) for all \(p,q\geq m_k\). The series \(y_{m_1}+\sum_{k\geq 1}(y_{m_{k+1}}-y_{m_k})\) is absolutely convergent, so it converges; its partial sums are \(y_{m_{k}}\), so the subsequence \((y_{m_k})\) converges to some \(y\in V\). A Cauchy sequence with a convergent subsequence converges to the same limit, so \(y_m\to y\).