P. Holick\'y, M. \v Sm\'\i dek, L. Zaj\'\i \v cek
Convex functions with non-Borel set of G\^ateaux differentiability points

Comment.Math.Univ.Carolinae 39,3 (1998) 469-482.

Abstract:We show that on every nonseparable Banach space which has a fundamental system (e.g. on every nonseparable weakly compactly generated space, in particular on every nonseparable Hilbert space) there is a convex continuous function $f$ such that the set of its G\^ateaux differentiability points is not Borel. Thereby we answer a question of J. Rainwater (1990) and extend, in the same time, a former result of M. Talagrand (1979), who gave an example of such a function $f$ on $\ell ^1(\frak c)$.

Keywords: convex function, G\^ateaux differentiability points, Borel set, fundamental system
AMS Subject Classification: Primary 46G05; Secondary 46B20