More of Dedekind: his series test in normed spaces (Q1751441)

The authors prove that a normed space \(X\) is complete if (and only if) Dedekind's series test holds in \(X,\) that is, if the series \(\sum_{n=1}^{\infty }a_{n}x_{n}\) converges in \(X\) whenever \(\sum_{n=1}^{\infty}a_{n}\) is a convergent series of real (or complex) numbers and \(\left( x_{n}\right) \) is a sequence of elements of \(X\) with \(\sum_{n=1}^{\infty}\left\Vert x_{n+1}-x_{n}\right\Vert <\infty.\) If \(X\) is a normed algebra with identity, then this result remains correct in the case where \(a_{n}\in X\) \(\left( n\in\mathbb{N}\right) .\)