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Keywords:
convex function; G\^ateaux differentiability points; Borel set; fundamental system
convex function; G\^ateaux differentiability points; Borel set; fundamental system
Summary:
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 $\ell^1(\frak c)$.
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