On certain biorthogonal systems related to flatness (Q1103155)

\textit{R. E. Harrell} and \textit{L. A. Karlovitz} have introduced and studied the case of flat Banach spaces [Trans. Am. Math. Soc. 192, 209-218 (1974; Zbl 0298.46016); Pac. J. Math. 59, 85-91 (1975; Zbl 0317.46016); Math. Ann. 202, 245-250 (1973; Zbl 0247.46031)]. Flatness is a geometric property; the dual of a flat space is flat, flat dual spaces are non-separable, therefore the dual of a flat space is non- separable. \textit{C. Stegall} has characterized the class of separable Banach spaces with non-separable dual by biorthogonal systems with a certain property [Trans. Am. Math. Soc. 206, 213-223 (1975; Zbl 0318.45056)]. The author introduces a geometric condition (P), which is easily shown to be stronger than Stegall's condition. He proves that for a separable Banach space X the following implications hold: X is flat \(\Rightarrow\) X has (P) \(\Rightarrow\) \(X^*\) is flat.