On Fréchet differentiability of convex functions on Banach spaces. (English) Zbl 0831.46045

Tang, Wee-Kee

Commentat. Math. Univ. Carol. 36, No. 2, 249-253 (1995).

Summary: Equivalent conditions for the separability of the range of the subdifferential of a given convex Lipschitz function \(f\) defined on a separable Banach space are studied. The conditions are in terms of a majorization of \(f\) by a \(C^1\)-smooth function, separability of the boundary for \(f\) or an approximation of \(f\) by Fréchet smooth convex functions.

MSC:

46G05	Derivatives of functions in infinite-dimensional spaces
49J50	Fréchet and Gateaux differentiability in optimization
46B03	Isomorphic theory (including renorming) of Banach spaces

Keywords:

separability of the range of the subdifferential; convex Lipschitz function; \(C^ 1\)-smooth function; Fréchet smooth convex functions