On the Kunen–Shelah properties in Banach spaces
Volume 157 / 2003
Abstract
We introduce and study the Kunen–Shelah properties ${\rm KS}_i$, $ i=0,1,\ldots ,7$. Let us highlight some of our results for a Banach space $X$: (1) $X^*$ has a $w^*$-nonseparable equivalent dual ball iff $X$ has an $\omega$-polyhedron (i.e., a bounded family $\{x_i\}_{i< \omega}$ such that $x_j\not\in \overline{\rm co} (\{x_i :i\in \omega\setminus \{j\}\})$ for every $j\in \omega$) iff $X$ has an uncountable bounded almost biorthogonal system (UBABS) of type $\eta$ for some $\eta\in [0,1)$ (i.e., a bounded family $\{(x_{\alpha},f_{\alpha})\}_{1\leq \alpha < \omega}\subset X\times X^*$ such that $f_{\alpha}(x_{\alpha})=1$ and $|f_{\alpha}(x_{\beta})|\leq\eta$ if $\alpha\neq\beta$); (2) if $X$ has an uncountable $\omega$-independent system then $X$ has an UBABS of type $\eta$ for every $\eta \in (0,1)$; (3) if $X$ does not have the property (C) of Corson, then $X$ has an $\omega$-polyhedron; (4) $X$ has no $\omega$-polyhedron iff $X$ has no convex right-separated $\omega$-family (i.e., a bounded family $\{x_i\}_{i< \omega}$ such that $x_j\not\in \overline{\rm co} (\{x_i: j< i< \omega\})$ for every $j\in \omega$) iff every $w^*$-closed convex subset of $X^*$ is $w^*$-separable iff every convex subset of $X^*$ is $w^*$-separable iff $\mu (X)=1$, $\mu (X)$ being the Finet–Godefroy index of $X$ (see [1]).
