Determining $c_0$ in $ C({\cal K})$ spaces

S. A. Argyros ¹   ; V. Kanellopoulos ¹

¹ Department of Mathematics National Technical University of Athens Athens 15780, Greece

Fundamenta Mathematicae, Tome 187 (2005) no. 1, pp. 61-93

Résumé

For a countable compact metric space $\mathcal{K}$ and a seminormalized weakly null sequence $(f_n)_n$ in $C(\mathcal{K})$ we provide some upper bounds for the norm of the vectors in the linear span of a subsequence of $(f_n)_n$. These bounds depend on the complexity of $\mathcal{K}$ and also on the sequence $(f_n)_n$ itself. Moreover, we introduce the class of $c_0$-hierarchies. We prove that for every $\alpha\omega_1$, every normalized weakly null sequence $(f_n)_n$ in $C(\omega^{\omega^\alpha})$ and every $c_0$-hierarchy $\mathcal{H}$ generated by $(f_n)_n$, there exists $\beta \leq\alpha$ such that a sequence of $\beta$-blocks of $(f_n)_n$ is equivalent to the usual basis of $c_0$.

Détail

DOI : 10.4064/fm187-1-3

Keywords: countable compact metric space mathcal seminormalized weakly null sequence mathcal provide upper bounds norm vectors linear span subsequence these bounds depend complexity mathcal sequence itself moreover introduce class hierarchies prove every alpha omega every normalized weakly null sequence omega omega alpha every hierarchy mathcal generated there exists beta leq alpha sequence beta blocks equivalent usual basis