Ordinal indices and Ramsey dichotomies measuring $c_0$-content and semibounded completeness

Vassiliki Farmaki¹

¹ Department of Mathematics University of Athens Panepistimiopolis 15784 Athens, Greece

Fundamenta Mathematicae, Tome 172 (2002) no. 2, pp. 153-179

Résumé

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.

Détail

DOI : 10.4064/fm172-2-4

Keywords: study content seminormalized basic sequence chi banach space ordinal indices taking values omega determine dichotomies every ordinal stage based ramsey type principle every countable ordinal obtained earlier author introduce indices index chi semibounded completeness index chi examine their relationship countable ordinal values these indices always form omega zeta these results extend countable ordinal level earlier result odell which stated only limiting first uncountable ordinal