Extracting long basic sequences from systems of dispersed vectors
The author studies general geometric properties of a non-separable Banach space \(X\) which allow to extract long basic sequences from systems of dispersed vectors. These properties are much weaker than WCG and closely related to Corson's property \((C)\). For example, the space \(X\) has property \((\mathcal{B})\) if, for each nested sequence of closed linear subspaces \((Z_{\alpha})_{\alpha<\kappa}\) of \(X\) such that \(\bigcap_{\alpha<\kappa}Z_{\alpha}=\{0\}\), the subspace \(\bigcup_{\alpha<\kappa}Z_{\alpha}^\bot\subset X^*\) is norming. Given a limit ordinal \(\kappa\), a sequence \((x_{\alpha})_{\alpha<\kappa}\) of \(X\) is strongly dispersed if \([x_\alpha]_{\alpha>\beta} \neq[x_\alpha]_{\alpha>\gamma}\) for \(\beta\neq\gamma\) and \(\bigcap_{\beta<\kappa}[x_\alpha]_{\alpha>\beta}=\{0\}\). One of the author's result says: If \(X\) satisfies \((\mathcal{B})\) and \((x_{\alpha})_{\alpha<\kappa}\) is strongly dispersed, then there is a long basic sequence \((x_{\alpha_\sigma})_{\sigma<\kappa}\).
scientific article; zbMATH DE number 6084092
scientific article