Boundedness of biorthogonal systems in Banach spaces (Q611025)

Let \(X\) be a Banach space. A biorthogonal system \(\{(x_\gamma, f_\gamma)\}_{\gamma \in \Gamma } \subset X \times X^*\) is said to be a Markushevich basis (M-basis) if the linear span of \(\{x_\gamma\}_{\gamma \in \Gamma }\) is dense in \(X\) and the linear span of \(\{f_\gamma\}_{\gamma \in \Gamma }\) is \(w^*\) dense in \(X^*\). An M-basis is said to be bounded if \(\sup_{\gamma \in \Gamma }\|x_\gamma\| \|f_\gamma\| < \infty\). \textit{A. N. Plichko} [Sov. Math., Dokl. 25, 386--389 (1982); translation from Dokl. Akad. Nauk SSSR 263, 543--546 (1982; Zbl 0531.46010)] proved in particular that every Banach space that admits a Markushevich basis also admits a bounded Markushevich basis. The authors of the paper under review have discovered an error in Plichko's proof: at some point a stronger property than that of being M-basis (called strong M-basis) was used implicitly. The aim of the article is to correct that error.