A subsequence characterization of sequences spanning isomorphically polyhedral Banach spaces

Let $(x_n)$ be a sequence in a Banach space $X$ which does not converge in norm, and let $E$ be an isomorphically precisely norming set for $X$ such that $\sum_n |x^*(x_{n+1}-x_n)|< \infty, \; \forall x^* \in E. \qquad (*)$ Then there exists a subsequence of $(x_n)$ which spans an isomorphically polyhedral Banach space. It follows immediately from results of V. Fonf that the converse is also true: If a separable Banach space $Y$ is a separable isomorphically polyhedral then there exists a non norm convergent sequence $(x_n)$ which spans $Y$ and there exists an isomorphically precisely norming set $E$ for $Y$ such that $(*)$ is satisfied. As an application of this subsequence characterization of sequences spanning isomorphically polyhedral Banach spaces we obtain a strengthening of a result of J. Elton, and an Orlicz-Pettis type result.

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