On existence of the converging ordering (Q1311183)

A sequence \(\{x_ n\}^ \infty_{n=1}\) in a topological vector space (t.v.s.) \(X\) is called orderable if \(\sum^ \infty_{n=1} x_{\pi(n)}\) converges for some permutation \(\pi\) of the positive integers. A theorem of Steinitz from 1913 characterizes the sequences in \(\mathbb{R}^ m\) \((m\geq 1)\) which are orderable as those \(\{x_ n\}^ \infty_{n=1}\) so that \(\{f(x_ n)\}^ \infty_{n=1}\) is orderable in \(\mathbb{R}^ 1\) for each \(f\) in \((\mathbb{R}^ m)^*\), and hence by Riemann's theorem those for which the series of positive and negative terms of \(\{f(x_ n)\}^ \infty_{n=1}\) both converge or both diverge for every \(f\) in \((\mathbb{R}^ m)^*\). In this paper the problem of extending Steinitz' theorem to an infinite- dimensional t.v.s. is studied, and the following definitive result proved: Theorem: The Steinitz characterization of orderable sequences holds in the infinite-dimensional separable Fréchet space \(X\) if and only if \(X\) is isomorphic to the space \(s\), the countable product of lines.