On shifts of sequences (Q1093843)

A sequence \(X=(x_ k)_{k\in {\mathbb{N}}}\) in a topological vector space E is called complete if \(\overline{span}\{x_ k\}=E\) and representing if every \(y\in E\) can be represented as \(x=\sum^{\infty}_{k=1}c_ kx_ k\) for suitable scalars \(c_ k.\) If E is separable it is shown that these two properties are inherited by the shifted sequence \((x_{k+1})_{k\in {\mathbb{N}}}\) iff \(E'=\{0\}\).