Weak-star compact subsets in the bidual of a Banach space (Q2772753)

The paper presents several interesting results on special bounded subsets in the bidual \(X^{**}\) of a Banach space \(X\). Given a set \(A\subset X\), denote by \(\widetilde A\) the weak\(^*\)-closure of \(A\) in \(X^{**}\) and put \(A^*= \bigcup\{\widetilde P: P\subset A\) countable\}.NEWLINENEWLINENEWLINELet \(A\) be bounded and \(B\) the closed convex hull of \(A\). Denote by \(Y\) be the weak\(^*\)-closure of the span of \(X\cup A^*\) in \(X^{**}\). Suppose that \(Y/X\) is separable. Then \(A\) is weak\(^*\)-sequentially dense in \(\widetilde A\), so that \(\widetilde A= A^*\). Moreover, \(\widetilde B\) is the norm closed convex hull of its strongly exposed points.NEWLINENEWLINENEWLINEThe sequence space \(c_0\) contains a closed, bounded, absolutely convex set \(B\) such that the standard unit vectors \(e_n\) form a shrinking basis in the Banach space \((c_0)_B\) generated by \(B\). Moreover, \((c_0)_B\) admits a weak\(^*\)-closed topological complement in its bidual \((c_0)^{**}_B\) which is an isomorphic copy of \(\ell_1\).NEWLINENEWLINENEWLINE\(c_0\) also contains, for each \(m\in\mathbb{N}\), a closed, bounded, absolutely convex set \(B_m\) such that \((e_n)_n\) is a shrinking basis in \((c_0)_{B_m}\) and \((c_0)_{B_m}\) is quasi-reflexive of order \(m\).NEWLINENEWLINEFor the entire collection see [Zbl 0972.00067].