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Abstract

We study the $c_0$-content of a seminormalized basic sequence $(\chi _n)$ in a Banach space, by the use of ordinal indices (taking values up to $\omega _1$) that determine dichotomies at every ordinal stage, based on the Ramsey-type principle for every countable ordinal, obtained earlier by the author. We introduce two such indices, the $c_0$-index $\xi ^{(\chi _n)}_0$ and the semibounded completeness index $\xi ^{(\chi _n)}_b$, and we examine their relationship. The countable ordinal values that these indices can take are always of the form $\omega ^{\zeta }$. These results extend, to the countable ordinal level, an earlier result by Odell, which was stated only for the limiting case of the first uncountable ordinal.