# Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

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

### Tom 187 / 2005

Fundamenta Mathematicae 187 (2005), 61-93 MSC: 46B03, 06A07. DOI: 10.4064/fm187-1-3

#### Streszczenie

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 $c_0$.

#### Autorzy

• S. A. ArgyrosDepartment of Mathematics
National Technical University of Athens
Athens 15780, Greece
e-mail
• V. KanellopoulosDepartment of Mathematics
National Technical University of Athens
Athens 15780, Greece
e-mail

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