Determining $c_0$ in $ C({\cal K})$ spaces
Volume 187 / 2005
Fundamenta Mathematicae 187 (2005), 61-93
MSC: 46B03, 06A07.
DOI: 10.4064/fm187-1-3
Abstract
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 $c_0$.
