Banach spaces which are isometric to subspaces of \(c_0(\Gamma)\) (Q2230560)

Characterizing those Banach spaces which can be found as subspaces of \(c_0\) is a long-standing problem in the mathematical literature; see, for instance, [\textit{S. A. Argyros} et al., Mathematika 62, No. 3, 685--700 (2016; Zbl 1359.46005); \textit{G. Godefroy} et al., Geom. Funct. Anal. 10, No. 4, 798--820 (2000; Zbl 0974.46023)]. In this article the authors focus on the isometric characterization of these spaces in terms of their extremal structure. One of the main results of the paper is the following: {Theorem.} Suppose that \(X\) is a separable Banach space. Then the following statements are equivalent: \begin{itemize} \item[1.] \(X\) is isometric to a subspace of \(c_0\). \item[2.] The set of all extreme points of \(B_{X^*}\) is a weak*-null sequence in \(X^*\). \item[3.] The set of all Gateâux derivatives of the norm \(\|\cdot \|\) of \(X\) is a weak*-null sequence. \item[4.] The set of all Fréchet derivatives of the norm \(\|\cdot \|\) of \(X\) is a weak*-null sequence and a boundary for \(X\). \item[5.] There is a weak*-null \(1\)-norming sequence in \(X^*\). \end{itemize} Notice that some of the equivalences of the previous theorem might belong to the folklore; see, e.g., Proposition III.1 in [\textit{G.~Godefroy}, Extr. Math. 16, No.~1, 1--25 (2001; Zbl 0986.46009)]. The authors also prove several properties of subspaces of \(c_0\) and deal with the nonseparable setting, i.e., they also study subspaces of \(c_0(\Gamma)\) for some uncountable set \(\Gamma\), providing several characterizations of such spaces.