On non-fragmentability of Banach spaces
The paper is concerned with fragmentability in Banach spaces, a notion due to \textit{J. E. Jayne} and \textit{C. Rogers} [J. Funct. Anal. 56, 279-299 (1984; Zbl 0581.28007)], and which has many interesting connections with other topics in Banach space theory as locally uniformly convex and Kadec renormings, generic differentiability of convex functions, Radon-Nikodým property. \textit{R. Haydon} and \textit{V. Zizler} [Can. Math. Bull. 32, No. 1, 122-128 (1989; Zbl 0694.46011)], have constructed a closed linear subspace \(X\) of \(\ell^\infty\) which is a tree-complete sublattice (see loc. cit.) of the lattice \(\ell^\infty\), contains the space \(c_0\) and does not contain any isomorphic copy of \(\ell^\infty\). The aim of the present paper is to exhibit new properties of the space \(X\), namely, the space \((X/c_0,\) weak) is not fragmented by any metric and, as a consequence, \(X/c_0\) admits no equivalent strictly convex renorming.
generic differentiability of convex functions
scientific article; zbMATH DE number 1334255
scientific article